On the relation between length functions and exact Sylvester rank functions

Abstract Inspired by the work of Crawley-Boevey on additive functions in locally finitely presented Grothendieck categories, we describe a natural way to extend a given exact Sylvester rank function on the category of finitely presented left modules over a given ring R, to the category of all left R-modules.


Introduction
The notion of Sylvester matrix rank function has been studied and applied in di erent areas of Algebra, especially in the context of C * -algebras and von Neumann regular rings. In fact, these functions were already considered by von Neumann, and they were investigated by I. Halperin, K. R. Goodearl and D. Handelman, among others, see [3] for an exposition of these results and classical references for the theory. For general rings, they were introduced by Malcolmson in order to characterize ring homomorphisms to division rings and simple artinian rings, we refer to [6] for results going in this direction. Let us remark that, in this theory, there is a special class of rank functions called exact Sylvester rank functions that correspond to at embeddings.
There has been a recent trend of studies on Sylvester rank functions, in connection, amongst other subjects, with the strong Atiyah Conjecture and the Lück approximation Conjecture, see [4] for a survey on these results.
A Sylvester matrix rank function ρ on a (associative and unitary) ring R is a function that assigns a non-negative real number to each matrix over R and satis es the following conditions: (Mat.rk.1) ρ(A) = if A is a zero matrix and ρ( ) = ; (Mat.rk.2) ρ(AB) ≤ min{ρ(A), ρ(B)} for any matrices A and B which can be multiplied; for any matrices A and B; for any matrices A, B and C, of appropriate sizes.
Another, closely related, approach to rank functions is that of Sylvester module rank functions. Indeed, such a function is a non-negative real invariant de ned on the category of nitely presented left R-modules δ : fp(R) → R ≥ , that satis es the following axioms: It is an easy but useful exercise to verify that, given a Sylvester matrix rank function ρ on R one obtains a Sylvester module rank function δρ on the same ring just letting δρ(P) := ρ(A P ), where A P is the matrix associated with a given presentation of P. On the other hand, given a Sylvester module rank function δ, one obtains a Sylvester matrix rank function ρ δ just letting ρ δ (A) := δ(R n /R m A), where we have interpreted the matrix A as a morphism R m → R n . One can show that these procedures induce a bijection between Sylvester matrix and module rank functions, so the two theories are essentially equivalent.
A Sylvester module rank function χ : fp(R) → R ≥ is said to be exact if it satis es the following condition: (EX) given a surjection X Y in fp(R), The present paper originates from a question asked me by Andrei Jaikin in 2017 during a visit to the ICMAT in Madrid. In particular, it is known for some special classes of rings (e.g., von Neumann regular rings) that any exact Sylvester module rank function δ : fpR → R ≥ can be extended to a very well-behaved (i.e., additive, see below) invariantδ : Thus, it is natural to ask for a characterization of the class of rings where any exact Sylvester module rank function is the restriction to nitely presented modules of an additive function de ned on all modules.
Some time after this question was asked to me I recalled that a similar problem, with di erent terminology, had been studied in the setting of locally nitely presented Grothendieck categories by Crawley-Boevey [1,2] and, in fact, minor modi cations of Crawley-Boevey's argument (essentially due to the fact that Crawley-Boevey's functions take values in N while our invariants are real-valued) allow us to prove the following Main Theorem. Given an exact Sylvester module rank function χ : fp(R) → R ≥ , there exists a function χ : R-Mod → R ≥ ∪ {∞} which extends χ (i.e., χ fp(R) = χ) and which is additive in the following sense: In fact, it will be clear from the construction that the function χ : R-Mod → R ≥ ∪ {∞} produced by the above theorem is a length function in the sense of [5,7,8]. As a nal remark, let us notice that all the results in this short paper can be trivially extended to locally nitely presented Grothendieck categories but we preferred to write this note in the setting of modules over a ring, as it is the setting where the most applications of this formalism are expected.

Proof of the Main Theorem
This entire section is devoted to the proof of our Main Theorem. This proof proceeds by steps: in Subsection 1.1 we extend our exact function from the class of nitely presented left R-modules fp(R) to the class of nitely generated left R-modules fg(R), while in Subsection 1.2 we extend it from fg(R) to R-Mod. Finally, in the last two subsections, we prove that the invariant of R-Mod obtained via this extension procedure is additive, so it gives us a length function.

. Extension to fg(R)
Let us de ne a functionχ : fg(R) → R ≥ on a nitely generated module F ∈ fg(R) by the following formula: Notice rst that, given X ∈ fp(R), then id X : X → X is a surjection from a nitely presented module, sō χ(P) = χ(P). This shows thatχ fp(R) = χ. We collect in the following lemma some easy observations aboutχ. In particular, we show thatχ is monotone under taking quotients and nitely generated submodules: Lemma 1.1. In the above setting, the following properties hold true: Proof. Part (1) follows directly from the de nition since, given a surjection X F, composition with F F gives a surjection X F . Now consider a short exact sequence as in part (2). The fact that F is nitely generated is an easy consequence of Schanuel's Lemma. But then, by exactness and the observation that χ fp(R) = χ, we obtain:χ Let us now verify part (3). Indeed, given ε > , consider a surjection X F, with X ∈ fp(R), such that χ(X) ≤χ(F) + ε, and complete X → F ← F to the following pullback diagram: K where rows and columns are exact. Then, since F is nitely generated, there exists a nitely generated submodule K ≤ K such that the restriction K → F is still surjective. Furthermore, after identifying K and K with submodules of X, we can notice that X/K is nitely presented. Now we conclude as follows: where the central equality holds by part (2). As the choice of ε was arbitrary, we deduce thatχ(F ) ≤χ(F).

. Extension to R-Mod
Let us de ne a function on a module M ∈ R-Mod by the following formula: Notice rst that, given F ∈ fg(R), then F ≤ F is a nitely generated submodule of itself, and so, by the monotonicity ofχ under taking nitely generated submodules, χ(F) =χ(F). This shows that χ fg(R) =χ and so, in particular, also that χ fp(R) =χ fp(R) = χ.

. χ is super-additive
Consider a short exact sequence in R-Mod as follows:  . χ is sub-additive Consider a short exact sequence in R-Mod as follows: In this subsection we are going to verify that χ(M) ≤ χ(N) + χ(L). We proceed by steps: Step 1. We start showing the statement in case L and N are nitely generated and the sequence splits (so M ∼ = L ⊕ N). Indeed, given ε > , consider two surjections L L and N N with L , N ∈ fp(R), and such that χ(L ) ≤ χ(L) + ε/ and χ(N ) ≤ χ(N) + ε/ . Then, there is a surjection L ⊕ N → L ⊕ N ∼ = M and L ⊕ N ∈ fp(R), so that where the equality is a consequence of (Mod.rk.2). We can conclude by the arbitrariness of ε. One now concludes by the arbitrariness of M .
Step 4. Let us prove the statement in case M ∈ fg(R) (so that also N ∈ fg(R)). Given ε > , take a surjection N N with N ∈ fp(R) and such that χ(N ) ≤ χ(N) + ε, and consider the following pullback diagram: One concludes by the arbitrariness of M .