Towards an Homological Generalization of the Direct Summand Theorem

We present a more general (parametric-) homological characterization of the Direct Summand Theorem. Specifically, we state two new conjectures: the Socle-Parameter conjecture (SPC) in its weak and strong forms. We give a proof for the week form by showing that it is equivalent to the Direct Summand Conjecture (DSC), now known to be true after the work of Y. Andr\'{e}, based on Scholze's theory of perfectoids. Furthermore, we prove the SPC in its strong form for the case when the multiplicity of the parameters is smaller or equal than two. Finally, we present a new proof of the DSC in the equicharacteristic case, based on the techniques thus developed.


Introduction
Fundamental work of Hochster, Peskine, Szpiro, and Serre during the second half of the twentieth century allowed us to develop a theory of multiplicities, along with the introduction of powerful prime characteristic methods in commutative algebra [1][2][3]. As a by-product of this effort for understanding general properties of commutative rings and their prime spectra in terms of homological and algebraic invariants, a collection of homological conjectures emerged [1]. Further research of Hochster's showed that most of these conjectures were essentially new ways of describing a quite central algebraic splitting phenomenon for a particular kind of ring extensions: most of these homological questions turned out to be equivalent to the direct summand conjecture (DSC) [1,[4][5][6][7].
Specifically, the DSC states that if ↪ R S is a finite extension of rings and if R is a regular ring, then R is a direct summand of S as R-module. Or equivalently, this extension splits as a map of R-modules, i.e. there exists a retraction → ρ S R : sending 1 S to 1 R . After many decades, the DSC (or direct summand theorem [DST]) was finally proved in the former general form by Y. André, by first reducing to the case of unramified complete regular local rings, as it had been suggested by the seminal work of Hochster's, and then by using Scholze's theory of perfectoid spaces [8,9]. Even though this fundamental question was settled, the former results (regarded as a whole) suggest some directions for future research in local homological algebra.
In this work, we present an equivalent form of the DSC given in terms of an estimate for the difference of the lengths of the first two Koszul homology groups of quotients of Gorenstein rings by principal zerodivisor ideals. Dutta and Griffith (see [10,Theorem 1.5]) had obtained similar results in an independent manner, although in a rather different context (for the case of complete and almost complete intersections).
In Sections 3 and 4, we state an equivalent form of the DSC conjecture in terms of the existence of annihilators of zero divisors on Gorenstein local rings not belonging to parameter ideals [11,12]. Based on these results, we find a new conjecture equivalent, in its weak form, to the DSC (Section 6). In its strong form, this conjecture states that if ( ) T η , is a Gorenstein local ring of dimension d and { … } ⊆ x x T , , d 1 is any system of parameters, and if we denote by Q the ideal generated by these parameters, then for any zero divisor ∈ z T, and for any lifting ∈ u T of a socle element in T/Q (i.e. ( ) = ( ) / η u Ann¯T Q ) it must hold that ∈ ( ) uz Q z . This rather technical condition allows for more flexibility when one tries to do computations in particular examples (see for instance the proof of Proposition 12). We call this conjecture the socle-parameter conjecture, strong form (SPCS). We obtain the weak form if we add the requirement that in the mixedcharacteristic case / = > T η p char 0 and = x p 1 . The SPCS is at the same time equivalent to a very general and homological condition involving the lengths of the Koszul homology groups: Evidently, this last condition only involves homological estimates. Then, a theorem of Ikeda (see [13,Corollary 1.4]) helps one to show that the SPCS is true when the multiplicities of the parameters are smaller than or equal to two, suggesting induction as a way of solving (Section 8).This approach has its origins in the work of the first author (J. D. Vélez). Specifically, the idea of proving the DSC by means of annihilators is first introduced in his thesis (see [14,Lemma 3.1.2.] and [15]). The reduction to the case = / S T J, where T is a Gorenstein local ring, and J is a principal ideal, was stated in a private communication from Vélez to Hochster, in 1996, and appears more explicitly in [11] and in [12]. Similar results related to the SPCS were obtained independently by Strooker and Stückrad [16].
Finally, it is worth noting that due to their simplicity and conceptual clarity, the results exposed in this paper are able to be used, in the context of modern artificial intelligence and pure mathematics, for creating pseudo-pre-code regarding fulfilling artificial mathematical intelligence in the specific mathematical subdiscipline of commutative algebra [17]. More specifically, this can be done based on the new cognitive foundations of mathematics program [ Proof. Since k is algebraically closed we can factor ( ) [18,Theorem 8.7 and proof] i , because translation and reduction mod m commutes. But that means exactly that , for any indices i, j. In conclusion, we get an isomorphism of rings between B and   and ∈ a m iαj . Furthermore, by the distributive law between tensor products and direct sums (see [20]) we get T J w w w has the desired form of our proposition, and then it is enough to prove the DSC in this case. □ , , be a regular local ring of dimension d, and , , , . Then T is a Gorenstein local ring with maximal ideal Proof. First, we see that T is a local Cohen-Macaulay (C-M) ring. In fact, let m 1 be any maximal ideal of T.
be the ideal generated by the system of parameters { … … } y y x x , , , , ,  forms a regular sequence. Now, due to the fact that R is Gorenstein, we see that [ … ] R y y , , r 1 so is. Moreover, one can also prove that when one mods out a Gorenstein ring by a regular sequence, the corresponding quotient ring remains Gorenstein. So, this proves that ) and due to the fact that all the maximal ideals of T contract to m by module-finiteness. □

DSC in terms of annihilators
Now we make preparations for the proof of the following fact: let , be a finite homomorphism of local rings, i.e. ( ) ⊆ h m η, where T is a local -R free ring, with T/mT Gorenstein, and let = / S T J, splits if and only if ⊈ J mT Ann T (by abuse of notation we denote by h again its composition with the natural projection → π T S : ).
Towards a generalization of the DST  1355 Proof. Clearly, we can assume that ⊆ I η. We know ( ) = A η nil ; therefore, there exists ∈ n such that = η 0 , be a local ring, T a finitely generated R-free module, and → θ T T : an R-homomorphism. Then θ is an isomorphism of R-modules if and only if , where Ā is the reduction of A mod m. Finally, since Ā is the matrix defining θ¯, the last condition is equivalent to saying that θ¯is an isomorphism ofk vector spaces.
is clearly an isomorphism ofk vector spaces. Since T is -R free, by Remark 4, θ is an isomorphism which means just that , which implies that / → ρ T J R : is the desired splitting -R homomorphism. □

Reduction to the case where J is principal
In the next proposition, we prove that (a proof of) the DST can be reduced to (a proof for) the case where J is a principal ideal generated by an element in mT.

Proof.
(1) In general, if R is regular, then so is the polynomial ring [ ] R T (see [20]). In particular, R m is a regular local ring and

Socle-parameter conjecture
In this section, and in the next section, we state two new conjectures the socle-parameter conjecture (SPC) in its strong (SPCS) and weak forms (SPCW), and we prove that the SPCW is equivalent to the DSC, and that the SPCS implies the SPCW. Besides, these two conjectures are equivalent in the equicharacteristic case and therefore both are equivalent to the DSC in the equicharacteristic case. However, as far as we know, the mixed characteristic case remains open. The new approach shows that the DSC is, in essence, a problem concerning algebraic and homological properties of Gorenstein local rings.
Let us start by reviewing some elementary notions. Let R be an -graded ring such that R 0 is an Artinian ring, and such that R is finitely generated as an R 0 -algebra. Let M be a finitely generated graded R-module of dimension d. Then, it is elementary to see that each homogeneous part M n is a finitely generated R 0 -module, and therefore, it has finite length (see [18,Proposition 6.5.]). It is well known in this case that there exists a unique polynomial  For a more technical reformulation of this notion due to Auslander and Buchsbaum, see [22]. Now, a theorem of Serre (see [22,   . Let ∈ z T be a zero divisor. Then ⋅ ∈ ⋅( ) u z Q z . Now, it holds that this strong form of the conjecture is equivalent to saying that ℓ( ( /( ))) − H x T z , 0 ℓ( ( /( ))) > H x T z , 0 1 . We prove the last equivalence: Proposition 9. In the situation of the SPCS the following are equivalent.

SPCS
2 Consider the following natural short exact sequence , by the isomorphism sending t to tz. After tensoring with T/Q we get Ann .

T T
Now, let us see that z Q z , and it is equivalent to = ∈ /( ( ) + ) u T Ann z Q 0 Proposition 10. Assume the same hypothesis as in SPCS and, in addition, that 1, then SPCS holds.
. It is a well-known fact that if ( /( )) H x T z , r denotes the Koszul homology and = { ( /( )) ≠ } q r H x T z sup : , 0 r , then = − a d q, therefore, = − ≤ q d a 1. In the case that = d 0, = Q 0, and is an Artinian ring. Hence, by previous results, we get Now, by previous comments and the fact that ≥ d 1 we see that ( /( )) > e Q T z, 0. □

SPCW
, and let ∈ u T be any lifting of a socle element in T/Q, i.e.
. Let z be a zero divisor. Then ⋅ ∈ ⋅( ) u z Q z , which is equivalent to the inequality ℓ( ( /( ))) − ℓ( ( /( ))) > H x T z H x T z , , 0 0 1 . Note that between the two forms of the SPC the only difference is the fact that in the mixed characteristic case one assume that = x p 1 . One needs this last condition in order to apply Cohen's structure theorem in mixed characteristic.
Remark 5. For proving any of the two versions of the SPC, it is enough to assume that ( ) T η , is complete. . Furthermore, T is also a Gorenstein ring (see [20,Theorem 18.3]).

Proof. Let
Finally, (( ′) + ( ′))/( ) ≅ (( ) + ))/( )) ⊗ ≠ QT z u QT z Q z u Q z T :ˆ: : QT ẑ which contradicts SPC in the complete case. □ 6 Equivalence to the DSC First, we review the notion of a coefficient ring: such that the natural projection ⊆ ↪ / = π K R R m k : 0 is an isomorphism. On the other hand, let ( ) R m k , , be a complete quasi-local (that means with a unique maximal ideal m but not necessarily Noetherian), mixed characteristic, and separated ring (i.e. ∩ = ( ) ∈ m 0 n n ). Then a coefficient ring for R is a sub-ring ( such that it is a local complete discrete valuation ring such that ∩ = m D η, and so that the inclusion induces an isomorphism / ≅ / D η R m. It is elementary to see that if = R char 0 and = > k p char 0, then D is a domain, and therefore one dimensional, which means exactly that ( ) D η , is a discrete valuation domain (DVD). A theorem of I. S. Cohen states that for complete local rings there always exists a coefficient ring (see [23,Theorem 9,Theorem 11] and [24, p. 24] , and then evaluating in Hence, ∉ p m 2 and then ≠ ∈ / p m m 0 2 is a part of a basis of / m m 2 as k-vector space, which is equivalent by the Lemma of Nakayama to the fact that p is a part of a minimal set of generators of m, say, R So T is a free R-module. Furthermore, z is contained in an associated prime of T because it is a zero divisor.
Since T is C-M, any associated prime is, in fact, a minimal prime. Thus, z is contained in a minimal prime ∈ P T Spec . Moreover, since T is C-M, T is equidimensional, which means, in particular, that In conclusion, , and then ∈ ⋅( ) uz Q z . Another way of proving this is using directly Proposition 9. □

SPCS for small multiplicities
A natural way one could attack the SPCS would be by induction on ( ) e T , the multiplicity of T. We note first that by Remark 5 one may assume that T is complete. In the case ( ) = e T 1, then, since T is a complete C-M ring, it is equicharacteristic and therefore unmixed. Hence, by the Criterion for multiplicity one (see [25]) T must be a regular local ring; in particular, this ring is an integral domain, which implies = z 0, from which the SPCS follows directly.
Suppose now that ( ) = e T 2. Since T is C-M, it must satisfy the S2 condition of Serre: for any ∈ P T Spec , ( ) ≥ ( ) P T T depth , min 2,dim P , due to the fact that ( ) = ( ) P T T depth , dim P . Hence, by a Theorem of Ikeda (see [13,Corollary 1.3.]) T is a hypersurface of the form /( ) B f , where B is a complete regular local ring. Now, we prove a more general result, namely, that the SPCS holds for residue class ring of local Gorenstein rings which are UFD and C-M, which implies, in particular, the case of multiplicity two because regular local rings are UFD and C-M (see [21]).  [21,Proposition 18.13]). Since T is equidimensional, , , ,  Proof. After tensoring with the completion of R, which is faithfully flat, we can assume, by previous comments, that R is complete. By Cohen's structure theorem (see [ is a system of parameters for the -R module /( ) T z , we know that