BY 4.0 license Open Access Published by De Gruyter October 16, 2021

Rank functions on triangulated categories

Joseph Chuang and Andrey Lazarev ORCID logo

Abstract

We introduce the notion of a rank function on a triangulated category π’ž which generalizes the Sylvester rank function in the case when π’ž=𝖯𝖾𝗋𝖿⁒(A) is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If π’ž=𝖯𝖾𝗋𝖿⁒(A) as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

1 Introduction

The dimension of a vector space V over a (possibly skew-)field K is a basic characteristic of V and it is an elementary fact that it is an invariant, i.e. does not depend on the choice of a basis in V. However, further generalizations with K replaced with a noncommutative ring A, and V replaced with an A-module, are not straightforward. Indeed, there are examples of rings such that their free modules do not have a well-defined dimension, or rank. One possibility to obviate this difficulty is to start with a homomorphism f:A→K where K is a skew-field (which allows one to associate to an A-module a K-module by tensoring up) and define the rank of an A-module as the rank of the corresponding K-module. Different homomorphisms f give rise to possibly different ranks. This suggests that ranks are closely related to homomorphisms into skew-fields. This was made precise by Cohn and Schofield [9, 30] by showing that maps f as above are in one-to-one correspondence with certain functions, called Sylvester rank functions, on finitely presented A-modules, defined intrinsically.

A crucial part of the Cohn–Schofield theory is the method of matrix localization. Given a ring A and a matrix M with entries in A, there exists another ring A⁒[M-1] supplied with a map Aβ†’A⁒[M-1] such that the matrix M becomes invertible over A⁒[M-1]; moreover, A⁒[M-1] is universal in the sense that any other ring having this property factorizes uniquely through A⁒[M-1]. If M is a 1Γ—1 matrix, i.e. an element s∈A, then A⁒[M-1] reduces to A⁒[s-1], the usual localization of A at s. Furthermore, if A is a commutative ring, then general matrix localization reduces to inverting a single element, namely the determinant of M; however in the general noncommutative case no such reduction is possible.

Our original motivation was to rework the Cohn–Schofield theory in a way intrinsic to the derived category of A. To this end, we formulate the notion of a rank function on an arbitrary triangulated category π’ž. Compared to the Sylvester rank function, our definition is simpler and, arguably, more natural. In the case π’ž=𝖯𝖾𝗋𝖿⁒(A) for a ring A, it subsumes that of the Sylvester rank function but does not reduce to it. The β€œexotic” rank functions on 𝖯𝖾𝗋𝖿⁒(A) (i.e. those that are not equivalent to Sylvester rank functions) are related to maps from A into graded skew-fields or graded simple Artinian rings in the homotopy category of differential graded (dg) rings. Recall that maps in the homotopy category of dg rings are computed by replacing the source with a cofibrant dg ring. This needs to be done even if the source is an ordinary ring. In other words, such maps are invisible on the classical level.

Our notion of a rank function on π’ž appears close, albeit certainly not equivalent, to the notion of cohomological functions on π’ž in the sense of Krause [22]. The precise relationship between the two notions is unclear at the moment and we hope to return to this question in future.

Apart from the purely algebraic motivation described above, Sylvester rank functions are of great relevance to geometric group theory and, in particular, to various versions of the Atiyah conjecture, cf. [19] for a survey of recent results in this direction. A natural question, that we also leave for future investigation, is whether our notion of a rank functions can be usefully exploited in that context.

One unexpected source of rank functions on triangulated categories turns out to be Bridgeland’s β€œstability conditions” [6]. Namely, it turns out that there is a continuous map from the space of stability conditions to that of rank functions on the same triangulated category. Here we limit ourselves with merely recording this observation but it undoubtedly deserves further study.

Next, we need to develop an analogue of matrix localization in a homotopy invariant way. Even when one is interested in inverting only one element in a ring, i.e. a 1Γ—1 matrix, care is needed since this operation is not exact in the noncommutative context. The notion of derived localization (in this restricted sense) was developed in the previous work of the authors [5]. In the present paper we build on this work to construct derived localization of a dg ring A with respect to an arbitrary thick subcategory of 𝖯𝖾𝗋𝖿⁒(A). This extends the notion of matrix localization since the latter is the nonderived version of the localization with respect to the thick subcategory generated by a collection of free complexes of length 2. In contrast with inverting a single element, even for commutative rings general derived localization may have nontrivial derived terms. However, for hereditary algebras, derived localization reduces to the nonderived notion.

A rank function ρ on a triangulated category π’ž has a kernel Ker⁑(ρ), the thick subcategory of π’ž consisting of objects of rank zero. We describe those rank functions for which the Verdier quotient π’ž/Ker⁑(ρ) is simple, i.e. equivalent to the perfect derived category of a graded skew-field. These are the so-called localizing rank functions. It turns out that localizing rank functions classify homotopy classes of homological epimorphisms from dg rings into dg fields and dg simple Artinian rings. We call such homological epimorphisms derived fraction fields. Strikingly, ordinary rings (even finite-dimensional algebras over fields) have nontrivial derived fraction fields.

There are, of course, exact functors from a given triangulated category π’ž to a simple one that are not Verdier quotients; these induce (non-localizing) rank functions on π’ž. It is an open question to which extent such functors are captured by rank functions.

The paper is organized as follows. In Section 2 the notion of a rank function on a triangulated category is introduced, in two equivalent ways: as a function on objects or on morphisms satisfying appropriate conditions, as well as its refinement, a d-periodic rank function where d=1,2,…,∞ (an ordinary rank function is then 1-periodic). We show how rank functions can be constructed using functors into simple triangulated categories and (very briefly) stability conditions.

In Section 3 we prove that ∞-periodic rank functions on the perfect derived category of ordinary rings are nothing other than Sylvester rank function whereas Section 4 establishes various further properties of rank functions reminiscent of those for Sylvester rank functions.

Section 5 develops the theory of derived localization for dg algebras at thick subcategories of their perfect derived categories and compares it with the nonderived notion. This section has an independent interest and can be read independently of the rest of the paper. Finally, Section 6 introduces the notion of a localizing rank function and shows that they describe Verdier quotients into simple triangulated categories and homotopy classes of homological epimorphisms from dg algebras into dg fields and dg simple Artinian rings.

1.1 Notation and conventions

Throughout this paper we work with homologically graded chain complexes over a fixed unital commutative ring 𝐀. Unadorned tensor products and Homs will be assumed to be taken over 𝐀. For a chain complex A we denote by Σ⁒A its suspension given by (Σ⁒A)i=Ai-1. The signs ≃ and β‰… will stand for quasi-isomorphisms and isomorphisms of chain complexes, respectively.

We will normally use the abbreviation β€œdg” for β€œdifferential graded”. By β€œdg algebra” we will mean β€œdg associative unital algebra” over 𝐀. The category of dg algebras 𝖽𝗀𝖠𝗅𝗀 has the structure of a closed model category (with weak equivalences being multiplicative quasi-isomorphisms) and so it makes sense to talk about homotopy classes of maps between dg algebras. A given dg algebra B together with a dg algebra map Aβ†’B (not necessarily central) will be referred to as an A-algebra. The category A↓𝖽𝗀𝖠𝗅𝗀 of A-algebras is likewise a closed model category. The category of right dg A-modules over a dg algebra A will be denoted by π–¬π—ˆπ–½β’-⁒A and we will refer to its objects as A-modules. The category π–¬π—ˆπ–½β’-⁒A is a closed model category whose homotopy category is denoted by 𝖣⁒(A), the derived category of A. We chose to focus (for notational convenience) on right modules but will occasionally use left modules as well, making sure that no confusion would arise.

For A-modules M,N we denote by HomA⁑(M,N) the chain complex of A-linear homomorphisms Mβ†’N. We write RHomA⁑(M,N) for the corresponding derived functor obtained by replacing M with a cofibrant A-module quasi-isomorphic to M. If M=N, we will write EndA⁑(M) and REndA⁑(M) for HomA⁑(M,N) and RHomA⁑(M,N), respectively.

For a (right) A-module M and a left A-module N we write MβŠ—AN and MβŠ—ALN for their tensor product and derived tensor product, respectively. For two dg algebras B and C supplied with a dg algebra maps Aβ†’B and Aβ†’C their free product or pushout will be denoted by B*AC, and its derived version – by B*ALC.

A description of the closed model categories of algebras and modules convenient for our purposes is given in [5], except for the minor difference that left modules are treated in that paper whereas right modules are emphasized here.

We will freely use the language of triangulated categories and their localizations, cf. [21] for an overview. If π’ž is a triangulated category with translation functor Ξ£ and Ξ£dβ‰…id, we say that π’ž has period d; if no such d exists, π’ž is said to have infinite period. Given a triangulated category π’ž, its full triangulated subcategory S is thick if it is closed with respect to taking retracts. In this situation one can form the Verdier quotientπ’ž/S, a triangulated category supplied with a (triangulated) functor j:π’žβ†’π’ž/S whose kernel is S and universal with respect to this property. A triangulated subcategory S of π’ž is called localizing if it contains all small coproducts; in that case the Verdier quotient often admits a right adjoint i:π’ž/Sβ†’π’ž, and then the endofunctor L:=i∘j:π’žβ†’C is called the (Bousfield) localization functor with respect to S. It is necessarily idempotent: L2β‰…L and for any Xβˆˆπ’ž the natural map Xβ†’L⁒(X) is called the localization of X (with respect to S). For a collection of objects S of π’ž we will denote by Loc⁑(S) the smallest localizing subcategory of π’ž containing S; we say that S generates Loc⁑(S).

We say that an object X is a (classical) generator of the triangulated category π’ž if π’ž is the smallest thick subcategory of π’ž containing X.

For a triangulated category C a perfect (or compact) object X is characterized by the property

Homπ’žβ‘(X,βŠ•i∈IXi)β‰…βŠ•i∈IHomπ’žβ‘(X,Xi)

for any collection of Xi,i∈I of A-modules indexed by a set I. The full subcategory of 𝖣⁒(A) consisting of perfect A-modules will be denoted by 𝖯𝖾𝗋𝖿⁒(A); note that A is a generator of 𝖯𝖾𝗋𝖿⁒(A).

A dg category is understood to be a category enriched over dg 𝐀-modules. Thus, for two objects X1 and X2 in a dg category C we have a dg space of homomorphisms Hom⁑(X1,X2) and composition is a dg map. The homotopy category H0⁒(C) of the dg category C has the same objects as C and for two objects X1,X2 in C we have

HomH0⁒(C)⁑(X1,X2):=H0⁒[HomC⁑(X1,X2)].

A dg functor F:Cβ†’Cβ€² between two dg categories is quasi-essentially surjective if H0⁒(F):H0⁒(C)β†’H0⁒(Cβ€²) is essentially surjective and quasi-fully faithful if F induces quasi-isomorphisms on the Hom-spaces; if both conditions are satisfied then F is called a quasi-equivalence.

2 Rank functions on triangulated categories

Let π’ž be a triangulated category.

Definition 2.1.

A rank function on π’ž is an assignment to each object X of π’ž of a nonnegative real number ρ⁒(X) such that the following conditions hold.

  1. Translation invariance: for any object X, we have

    (O1)ρ⁒(Σ⁒X)=ρ⁒(X).
  2. Additivity: for any objects X and Y, we have

    (O2)ρ⁒(XβŠ•Y)=ρ⁒(X)+ρ⁒(Y).
  3. Triangle inequality: for any exact triangle Xβ†’Yβ†’Z↝, we have

    (O3)ρ⁒(Y)≀ρ⁒(X)+ρ⁒(Z).

Remark 2.2.

Conditions (O1) and (O3) may be replaced by the statement that for any exact triangle Xβ†’Yβ†’Z↝, the triple (ρ⁒(X),ρ⁒(Y),ρ⁒(Z)) is triangular, i.e. it is composed of the side lengths of a (possibly degenerate) planar triangle. Indeed, one only needs to show that the latter condition together with (O2) implies (O1). Consider the exact triangle Xβ†’0Xβ†’XβŠ•Ξ£β’X↝. Then the triangularity condition together with (O2) implies that

ρ⁒(Σ⁒XβŠ•X)=ρ⁒(Σ⁒X)+ρ⁒(X)≀2⁒ρ⁒(X)

so that ρ⁒(Σ⁒X)≀ρ⁒(X). Similarly, the exact triangle Σ⁒XβŠ•X→Σ⁒Xβ†’0Σ⁒X↝ implies that ρ⁒(X)≀ρ⁒(Σ⁒X) and it follows that ρ⁒(X)=ρ⁒(Σ⁒X).

Rank functions on triangulated categories may alternatively be defined as functions on morphisms, as follows.

Definition 2.3.

A rank function on π’ž is an assignment to each morphism f in π’ž of a nonnegative real number ρ⁒(f), such that the following conditions hold.

  1. Translation invariance: for any morphism f, we have

    (M1)ρ⁒(Σ⁒f)=ρ⁒(f).

  2. Additivity: for any morphisms f and g, we have

    (M2)ρ⁒(fβŠ•g)=ρ⁒(f)+ρ⁒(g).

  3. Rank-nullity condition: for any exact triangle X→𝑓Y→𝑔Z↝, we have

    (M3)ρ⁒(f)+ρ⁒(g)=ρ⁒(idY).

The translation between the two definitions is given by the following formulas:

(2.1)ρ⁒(X)=ρ⁒(idX),
(2.2)ρ(f:Xβ†’Y)=ρ⁒(X)+ρ⁒(Y)-ρ⁒(cone⁑(f))2.

Proposition 2.4.

Definitions 2.3 and 2.1 are equivalent.

Proof.

Given a nonnegative function ρ on morphisms in π’ž satisfying the three conditions of Definition 2.3, the rule (2.1) defines a function on objects which is clearly nonnegative, and translation invariance (O1) and additivity (O2) follow immediately from the corresponding properties (M1) and (M2). Finally, take an exact triangle

X→𝑓Y→𝑔Z↝

in π’ž, with rotated exact triangles Y→𝑔Zβ†’β„ŽΞ£β’X↝ and Ξ£-1⁒Zβ†’Ξ£-1⁒hX→𝑓Y↝. We have

ρ⁒(X)+ρ⁒(Z)-ρ⁒(Y)=ρ⁒(idX)+ρ⁒(idZ)-ρ⁒(idY)
=(ρ⁒(Σ-1⁒h)+ρ⁒(f))+(ρ⁒(g)+ρ⁒(h))-(ρ⁒(f)+ρ⁒(g))
=2⁒ρ⁒(h)β‰₯0,

by (M1) and (M3), which establishes (O3).

Conversely, given a nonnegative function ρ on objects of π’ž satisfying the three conditions of Definition 2.1, define a function on morphisms by formula (2.2). For f:Xβ†’Y, we have an exact triangle Yβ†’cone⁑(f)→Σ⁒X↝, so (O1) and (O3) imply that ρ⁒(f)β‰₯0. Properties (M1) and (M2) are consequences of (O1) and (O2), respectively, and, finally, given an exact triangle

X→𝑓Y→𝑔Z↝,

we confirm property (M3):

ρ⁒(f)+ρ⁒(g)=ρ⁒(X)+ρ⁒(Y)-ρ⁒(Z)2+ρ⁒(Y)+ρ⁒(Z)-ρ⁒(Σ⁒X)2
=ρ⁒(Y)
=ρ⁒(idY),

using (O1). ∎

In view of Proposition 2.4, we regard a rank function on π’ž as a function on both objects and morphisms, related by equations (2.1) and (2.2).

We call a rank function ρ

  1. β€’

    object-faithful if for all nonzero objects X we have ρ⁒(idX)β‰ 0,

  2. β€’

    morphism-faithful if for all nonzero maps f we have ρ⁒(f)β‰ 0,

  3. β€’

    integral if ρ⁒(f)βˆˆβ„€ for all f,

  4. β€’

    prime if ρ is integral and π’ž admits a generator X such that ρ⁒(idX)=1.

Remark 2.5.

In the case π’ž is a tensor triangulated category (i.e. it has a symmetric monoidal structure compatible with its triangulation, cf. [17, Appendix A]), it makes sense to require additionally that a rank function ρ is multiplicative in the sense that

ρ⁒(idXβŠ—idY)=ρ⁒(idX)⁒ρ⁒(idY)

for any two objects X,Yβˆˆπ’ž. Multiplicative rank functions are likely to be of relevance to tensor triangular geometry, [1] but will not be considered in this paper.

2.1 Periodic rank functions

We will consider a certain refinement of the notion of a rank function defined above.

Let R be an ordered commutative ring and fix d∈{1,2,…}βˆͺ{∞}. Put

R⁒(d):={R⁒[q,q-1]if ⁒d=∞,R⁒[q]/(qd-1)if ⁒d<∞,

and

Rβ‰₯0⁒(d):={Rβ‰₯0⁒[q,q-1]if ⁒d=∞,Rβ‰₯0⁒[q]/(qd-1)if ⁒d<∞.

For Ο•,ψ∈R⁒(d), we write Ο•β‰₯ψ to mean Ο•-ψ∈Rβ‰₯0⁒(d). Given two integers q,qβ€² where d is divisible by dβ€² or d=∞, there is an obvious reduction map Ο€d,dβ€²:R⁒(d)β†’R⁒(dβ€²). We will mostly be interested in the case R=ℝ or R=β„€.

Definition 2.6.

A d-periodic rank function on a triangulated category π’ž is an assignment to each morphism f in π’ž of ρ⁒(f)βˆˆβ„β‰₯0⁑(d) such that axioms (M2) and (M3) hold, axiom (M1) gets modified as follows:

  1. Translation invariance: for any morphism f, we have

    (Mp1)ρ⁒(Σ⁒f)=q⁒ρ⁒(f),

    and, in addition, the following axioms hold:

  2. Triangular inequality: for all morphisms f, g and h in π’ž such that g and h share the same domain and f and h share the same codomain, we have

    (M4)ρ⁒(fh0g)β‰₯ρ⁒(f)+ρ⁒(g).

  3. Ideal condition: for any morphisms f and g for which the composition gf is defined, we have

    (M5)ρ⁒(g⁒f)≀ρ⁒(f) and ρ⁒(g⁒f)≀ρ⁒(g).

Remark 2.7.

Note that given two integers d and dβ€² where dβ€² divides d or d=∞, a d-periodic rank function on π’ž determines a dβ€²-periodic rank function on π’ž via the reduction map ℝ⁑(d)→ℝ⁑(dβ€²). In particular, any d-periodic rank function gives rise to a 1-periodic rank function. Under this reduction, axiom (Mp1) becomes (M1) and we will see that often (e.g. when d=1) the axioms (M4) and (M5) are consequence of axioms (M1), (M2) and (M3). In particular, the terms β€œrank function” and β€œ1-periodic rank function” are synonymous. Furthermore, often a periodic rank function can be defined as a function on objects (e.g. we saw that it holds for d=1).

A d-periodic rank function taking values in β„€β‰₯0⁑(d) will be called integral. The notions of a prime, object-faithful and morphism-faithful rank function obviously make sense in the d-periodic case.

Proposition 2.8.

Let ρ be a d-periodic rank function on C. For all objects X in C, put

ρ⁒(X):=ρ⁒(idX)βˆˆβ„β‰₯0⁑(d).

Then ρ satisfies the additivity axiom (O2) of the rank function whereas axioms (O1), (O3) get refined as follows:

  1. Translation invariance: for any object X in π’ž, we have

    (Op1)ρ⁒(Σ⁒X)=q⁒ρ⁒(X).

  2. Triangle inequality: for any exact triangle Xβ†’Yβ†’Z↝ in π’ž we have

    (Op3)ρ⁒(X)-ρ⁒(Y)+ρ⁒(Z)=(q+1)⁒ϕ

    for some Ο•βˆˆβ„β‰₯0⁑(d). More precisely, we can take Ο•=ρ⁒(f), where f:Ξ£-1⁒Z→𝑓X is the connecting morphism for the triangle.

Moreover, if ρ is integral, i.e. ρ⁒(f)∈Zβ‰₯0⁑(d) for all morphisms f in C, then ρ⁒(X)∈Zβ‰₯0⁑(d) for any X∈C.

Proof.

This is a straightforward modification of the first part of proof of Proposition 2.4. In particular, conditions (Op1) and (O2) follow immediately from (Mp1) and (M2). Furthermore, given an exact triangle

Ξ£-1⁒Z→𝑓X→𝑔Yβ†’β„ŽZ,

we have, using (M3) and (Mp1),

ρ⁒(X)-ρ⁒(Y)+ρ⁒(Z)=(ρ⁒(f)+ρ⁒(g))-(ρ⁒(g)+ρ⁒(h))+(ρ⁒(h)+ρ⁒(Σ⁒f))
=(q+1)⁒ρ⁒(f),

as desired. The claim about integrality is likewise clear. ∎

Remark 2.9.

Note that in order to obtain axioms (Op1), (O2) and (Op3), we only make use of conditions (Mp1), (M2) and (M3) of Definition 2.6.

The following useful observation relates the rank of morphism with that of its source and target.

Lemma 2.10.

Let ρ be a d-period rank function on C. For any morphism f:Xβ†’Y, we have ρ⁒(f)≀ρ⁒(X) and ρ⁒(f)≀ρ⁒(Y).

Proof.

Consider the factorizations f=f∘idX and f=idY∘f in the ideal condition (M5). ∎

To check condition (Op3) when d=∞, the following obvious lemma is useful.

Lemma 2.11.

A real Laurent polynomial f⁒(q)=βˆ‘n∈Zan⁒qn can be written as

f⁒(q)=(1+q)⁒ϕ,

where Ο• has nonnegative coefficients if and only if for any n∈Z it holds that

βˆ‘i=0∞(-1)i⁒an+iβ‰₯0.

It is natural to ask whether a rank function is determined by its values on objects (so as to obtain a periodic analogue Proposition 2.4). The following result gives a partial answer to that.

Proposition 2.12.

Assume that d=∞ or d is odd and there is given an assignment to any object X of C of an element ρ⁒(X)∈R⁒(d) satisfying conditions (Op1), (O2) and (Op3). Then, for any f:Xβ†’Y in C the formula

(2.3)ρ⁒(f):=ρ⁒(Y)-ρ⁒(cone⁑(f))+q⁒ρ⁒(X)q+1

determines a d-periodic rank function on C. If, in addition, ρ⁒(X)∈Z⁑(d) and the polynomial Ο• figuring in triangle inequality (Op3), belongs to Z⁑(d), then the obtained rank function is integral.

Proof.

We will only treat the case of real rank functions; the integral case is obtained completely analogously.

Note that q+1 is invertible in ℝ⁑(d) if d is odd, and a nonzero-divisor (like all nonzero elements) if d=∞. By Proposition 2.8, the element ρ⁒(Y)-ρ⁒(cone⁑(f))+q⁒ρ⁒(X) is divisible by q+1 and it follows that it is uniquely divisible. Thus, ρ⁒(f)βˆˆβ„β‘(d) for any Xβˆˆπ’ž. Conditions (Mp1), (M2) and (M3) of Definition 2.6 are proved by the same argument as the second part of Proposition 2.4.

To check condition (M5), suppose we are given morphisms f:X→Y and g:Y→Z. We have

ρ⁒(f)-ρ⁒(g⁒f)=q⁒ρ⁒(X)+ρ⁒(Y)-ρ⁒(cone⁑(f))q+1-q⁒ρ⁒(X)+ρ⁒(Z)-ρ⁒(cone⁑(g⁒f))q+1
=ρ(Y)-ρ(cone(f)βŠ•Z)+ρ(cone(gf)))q+1
β‰₯0,

where the inequality is deduced from Lemma 2.13 below and (Op3).

Finally, we check condition (M4). Suppose we have morphisms f:Xβ†’Y, g:Zβ†’W and h:Zβ†’Y in π’ž. There is an exact triangle in π’ž of the form

cone⁑(g)β†’cone⁑(fh0g)β†’cone⁑(f)↝.

Hence, by (O2) and (Op3),

ρ⁒(fh0g)=ρ⁒(YβŠ•W)-ρ⁒(cone⁑(fh0g))+q⁒ρ⁒(XβŠ•Z)q+1
β‰₯ρ⁒(Y)+ρ⁒(W)-ρ⁒(cone⁑(f))-ρ⁒(cone⁑(g))+q⁒ρ⁒(X)+q⁒ρ⁒(Z)q+1
=ρ⁒(f)+ρ⁒(g).∎

Lemma 2.13.

For any pair of composable morphisms f:X→Y and g:Y→Z in a triangulated category, there exists an exact triangle of the form

Yβ†’cone⁑(f)βŠ•Zβ†’cone⁑(g⁒f)↝.

Proof.

There exists the following commutative diagram where the right square is a homotopy pushout and the rows are exact triangles:

By a version of the octahedral axiom the dotted arrow exists making the whole diagram commutative and the bottom row into an exact triangle. It follows that Wβ‰…cone⁑(g⁒f) and the desired result follows. ∎

2.2 Stability conditions and rank functions

Recall that a stability condition on a triangulated category π’ž, cf. [6] is a pair (𝒫,Z) where 𝒫=𝒫⁒(Ο•),Ο•βˆˆβ„ is a slicing on π’ž, a collection of subcategories of π’ž with properties modelled on the Postnikov truncations in the category of chain complexes and a central charge, that is a homomorphism Z:K0⁒(π’ž)β†’β„‚ compatible with the slicing in a suitable way. Every nonzero object E of π’ž has a filtration

0=E0→…→En=E

so that Ai:=cone⁑(Ei-1β†’Ei) belongs to P⁒(Ο•i) and Ο•1>…>Ο•n; one sets ϕ𝒫-⁒(E)=Ο•1 and ϕ𝒫+⁒(E)=Ο•n. The mass of E is defined as

mσ⁒(E)=βˆ‘i=1n|Z⁒(Ai)|.

More generally, one can introduce a parameter and define

mΟƒ,t⁒(E)=βˆ‘|Z⁒(Ai)|⁒eΟ•i⁒t

cf.[10, Section 4.5].

Proposition 2.14.

Given a stability condition σ=(P,Z) on a triangulated category C, the mass mσ defines an object faithful rank function on C.

Proof.

Translation invariance (O1), additivity (O2) and object-faithfulness are obvious. The triangle inequality for mΟƒ,t⁒(E) is proved in [18, Proposition 3.3] for all t and (O3) follows by taking t=0. ∎

Remark 2.15.

The last result suggests that a rank function may serve as a replacement for a stability condition on π’ž which exists even in the absence of a t-structure (e.g. when π’ž is periodic).

The set Stab⁑(π’ž) of stability conditions on π’ž is a topological space. The topology may be induced by the generalized metric (allowed to assume infinite values):

d⁒(Οƒ1,Οƒ2)=sup0β‰ Eβˆˆπ’žβ‘{|ϕσ2-⁒(E)-ϕσ1-⁒(E)|,|ϕσ2+⁒(E)-ϕσ1+⁒(E)|,|log⁑mΟƒ2⁒(E)mΟƒ1(E))|}

(cf. [6, p. 341]). In a similar (albeit much more obvious) way, the set FRank⁑(π’ž) of object faithful rank functions on π’ž is topologized by the generalized metric

d⁒(ρ1,ρ2)=sup0β‰ Eβˆˆπ’žβ‘|log⁑ρ1⁒(E)ρ2⁒(E)|.

On the set Rank⁑(π’ž) of all rank functions on π’ž, it is more useful, following Schofield [30, Chapter 7], to consider the topology of pointwise convergence of real-valued functions on objects of π’ž.

The following result is immediate:

Proposition 2.16.

The maps

Stab⁑(π’ž)β†’FRank⁑(π’ž)β†ͺRank⁑(π’ž),
σ↦mΟƒ

are continuous.

In [3], it is suggested that a compactification of the quotient Stab⁑(π’ž)/β„‚ of Stab⁑(π’ž) by a natural action of β„‚ may be constructed as the closure of its image in Rank⁑(π’ž)/ℝ×.

2.3 Simple triangulated categories

The most basic examples of rank functions come from simple triangulated categories.

Definition 2.17.

We say that a triangulated category π’ž is simple if every exact triangle in π’ž is split (by which we mean that it is isomorphic to a direct sum of triangles having an isomorphism as one of their arrows) and π’ž is generated as a triangulated category by one nonzero indecomposable object.

Simple triangulated categories are in one-to-one correspondence with graded skew-fields, i.e. graded rings whose nonzero homogeneous elements are invertible.

Proposition 2.18.

A triangulated category C is simple if and only if C is equivalent to Perf⁒(K) where K is a graded skew-field (i.e. the category of finite-dimensional vector K-spaces).

Proof.

Let π’ž be a simple triangulated category with an indecomposable generator X and let f:Xβ†’Ξ£n⁒X be a morphism. Then the exact triangle

X⁒→𝑓⁒Σn⁒Xβ†’cone⁑(f)↝

is contractible, that is one of its arrows contains an isomorphism as a direct summand. Since X is indecomposable, this can only happen if f is an isomorphism or zero. We conclude that the graded ring End*⁑(X) is a graded skew-field and π’ž is equivalent to the category of graded vector spaces over it. Conversely, all monomorphisms and epimorphisms of graded modules over a graded field K split and it follows that 𝖯𝖾𝗋𝖿⁒(K) is a simple triangulated category. ∎

Remark 2.19.

Graded skew-fields can easily be classified in terms of ordinary skew-fields. Indeed, let K be a graded skew-field; then K0, its zeroth component is an (ungraded) skew-field. Suppose that K0β‰ K and let d be the smallest positive integer such that K has a nonzero, hence invertible, element of degree d; denote this element by t. Then K is isomorphic as a K0-module to K0⁒[t,t-1] and if t is central then this isomorphism is multiplicative. In general K will be isomorphic to a skew Laurent polynomial ring K0⁒[t,t-1]Οƒ where Οƒ is an automorphism of K0 and the multiplication is determined by the rule t⁒a=σ⁒(a)⁒t for a∈K0.

The graded skew-field of the form K=K0⁒[t,t-1]Οƒ with |t|=d will be called d-periodic and d is called the period of K; furthermore we adopt the convention that an ungraded skew-field K=K0 has infinite period. The corresponding simple triangulated category 𝖯𝖾𝗋𝖿⁒(K) is d-periodic, i.e. the functor Ξ£d on 𝖯𝖾𝗋𝖿⁒(K) is naturally isomorphic to the identity.

Periodic rank functions on simple triangulated categories are essentially unique. More precisely, we have the following result.

Proposition 2.20.

The space of dβ€²-periodic rank functions on a d-periodic simple triangulated category is isomorphic to Rβ‰₯0⁑(dβ€²) if d=∞ or dβ€² is a divisor of d; otherwise it is {0}. If the space is nonzero, there exists a unique (up to multiplication by a power of q) prime rank function.

Proof.

Let π’ž be a simple d-periodic triangulated category with a generator X and let ρ be a nonzero dβ€²-periodic rank function on π’ž; assume that dβ€²<∞. Then ρ⁒(X)βˆˆβ„β‰₯0⁑(dβ€²) so that dβ€² is the smallest integer with qdβ€²=1. If d<∞, then Xβ‰…Ξ£d⁒X and

ρ⁒(X)=ρ⁒(Σd⁒X)=qd⁒ρ⁒(X)

so that qd=1 from which it follows that dβ€² divides d. Furthermore, any other dβ€²-periodic rank function can be obtained from ρ by multiplying it with an arbitrary element of ℝβ‰₯0⁑(dβ€²) and rank functions corresponding to different elements in ℝβ‰₯0⁑(dβ€²) will be different. The argument with dβ€²=∞ is similar. Finally, if a nonzero rank function on π’ž exists, then the condition ρ⁒(X)=1 specifies it uniquely. ∎

Corollary 2.21.

A simple triangulated category admits a unique prime rank function.

Proof.

This follows from Proposition 2.20 by specializing q=1. ∎

Remark 2.22.

Recall that by Proposition 2.18 a simple triangulated category is equivalent to the category of finite-dimensional vector spaces over a (possibly graded) skew-field. Clearly the unique rank function of Corollary 2.21 is just the dimension of a finite-dimensional vector space.

Proposition 2.23.

If a triangulated category admits a morphism-faithful prime rank function, then it is simple.

Proof.

Let π’ž be a triangulated category having a generator X and a morphism-faithful rank function ρ with ρ⁒(X)=1. Let

f:Ξ£n⁒Xβ†’X

be any nonzero morphism of some degree n; by (M3), ρ⁒(f) is either 1 or 0, by morphism-faithfulness the second possibility is ruled out so ρ⁒(f)=1. Considering the exact triangle

Ξ£-1⁒Yβ†’β„ŽΞ£d⁒X→𝑓X→𝑔Y

and using (M3) again, we conclude that ρ⁒(h)=ρ⁒(g)=0 and then by morphism-faithfulness g=h=0 and f is an isomorphism. Thus, any nonzero element in End⁑(X) is invertible, i.e. End⁑(X) is a graded skew-field. ∎

Definition 2.24.

A dg algebra A is called a dg skew-field if its homology H⁒(A) is a graded skew-field. Similarly, A is a simple Artinian dg algebra if H⁒(A) is a graded simple Artinian algebra (i.e. it is isomorphic to the graded matrix algebra Mn⁒(K) over some graded skew-field K).

The derived categories of dg skew-fields and dg Artinian simple algebras are all simple:

Proposition 2.25.

The following statements hold.

  1. (1)

    Let A be a simple Artinian dg algebra so that H⁒(A)β‰…Mn⁒(K), where K is a graded skew-field of period d and n is some integer. Then 𝖯𝖾𝗋𝖿⁒(A) is a simple triangulated category of period d.

  2. (1)

    Conversely, if A is a dg algebra for which 𝖯𝖾𝗋𝖿⁒(A) is simple, then A is a simple dg Artinian algebra.

Proof.

The homology graded ring H⁒(A) of A is a graded matrix algebra over some graded skew-field K. A primitive idempotent e∈H⁒(A) determines a retract X:=e⁒A in the triangulated category 𝖯𝖾𝗋𝖿⁒(A). The H⁒(A)-module H⁒(X) is a simple generator of the category of H⁒(A)-module. Given a perfect dg A-module M, its homology H⁒(M) is a graded H⁒(A)-module that is a finite direct sum of H⁒(A)-modules H⁒(X) and it is clear that this decomposition lifts to 𝖯𝖾𝗋𝖿⁒(A) so that M is a direct sum of copies of X in 𝖯𝖾𝗋𝖿⁒(A). Similarly, a map Mβ†’N of A-modules is determined by the map H⁒(M)β†’H⁒(N) of H⁒(A)-modules and it follows that any exact triangle in 𝖯𝖾𝗋𝖿⁒(A) splits. Thus, 𝖯𝖾𝗋𝖿⁒(A) is simple and its period clearly coincides with that of K. This proves (1).

For (2) let A be a dg algebra for which 𝖯𝖾𝗋𝖿⁒(A) is simple; denote by X an indecomposable generator of 𝖯𝖾𝗋𝖿⁒(A) and assume without loss of generality that X is a cofibrant A-module. Since every nonzero map Xβ†’Ξ£n⁒X is invertible in 𝖯𝖾𝗋𝖿⁒(A), the dg algebra B:=EndA⁑(X) is a dg skew-field. Then the functor F:M↦MβŠ—AX is a dg equivalence between the categories of cofibrant left A-modules and cofibrant B-modules and therefore the dg algebras Aβ‰…EndA⁑(A) and EndB⁑(F⁒(A))=EndB⁑(X) are quasi-isomorphic. Since B is a dg skew-field, the B-module X is a direct sum of simple B-modules and thus, EndB⁑(X) is a matrix algebra of a dg skew-field, i.e. a dg simple Artinian ring. ∎

Remark 2.26.

Another notion of a simple dg algebra was considered in the recent paper [28]. Orlov’s notion is much stronger than ours, in that it is quasi-isomorphic to an ordinary simple algebra. A simple Artinian dg algebra in our sense or even a dg skew-field of finite period is not determined up to quasi-isomorphism by its homology algebra. Examples of non-formal A∞ algebras whose homology algebras are skew-fields are not hard to construct (cf. for example even Moore algebras of [23]). This situation arises also in stable homotopy theory where Eilenberg–MacLane spectra of graded fields and Morava K-theories have equivalent simple homotopy categories of modules yet are very different in many ways, e.g. they have inequivalent homotopy categories of bimodules.

3 Rank functions on perfect derived categories of ordinary rings

We start with a short review of ordinary Sylvester rank functions, following [30]. This theory and its applications motivated us to develop its derived analogue.

3.1 Sylvester rank functions

Let A be a ring. Denote by fp⁑(A) the category of finitely presented A-modules, and by π–―π—‹π—ˆπ—ƒβ‘(A) the subcategory of finitely generated projective A-modules.

Definition 3.1.

A Sylvester morphism rank function on A associates to each morphism f of π–―π—‹π—ˆπ—ƒβ‘(A) a rankρ⁒(f)βˆˆβ„β‰₯0. It is required to satisfy the following conditions.

  1. Normalization: it holds

    (m1)ρ⁒(idA)=1.

  2. Additivity: for any morphisms f and g, we have

    (m2)ρ⁒(fβŠ•g)=ρ⁒(f)+ρ⁒(g).

  3. Triangular inequality: for all morphisms f, g and h such that g and h share the same domain and f and h share the same codomain, we have

    (m3)ρ⁒(fh0g)β‰₯ρ⁒(f)+ρ⁒(g).

  4. Ideal condition: for any morphisms f and g for which the composition gf is defined, we have

    (m4)ρ⁒(g⁒f)≀ρ⁒(f) and ρ⁒(g⁒f)≀ρ⁒(g).

Definition 3.2.

A Sylvester module rank function on A associates to each finitely presented A-module M a rankρ⁒(M)βˆˆβ„β‰₯0 such that

  1. Normalization: it holds

    (o1)ρ⁒(A)=1.

  2. Additivity: for all finitely presented modules M and N, we have

    (o2)ρ⁒(MβŠ•N)=ρ⁒(M)+ρ⁒(N).

  3. Triangle inequality: for any exact sequence

    L→M→N→0

    of finitely presented A-modules, we have

    (o3)ρ⁒(N)≀ρ⁒(M)≀ρ⁒(L)+ρ⁒(N).

Let

P→𝑓Qβ†’Mβ†’0

be an exact sequence of A-modules, with P,Qβˆˆπ–―π—‹π—ˆπ—ƒβ‘(A), so that M∈fp⁑(A). The formulas

ρ⁒(f)=ρ⁒(Q)-ρ⁒(M) and ρ⁒(M)=ρ⁒(idQ)-ρ⁒(f),

yield a one-to-one correspondence between Sylvester morphism rank functions on A and Sylvester object rank functions on A. So we just call the resulting function defined on both morphisms and objects a Sylvester rank function on A.

Example 3.3.

Let A be a skew-field; then the category fp⁑(A) is just the category of finite-dimensional vector spaces over A. It is clear that fp⁑(A) has a unique Sylvester rank function given by the dimension of a vector space; a triangulated analogue of this obvious result is Corollary 2.21.

Remark 3.4.

A Sylvester rank function on A may be recovered from its values on homomorphisms f:Am→An between free modules, since every finitely presented module is isomorphic to the cokernel of such a map. By additivity, one may even restrict to the case of square matrices, i.e. the case m=n.

Let S be a simple Artinian ring. Then S is isomorphic to a matrix algebra Mn⁒(K) over a skew-field K. Every object of fp⁑(S)=π–―π—‹π—ˆπ—ƒβ‘(S) is a finite direct sum of copies of the simple module V=Kn, and the unique Sylvester rank function on fp⁑(S) takes the value 1n on V.

Given a homomorphism A→B, we obtain a right exact functor

fp⁑(A)β†’fp⁑(B),M↦BβŠ—AM.

It is easy to confirm that any Sylvester rank function on B pulls back via this functor to a Sylvester rank function on A. In particular, any homomorphism Aβ†’S into a simple Artinian ring determines a canonical Sylvester rank function on A. Two homomorphisms Aβ†’S and Aβ†’Sβ€² into simple Artinian rings are considered equivalent if they become equal after composing with homomorphisms Sβ†’Sβ€²β€² and Sβ€²β†’Sβ€²β€² into a third simple Artinian ring. Equivalent homomorphisms clearly determine the same Sylvester rank function on fp⁑(A).

Theorem 3.5.

There is a one-to-one correspondence between equivalence classes of homomorphisms of A into simple Artinian rings S and Q-valued Sylvester rank functions on A taking the value 0 or 1 on A/r⁒A for any integer r. In this correspondence S is a skew-field if and only if the corresponding rank function is Z-valued.

Proof.

See [30, Chapter 7, Theorems 7.12 and 7.14]. ∎

Remark 3.6.

Let ρ be a Sylvester rank function on A. Since 0≀ρ⁒(A/r⁒A)≀1 for any integer r, the condition on ρ⁒(A/r⁒A) is automatically satisfied if ρ is β„€-valued. The same is true for an arbitrary Sylvester rank function on A when A is an algebra over a field, for then A/r⁒A is isomorphic to either A or 0. See [30, p. 121] for a discussion on this issue.

We already explained how to obtain a rank function from a homomorphism into a simple Artinian ring S, and since S is an algebra over a field, namely its center, it will satisfy the additional condition, cf. Remark 3.6. The reverse construction in the case of an integral Sylvester rank function ρ is described as follows. The Cohn localization of Aρ of A with respect to all morphisms f:Pβ†’Q in π–―π—‹π—ˆπ—ƒβ‘(A) such that ρ⁒(f)=ρ⁒(P)=ρ⁒(Q), can be shown to be local with residue skew-field, say, K, cf. [30, Theorem 7.5]. The desired homomorphism is the composition Aβ†’Aρ→K.

3.2 Derived rank functions

Let A be an ordinary ring. We consider periodic rank functions on the triangulated category 𝖯𝖾𝗋𝖿⁒(A). A d-periodic rank function ρ on 𝖯𝖾𝗋𝖿⁒(A) is called normalized if ρ⁒(A)=1. We will denote by NRkd⁑(𝖯𝖾𝗋𝖿⁒(A)) and Syl⁑(A) the set of normalized d-periodic rank functions on 𝖯𝖾𝗋𝖿⁒(A) and Sylvester rank functions on fp⁑(A), respectively. In this subsection we give a comparison between NRkd⁑(𝖯𝖾𝗋𝖿⁒(A)) and Syl⁑(A).

Theorem 3.7.

Let A be an ordinary ring and d∈{1,2,…}βˆͺ{∞}. The restriction of any normalized d-periodic rank function on Perf⁒(A) to Proj⁑(A) is a Sylvester morphism rank function on A. In the case d=∞, this determines a one-to-one correspondence between normalized ∞-periodic rank functions on Perf⁒(A) and Sylvester rank functions on A.

Proof.

By additivity, any normalized periodic rank function takes values in ℝ on any object in π–―π—‹π—ˆπ—ƒβ‘(A), and therefore on any morphism in π–―π—‹π—ˆπ—ƒβ‘(A), by Lemma 2.10. The first statement then follows from a straightforward unraveling of definitions. The stronger statement for d=∞ is a consequence of the Proposition 3.9 below. ∎

Lemma 3.8.

Let ρ be a normalized d-periodic rank function on Perf⁒(A). Let X be a bounded complex of A-modules with finitely generated projective terms Xn. Then

ρ⁒(X)β‰€βˆ‘nβˆˆβ„€Οβ’(Xn)⁒qn.

Proof.

This follows by induction on the length of X from (Op3). ∎

Proposition 3.9.

Let A be an ordinary ring. Any normalized ∞-periodic rank function ρ on Perf⁒(A) is determined by its restriction to Proj⁑(A). More precisely, given a bounded complex X={Xn} of finitely generated projective A-modules with a differential dn:Xnβ†’Xn-1, we have

(3.1)ρ⁒(X)=βˆ‘nβˆˆβ„€(ρ⁒(Xn)-ρ⁒(dn)-ρ⁒(dn+1))⁒qn.

Moreover, any Sylvester rank function on A extends (uniquely) via this formula to a rank function ρ on Perf⁒(A). This extension will be referred to as the derived rank function on Perf⁒(A).

Proof.

Let us prove that given a Sylvester rank function ρ, formula (3.1) gives a ∞-periodic rank function on 𝖯𝖾𝗋𝖿⁒(A). Note that a bounded exact complex

X={⋯←dnXn←dn+1β‹―}

of finitely generated projective A-modules splits as a direct sum of elementary exact complexes of length two. It is clear that for such a complex one has ρ⁒(Xn)=ρ⁒(dn)+ρ⁒(dn+1) and it follows that ρ⁒(X)=0.

Given a quasi-isomorphism f:Xβ†’Y of bounded complexes of finitely generated projective A-modules, we can factor it as X⁒→𝑖⁒Z⁒→𝑝⁒Y where Z is the mapping cylinder of f. Since X and Y are chain deformation retracts of Z, it follows that ρ⁒(X)=ρ⁒(Z)=ρ⁒(Y). Therefore, ρ is well-defined on 𝖯𝖾𝗋𝖿⁒(A).

It is clear that the function ρ on 𝖯𝖾𝗋𝖿⁒(A) given by (3.1) satisfies axioms (Op1) and (O2) of the rank function. To prove (Op3), consider an exact triangle Xβ†’Yβ†’Z↝ in 𝖯𝖾𝗋𝖿⁒(A). Without loss of generality, we can assume that the map Xβ†’Y is a cofibration of complexes of A-modules, in other words, it is a degree-wise split injection, and Zβ‰…X/Y is a bounded complex of finitely generated projective A-modules. By property (o2) of a Sylvester rank function, we have for all nβˆˆβ„€,

(3.2)ρ⁒(Yn)=ρ⁒(Xn)+ρ⁒(Zn).

Denote by dnX,dnY,dnZ,nβˆˆβ„€ the differentials in the complexes X,Y and Z, respectively. Then by (m3) we have for all nβˆˆβ„€,

(3.3)ρ⁒(dnY)β‰₯ρ⁒(dnX)+ρ⁒(dnZ).

From (3.2) we deduce

ρ(X)-ρ(Y)+ρ(Z)=βˆ‘nβˆˆβ„€(ρ(dnY)+ρ(dn+1Y)-ρ(dnX)-ρ(dn+1X)
   -ρ(dnZ)-ρ(dn+1Z))qn.

Using (3.3), we conclude that the polynomial ρ⁒(X)-ρ⁒(Y)+ρ⁒(Z) satisfies the condition of Lemma 2.11. It follows that ρ⁒(X)-ρ⁒(Y)+ρ⁒(Z)=(q+1)⁒ϕ with Ο•βˆˆβ„β‰₯0⁒(d) as required.

Given a morphism f:Pβ†’Q in π–―π—‹π—ˆπ—ƒβ‘(A), (3.1) gives

ρ⁒(cone⁑(f))=(ρ⁒(P)-ρ⁒(f))⁒q+(ρ⁒(Q)-ρ⁒(f)),

consistent with (2.3). It follows that the ∞-periodic rank function on 𝖯𝖾𝗋𝖿⁒(A) defined by (3.1) does in fact extend the original Sylvester rank function ρ on π–―π—‹π—ˆπ—ƒβ‘(A).

We will now prove that, conversely, any ∞-periodic rank function ρ on 𝖯𝖾𝗋𝖿⁒(A) is necessarily given by formula (3.1). Given an interval I of integers, denote by XIβˆˆπ–―π–Ύπ—‹π–Ώβ’(A) the brutal truncation of X concentrated in degrees n∈I, i.e. (XI)n=Xn for n∈I, and (XI)n=0 otherwise, with differential dn equal to that of X for all n such that {n,n-1}∈I. Since the claimed formula for ρ⁒(X) is consistent with condition (Op1), it suffices to prove that the constant term of ρ⁒(X) is equal to ρ⁒(X0)-ρ⁒(d0)-ρ⁒(d1).

From the standard exact triangle

Ξ£-1⁒X[2,∞)⁒→ϕ⁒X(-∞,1]β†’Xβ†’X[2,∞)↝

we obtain by (Op3) that

(3.4)ρ⁒(X(-∞,1])-ρ⁒(X)+ρ⁒(X[2,∞))=(q+1)⁒ρ⁒(Ο•).

Next note that by Lemma 3.8, the constant terms of ρ⁒(X[2,∞)) and of ρ⁒(Ξ£-1⁒X[2,∞)) are zero, and then the constant term of ρ⁒(Ο•) is likewise zero since ρ⁒(Ο•)≀ρ⁒(Ξ£-1⁒X[2,∞)) by Lemma 2.10. It follows from equation (3.4) that the constant terms of ρ⁒(X) and ρ⁒(X(-∞,1]) coincide.

Arguing similarly with the exact triangle

β†’X(-∞,1]β†’X[-1,1]→Σ⁒X(-∞,-2]↝,

we find that the constant terms of ρ⁒(X[-1,1]) and ρ⁒(X(-∞,1]) coincide and thus, also coincide with the constant term of ρ⁒(X).

Write

ρ⁒(X[-1,1])=a⁒q-1+b+c⁒q,
ρ⁒(X[-1,0])=a′⁒q-1+bβ€²,
ρ⁒(X[0,1])=bβ€²β€²+c′′⁒q.

From the two exact triangles

X[-1,0]β†’X[-1,1]β†’Ξ£X1↝ and Σ-1X-1β†’X[-1,1]β†’X[0,1]↝,

and the triangle inequality (Op3) we find that a=aβ€², c=cβ€²β€² and

ρ⁒(X[-1,1])q=-1-ρ⁒(Σ⁒X-1)q=-1+ρ⁒(X[0,1])q=-1=0.

Furthermore, from the exact triangle

X0β†’X[0,1]→Σ⁒X1↝

we obtain ρ⁒(X[0,1])q=0=ρ⁒(X0)-ρ⁒(d1) and similarly

q⁒ρ⁒(X[-1,0])q=0=ρ⁒(X-1)-ρ⁒(d0).

We then calculate

b=ρ⁒(X[-1,1])q=-1+a+c
=ρ⁒(Σ⁒X-1)q=-1+ρ⁒(X[0,1])q=-1+aβ€²+cβ€²β€²
=-ρ⁒(X-1)+bβ€²β€²+aβ€²
=ρ⁒(X[0,1])q=0+(q⁒ρ⁒(X[-1,0]))q=0-ρ⁒(X-1)
=ρ⁒(X0)-ρ⁒(d1)+ρ⁒(X-1)-ρ⁒(d0)-ρ⁒(X-1)
=ρ⁒(X0)-ρ⁒(d1)-ρ⁒(d0),

as desired. ∎

Remark 3.10.

Let A be an ordinary ring. Let d<∞. Recall from Section 2.1 that Ο€βˆž,d:℀⁒(∞)→℀⁒(d) is the natural reduction map. It allows one to assign to any normalized ∞-periodic rank function ρ on 𝖯𝖾𝗋𝖿⁒(A) the normalized d-periodic rank function Ο€βˆž,d∘ρ on 𝖯𝖾𝗋𝖿⁒(A). This is a injective map with a canonical splitting: given a normalized d-periodic rank function ρ, restrict it to a Sylvester rank function on A, and then take the unique extension to a normalized (∞-periodic) rank function ρ^ on 𝖯𝖾𝗋𝖿⁒(A) provided by Proposition 3.9:

Suppose that Ξ²:Aβ†’K is a map in the homotopy category of dg rings from a ring A to a d-periodic skew-field K. Let ρ=ρβ be the normalized d-periodic rank function on 𝖯𝖾𝗋𝖿⁒(A) obtained via pullback along Ξ² of the unique normalized d-periodic rank function on the perfect derived category of K. Then ρ^ may be described as follows. The homomorphism H⁒(Ξ²):Aβ†’K obtained by passing to homology algebras factors as the composition of a homomorphism Ξ²0:Aβ†’K0 and the inclusion K0β†ͺK, and we have ρ^=ρβ0, the pullback along Ξ²0 of the unique normalized (∞-periodic) rank function on 𝖯𝖾𝗋𝖿⁒(K0). Moreover, ρ=Ο€βˆž,d∘ρ^ if and only if (the homotopy class of maps) Ξ² is realized as an actual ring homomorphism Aβ†’K (and then, it must necessarily be H⁒(Ξ²)).

4 Further properties of rank functions

As before, we assume that π’ž is a triangulated category with a d-periodic rank function ρ. We will examine the behavior of ρ with respect to functors in or out of π’ž. Given another triangulated category π’žβ€² and an exact functor F:π’žβ€²β†’π’ž, the pullback ρ′=F*⁒(ρ), assigning to each morphism f of π’žβ€² the rank ρ′⁒(f)=ρ⁒(F⁒(f)), is clearly also a d-periodic rank function on π’žβ€². The pullback of an integral rank function is integral.

On the other hand, a pushforward of ρ is not always possible; later on we will consider the case of a pushforward along a Verdier quotient.

Lemma 2.10 motivates the following definitions.

Definition 4.1.

Let ρ be a d-periodic rank function on π’ž. We say that a morphism f:Xβ†’Y in π’ž is

  1. β€’

    left ρ-full if ρ⁒(f)=ρ⁒(X),

  2. β€’

    right ρ-full if ρ⁒(f)=ρ⁒(Y),

  3. β€’

    ρ-full if it is left full and right full.

Lemma 4.2.

Let ρ be a d-periodic rank function on C.

  1. (i)

    Let

    X→𝑓Y→𝑔Z↝

    be an exact triangle in π’ž, with a connecting homomorphism h:Ξ£-1⁒Zβ†’X. Then:

    1. –

      f if left ρ -full if and only if g is right ρ -full if and only if ρ⁒(h)=0,

    2. –

      f is ρ -full if and only if ρ⁒(Z)=0.

  2. (ii)

    Let f:Xβ†’Y and g:Yβ†’Z be morphisms in π’ž. If f is ρ -full, then ρ⁒(g⁒f)=ρ⁒(g). If g is ρ -full, then ρ⁒(g⁒f)=ρ⁒(f).

Proof.

The first part follows directly from Axiom (M3). For the second part, consider the exact triangle

cone⁑(f)β†’cone⁑(g⁒f)β†’cone⁑(g)↝,

given by the octahedral axiom. Suppose that f is ρ-full. Then ρ⁒(cone⁑(f))=0 and therefore ρ⁒(cone⁑(g⁒f))=ρ⁒(cone⁑(g)). We deduce that

(q+1)⁒(ρ⁒(g⁒f)-ρ⁒(g))=(ρ⁒(X)-ρ⁒(cone⁑(g⁒f))+q⁒ρ⁒(Z))
-(ρ⁒(Y)-ρ⁒(cone⁑(g))+q⁒ρ⁒(Z))=0.

Since, by (M5), ρ⁒(g⁒f)-ρ⁒(f)β‰₯0, we deduce that ρ⁒(g⁒f)-ρ⁒(g)=0, as desired. The argument when g is ρ-full is similar. ∎

Corollary 4.3.

Let Ξ± be an invertible element in k and f:Xβ†’Y is a morphism in C. Then ρ⁒(Ξ±β‹…f)=ρ⁒(f).

Proof.

The multiplication by Ξ± is an automorphism of X and so, it is a ρ-full map. Then the conclusion follows from Lemma 4.2 (2). ∎

Corollary 4.4.

The full subcategory on

Ker⁑(ρ):={Xβˆˆπ’ž:ρ⁒(X)=0}

is a thick subcategory of C. Moreover, ρ descends to a d-periodic rank function ρ¯ on the Verdier quotient C/U by any thick subcategory UβŠ†Ker⁑(ρ), and the pullback of ρ¯ to C is ρ. The obtained d-periodic rank function on C/Ker⁑(ρ) is object-faithful.

Proof.

The category π’ž/𝒰 is the localization of π’ž at a set of ρ-full morphisms, and so any morphism f:Mβ†’N in C/𝒰 can be written as a roof

Mβ†’β„ŽL←𝑔N,

where h and g are morphisms in C and cone⁑(g) is an object of 𝒰 (so, in particular, g is ρ-full). We set ρ¯⁒(f):=ρ⁒(h); then part (2) of Lemma 4.2 ensures that ρ¯ is a well-defined function on morphisms of π’ž/𝒰. Clearly, ρ¯ pulls back to ρ, and satisfies (Mp1) and (M2). That ρ¯ obeys (M3)–(M5) follows from the fact that every triangle or diagram of the form Xβ†’Yβ†’Z or Xβ†’Y←Zβ†’W in π’ž/𝒰 is isomorphic to the image of a triangle or diagram of the same form in π’ž.

The object-faithfulness of ρ¯ on π’ž/Ker⁑(ρ) is clear. ∎

Remark 4.5.

Let d and dβ€² be positive integers with d divisible by dβ€²; clearly the only element in ℝ⁑(d) mapping to zero under the reduction map ℝβ‰₯0⁑(d)→ℝβ‰₯0⁑(dβ€²) is zero. Thus, given a d-periodic rank function ρ and the corresponding dβ€²-periodic rank function ρ′, their kernels coincide. In particular, if one is interested in the thick subcategories that are kernels of rank functions, it results in no loss of generality to consider only ordinary (i.e. 1-periodic) rank functions.

Lemma 4.6.

Let f and g be morphisms in C with the same domain and codomain. Then

ρ⁒(f+g)≀ρ⁒(f)+ρ⁒(g).

Proof.

We have a factorization

(f+g)=(11)⁒(f00g)⁒(11).

The desired inequality follows from axioms (M5) and (M2). ∎

Corollary 4.7.

For any pair of objects X,Y of C, define

Homπ’žΟβ‘(X,Y)={f∈Homπ’žβ‘(X,Y):ρ⁒(f)=0}.

This is an ideal of morphisms in C, by (M5) and by Lemma 4.6.

Lemma 4.8.

Let f and g be morphisms of C with ρ⁒(g)=0. Then ρ⁒(f+g)=ρ⁒(f).

Proof.

By Lemma 4.6 we have ρ⁒(f+g)≀ρ⁒(f)+ρ⁒(g)=ρ⁒(f). Taking into account that ρ⁒(-g)=ρ⁒(g) by Corollary 4.3 we have

ρ⁒(f)=ρ⁒(f+g-g)≀ρ⁒(f+g)+ρ⁒(g)=ρ⁒(f+g)

which implies the desired equality. ∎

Lemma 4.9.

Let ρ be a prime rank function on a triangulated category C supplied with a generator X and denote by Lρ⁒(X) the image of X in the Verdier quotient C/Ker⁑(ρ). Then the graded algebra Endβˆ™β‘(Lρ⁒(X)) is local.

Proof.

Recall that the induced rank function ρ¯ on π’ž/Ker⁑(ρ) is object-faithful. The set

I:={f∈End⁑(Lρ⁒(X)):ρ¯⁒(f)=0}

is an ideal in Endβˆ™β‘(Lρ⁒(X)) by (M5). If f∈Endβˆ™β‘(Lρ⁒(X)) is not in I, then ρ¯⁒(f)=1, in other words f:Lρ⁒(X)β†’Lρ⁒(X) is a ρ¯-full map. By Lemma 4.2 the cone of f has zero rank and so is zero in π’ž/Ker⁑(ρ); thus f is invertible. Since any element in Endβˆ™β‘(Lρ⁒(X)) not in I is invertible, it follows that I is a maximal ideal and Endβˆ™β‘(Lρ⁒(X)) is local. ∎

So any prime rank function on a triangulated category with a generator X gives rise to a map Endβˆ™β‘(X)β†’Fρ, where Fρ is the (graded, skew) residue field of Endβˆ™β‘(Lρ⁒(X)).

Recall that the idempotent completion of an additive category π’ž is a category π’ž~ whose objects are pairs (X,e) where X is an object of π’ž and e is an idempotent endomorphism of X; a morphism between two such pairs (X,e) and (Y,t) is a morphism f:Xβ†’Y in π’ž such that e⁒f=f=f⁒t. The idempotent completion of a triangulated category is known to be triangulated and it possesses a universal property with respect to exact functors into idempotent complete triangulated categories [2].

Lemma 4.10.

Any d-periodic rank function ρ:Cβ†’R⁒(d) on a triangulated category C extends uniquely to a d-periodic rank function on its idempotent completion.

Proof.

Given a map f:(X,e)β†’(Y,t) in π’ž~, define its rank as the rank of f:Xβ†’Y in π’ž. Note that under this definition the rank of the identity morphism of (X,e) is ρ⁒(e). All axioms of the rank function except (M3) are obvious. To check (M3), consider an exact triangle in π’ž~:

(X,e)⁒→𝑓⁒(Y,t)⁒→𝑔⁒(Z,k)↝

and note that it is a direct summand in π’ž~ of the following exact triangle in π’ž

(4.1)(X,1)⁒→𝑓⁒(Y,1)⁒→g′⁒(Zβ€²,1)↝.

Moreover, there is the following isomorphism in π’ž~:

(Zβ€²,1)β‰…(Z,k)βŠ•(Σ⁒X,1-e)βŠ•(Y,t)

and the morphism gβ€²:(Y,1)β†’(Zβ€²,1) can be represented as the composite map

from which it follows that ρ⁒(gβ€²)=ρ⁒(g)+ρ⁒(1-t). The exact triangle (4.1) gives

ρ⁒(t)+ρ⁒(1-t)=ρ⁒((Y,t)βŠ•(Y,1-t))
=ρ⁒(Y)
=ρ⁒(f)+ρ⁒(gβ€²)
=ρ⁒(f)+ρ⁒(g)+ρ⁒(1-t)

and it follows that ρ⁒(t)=ρ⁒(f)+ρ⁒(g) as desired. ∎

Assume now that π’ž has a generator X and ρ is a prime rank function. An object of π’ž which is a finite coproduct of shifted copies of X will be called graded free. Note that a map between graded free objects could be written as a rectangular matrix M whose entries are graded endomorphisms of X, so we can speak of the rank ρ⁒(M) of M, generalizing the familiar notion in linear algebra. We will denote by MΒ― the matrix obtained from M by passing to the graded residue field Fρ of the graded local algebra EndLρ⁑(X) and its usual graded rank by rankFρ⁑(MΒ―).

Proposition 4.11.

We have ρ⁒(M)=rankFρ⁑(M¯).

Proof.

We may assume that ρ is object-faithful, so that Endβˆ™β‘(X) is graded local, otherwise replace π’ž with the Verdier quotient π’ž/Ker⁑(ρ). By the usual row reduction process, M=E⁒R, where E is an invertible matrix and R is a rectangular matrix such that RΒ― is of the form (I000), where I is an identity matrix. Here E is a product of permutation matrices and upper and lower triangular matrices with invertible elements on the diagonal. So it suffices to prove that ρ⁒(R)=rankFρ⁑(RΒ―). This follows from Lemma 4.8. ∎

Corollary 4.12.

Let M be a matrix over Endβˆ™β‘(X) representing a morphism between graded free objects. The M has a (square) ρ-full submatrix N with ρ⁒(N)=ρ⁒(M).

Proof.

This follows from the corresponding result for the graded ranks of matrices over a graded skew-field. ∎

5 Derived localization of differential graded algebras

Let A be a dg algebra assumed, without loss of generality, to be cofibrant, and let Ο„ be a thick subcategory of 𝖯𝖾𝗋𝖿⁒(A).

Definition 5.1.

A dg A-module M is called Ο„-local if RHomA⁑(X,M)=0 for any XβˆˆΟ„.

Lemma 5.2.

Let B be a dg algebra supplied with a map Aβ†’B. Then B is Ο„-local if and only if BβŠ—ALX≃0 for any XβˆˆΟ„.

Proof.

Note that

RHomA⁑(X,B)≃RHomB⁑(XβŠ—ALB,B),

thus HomB⁑(XβŠ—ALB,B)≃0 implies RHomA⁑(X,B)≃0. Conversely, since X is a perfect dg A-module, XβŠ—ALB is a perfect dg B-module. Note that a perfect dg B-module (being quasi-isomorphic to its double B-dual) is quasi-isomorphic to zero if and only if its B-dual is quasi-isomorphic to zero. Therefore 0≃RHomA⁑(X,B)≃RHomB⁑(XβŠ—ALB,B) implies that XβŠ—ALB≃0. ∎

Definition 5.3.

The derived localization of A with respect to Ο„ is a dg algebra Lτ⁒(A) together with a map Aβ†’Lτ⁒(A) making it a Ο„-local A-module and such that for any dg algebra B and a map f:Aβ†’B making B a Ο„-local A-module, there is a unique up to homotopy map Lτ⁒(A)β†’B making the following diagram commutative in the homotopy category of dg algebras:

Remark 5.4.

Let s∈A be an n-cycle of a dg algebra A for nβˆˆβ„€, and let A/s be the homotopy cofiber of the left multiplication map Ξ£n⁒Aβ†’A. Then A/s is a perfect A-module and the localization with respect to the thick subcategory γ€ˆA/s〉 generated by it, exists and is given explicitly by the formula

Ls⁒A:=Lγ€ˆA/s〉⁒(A)=A*𝐀⁒[s]Lπ€β’γ€ˆs,s-1〉,

cf. [5]. We will now generalize this result to an arbitrary thick subcategory.

Theorem 5.5.

For any dg algebra A and any thick subcategory Ο„ of Perf⁒(A), the derived localization Lτ⁒(A) exists and is unique up to homotopy.

Proof.

Let us first suppose that Ο„ is generated by a single perfect object Xβˆˆπ–¬π—ˆπ–½β’-⁒A, assumed without loss of generality to be cofibrant. Denote by E the dg algebra of endomorphisms of the A-module AβŠ•X. Let e be the element in E given by the projection Eβ†’A along X followed by the inclusion of A into E. Then e is a zero-cocycle and an idempotent of E. We will show that the desired (derived) localization Lτ⁒(A) may be constructed as Le⁒E, the (derived) localization of E at e, cf. Remark 5.4. We have the following commutative diagram of dg categories:

(5.1)

Here F⁒(M)=HomA⁑(AβŠ•X,M), for Mβˆˆπ–¬π—ˆπ–½β’-⁒A, E/e is the cofiber of the left multiplication by e on E, {X} the full dg subcategory of π–¬π—ˆπ–½β’-⁒A containing X and closed with respect to arbitrary coproducts, homotopy cofibers and passing to quasi-isomorphic modules and similarly {E/e} is the full dg subcategory of π–¬π—ˆπ–½β’-⁒A containing E/e, having arbitrary coproducts, homotopy cofibers and closed with respect to passing to quasi-isomorphic modules. Note that E/e is quasi-isomorphic as an E-module to (1-e)⁒EβŠ•Ξ£β’(1-e)⁒E and

F⁒(X)=HomA⁑(AβŠ•X,X)β‰…XβŠ•HomA⁑(X,X)β‰…(1-e)⁒X.

Therefore the image of {X} under F is {E/e} and the commutativity of 5.1 indeed holds.

Since AβŠ•X is a compact generator of 𝖣⁒(A), the functor F is a quasi-equivalence, as well as its restriction {X}β†’{E/e}.

Let us construct (a homotopy class of) a map of dg algebras Aβ†’Le⁒E. Note that the functor F could be viewed as tensoring over A with the left A-module HomA⁑(AβŠ•X,A)β‰…e⁒E. Composing F with the localization functor into π–¬π—ˆπ–½β’-⁒Le⁒E, we obtain the functor

G:π–¬π—ˆπ–½β’-⁒Aβ†’π–¬π—ˆπ–½β’-⁒Le⁒E,M↦MβŠ—A(e⁒EβŠ—ELe⁒E)β‰…MβŠ—Ae⁒Le⁒E.

Since G is a dg functor, there is an induced (homotopy class of a) map of dg algebras

f:Aβ‰…EndA⁑(A)β†’EndLe⁒E⁑(e⁒Le⁒E)≃EndLe⁒E⁑(Le⁒E)β‰…Le⁒E.

The commutative diagram (5.1) implies

XβŠ—ALe⁒E≃G⁒(X)≃F⁒(X)βŠ—ELe⁒E≃0,

meaning that Le⁒E is γ€ˆX〉-local.

Let us now prove that Le⁒E has the required universal property with respect to maps from A into γ€ˆX〉-local algebras. To this end, let A~:=A×𝐀. Then A~ is a dg algebra with the differential d⁒(a,x)=dA⁒(a) and the product (a,x)β‹…(aβ€²,y)=(a⁒aβ€²,x⁒y) for a,aβ€²βˆˆA,x,y∈𝐀. There is a dg algebra map f~:A~β†’E given by the diagonal embedding (a,x)↦(la,1-e) where la is the action of A on itself by the left multiplication by a. Note that f~ could also be described as a map

A~β‰…EndA~⁑(A~)β†’EndLe⁒E⁑(Le⁒E)β‰…E

induced by the functor -βŠ—A~E:π–¬π—ˆπ–½-A~β†’π–¬π—ˆπ–½-E.

The idempotent (1,0)∈A~ is mapped to e∈E under f~, and, slightly abusing notation, we will denote it also by e. We obtain an induced map on derived localizations Le⁒(A~)β†’Le⁒(E). This produces a homotopy class of dg algebra maps Aβ†’Le⁒(E) since clearly Le⁒(A~)≃A. This is the same map (up to homotopy) as f as implied by the following diagram of dg categories and dg functors, commutative up to a natural isomorphism:

Let B be a dg algebra supplied with a map g:Aβ†’B and γ€ˆX〉-local (so that XβŠ—AB≃0); we need to show that g extends uniquely (up to homotopy) to a map Lτ⁒(A):=Le⁒Eβ†’B. Consider the following diagram of dg algebras, commutative up to homotopy (excluding the dotted arrow):

(5.2)

Here the arrow from E to B is induced by the functor -βŠ—AB:π–¬π—ˆπ–½-Aβ†’π–¬π—ˆπ–½-B. By the universal property of the localization Le⁒E the dotted arrow, making commutative the upper triangle of (5.2), exists. It follows that the lower triangle of (5.2) commutes upon restriction to A~. However since the map A~β†’A≃Le⁒E becomes an isomorphism upon inverting e, and the dg algebra B is e-inverting, we conclude that the lower triangle of (5.2) is commutative. Conversely, a similar argument shows that any dotted arrow, making the lower triangle commutative, makes the upper triangle commutative and is, therefore, unique.

So, the theorem is proved under the assumption that Ο„ is generated by a single perfect object X. It is easy to see that for two perfect objects X,Yβˆˆπ–¬π—ˆπ–½β’-⁒A, we have

Lγ€ˆXβŠ•Y〉⁒(A)≃Lγ€ˆX〉⁒(A)*ALLγ€ˆY〉⁒(A)

as both sides satisfy the required universal property with respect to any dg algebra B local with respect to both X and Y. Letting Xs,s∈S be a collection of compact generators of Ο„ indexed by a set S, we can set AΟ„:=∐A,s∈SLLγ€ˆXS〉⁒(A), the (derived) coproduct over A of derived localizations Lγ€ˆXS〉⁒(A). ∎

5.1 Derived localization of A-algebras

Recall that if C is a closed model category and X is an object of C, then the undercategory of X is the category X↓C with objects Y∈C supplied with a map Xβ†’Y and morphisms being obvious commutative triangles in C. The undercategory of X inherits the structure of a closed model category from C; in the case when X is cofibrant, this undercategory is homotopy invariant in the sense that for any other cofibrant Xβ€² and a weak equivalence Xβ†’Xβ€² the undercategories X↓C and X′↓C are Quillen equivalent.

Lemma 5.6.

For a Ο„-local A-algebra B there exists a map of A-algebras Lτ⁒(A)β†’B, unique up to homotopy in the undercategory A↓dgAlg.

Proof.

As in the proof of Theorem 5.5 we first consider the case when Ο„ is generated by a single perfect A-module X and arguing similarly with diagram (5.2), we conclude that there exists a map g:LΟ„β†’B making the diagram of dg algebras (which is a fragment of the diagram (5.2))

(5.3)

homotopy commutative in the category of dg algebras. Thus, g could be viewed as a map in the undercategory A↓𝖽𝗀𝖠𝗅𝗀. The uniqueness of g is proved similarly: suppose that there exists another map of A-algebras gβ€²:Lτ⁒(A)β†’B making the (5.3) homotopy commutative. Then g and gβ€² are homotopic in A↓𝖽𝗀𝖠𝗅𝗀 if and only if they are homotopic in A~↓𝖽𝗀𝖠𝗅𝗀 and this, in turn, is equivalent to them being homotopic in E↓𝖽𝗀𝖠𝗅𝗀. But they are indeed homotopic in E↓𝖽𝗀𝖠𝗅𝗀 by the defining property of the derived localization Le⁒E, cf. [5, Definition 3.3]. The proof is finished as that of Theorem 5.5. ∎

Remark 5.7.

Another way to formulate Lemma 5.6 is to say that Lτ⁒(A) is the initial object of the subcategory of the homotopy category of A↓𝖽𝗀𝖠𝗅𝗀 consisting of Ο„-local A-algebras. It may appear that this statement and its proof are just rephrasing the corresponding parts of the statement and proof of Theorem 5.5. The substantive difference is that two maps of A-algebras may be homotopic as maps of dg algebras but not as maps in the undercategory A↓𝖽𝗀𝖠𝗅𝗀. We will discuss this discrepancy in more detail below.

Let now B be an A-algebra, assumed, without loss of generality, to be cofibrant in A↓𝖽𝗀𝖠𝗅𝗀 (meaning that the given map Aβ†’B is a cofibration of dg algebras). The induction functor -βŠ—AB:π–¬π—ˆπ–½-Aβ†¦π–¬π—ˆπ–½-B takes 𝖯𝖾𝗋𝖿⁒(A) into 𝖯𝖾𝗋𝖿⁒(B) and (abusing the notation slightly) we will denote by the image of the thick subcategory Ο„βˆˆπ–―π–Ύπ—‹π–Ώβ’(A) under this functor, by the same symbol Ο„.

Proposition 5.8.

We have a natural isomorphism

Lτ⁒(B)β‰…Bβˆ—ALLτ⁒(A)

in the homotopy category of A↓dgAlg.

Proof.

This is similar to [5, Lemma 3.7]. There is a Quillen adjunction

A↓𝖽𝗀𝖠𝗅𝗀⇄B↓𝖽𝗀𝖠𝗅𝗀

with the left adjoint -βˆ—AB and the right adjoint being the restriction functor. It is easy to see that this adjunction restricts to an adjunction between Ο„-local A-algebras and Ο„-local B-algebras and the corresponding homotopy categories. Since left adjoints preserve initial objects, the desired statement follows from Lemma 5.6. ∎

Corollary 5.9.

The map of dg algebras Lτ⁒(A)β†’Lτ⁒(A)*ALLτ⁒(A) given by the inclusion of the either factor is a quasi-isomorphism.

Recall from [16, Chapter 5], for any closed model category C, the notion of a derived mapping space, a simplicial set Map⁑(X,Y), where X,Y∈C, generalizing the usual simplicial mapping space in a simplicial model category. Recall also from [26] that a morphism f:Xβ†’Y in a model category is said to be a homotopy epimorphism if for any object Z, the induced morphism Map⁑(Y,Z)β†’Map⁑(X,Z) is an injection on connected components of the corresponding simplicial sets and an isomorphism on homotopy groups for any choice of a base point. Then we have the following generalization of [5, Proposition 3.17], which follows, as in op.cit. from Corollary 5.9.

Corollary 5.10.

The localization map Aβ†’Lτ⁒(A) is a homotopy epimorphism.

5.2 Homotopy coherence

We defined the derived localization Lτ⁒(A) of a dg algebra A through a certain universal property formulated in the homotopy category of dg algebras. It makes sense to ask whether more structured notions, taking into account the ∞-structure of the category of dg algebras (i.e. the homotopy type of mapping spaces) can reasonably be considered. We will show that such, apparently more refined, versions of derived localization, nevertheless, turn out to be equivalent to the one defined above.

Proposition 5.11.

Let C be an Ο„-local A-algebra. The following statements are equivalent:

  1. (1)

    C is isomorphic in the homotopy category of A↓𝖽𝗀𝖠𝗅𝗀 to Lτ⁒(A).

  2. (1)

    For any Ο„ -local A-algebra B there is a unique map Cβ†’B in the homotopy category of A↓𝖽𝗀𝖠𝗅𝗀.

  3. (1)

    For any Ο„ -local A-algebra B the map Map⁑(Lτ⁒(A),B)β†’Map⁑(A,B) induced by the localization map Aβ†’Lτ⁒(A) is a weak equivalence.

  4. (1)

    For any Ο„ -local A-algebra B the mapping space MapA⁑(Lτ⁒(A),B) in A↓𝖽𝗀𝖠𝗅𝗀 is contractible.

Proof.

There is the following homotopy fiber sequence of simplicial sets:

(5.4)MapA⁑(Lτ⁒(A),B)β†’Map⁑(Lτ⁒(A),B)β†’Map⁑(A,B);

here MapA⁑(Lτ⁒(A),B) is the homotopy fiber over the map Aβ†’B that determines B as an A-algebra. The second map in (5.4) is a weak equivalence if and only if its homotopy fiber is contractible over every point. Thus, (3) and (4) are equivalent. Implication (2) ⇒ (1) is implied by the following fragment of the long exact sequence of the fibration (5.4):

β†’Ο€0⁒MapA⁑(Lτ⁒(A),B)β†’Ο€0⁒Map⁑(Lτ⁒(A),B)β†’Ο€0⁒Map⁑(A,B),

and the reverse implication (1) ⇒ (2) is Lemma 5.6. Implication (3) ⇒ (1) is obvious. Finally, implication (1) ⇒ (3) follows from Corollary 5.10. ∎

Remark 5.12.

Let us call a map Bβ†’C in A↓𝖽𝗀𝖠𝗅𝗀 a Ο„-local equivalence if for any Ο„-local A-algebra X there is a weak equivalence MapA⁑(C,X)β†’MapA⁑(B,X). Then Proposition 5.11 implies that for an A-algebra B its derived localization Lτ⁒(B)≃Lτ⁒(A)*ALB is the Bousfield localization of B in A↓𝖽𝗀𝖠𝗅𝗀 with respect to Ο„-local equivalences, cf. [15] regarding this notion.

5.3 Module localization

We will now relate the notion of derived localization of algebras to the Bousfield localization of 𝖣⁒(A). Localization functors exist for a large class of triangulated categories π’ž and thick subcategories S. For example, such is the case when π’ž=𝖣⁒(A), the derived category of a dg algebra and S=Loc⁑(Ο„) where Ο„ is a perfect thick subcategory of 𝖣⁒(A).

The localization of an object Mβˆˆπ–¬π—ˆπ–½β’-⁒A with respect to a thick subcategory Ο„βˆˆπ–―π–Ύπ—‹π–Ώβ’(A) is a Ο„-local A-module N together with a map f:Mβ†’N that is a local Ο„-equivalence, i.e. for any Ο„-local A-module L the induced map f*:RHom⁑(N,L)β†’RHom⁑(M,L) is a quasi-isomorphism. A localization of an A-module is clearly defined up to a quasi-isomorphism and, (slightly blurring the distinction between the category π–¬π—ˆπ–½β’-⁒A and 𝖣⁒(A)) we will refer to it as the localization of M and denote by LΟ„π–¬π—ˆπ–½β’-⁒A⁒(M). The following results connect module localization and (derived) algebra localization, generalizing the corresponding results in [5].

Proposition 5.13 ([11, Proposition 2.5], [5, Theorem 4.12]).

There is a dg algebra X supplied with a dg algebra map Aβ†’X such that X≃LΟ„Mod⁒-⁒A⁒(M) as an A-module.

Proof.

Let Ο„βŠ—1 be the thick subcategory in 𝖯𝖾𝗋𝖿⁒(AβŠ—Aop) (i.e. perfect A-bimodules) generated by A-bimodules of the form XβŠ—Aop with XβˆˆΟ„. The argument of [5, Theorem 4.12] shows that LΟ„π–¬π—ˆπ–½β’-⁒A⁒(M) is quasi-isomorphic as an A-module to REndAop⁑(LΟ„βŠ—1π–¬π—ˆπ–½β’-⁒AβŠ—Aop⁒(A)) and the latter is clearly an A-algebra. ∎

The following result is proved in [11, Proposition 2.10].

Proposition 5.14.

Let M be an A-module. Then M≃MβŠ—ALAβ†’MβŠ—ALLΟ„Mod⁒-⁒A⁒A is the localization LΟ„Mod⁒-⁒A⁒(M) of M.

Corollary 5.15.

The Quillen adjunction

π–¬π—ˆπ–½β’-⁒Aβ†’π–¬π—ˆπ–½β’-⁒LΟ„π–¬π—ˆπ–½β’-⁒A⁒A

with left adjoint given by extension of scalars M↦MβŠ—ALLΟ„Mod⁒-⁒A⁒A and right adjoint given by restriction along Aβ†’LΟ„Mod⁒-⁒A⁒A induces an equivalence between D⁒(LΟ„Mod⁒-⁒A⁒A) and the full subcategory of D⁒(A) of Ο„-local modules.

Proof.

By Proposition 5.14 the functor M↦MβŠ—ALLΟ„π–¬π—ˆπ–½β’-⁒A⁒A is the Ο„-localization of the A-module M; thus if M is already Ο„-local, then

Mβ†’MβŠ—ALLΟ„π–¬π—ˆπ–½β’-⁒A⁒A

is a quasi-isomorphism. Moreover, since LΟ„π–¬π—ˆπ–½β’-⁒A⁒A, and LΟ„π–¬π—ˆπ–½β’-⁒A⁒A is Ο„-local, any LΟ„π–¬π—ˆπ–½β’-⁒A⁒A-module is also Ο„-local as lying in the localizing subcategory generated by any LΟ„π–¬π—ˆπ–½β’-⁒A⁒A. Therefore, for any LΟ„π–¬π—ˆπ–½β’-⁒A⁒A-module M the map MβŠ—ALLΟ„π–¬π—ˆπ–½β’-⁒A⁒Aβ†’M is a quasi-isomorphism. ∎

Theorem 5.16.

If LΟ„Mod⁒-⁒A⁒(A) is a dg A-algebra which is the localization of A as an A-module, then it is also the localization of A as a dg algebra (so that LΟ„Mod⁒-⁒A⁒(A) and Lτ⁒(A) are isomorphic in the homotopy category of A-algebras).

Proof.

This theorem is proved in [5] in the special case when Ο„ is generated by a set of A-modules having the form of a cofiber of an endomorphism of A, however the proof continues to hold for arbitrary Ο„. For the reader’s convenience we will repeat the main points. First, we prove that for any A-algebra C that is Ο„-local as an A-module there is a (homotopy class of a) map of A-algebras LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A)β†’C. This is [5, Lemma 41.7] and the proof applies verbatim. Since the algebra localization Lτ⁒(A) is Ο„-local, there is an A-algebra map f:LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A)β†’Lτ⁒(A). Next, by the universal property of Lτ⁒(A) and since LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A) is a Ο„-local A-algebra, there is a map g:Lτ⁒(A)β†’LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A). The composition

f∘g:Lτ⁒(A)β†’Lτ⁒(A)

is an endomorphism of Lτ⁒(A) as an A-algebra and should, therefore, be homotopic to the identity. Similarly, the composition g∘f is an endomorphism of LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A) that is homotopic to the identity. Thus, f and g are mutually inverse quasi-isomorphisms of dg algebras LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A) and Lτ⁒(A). ∎

Definition 5.17.

A map of dg algebras Aβ†’Lτ⁒(A) is called a finite homological epimorphism corresponding to a thick subcategory Ο„βˆˆπ–―π–Ύπ—‹π–Ώβ’(A).

Remark 5.18.

A dg algebra map Aβ†’B is said to be a homological epimorphism (cf. [29]) if the map BβŠ—ALBβ†’B, induced by the multiplication on B, is a quasi-isomorphism. Clearly the derived localization map Aβ†’Lτ⁒A is a homological epimorphism (since the map Lτ⁒(A)βŠ—ALLτ⁒(A)β†’A is the Ο„-localization of Lτ⁒(A) but Lτ⁒(A) is already Ο„-local) but not every homological epimorphism is of this form, owing to the failure of the so-called telescope conjecture, cf. [20].

Corollary 5.19.

Let A be a dg algebra. Then there is a one-to-one correspondence between:

  1. β€’

    thick subcategories in 𝖯𝖾𝗋𝖿⁒(A),

  2. β€’

    equivalence classes of homotopy classes of finite homological epimorphisms from A where two such A→B and A→B′ are equivalent if there is a homotopy commutative diagram

    where B→B′ is a quasi-isomorphism.

Proof.

Given a thick subcategory Ο„ we construct a finite homological epimorphism Aβ†’Lτ⁒(A). Conversely, associate to a finite homological epimorphism Aβ†’B the kernel of the functor -βŠ—AB:𝖯𝖾𝗋𝖿(A)→𝖯𝖾𝗋𝖿(B). The universal property of the derived localization Lτ⁒(A) implies that these constructions are mutually inverse, as claimed. ∎

5.4 Derived localization of ordinary rings

Let A be an ordinary algebra and let Ο„ be a thick subcategory in 𝖯𝖾𝗋𝖿⁒(A) generated by a collection of objects represented by complexes of finitely generated projective A-modules of length two. In other words, the localization map 𝖯𝖾𝗋𝖿⁒(A)→𝖯𝖾𝗋𝖿⁒(A)/Ο„ inverts some maps between finitely generated projective A-modules. If these modules are free, such maps are represented by matrices with entries in A and derived localization map Aβ†’Lτ⁒A is the derived version of Cohn’s matrix localization [9]. The more general (still underived) version belongs to Schofield [30].

Definition 5.20.

Let S be a collection of maps between finitely generated projective A-modules. An algebra B supplied with a map Aβ†’B is called S-inverting if the functor ?↦BβŠ—A? carries every morphism in S into an isomorphism of B-modules. The localization of A at S is the S-inverting algebra A⁒[S-1] such that for any other S-inverting S-algebra B the map Aβ†’B factors uniquely through A⁒[S-1].

It is clear that A⁒[S-1] is unique if it exists. The existence of A⁒[S-1] is [30, Theorem 4.1]. The following result establishes a precise relationship between the derived and underived notions (and simultaneously gives an independent proof of the existence of A⁒[S-1]).

Theorem 5.21.

Let A, S and Ο„ be as above. Then Lτ⁒A is a connective dg algebra, i.e. Hn⁒(Lτ⁒(A))=0 for n<0 and H0⁒(Lτ⁒(A))=A⁒[S-1].

Proof.

By Theorem 5.16 we can identify Lτ⁒(A) with LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A). With this, the desired result is the combination of [11, Proposition 3.1] and [11, Proposition 3.2] (note that op. cit. works with left modules but this difference is, of course, unimportant). ∎

It is natural to ask when Cohn–Schofield localization coincides with derived localization. The following result answers this question.

Proposition 5.22.

Let A, S and Ο„ be as above. Then the canonical map Lτ⁒Aβ†’A⁒[S-1] is a quasi-isomorphism if an only if A⁒[S-1] is stably flat over A, i.e. TornA⁑(A⁒[S-1],A⁒[S-1])=0 for n>1.

Proof.

After the identification of Lτ⁒(A) and LΟ„π–¬π—ˆπ–½β’-⁒A⁒(A), this is [11, Proposition 3.3]. ∎

Corollary 5.23.

Let A be right-hereditary, i.e. having right global dimension ≀1. Then the derived localization LΟ„ at any thick subcategory Ο„ of Perf⁒(A) is (quasi-isomorphic to) its Cohn–Schofield localization.

Proof.

This follows at once from Proposition 5.22 since higher Tor functors over a hereditary algebra vanish. ∎

5.5 Commutative rings

Assume that A is an ordinary commutative ring. Derived localizations corresponding to two-term complexes Aβ†’A given by multiplications by elements of A are ordinary localizations of A as a commutative ring at a multiplicatively closed subset. However for more general thick subcategories Ο„, it is possible for Lτ⁒(A) to be a genuine dg algebra. Recall that the well-known Hopkins–Neeman–Thomason theorem [31] gives the classification of all thick subcategories in 𝖯𝖾𝗋𝖿⁒(A): these correspond bijectively to the unions of closed subsets in Spec⁑(A) having quasi-compact complement (such subsets are sometimes called Hochster open sets). Specifically, to any such subset S one associates the thick subcategory of 𝖯𝖾𝗋𝖿⁒(A) consisting of perfect A-modules having support on S. Then one has the following result, in which we π’œ stands for the structure sheaf on Spec⁑A and we identify 𝖣⁒(A) with the category of complexes of quasi-coherent sheaves of π’œ-modules.

Proposition 5.24.

Let A be a Noetherian commutative ring and let Ο„S be the thick subcategory of dgA-modules with support on a closed subset S of Spec⁑A and denote by i:Spec⁑(A)βˆ–Sβ†ͺSpec⁑(A) the corresponding inclusion map. Then Lτ⁒(A)β‰…R⁒i*⁒i*⁒(A) as objects of D⁒(A).

Proof.

Let 𝖣S⁒(A) be the subcategory of 𝖣⁒(A) consisting of complexes of quasi-coherent sheaves on Spec⁑(A) supported on S. It is clear that the (homotopy) fiber of the natural map π’œβ†’R⁒i*⁒i*β’π’œ is supported on S. On the other hand, the sheaf R⁒i*⁒i*β’π’œ is Ο„-local; indeed for any π’œ-module sheaf β„± supported on S we have i*⁒ℱ≃0 and so

RHom⁑(R⁒i*⁒i*β’π’œ,β„±)≃RHom⁑(i*⁒(π’œ),i*⁒ℱ)≃0.

It follows that R⁒i*⁒i*β’π’œ is the localization of π’œ with respect to the localizing subcategory 𝖣S⁒(A). Now the desired statement follows from Theorem 5.16. ∎

Remark 5.25.

The assumptions that A is Noetherian and S is closed are imposed in order for the locally ringed space Spec⁑(A)βˆ–S to be a scheme and for the direct image functor R⁒i* to land in the derived category of quasi-coherent sheaves on Spec⁑(A).[1]

Example 5.26.

Let A:=𝐀⁒[x,y] be the polynomial algebra in two variables and let Ο„ be the thick subcategory generated by the 1-dimensional A-module 𝐀⁒[x,y]/(x,y). It is easy to see (e.g. using the Koszul complex) that Lτ⁒(A) is a dg algebra whose homology is concentrated in degrees 0 and -1.

It is natural to ask whether for a commutative dg algebra A its derived localization Lτ⁒(A) is also such. As usual, it is better to consider E∞ algebras rather than strictly commutative ones. Then a positive answer to this question could be derived by combining the results of [13] and [24]. Since E∞ algebras are rather tangential to the main themes of the present paper, we will only sketch the proof and omit all topological and operadic prerequisites, referring to the two above mentioned sources for details.

Proposition 5.27.

If A is a (dg) E∞ algebra, then Lτ⁒(A) is also a dg E∞ algebras and the localization Aβ†’Lτ⁒(A) can be constructed as a map of dg E∞ algebras.

Proof.

Let H⁒𝐀 be the Eilenberg–MacLane spectrum corresponding to the ring 𝐀; it is known to be a commutative S-algebra. According to [24, Theorem 7.11], there is a functor Ξ:Bβ†¦Ξžβ’(B) from the homotopy category of commutative H⁒𝐀 algebras to the homotopy category of dg E∞ algebras and another one 𝐑:M↦𝐑⁒(M) from the homotopy category of Ξ⁒(B)-modules to the homotopy category of B-modules. Moreover, both Ξ and 𝐑 are equivalences. Additionally, the B-module 𝐑⁒(Ξ⁒(B)) is weakly equivalent to B for any S-algebra B; this property is not stated explicitly in op.cit. but follows readily from the construction.

It suffices to show that LΟ„π–¬π—ˆπ–½β’-⁒A is quasi-isomorphic to a dg E∞ algebra and, by the setup described above this is equivalent to showing that the Bousfield localization of (Ξ)-1⁒(A) is a commutative S-algebra (and the map into it from (Ξ)-1⁒(A) is that of commutative S-algebras). But this is proved in [13, Chapter 8, Theorem 2.2]. ∎

5.6 Small example

The following is the smallest example of a finite-dimensional algebra possessing nontrivial derived localization. Let A be the algebra with a basis e1, e2, α1, α2 so that e12=e1,e22=e2,α1⁒e1=α1=e2⁒α1,e1⁒α2=α2=α2⁒e2 and the rest of the products are zero. The algebra A is the path algebra of a quiver with two vertices and two arrows between them running in the opposite directions, subject to the relations above. The elements αi,i=1,2 correspond to the two arrows and the elements ei,i=1,2 correspond to the trivial loops at each vertex.

Set B:=REndA⁑(𝐀,𝐀), where A acts on 𝐀 via e2=Ξ±i=0 for i=1,2, and also set Bβ€²:=REndA⁑(𝐀×𝐀,𝐀×𝐀), where A acts on 𝐀×𝐀 via Ξ±i=0,i=1,2. It is immediate that Bβ€² is (quasi-isomorphic to) the path algebra of the same quiver with arrows marked by the generators Ξ±iβ€², i=1,2 with |Ξ±iβ€²|=-1 and no relations. It follows that B≃e1⁒B′⁒e1 is the polynomial algebra on one generator Ξ²=Ξ±2′⁒α1β€² with |Ξ²|=-2. It further follows that

REndB⁑(𝐀,𝐀)≃𝐀⁒[Ξ±]/(Ξ±2),

the exterior algebra on one generator Ξ± with |Ξ±|=1.

On the other hand, it is clear that Le1⁒A is the localization of the algebra A as an A-module with respect to the functor RHomA⁑(-,e1⁒A⁒e1)≃RHomA⁑(-,𝐀) and the latter localization is (quasi-isomorphic to) REndB⁑(𝐀,𝐀), cf. [12, Theorem 2.1 and Proposition 4.8] for this kind of statement.

All told, we conclude that Le1⁒A≃𝐀⁒[Ξ±]/(Ξ±2). The nonderived localization A⁒[e1-1] of A is, of course, the ground ring 𝐀.

Next, consider the projective A modules e1⁒A and e2⁒A; then the left multiplication with Ξ±2 determines a map e1⁒Aβ†’e2⁒A which we will regard as an object in 𝖯𝖾𝗋𝖿⁒(A). Denote by Ο„ the thick subcategory generated by this object. Then Lτ⁒A is isomorphic to M2⁒(𝐀), the 2Γ—2 matrix algebra over 𝐀 and since M2⁒(𝐀) is flat over A, we conclude that no higher derived terms are present, i.e. Lτ⁒(A)≃M2⁒(𝐀). Similar conclusions can be made regarding derived localizations of A at e2 and at the object e2⁒Aβ†’e1⁒A determined by the left multiplication by Ξ±1.

5.7 Group completion

Let M be a discrete monoid and let BM be its classifying space. Then the based loop space Ω⁒B⁒M is the so-called group completion of M and according to McDuff’s theorem, any topological space X is weakly equivalent to some Ω⁒B⁒M [25]. The chain algebra C*⁒(Ω⁒B⁒M) is quasi-isomorphic to the derived localization of the monoid algebra 𝐀⁒[M] at all monoid elements by [5, Theorem 10.3]. It follows that the derived localization of an ordinary (ungraded) algebra, such as 𝐀⁒[M] can be a fairly arbitrary dg algebra. Here is a particularly striking example due to Fiedorowicz [14].

Example 5.28.

Let M be the monoid with five elements {1,xi⁒j,i,j=1,2} which multiply according to the rule xi⁒j⁒xk⁒l=xi⁒l. It is easy to see that the nonderived group completion of M is trivial and that H*⁒(M,𝐀):=Tor*𝐀⁒[M]⁑(𝐀,𝐀) coincides with the homology of S2, the 2-dimensional sphere. It follows that BM is weakly equivalent to S2 and, therefore Ω⁒B⁒M is weakly equivalent to Ω⁒S2. The homology of Ω⁒S2 is 𝐀⁒[x] with |x|=1, and this dg algebra is clearly formal. Thus, the derived localization of 𝐀⁒[M] is (quasi-isomorphic to) 𝐀⁒[x]. Note that the localization map 𝐀⁒[M]β†’C*⁒(Ω⁒B⁒M)≃𝐀⁒[x] is highly nontrivial in the homotopy category of dg algebras (e.g. it induces a Verdier quotient on the level of derived categories) and it is not the one that factors through 𝐀.

5.8 Rank functions for derived localization algebras

Let A be a dg algebra and ρ be a rank function on 𝖯𝖾𝗋𝖿⁒(A). Consider the map Aβ†’Lρ⁒(A) from A into its derived localization at Ker⁑(ρ). Then the following result holds.

Theorem 5.29.

The rank function ρ descends to an object-faithful rank function ρ¯ on Perf⁒(Lρ⁒(A)) (so that the pullback of ρ¯ under the direct image functor Perf⁒(A)β†’Perf⁒(Lρ⁒(A)) is ρ).

Proof.

By Corollary 4.4, the rank function ρ descends to an object-faithful rank function on the Verdier quotient 𝖯𝖾𝗋𝖿⁒(A)/Ker⁑(ρ). Note that 𝖯𝖾𝗋𝖿⁒(Lρ⁒(A)) is the idempotent completion of 𝖯𝖾𝗋𝖿⁒(A)/Ker⁑(ρ) by [27, Theorem 2.1] and so the obtained rank function on 𝖯𝖾𝗋𝖿⁒(A)/Ker⁑(ρ) extends further to ρ¯ on 𝖯𝖾𝗋𝖿⁒(Lρ⁒(A)) by Lemma 4.10. Clearly, ρ¯ has the required properties. ∎

Remark 5.30.

The above theorem is an extension of the corresponding statement for Sylvester rank functions, [30, Theorem 7.4]. This is a key result in theory of Sylvester rank functions and its proof in op.cit. is very involved. The almost trivial proof of the much more general result above demonstrates the advantage of the notion of a rank function for triangulated categories over the classical notion.

5.9 Loops on p-completions of topological spaces

Another example of derived localization in topology comes from the study of chain algebras of based loops on completed classifying spaces of finite groups, cf. [8, 4]. Let X be a topological space such that Ο€1⁒(X) is finite (e.g. the classifying space of a finite group) and Xp∧ is the p-completion of X. Then it is proved in [7] that there is an idempotent eβˆˆπ”½p⁒[Ο€1⁒(X)] such that the derived localization Le⁒𝔽p⁒[Ο€1⁒(X)] is quasi-isomorphic, as a dg algebra, to C*⁒Ω⁒(Xp∧), the chain algebra of the based loop space of Xp∧.

Remark 5.31.

When X is the classifying space of a finite group, this result (in a somewhat different formulation) was proved in the paper [32] where a good portion of derived localization theory was also developed. Unfortunately, the present authors had not been aware of this earlier work and did not make a proper attribution to it in [5, 7].

6 Localizing rank functions and fraction fields

We will start with the following almost obvious result.

Proposition 6.1.

Let f:Xβ†’Y be a morphism in a triangulated category C supplied with a rank function ρ such that ρ⁒(f)=0. Then the following conditions are equivalent:

  1. (1)

    The morphism f factors through an object of rank zero.

  2. (1)

    There exists an object Z in π’ž and ρ -full morphism g:Zβ†’X such that f∘g=0.

  3. (1)

    There exists and object W in π’ž and a ρ -full morphism h:Yβ†’W such that h∘f=0.

  4. (1)

    The morphism f maps to zero under the Verdier quotient map π’žβ†’π’ž/Ker⁑(ρ).

Proof.

The equivalence of (1) with (2) and (3) follows from Lemma 4.2 and the equivalence of (2) and (3) with (4) follows from the characterization of morphisms in a Verdier quotients in terms of left or right fractions. ∎

Definition 6.2.

An integral rank function on a triangulated category π’ž is localizing if any morphism in π’ž of rank zero satisfies either of the equivalent conditions of Proposition 6.1.

Remark 6.3.

The notion of a localizing rank function is motivated by constructing derived localization of dg rings to dg simple Artinian rings. If one is interested in maps into dg algebras more general than dg simple Artinian rings (e.g. dg analogues of von Neumann regular rings), one can speculate that real-valued rank functions will be relevant. In this context perhaps it is more natural to require that any rank 0 morphism factors through objects of arbitrarily small rank. We will not elaborate on this more subtle notion in the present paper however.

Theorem 6.4.

Let C be a triangulated category admitting a generator. Then there is a bijection between the following two sets:

  1. β€’

    localizing prime rank functions,

  2. β€’

    thick subcategories of π’ž with a simple Verdier quotient.

Proof.

Let Ο„ be a thick subcategory of π’ž such that π’ž/Ο„ is simple. Fix an indecomposable object X in π’ž/Ο„. Then there is a unique morphism-faithful rank function on π’ž/Ο„ taking value 1 on X, see Corollary 2.21. The pullback of this rank function to π’ž is a localizing rank function on π’ž.

Conversely, given a localizing rank function ρ:π’žβ†’β„€, let ρ¯ be the rank function on π’ž/Ker⁑(ρ) induced by ρ; since ρ is localizing, ρ¯ is morphism faithful and ρ¯⁒(X)=1 for some generator X. By Proposition 2.23, π’ž/Ker⁑(ρ) is simple.

It is clear that the two processes described define mutually inverse maps between the two sets in the statement of the theorem. ∎

The following result gives a complete description of homotopy classes of derived localizations of dg algebras into dg skew-fields or, more generally, dg simple Artinian rings, in terms of rank functions. Classically, only partial results of this sort were available (e.g. for a very specific class of rings or a particular class of localizations), cf. [30, Theorems 5.4 and 5.5] and [9, Theorem 4.6.14].

Theorem 6.5.

Let A be a dg algebra. Then there is a bijection between the following two sets:

  1. β€’

    localizing prime rank functions on 𝖯𝖾𝗋𝖿⁒(A),

  2. β€’

    equivalence classes of homotopy classes of finite homological epimorphisms A→B into simple Artinian dg algebras B.

Moreover, ρ⁒(A)=1 if and only if B is a dg skew-field.

Proof.

By Theorem 6.4 localizing prime rank functions ρ on 𝖯𝖾𝗋𝖿⁒(A) correspond bijectively to thick subcategories on 𝖯𝖾𝗋𝖿⁒(A) with a simple Verdier quotient. For such a thick subcategory Ο„ the image of A in 𝖯𝖾𝗋𝖿⁒(A) is a generator M of 𝖯𝖾𝗋𝖿⁒(A)/Ο„ that we can assume without loss of generality to be cofibrant over A. Since 𝖯𝖾𝗋𝖿⁒(Lτ⁒(A)) is the idempotent completion of 𝖯𝖾𝗋𝖿⁒(A)/Ο„ and 𝖯𝖾𝗋𝖿⁒(A)/Ο„ is simple, 𝖯𝖾𝗋𝖿⁒(Lτ⁒(A)) is also simple and it follows by Proposition 2.25 that Lτ⁒(A) is a simple Artinian dg algebra. By Corollary 5.19 such thick subcategories correspond bijectively with the equivalence classes of homotopy classes of finite homological epimorphisms Aβ†’Lτ⁒(A). Finally, the condition ρ⁒(A)=1 means that M is an indecomposable object of 𝖯𝖾𝗋𝖿⁒(A)/Ο„ and in this case H⁒(Lτ⁒(A))β‰…End𝖯𝖾𝗋𝖿⁒(A)/τ⁑(M,M) must be a graded skew-field. ∎

Example 6.6.

Let A be the 4-dimensional algebra of Section 5.6. The localization map Aβ†’Lτ⁒(A)≃M2⁒(𝐀) induces a functor 𝖯𝖾𝗋𝖿⁒(A)→𝖯𝖾𝗋𝖿⁒(M2⁒(𝐀)) where the target triangulated category is simple. The induced rank function on 𝖯𝖾𝗋𝖿⁒(A) is localizing.

On the other hand, consider the localization map Aβ†’Le1⁒(A)≃𝐀⁒[Ξ±]/Ξ±2. The category 𝖯𝖾𝗋𝖿⁒(𝐀⁒[Ξ±]/Ξ±2) is not simple, but composing with the augmentation 𝐀⁒[Ξ±]/Ξ±2→𝐀, we obtain a map A→𝐀. Note that the latter map is the Cohn–Schofield (nonderived) localization of A. The induced rank function (which corresponds to a certain Sylvester rank function for A) is not localizing.

6.1 Localizing rank functions for hereditary rings

Localizing rank functions are easiest to construct for perfect derived categories of (right)-hereditary algebras in which case they essentially reduce to the nonderived notion.

Let A be an ordinary 𝐀-algebra and let K0⁒(A) be its Grothendieck group of its category of finitely generated projective modules. The abelian group K0⁒(A) has a pre-order specified by the declaring the classes of finitely generated projective modules in K0⁒(A) to be positive. Then we have the notion of a projective rank function on A cf. [30].

Definition 6.7.

A projective rank function on A is homomorphism of pre-ordered groups ρ:K0⁒(A)→ℝ for which ρ⁒[A]=1.

A Sylvester rank function on A (which, by Theorem 3.7 is equivalent to a normalized ∞-periodic rank function on 𝖯𝖾𝗋𝖿⁒(A)) restricts to a function on the positive cone of K0⁒(A) with values in nonnegative real numbers and thus, determines a projective rank function on A. One can ask whether, conversely, one can associate to a projective rank function on A a Sylvester rank function on A. For this, a rank needs to be assigned to any map between two finitely generated projective A-modules. One can attempt the following definition.

Definition 6.8.

Let A be a ring with a projective rank function ρ. Given f:Pβ†’Q, a map between two finitely generated projective A-modules, its inner rank ρ⁒(f) is defined as ρ⁒(f):=inf⁑(ρ⁒(S)), where S ranges through finitely generated projective A modules through which f factors.

There is no reason for an inner rank function to be Sylvester in general. However, this holds in one important special case.

Proposition 6.9.

Let A be a hereditary algebra with a projective rank function ρ. Then the associated inner rank function is Sylvester.

Proof.

We prove the desired statement by quoting relevant results of [30] (all of which are elementary and easy). Firstly, by [30, Theorem 1.11] the Sylvester law of nullity holds for ρ; that is if Ξ±:P0β†’P1 and Ξ²:P1β†’P2 are two maps between finitely generated projective A-modules for which β∘α=0, then ρ⁒(Ξ±)+ρ⁒(Ξ²)≀ρ⁒(P1).

Next, the law of nullity implies axioms (m2) and (m3) of the Sylvester morphism rank functions, by [30, Lemmata 1.14 and 1.15]. Finally, axioms (m1) and (m4) are immediate from the definition. ∎

Remark 6.10.

As is clear from the above proof, Proposition 6.9 is essentially contained in [30], although it is not explicitly formulated in op. cit. as such. Moreover, the statement of the proposition holds under the weaker assumption that A be weakly semihereditary (as opposed to hereditary). We will not need this stronger result.

Corollary 6.11.

Let A be a hereditary algebra with a projective rank function ρ. Then ρ extends uniquely to a rank function on Perf⁒(A).

Proof.

The inner rank function associated to ρ is Sylvester by Proposition 6.9. Next, Proposition 3.9 implies that a Sylvester rank function gives rise to an ∞-periodic rank function on 𝖯𝖾𝗋𝖿⁒(A) and the latter determines, by reduction, a (1-periodic) rank function on 𝖯𝖾𝗋𝖿⁒(A). ∎

Remark 6.12.

For a hereditary algebra A there is a one-to-one correspondence between rank functions and ∞-periodic rank functions on 𝖯𝖾𝗋𝖿⁒(A). Indeed, this is straightforward to check on complexes of finitely generated projective A-modules of length 2 using formula (3.1) and the hereditary property implies that such complexes generate the whole triangulated category 𝖯𝖾𝗋𝖿⁒(A).

Proposition 6.13.

Let A be a hereditary algebra and let ρ be a rank function on Perf⁒(A) associated, as above, with a projective rank function on A. Then ρ is localizing.

Proof.

The rank function ρ is localizing if and only if the derived localization Lρ⁒A of A at Ker⁑(ρ) is a dg skew-field by Theorem 6.5 and by Corollary 5.23, Lρ⁒A is quasi-isomorphic to the underived localization of A at the collection of ρ-full maps between finitely generated A-modules. By [30, Theorem 5.4] this underived localization is a skew-field and we are done. ∎

A standard application of this result is the construction of the skew-field of fractions of a free algebra.

Example 6.14.

Let π€β’γ€ˆS〉, the free algebra on a set S. Then there exists a unique homological epimorphism Aβ†’K⁒(S) where K⁒(S) is a skew-field. Indeed, A is a hereditary algebra with K0⁒(A)=β„€, thus there exists a unique projective rank on A that uniquely extends to a rank ρ function on 𝖯𝖾𝗋𝖿⁒(A). The corresponding derived localization Lρ⁒(A) (which coincides with the Cohn–Schofield localization since A is hereditary) is therefore a skew-field, commonly known as the free field on S, [9].

6.2 Derived fields of fractions

Given an ordinary ring A we will understand a classical field of fractions of A to be a (nonderived) localization A→K, where K is a skew-field. A derived field of fractions of A is a dg skew-field K together with a derived localization map A→K.

Remark 6.15.

This definition of a classical field of fractions agrees with the standard notion that ordinarily assumes that A is a commutative domain. In the noncommutative case the definition accepted e.g. in [9] is different, in particular the map A→K is assumed to be an embedding. Our definition is one that extends most naturally to the dg context.

Corollary 6.5 classifies derived fields of fractions of a ring A in terms of localizing rank functions on 𝖯𝖾𝗋𝖿⁒(A). The following examples demonstrate the existence of such (genuinely derived) fields of fractions and thus, localizing rank functions (that cannot be reduced to classical Sylvester rank functions) even for finite-dimensional algebras over fields.

Example 6.16.

(1) Let A=𝐀⁒[M] be the 5-dimensional algebra of Example 5.28. We saw that there is a derived localization map 𝐀⁒[M]→𝐀⁒[x], where |x|=1. Composing this with inverting x, we obtain a derived localization map 𝐀⁒[M]→𝐀⁒[x,x-1], where the target is a graded skew-field of period 1, i.e. a derived fraction field of A. The β€œclassical” part of this map is the augmentation map 𝐀⁒[M]→𝐀; this is an underived localization and thus, a classical fraction field in the sense understood above.

(2) This example is essentially contained in [7, Example 6.13]. Let 𝐀 be a field of characteristic 3 and consider A:=𝐀⁒[S3], the group algebra of the symmetric group on three symbols. Then A contains an idempotent e such that Le⁒A is (quasi-isomorphic to) the graded algebra generated by two indeterminates y and z with |z|=2,|y|=3 and z3=y2. The latter algebra has a fraction field k⁒[x,x-1] with x=z-1⁒y so |x|=1. Thus, k⁒[x,x-1] is a derived fraction skew-field for A. Again, the β€œclassical” part of the latter derived fraction field is the augmentation map of the group ring 𝐀⁒[S3].

Funding statement: This work was partially supported by EPSRC grants EP/T029455/1 and EP/T030771/1.

Acknowledgements

We are grateful to Marc Stefan and the anonymous referee for pointing out various inaccuracies and a host of useful comments.

References

[1] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. reine angew. Math. 588 (2005), 149–168. Search in Google Scholar

[2] P. Balmer and M. Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819–834. Search in Google Scholar

[3] A. Bapat, A. Deopurkar and A. Licata, A Thurston compactification of the space of stability conditions, preprint (2020), http://arxiv.org/abs/2011.07908. Search in Google Scholar

[4] D. Benson, An algebraic model for chains on Ω⁒B⁒Gp∧, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2225–2242. Search in Google Scholar

[5] C. Braun, J. Chuang and A. Lazarev, Derived localisation of algebras and modules, Adv. Math. 328 (2018), 555–622. Search in Google Scholar

[6] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. Search in Google Scholar

[7] J. Chuang and A. Lazarev, Homological epimorphisms, homotopy epimorphisms and acyclic maps, Forum Math. 32 (2020), no. 6, 1395–1406. Search in Google Scholar

[8] F. R. Cohen and R. Levi, On the homotopy theory of p-completed classifying spaces, Group representations: Cohomology, group actions and topology (Seattle 1996), Proc. Sympos. Pure Math. 63, American Mathematical Society, Providence (1998), 157–182. Search in Google Scholar

[9] P. M. Cohn, Skew fields, Encyclopedia Math. Appl. 57, Cambridge University, Cambridge 1995. Search in Google Scholar

[10] G. Dimitrov, F. Haiden, L. Katzarkov and M. Kontsevich, Dynamical systems and categories, The influence of Solomon Lefschetz in geometry and topology, Contemp. Math. 621, American Mathematical Society, Providence (2014), 133–170. Search in Google Scholar

[11] W. G. Dwyer, Noncommutative localization in homotopy theory, Non-commutative localization in algebra and topology, London Math. Soc. Lecture Note Ser. 330, Cambridge University, Cambridge (2006), 24–39. Search in Google Scholar

[12] W. G. Dwyer and J. P. C. Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002), no. 1, 199–220. Search in Google Scholar

[13] A. D. Elmendorf, I. Kriz, M. A. Mandell and J. P. May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys Monogr. 47, American Mathematical Society, Providence 1997. Search in Google Scholar

[14] Z. Fiedorowicz, Classifying spaces of topological monoids and categories, Amer. J. Math. 106 (1984), no. 2, 301–350. Search in Google Scholar

[15] P. S. Hirschhorn, Model categories and their localizations, Math. Surveys Monogr. 99, American Mathematical Society, Providence 2003. Search in Google Scholar

[16] M. Hovey, Model categories, Math. Surveys Monogr. 63, American Mathematical Society, Providence 1999. Search in Google Scholar

[17] M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, 1–114. Search in Google Scholar

[18] I. Ikeda, Mass growth of objects and categorical entropy, preprint (2016), http://arxiv.org/abs/1612.00995. Search in Google Scholar

[19] A. Jaikin-Zapirain, L2-Betti numbers and their analogues in positive characteristic, Groups St Andrews 2017 in Birmingham, London Math. Soc. Lecture Note Ser. 455, Cambridge University, Cambridge (2019), 346–405. Search in Google Scholar

[20] B. Keller, A remark on the generalized smashing conjecture, Manuscripta Math. 84 (1994), no. 2, 193–198. Search in Google Scholar

[21] H. Krause, Localization theory for triangulated categories, Triangulated categories, London Math. Soc. Lecture Note Ser. 375, Cambridge University, Cambridge (2010), 161–235. Search in Google Scholar

[22] H. Krause, Cohomological length functions, Nagoya Math. J. 223 (2016), no. 1, 136–161. Search in Google Scholar

[23] A. Lazarev, Hoschschild cohomology and moduli spaces of strongly homotopy associative algebras, Homology Homotopy Appl. 5 (2003), no. 1, 73–100. Search in Google Scholar

[24] M. A. Mandell, Topological André–Quillen cohomology and E∞ André–Quillen cohomology, Adv. Math. 177 (2003), no. 2, 227–279. Search in Google Scholar

[25] D. McDuff, On the classifying spaces of discrete monoids, Topology 18 (1979), no. 4, 313–320. Search in Google Scholar

[26] F. Muro, Homotopy units in A-infinity algebras, Trans. Amer. Math. Soc. 368 (2016), no. 3, 2145–2184. Search in Google Scholar

[27] A. Neeman, The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Γ‰c. Norm. SupΓ©r. (4) 25 (1992), no. 5, 547–566. Search in Google Scholar

[28] D. Orlov, Finite-dimensional differential graded algebras and their geometric realizations, Adv. Math. 366 (2020), Article ID 107096. Search in Google Scholar

[29] D. Pauksztello, Homological epimorphisms of differential graded algebras, Comm. Algebra 37 (2009), no. 7, 2337–2350. Search in Google Scholar

[30] A. H. Schofield, Representation of rings over skew fields, London Math. Soc. Lecture Note Ser. 92, Cambridge University, Cambridge 1985. Search in Google Scholar

[31] R. W. Thomason, The classification of triangulated subcategories, Compos. Math. 105 (1997), no. 1, 1–27. Search in Google Scholar

[32] P. Vogel, Cohn localization of finite group rings, J. Knot Theory Ramifications 26 (2017), no. 2, Article ID 1740017. Search in Google Scholar

Received: 2021-08-30
Published Online: 2021-10-16
Published in Print: 2021-12-01

Β© 2021 Walter de Gruyter GmbH, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.