EXACT SUBCATEGORIES, SUBFUNCTORS OF Ext , AND SOME APPLICATIONS

. Let ( A , E ) be an exact category. We establish basic results that allow one to identify sub(bi)functors of Ext E ( − , − ) using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and ﬁnd a range of applications from detecting regularity to understanding Ulrich modules.


Introduction
The Yoneda characterization of Ext is familiar to most students of homological algebra.Let A, B be two objects in an abelian category A. Then Ext A (A, B) is the set of all equivalence classes of sequences of the form 0 → B → C → A → 0, where two sequences α, β are equivalent if we have the following commutative diagram: ) can be given an abelian group structure by the well-known Baer sum as described in, for instance, [38,Tag 010I].This consideration can be carried out more generally in any exact category (see [14,Subsection 1.2]).
The purpose of this note is to study the following rather natural questions: what if we place additional constraints on the short exact sequences?When do we get a subfunctor of Ext 1 ?Can one apply such functors to study ring and module theory, similar to the ways homological algebra has been very successfully applied in the last decades?
Let us elucidate our goals with a concrete example.Let (R, m, k) be a Noetherian local ring.Let A, B be finitely generated R-modules.We consider exact sequences 0 → B → C → A → 0 of R-modules, with the added condition that µ(C) = µ(A) + µ(B), where µ(−) denotes the minimal number of generators.As we shall see later, the equivalence classes of such sequences do form a subfunctor of Ext 1 R , which we denote by Ext 1 R (−, −) µ .More surprisingly, the vanishing of a single module Ext 1 R (−, −) µ can be used to characterize the regularity of R, a feature that is lacking with the classical Ext 1 .
Analogous versions of classical homological functors have been studied by various authors, notably starting with Hochschild ( [23]) who studied relative Ext and Tor groups of modules over a ring with respect to subrings of the original ring.This work is further developed by Butler-Horrocks as well as Auslander-Solberg, where the point of view is switched to allowable exact sequences that give rise to sub(bi)functors of Ext 1 .The whole circle of ideas is now thriving on its own under the name "relative homological algebra", with exact structures playing a fundamental role, see [4,5,8,16,14,17,18,29,33,40] for an incomplete list of literature and [34,6] for some excellent surveys.In commutative algebra, as far as we know, this line of inquiry has not been exploited thoroughly, however traces of it can be found in [31,35] and [15, Section 1,2].
Although the existing literature provides excellent starting points and inspiring ideas for this present work, it is not always easy to extract the precise results needed for our intended applications.For instance, while the connections between subfunctors of Ext 1 and certain sub-exact structures on a fixed category are well known ([14, Section 1.2]), checking the conditions of substructures in each case can be time-consuming.
We are able to find criteria that can be applied in broad settings to identify exact subcategories and subfunctors.Here is a sample result applicable to our motivating example above, which follows from Theorem 4.8 and Proposition 3.8.Theorem 1.1.Let (A, E) be an exact category.Let φ : A → Z be a function such that φ is constant on isomorphism classes of objects in A, φ is additive on finite biproducts, and φ is sub-additive on kernelcokernel pairs in E (i.e., if M N L is in E, then φ(N ) ≤ φ(M ) + φ(L)).Set E φ := {kernel-cokernel pairs in E on which φ is additive}.Then E φ gives rise, via the Yoneda construction, to a subfunctor of Ext E (−, −).
Our above Theorem is partly motivated by, and can be used to recover and extend recent interesting work of Puthenpurakal in [31], see Theorem 4.17.
Another situation we would like to have convenient criteria for subfunctors is when one "restricts" to a certain subcategory.A concrete example we have in mind concerns Ulrich modules and their recent generalizations.These modules form a subcategory of Cohen-Macaulay modules over a commutative ring and have been receiving increasing attention over the years due to very interesting and useful algebraic and geometric properties that their existence or abundance imply.We are able to show that even the generalized notion of "I-Ulrich modules", recently introduced in [11], induces subfunctors of Ext 1 R (−, −).See Proposition 4.1 and Corollary 4.7 in this regard.
While this work is mainly concerned with foundational results, we also study the properties of some chosen new subfunctors, just to see if they are worth our efforts to show their existence!The early returns seem promising: these functors can be used to detect a wide range of ring and module-theoretic properties.Below we shall describe the organization of the paper and describe the most interesting findings in more detail.
Section 2 is devoted to preliminary results on subfunctors of additive functors.While these results are perhaps not new, we were unable to locate convenient references, hence their inclusion.
Section 3 establishes various foundational results on exact subcategories, which form the cornerstone of the theory.As mentioned above, our applications require some extra care in preparation, and we try to give complete proofs whenever possible.
Section 4 concerns our first main application.We study sub-additive numerical functions φ on an exact category and show that under mild conditions they induce exact subcategories and hence subfunctors of Ext 1 (−, −), which we denote by Ext 1 (−, −) φ .See Theorem 4.8.A key consequence, Theorem 4.17 is inspired by, as well as extends, [31,Theorem 3.11,Theorem 3.13].We also give similar results about subfunctors of Ext 1 induced by half-exact functors, in the spirit of [1], see Corollaries 4.13 and 4.14.
In Section 5 we focus on two special types of subfunctors, which arise from simple applications of previous sections.Already these cases appear to be interesting and useful.The first one is Ext 1 R (−, −) µ , where µ is the minimal number of generators function mentioned above.We compute the values of this subfunctor on all (pair of) finitely generated modules over a DVR (Corollary 5.1.14),as well as for certain pairs of modules over a Cohen-Macaulay ring of minimal multiplicity (Proposition 5. 1.22).
Using this subfunctor, we prove the following characterization of the regularity of local rings, which is the combination of Theorem 5.1.1,Theorem 5.1.12,Proposition 5.1.14and Corollary 5.2.7.Note that the regularity can be detected by the vanishing of a single Ext 1 R (−, −) µ -module.In the following statement, we mention that for a finitely generated module M over the local ring R, Ω i R M denotes the i-th syzygy in a minimal free resolution of M .Theorem 1.2.Let (R, m, k) be a local ring of depth t > 0.Then, the following are equivalent: (1) R is regular.
(3) Ext 1 R (k, M ) µ = 0 for some finitely generated R-module M of projective dimension t − 1. (4) R is Cohen-Macaulay, and Ext 1 R (Ω t−1 R k, N ) µ = 0 for some finitely generated non-zero R-module N of finite injective dimension.Moreover, if t = 1, then the above are also equivalent to each of the following.
) for all finitely generated R-modules M and N .
(6) R is Cohen-Macaulay and there exist Ulrich modules M, N such that N is faithful, M = 0 and for every R-module X that fits into a short exact sequence 0 → N → X → M → 0 one has X is also an Ulrich module.
We note here that the usual Ext-modules (without the µ) in ( 2), ( 3) and (4) of the above theorem are always non-zero.Moreover, the statement of part (6) of Theorem 1.2 apparently has nothing to do with subfunctor of Ext 1 , but we do not know a proof of (1) ⇐⇒ (6) (which is contained in Corollary 5.2.7) without resorting to Ext 1 Ul(R) (−, −) : Ul(R) op × Ul(R) → mod R. It is worth mentioning that one can also use Ext 1 R (−, −) µ to detect the property of R being a hypersurface of minimal multiplicity (Corollary 5. 1.23) or the weak m-fullness of a submodule (Proposition 5. 1.11).
The second type arises from I-Ulrich modules, where I is any m-primary ideal in a Noetherian local ring (R, m).See 4.5 and 4.7 for the definition of I-Ulrich modules and the fact that they form an exact category.In Theorem 5.2.14, using a subfunctor of Ext 1 corresponding to Ulrich modules over 1-dimensional local Cohen-Macaulay rings, we give some characterizations of modules belonging to add R (B(m)) and also characterize when B(m) is a Gorenstein ring in terms of annihilator of Ext 1 R of Ulrich modules.Here, B(−) denotes the blow-up.We give some applications of Theorem 5.2.14, one of which relates annihilation of Ext 1 R (Ul(R), m) with that of Ext 1 R (Ul ω (R), B(ω)) (see Corollary 5.2.17).Finally, we should mention that one of the main applications of our results has appeared in a separate work, where we study the splitting of short exact sequences of Ulrich modules and connections to other properties of singularities ([9]).

Preliminaries on subfunctors of additive functors
Unless otherwise stated, all rings in this paper are assumed to be commutative, Noetherian and with unity.For a ring R, Q(R) will denote its total ring of fractions.For an R-module M , λ R (M ) will denote its length.For a finitely generated R-module M , and i ≥ 1, by Ω i R M we mean Imf i , where we have an exact sequence with each F j being a finitely generated projective R-module.When R is moreover local, we choose this so that Im(f j ) ⊆ mF j−1 for each j ([3, Proposition 1.3.1]).
For definitions, and basic properties of additive categories, additive functors, R-linear categories, and R-linear functors, we refer the reader to [38, Tag 09SE, Tag 010M, Tag 09MI].We now recall the definition of subfunctors as in [27].Definition 2.1.Let A, B be two categories.Let F : A → B be a covariant (resp.contravariant) functor.A covariant (resp.contravariant) functor G : A → B is called a subfunctor of F if for every M ∈ A, there exists a monomorphism j M : G(M ) → F (M ), and moreover, for every M, N ∈ A, and f ∈ Mor A (M, N ), the following diagrams are commutative, where the left one stands for the covariant case and the right one for the contravariant case: 2.2.For our purposes, we will always take B to be either the category of abelian groups Ab, Mod R or mod R (hence monomorphisms are just injective morphisms) for some commutative ring R, and j M will usually be just the inclusion map.
The following Lemma is probably well-known, but we could not find an appropriate reference, hence we include a proof.This will be used throughout the remainder of the article, possibly without further reference.
Proof.We will only prove the covariant case of both, since the contravariant case is similar.
(1) Let A, B be two additive categories and let F : A → B be an additive functor.Also let G : A → B be a subfunctor of F .Then we have to show that the map G : Mor A (X, Y ) → Mor B (G(X), G(Y )) is a homomorphism of abelian groups for all X, Y ∈ A. Fix two objects X, Y ∈ A. Then we need to prove that G(f + g) = G(f ) + G(g) for all f, g ∈ Mor A (X, Y ).Since G is a subfunctor of F , we have the following commutative diagrams: Then we get where the third and the fifth equalities follow from the fact that B is an additive category.Now since j Y is a monomorphism, so (2) Let A, B be two R-linear categories and let F : A → B be an R-linear functor.Also let G : A → B be a subfunctor of F .Then we have to show that the map G : Then by part (1) we already have the additivity of G, so we only need to prove that G(rf ) = rG(f ) for all f ∈ Mor A (X, Y ) and for all r ∈ R. Since G is a subfunctor of F , we have the following commutative diagrams: Then we get where the third and the fifth equalities follow from the fact that B is an R-linear category.Now since j Y is a monomorphism, so We finish this section with a submodule inclusion result relating subfunctors of R-linear functors which will be applied in Section 5.
Lemma 2.4.Let A be an additive R-linear category, and G : A → Mod R be a subfunctor of an R-linear functor F : A → Mod R, and for every object A ∈ A, let j A : G(A) → F (A) be the monomorphism as in the definition of a subfunctor.Let I be an ideal of R.
Proof.We will only do the covariant case, the contravariant case being similar.For every i, we have π i : X → A i , which gives rise to the following commutative diagram Consequently, we get the following commutative diagram where the horizontal arrows are isomorphisms, since F and consequently G are additive functors (Lemma 2.3).Call the top horizontal map θ, and the bottom one α, so that α ), where we have used α −1 (IM ) = Iα −1 (M ), since α is an R-linear map.

Some generalities about exact subcategories
In this section, we record some generalities about exact subcategories of an exact category that we will later use for subcategories of mod R when R is a commutative Noetherian ring.All our sub-categories are strict (closed under isomorphism classes) and full, and we often abbreviate this as strictly-full.We will follow the definition of an exact category described in [6,Definition 2.1].We try to provide complete proofs whenever possible.
Given an exact category (A, E), we call a monomorphism X i − → Y to be an E-inflation (also, called an admissible monic) if it is the part of a kernel-cokernel pair X i − → Y − → Z, which lies in E. Dually, we call an epimorphism Y p − → Z to be an E-deflation (also, called an admissible epic) if it is the part of a kernel-cokernel pair X − → Y p − → Z, which lies in E. We will often denote an admissible monic (resp.an admissible epic) by (resp. ).We begin by stating a lemma on morphisms and kernel-cokernel pairs, which we will use frequently while proving that certain structures are closed under isomorphism classes of kernel-cokernel pairs.This should be standard and well-known, but we could not find an appropriate reference, hence we include a proof.Lemma 3.1.Let A be an additive category.Let M, N, L ∈ A be such that where the vertical arrows φ 1 , φ 2 , φ 3 are isomorphisms, then Proof.We will only prove that, ker(d ′ ) = i ′ , since the proof of Coker(i ′ ) = d ′ can be given by a dual argument.
From the above commutative diagram we have, Then by Equation 3.0.0.1 we have, • 0 = 0. Now, since ker(d) = i, by the universal property of a kernel of a map, there exists a morphism u : K → M such that i • u = f .Now define u ′ := φ 1 • u : K → M ′ .Then by Equation 3.0.0.1 we get that, We now record a Lemma on the intersection of exact subcategories.Note that, this is slightly different (and in view of [33, two morphisms in A λ for all λ.Hence by Lemma 3.1 we get that, M ′ → N ′ → L ′ is a kernel-cokernel pair in A λ for all λ.Since (A λ , E λ ) is an exact category and M N L is in E λ for all λ, we have Since (A λ , E λ ) is an exact category for all λ, by [6, Lemma 2.7] we get that is in E λ for all λ.As A λ is an additive subcategory of A for all λ, so 0 A = 0 A λ for all λ.Hence by ) is an exact category for all λ, by [6, Lemma 2.7] we get that is in E λ for all λ.As A λ is an additive subcategory of A for all λ, so 0 A = 0 A λ for all λ.Hence by definition 0 is an admissible epic in E λ for all λ.Hence there exist objects D λ ∈ A λ and kernel-cokernel is in E for all λ.Hence σ λ is a kernel-cokernel pair in A for all λ, so by the universal property of kernels we get that the kernelcokernel pairs σ λ 's are all isomorphic to each other.Hence all the D λ 's are isomorphic to each other.Now fix a λ, say λ 0 .Then D λ0 ∼ = D λ for all λ.Since A λ is a strict subcategory of A for all λ, we have D λ0 ∈ ∩ λ A λ .Now A λ is a strictly full subcategory of A for all λ, so D λ0 ∈ A λ implies that σ λ0 is a kernel-cokernel pair in A λ for all λ.Since E λ is closed under isomorphisms of kernel-cokernel pairs in A λ and σ λ 's are all isomorphic to each other, we get σ λ0 ∈ E λ for all λ.So, σ λ0 ∈ ∩ λ E λ .Hence Since (A λ , E λ ) is an exact subcategory of (A, E), by [6, Proposition 5.2] we get that the square is a pushout square in E for all λ.Then by the universal property of pushout we get that the kernel-cokernel pairs β λ 's are all isomorphic to each other, so all the B ′ λ 's are isomorphic to each other.Now fix a λ, say λ 0 .Then B ′ λ0 ∼ = B ′ λ for all λ.Since A λ is a strict subcategory of A for all λ, we have B ′ λ0 ∈ ∩ λ A λ .Now A λ is a strictly full subcategory of A for all λ, so B ′ λ0 ∈ A λ implies that β λ0 is a kernel-cokernel pair in A λ for all λ.Since E λ is closed under isomorphisms of kernel-cokernel pairs in A λ and β λ 's are all isomorphic to each other, we have β λ0 ∈ E λ for all λ.So, Next, we record a useful lemma for proving the exactness of certain structures on additive subcategories of a given exact category.Lemma 3.3.Let (A, E) be an exact category.Let A ′ be a strictly full additive subcategory of A. Define If the pullback (resp.pushout) in A of every Edeflation (resp.inflation) of sequences in E| A ′ along every morphism in A ′ is again in A ′ , then (A ′ , E| A ′ ) is an exact subcategory of (A, E).
The proof of Lemma 3.3 depends on Lemma 3.1 and the following proposition: Proposition 3.4.Let (A, E) be an exact category.Suppose we have the following diagram: where the square commutative diagram is a pullback diagram and e, p are admissible epics and i is a kernel of p. Then i ′ is an admissible monic with a cokernel given by B ′ C p•e .
Proof.The proof follows from the construction in the proof of [6,Proposition 2.15].The only missing point in the proof of [6, Proposition 2.15], towards showing i ′ is a kernel of p • e, is the following: it was not shown that (p • e) • i ′ = 0.This can be easily checked as follows: Proof of Lemma 3.3: Clearly, each kernel-cokernel pair in E| A ′ is a kernel-cokernel pair in A, hence also a kernel-cokernel pair in A ′ .Thus E| A ′ consists of kernel-cokernel pairs in A ′ .Now we will show that, E) is an exact category, by [6, Proposition 2.12(iv)] we have the following pushout commutative diagram with rows being kernel-cokernel pairs in E: Now by the assumption we have, is in E. As A ′ is an additive subcategory of A, so From now on, given a subcategory A ′ of an exact category (A, E), the notation E| A ′ will stand for as defined in Lemma 3.3.Using Lemma 3.3, we now record two quick consequences, which give a sufficient condition on a subcategory A ′ such that (A ′ , E| A ′ ) is an exact subcategory of (A, E).The first of which we state below now is well-known, see for instance [6,Lemma 10.20].However, due to the absence of a proof in [6,Lemma 10.20], we give a proof using our Lemma 3.3.Proposition 3.5.(c.f.[6,Lemma 10.20]) Let (A, E) be an exact category.Let A ′ be a strictly full additive subcategory of A. Assume that for every Proof.By Lemma 3.3, it is enough to show that the pullback (resp.pushout) in A of every E-deflation Then by [6, Proposition 2.12(iv)] we get the following pushout commutative diagram: , by assumption and the bottom row of the above diagram we get that B ′ ∈ A ′ .The pullback case follows by a dual argument.Hence (A ′ , E| A ′ ) is an exact subcategory of (A, E).
Theorem 3.6.Let (A, E) be an exact category.Let A ′ be a strictly full additive subcategory of A. Assume that A ′ is closed under kernels and co-kernels of admissible epics and monics in E respectively.Then, (A ′ , E| A ′ ) is an exact subcategory of (A, E).
Proof.By Lemma 3.3, it is enough to show that the pullback (resp.pushout) in A of every E-deflation be an inflation in E| A ′ and f : A → A ′ be a morphism in A ′ .Now we have the following pushout commutative diagram in (A, E): Then by [6, Proposition 2.12(ii)] we have the following kernel-cokernel pair in The pullback case follows by a dual argument.Hence (A ′ , E| A ′ ) is an exact subcategory of (A, E).
Given any exact category (A, E) and C, A ∈ A, one can define the Yoneda Ext group Ext E (C, A), which has an abelian group structure by Baer sum (see the beginning of [14, Section 1.2] for a description of Ext E (−, −)).When (A, E) is moreover an R-linear category, then Ext E (C, A) can be given an R-linear structure via either of the following constructions, both of which yield equivalent triples in Ext E (C, A): Given a kernel-cokernel pair σ : A B C in Ext E (C, A), the multiplication r • σ is either given by the following pullback diagram:

PB
or by the following pushout diagram: For this, we first record a remark.Remark 3.7.Let (A ′ , E ′ ) be a strictly full exact subcategory of (A, E).Then, for σ Hence there exists f ∈ Mor A (B, B ′ ) such that we have the following commutative diagram in A: Proposition 3.8.Let (A, E) be an exact category.Let (A ′ , E ′ ) be a strictly full exact subcategory of From now on, we will call this map φ.The well-definedness and injectivity of φ follow from Remark 3.7.Next, we will show that φ is a group homomorphism. and we get φ is a group homomorphism.Let A, B, C ∈ A ′ and let f ∈ Mor A ′ (A, B).First, we will show that Ext E ′ (C, −) is a subfunctor of Ext E (C, −), so we need to prove that the following diagram commutes: where for every . This shows that the natural inclusion map φ is R-linear.
In the following corollary, we denote Hom A (−, −) just by Hom(−, −) (we completely ignore the subcategory, since all our subcategories are full).In view of Definition 4.10, and the discussion following 4.11, the following corollary is crucial for recovering [35, Proposition 1.37, Proposition 1.38].
Corollary 3.9.Let (A, E) be an exact category.Let (A ′ , E ′ ) be a strictly full exact subcategory of (A, E).Let σ : M N L i p be a kernel-cokernel pair in E ′ .Then, for every A ∈ A ′ , we have the following commutative diagrams of long exact sequences: and Proof.We will only prove the covariant version (the first diagram above), since the proof of the contravariant version is given by the dual argument.
In the first diagram above, the commutativity of the first two squares is obvious.Also, the commutativity of the last two squares follows directly from the proof of Proposition 3.
. So, it is enough to prove that the following square is commutative: Let (A, E) be an exact category.Let A ′ be a strictly full subcategory of A.
Proposition 3.10.Let (A, E) be an exact category.Let A ′ be a strictly full additive subcategory of A.
Proof.By Lemma 3.3, it is enough to show that the pullback (resp.pushout) in A of every E-deflation (resp.inflation) of sequences in E| A ′ along every morphism in A ′ is again in A ′ .We will only prove the pullback case, since the proof of the pushout case can be given by a similar argument.Let B C be an admissible epic in E such that it is in E| A ′ , which means that there exist a kernel-cokernel pair where the columns are natural inclusion maps.So, for every is the pullback of β by f in (A, E).Now consider the pullback of γ by f in (A, E) as follows: Hence the pullback in A of every E-deflation of sequences in E| A ′ along every morphism in A ′ is again in A ′ .Hence by Lemma 3.3 we get that, (A ′ , E| A ′ ) is an exact subcategory of (A, E).

subcategories and subfunctors of Ext 1 from numerical functions and applications to module categories
In this section, we present tools to identify exact subcategories of an exact category coming from certain numerical functions.Consequently, due to Proposition 3.8, we are also able to identify subfunctors of Ext 1 associated with certain numerical functions.Our first result in this direction is an application of Theorem 3.6.Proposition 4.1.Let (A, E) be an exact category.Let φ : A → Z ≤0 be a function such that φ is constant on isomorphism classes of objects in A, φ is additive on finite biproducts, and φ is sub-additive on kernelcokernel pairs in . Let A ′ be the strictly full subcategory of A, whose objects are given by {M ∈ A : Hence 0 A ∈ A ′ .Since A is an additive category and φ is additive on finite biproducts, by definition of A ′ we get that for every X, Y ∈ A ′ , the biproduct of X and Y in A also belongs to A ′ .Hence A ′ is an additive subcategory of A. Let B ∈ A ′ and A 1 B A 2 i p be a kernel-cokernel pair in E. We will show that A 1 , A 2 ∈ A ′ .Since φ is sub-additive on kernel-cokernel pairs in E and B ∈ A ′ , we get 0 = φ(B) ≤ φ(A 1 ) + φ(A 2 ) ≤ 0. Hence φ(A 1 ) + φ(A 2 ) = 0. Since we know φ always takes non-positive values, we get φ(A 1 ) = φ(A 2 ) = 0, hence A 1 , A 2 ∈ A ′ .Thus, A ′ is closed under kernels and co-kernels of admissible epics and monics in E respectively.Then by Theorem 3.6 we get that, (A ′ , E| A ′ ) is an exact subcategory of (A, E.
We now proceed to give our main example of Proposition 4.1.First, we recall some terminologies and prove some preliminary lemmas.For any unexpected concepts and notations, we refer the reader to [3] and [28].
Let R be a commutative ring, and let S R be the collection of all short exact sequences of R-modules, which gives the standard exact structure on Mod R. When the ring in question is clear, we drop the suffix R and write only S. Note that, since mod R is extension closed in Mod R, hence the collection of all short exact sequences in mod R gives an exact subcategory of Mod R, and we take this as the standard exact structure of mod R. Note that, if (X , E) is an exact subcategory of Mod R and X ⊆ mod R, then (X , E) is also an exact subcategory of mod R. Now for a Noetherian local ring (R, m) of dimension d and for an integer s ≥ 0, let CM s (R) denote the full subcategory of mod R consisting of the zero-module, and all Cohen-Macaulay R-modules ([3, Definition 2.1.1]) of dimension s.Note that, when s = d, CM d (R) is just the category of all maximal Cohen-Macaulay modules, which we will also denote by CM(R).4.2.We quickly note that for each s ≥ 0, CM s (R) is closed under finite direct sums, direct summands and closed under extensions in Mod R. Indeed, let 0 → L → M → N → 0 be a short exact sequence with L, N ∈ CM s (R).If L or N is zero, then there is nothing to prove.So, assume L, N are non-zero.Now M ∈ mod R, and we also have the following calculation for dim Hence, if S is the standard exact structure on Mod R, then (CM s (R), S| CM s (R) ) is an exact subcategory of Mod R by Proposition 3.5.Now let I be an m-primary ideal and φ I : CM s (R) → Z be the function defined by φ I (M ) := λ R (M/IM ) − e R (I, M ), where e R (I, M ) is the multiplicity of M with respect to I ([3, Definition 4.6.1]).We first show that φ I satisfy all the hypothesis of Proposition 4.1.To prove this, we will need to pass to the faithfully flat extension S = R[X] m[X] whose unique maximal ideal is mS.That we can harmlessly pass to this extension, is discussed in the following 4.3.Let (R, m) be a local ring and consider S = R[X] m[X] which is a faithfully flat extension of R and the unique maximal ideal of S is mS, whose residue field S/mS is infinite (see [24,Section 8.4]).Let M ∈ mod(R).By [3, Theorem 2.1.7](and the sentences following it), we have M ∈ CM s (R) if and only if M ⊗ R S ∈ CM s (S).Let I be an m-primary ideal of R. As mS is the maximal ideal of S, so , where the last equality is by [38, Tag 02M1] remembering that S/mS is the residue field of S. Also, e R (I, Proof.Since it has been noticed that (CM s (R), S| CM s (R) ) is an exact subcategory of Mod R, by Lemma 4.4 and Proposition 4.1 it follows that (Ul s I (R), S| Ul s I (R) ) is an exact subcategory of (CM s (R), S| CM s (R) ), and hence of Mod R. The subfunctor part now follows from Proposition 3.8.
Next, we record a proposition for constructing a special kind of exact substructure of an exact structure E on a category A, without shrinking the category, that also comes from certain kinds of numerical functions.This will be used in the next section to recover and improve some results on module categories.Theorem 4.8.Let (A, E) be an exact category.Let φ : A → Z be a function such that φ is constant on isomorphism classes of objects in A, φ is additive on finite biproducts, and φ is sub-additive on kernelcokernel pairs in . Set E φ := {kernel-cokernel pairs in E on which φ is additive}.Then, (A, E φ ) is an exact subcategory of (A, E).

Proof
is in E. Since φ is additive on finite biproducts, φ is additive on the kernel-cokernel pair 0 . Hence by definition Now consider the pullback square of Proposition 3.4 By the dual (pullback version) of [6, Proposition 2.12(i) =⇒ (iv)] we get a commutative diagram in E as follows: The top row gives is in E and φ(−) is constant on isomorphism classes of objects, we have φ is an admissible epic in E φ , the bottom row of the in E is actually in E φ .This shows (A, E φ ) satisfies the axiom [E1 op ].

Now we will show that (A, E φ ) satisfies the axiom [E2]. Let A B
i be an admissible monic in E φ and A f − → A ′ be an arbitrary morphism in A. Then we have the following pushout commutative square in E: Then by [6, Proposition 2.12(ii)] we have the following kernel-cokernel pair in E: ).Also, by [6, Proposition 2.12(iv)] we have the following commutative diagram with rows being kernel-cokernel pairs in E: is an admissible monic in E φ .Thus (A, E φ ) satisfies the axiom [E2].A dual argument will show that (A, E φ ) satisfies the axiom [E2 op ].Hence (A, E φ ) is an exact subcategory of (A, E).
Moving forward, we record some applications of Theorem 4.8.4.9.Let X be a subcategory of Mod(R) such that (X , S| X ) is an exact subcategory of Mod R. In this case, for M, N ∈ X , by Ext 1 X (M, N ) we will mean Ext S|X (M, N ), i.e., Ext 1 X (M, N ) = {[σ] : the middle object of σ is in X } (Note that, it does not matter whether the equivalence class is taken in S| X or S by Remark 3.7).Note that, Ext 1 X (−, −) : X op × X → Mod R is a subfunctor of Ext 1 R (−, −) : X op × X → Mod R by Proposition 3.8.For example, we can apply this discussion to X = Ul s I (R) due to Corollary 4.7.Note that, if X is closed under taking extensions in Mod R, then (X , S| X ) is exact subcategory of Mod R by Proposition 3.5.In this case, Ext 1 X (M, N ) = Ext 1 R (M, N ) for all M, N ∈ X .Definition 4.10.Let X be a subcategory of Mod R such that (X , S| X ) is an exact subcategory of Mod R, and φ : X → Z be a function satisfying the hypothesis of Theorem 4.8, i.e., φ is constant on isomorphism classes of modules in X , φ is additive on finite direct sums, and sub-additive on short exact sequences of modules in X .Then, in the notation of Theorem 4.8, (X , S| X φ ) is an exact subcategory of (X , S| X ), hence an exact subcategory of Mod R. For M, N ∈ X , define Ext 1 Before we apply our discussion to more certain special cases, we record one preliminary lemma.Lemma 4.12.Let (X , E) be an exact subcategory of Mod R. Let G : X → fl(R) be an additive half-exact functor, where fl(R) denotes the full subcategory of mod R consisting of finite length modules.Then, φ(−) := λ R (G(−)) : X → Z satisfies the hypothesis of Theorem 4.8.Moreover, given a short exact sequence σ with objects in X , G(σ) is a short exact sequence of modules if and only if φ is additive on σ.

and in this case, we denote it just by Ext
Proof.We will only prove the statement when G is a covariant functor, since the proof for a contravariant functor is similar.Since G is a functor and λ R is constant on isomorphism classes of objects in Mod R, φ is constant on isomorphism classes of objects in Hence φ is subadditive on a short exact sequence with objects in X .Thus φ satisfies the hypothesis of Proposition 4.8.Next let, σ : 0 where X is a fixed R-module of finite length and i ≥ 0 is an integer.
Given a collection C ⊆ mod R, in [1], the authors considered the following: Let S C be the collection of all short exact sequences σ of finitely generated R-modules such that Hom R (X, σ) is short exact for all X ∈ C. We first prove that (mod R, S C ) is an exact subcategory of mod R. Since S C = ∩ C∈C S C , it is enough to prove that for each C ∈ C, S C gives an exact substructure on mod R (intersection of exact substructures is again an exact substructure by Lemma 3.2).Now note that, S C = {σ : G(σ) is short exact}, where G : mod R → fl(R) is given by G(−) := Hom R (C, −), so by Corollary 4.13 we get that (mod R, S C ) is an exact subcategory of mod R. Since Ext S C (−, −) = F C (−, −), by Proposition 3.8 we are done.
Next, we recover and improve upon [31,Theorem 3.11,Theorem 3.13].For a Noetherian local ring (R, m), let MD(R) denote the full subcategory of mod R consisting of all modules M such that either M = 0 or dim M = dim R. We recall that, if I is an m-primary ideal, then e R (I, −) is additive on MD(R).We also recall that, a finitely generated module M satisfy Serre's condition (S n ) if depth M p ≥ inf{n, dim R p } for all p ∈ Spec R. The full subcategory of mod R consisting of all modules satisfying (S n ) is denoted by S n (R).Note that, S n (R) is closed under extensions in mod R. Lemma 4.15.Let R be an equidimensional local ring.Let S 1 (R) be the collection of all modules in mod R satisfying Serre's condition (S 1 ).Then, S 1 (R) ⊆ MD(R).
Proof.The hypothesis on R implies that, dim(R/p) = dim R for all p ∈ Min(R).So, if 0 = M satisfy (S 1 ), then there exists p ∈ Ass(M ) ⊆ Min(R).Hence dim(M ) ≥ dim(R/p) = dim R, and we are done.Now inspecting the proof of [30,Proposition 17], the only place, where R is Cohen-Macaulay and M is maximal Cohen-Macaulay are required, is to ensure that e R (I, −) is additive on 0 → Ω R M → F → M → 0 and that M, Ω R M ∈ MD(R).But if we assume R is equidimensional and satisfy (S 1 ), and (What [31] denotes by e T R (M ) is exactly same as e T m (M ) in our notation).The following theorem recovers and improves upon [31,Theorem 3.11,Theorem 3.13] (since a local Cohen-Macaulay ring R is equidimensional, satisfies (S 1 ) and CM(R) ⊆ S 1 (R)).Theorem 4.17.Let (R, m) be an equidimensional local ring satisfying (S 1 ).Let X = S 1 (R) be the subcategory of mod R consisting of all modules satisfying (S 1 ).Let I be an m-primary ideal.Then, Since S 1 (R) is an extension closed subcategory of mod R, we have (S 1 (R), S| S1(R) ) is an exact subcategory of mod R. Since for each n, the function mod R → Z given by M → λ(Tor R 1 (M, R/I n+1 )) is subadditive on short exact sequences of mod R, we have e T I : S 1 (R) → Z is also subadditive on short exact sequences of S 1 (R).Thus (S 1 (R), S| e T I S1(R) ) is an exact subcategory of (S 1 (R), S| S1(R) ) by Theorem 4.8.Hence we are done by Proposition 3.8.

Special subfunctors of Ext 1 and applications
Throughout this section, R will denote a Noetherian local ring with unique maximal ideal m and residue field k.We will denote by µ R (−) the minimal number of generators function for finitely generated modules, i.e., µ R (M ) = λ R (M ⊗ R R/m) for all M ∈ mod R. We drop the subscript R when the ring in question is clear.We shall study properties and applications of a number of subfunctors of Ext 1 , whose existence follow from results in previous sections.When R is regular, we show Ext 1 R (Ω d−1 R k, R) µ = 0, which shows one direction of Theorem 5.1.1.For this, we first need two preliminary lemmas.
Then, y is part of a minimal system of generators of X, but X is cyclic, so X = Ry.Also, Ry ⊆ Y ⊆ X.Hence, Y = X.Lemma 5.1.3.Let M be a finitely generated R-module with first Betti number 1. Then, By hypothesis, we have Ω R M ∼ = R/I for some ideal I = R, and we have an exact sequence 0 → R/I → F → M → 0, for some free R-module F .So, in particular, ).Now, one direction of Theorem 5.1.1 is the following.
The other direction of Theorem 5.1.1 requires more work.First, we begin with the following setup.
for some unit r ∈ R. So, we have the following pushout diagram: − → ω R is an isomorphism.Hence by five lemma, X α ∼ = X β .So, there exists a unique module (up to isomorphism), call it E R , such that the middle term of every non-split exact sequence 0 Moving forward, we denote Hom R (−, ω R ) by (−) † .We need to collect some properties of E R to prove the other direction of Theorem 5.1.1.
We record the following lemma, which will be used to deduce further properties of the module E R .In the following, r(−) will denote the type of a module, i.e., r( We prove this by induction on d.First, let d = 1.By Lemma 5.1.6we have This finishes the inductive step, and hence the proof.Since µ(ω R ) = r(R), Lemma 5.1.8says that the sequence 0 As a consequence of this, we get the following: Proposition 5.1.9.Let (R, m, k) be a non-regular local Cohen-Macaulay ring of dimension d ≥ 1, and N be a finitely generated non-zero R-module with finite injective dimension.Then ).We first consider the case where R is complete, hence admitting a canonical module ω R .Consider the µ-additive (by Lemma 5.1.8)exact sequence σ : 0 , by applying Hom R (−, N ) to σ, we get the following part of a commutative diagram of exact sequences by Corollary 3.9: R (E R , N ) µ = 0 as well.Hence we get the following commutative diagram: R k → 0 be a short exact sequence.We need to show that, σ is µ-additive.Now consider the completion σ : 0 Since R is non-regular, Cohen-Macaulay of dimension d and admits a canonical module and N ∈ mod R has finite injective dimension over R, by the first part of the proof we get Ext Since number of generators does not change under completion, we get µ , where g : k → N + Rf is the splitting map, so k ∼ = g(k).Now f ∈ N + Rf = N ⊕ g(k), so f = x + y for some x ∈ N and y ∈ g(k) ⊆ M .Since k ∼ = g(k), we have mg(k) = 0. Now my ∈ mg(k) = 0, so y ∈ (0 : M m) = Soc(M ).Hence f = x + y ∈ N + Soc(M ).This finally shows that (mN : M m) ⊆ N +Soc(M ), so (mN : M m) = N +Soc(M ).Since m(N +Soc(M )) = mN , we get (m(N + Soc(M )) : M m) = N + Soc(M ), which implies N + Soc(M ) is a weakly m-full submodule M .Now note that, if depth(M ) > 0, then Soc(M ) = 0. Hence N + Soc(M ) = N is a weakly m-full submodule of M .Now we give another characterization of regular local rings in terms of vanishing of certain Ext 1 R (k, −) µ .Theorem 5.1.12.Let (R, m, k) be a local ring of depth t > 0.Then, the following are equivalent: (1) R is regular.
(3) Ext 1 R (k, M ) µ = 0 for some finitely generated R-module M of projective dimension t − 1. Proof.(1) =⇒ (2) Since R is regular, we have t = dim R. Since R is regular, we get that m is generated by a regular sequence x 1 , ..., x t .Now R/(x 1 , ..., x t−1 )R has projective dimension t − 1 over R and the (t − 1)-th Betti number of this module is 1 (by looking at the Koszul complex).Hence, Ext 1 R (k, R/(x 1 , ..., x t−1 )R) ∼ = Tor R t−1 (k, R/(x 1 , ..., x t−1 )R) ∼ = k by [3,Exercise 3.3.26].Now, we have an exact sequence 0 → R/(x 1 , ..., x t−1 )R  Proof.Let m = xR.Then, for every finitely generated R-module X, we have X ∼ = R ⊕a ⊕ (⊕ n i=1 R/x ai R) for some non-negative integers (depending on X) a, a i .Now, fix arbitrary N ∈ mod R. Applying Lemma 2.4 to the subfunctor Ext ) for every integer l ≥ 1, first fix an l ≥ 1, and look at the subfunctor Ext Again by the structure of finitely generated R-modules and Lemma 2.4, it is enough to prove that Ext for every integer b ≥ 1.Now these equalities follow from Lemma 5.1.13,since x is a non-zero-divisor.
When restricting to short exact sequences in Ext 1 , on which certain subadditive function, other than µ(−), is additive, one obtains vanishing of the corresponding submodule of Ext 1 R (M, F ) for any free R-module F .We make this precise in Proposition 5.1.16,whose proof uses the following lemma.
→ 0 be a short exact sequence, where a, b are non-negative integers and F 1 , F 2 are free R-modules.Let c be an integer such that c ≥ b and σ ⊗ R x c R is short exact.Then a = b and rank(F 2 ) = rank(F 1 ).So, in particular, σ is split exact.
Proof.Since x ∈ R is a non-zero-divisor, so R x a R , R x b R are torsion modules, i.e., have constant rank 0. Hence calculating rank along σ, we get rank(F 2 ) = rank(F 1 ).Next, we will show that a ≤ b.Dualizing σ by R, we get the following part of a long exact sequence: Ext , and dualizing by R, and calculating the cohomology we get that, Ext as well, which implies R is short exact, we have the following short exact sequence R are m-primary ideals.Hence by calculating the length along the short exact sequence σ R (L, F ) φL = 0. Proof.Note that, R is not a field, since dim R = 1.Since R is a regular local ring, we have L ∼ = G ⊕ L ′ , where G is a finite free R-module and L ′ is an R-module of finite length.We have H 0 ) φL , we may replace L by L ′ , and assume without loss of generality that L has finite length, and c ≥ ℓℓ(L).For simplicity, we denote φ L = φ.Now it is enough to show that, Ext 1 R (L, R) φ = 0. Since (R, m) is a regular local ring of dimension 1, R is a PID.Hence m = xR for some x = 0. Since L is a finite length module, by the structure theorem of modules over a PID we have, By the structure theorem of modules over a PID, we also have x a j R , where a j > 0. As R is an integral domain and R x b i R and R x a j R are all torsion R-modules, so calculating rank along the short exact sequence σ, we obtain s = 1.Moreover, Since σ is φ-additive, by using Lemma 4.12 with the functor we obtain a contradiction from Lemma 5.1.15.Thus t = 1.So, we get X ∼ = R ⊕ R x a R for some a = a j .Thus we have the short exact sequence σ : 0 Hence by applying Lemma 5.1.15on σ we get that, σ is split exact.Since σ is an arbitrary element of Ext Taking L = k, c = 1 = ℓℓ(k), we see that φ L (−) = µ(−) in Proposition 5.1.16.So, we also get another proof of Corollary 5.1.4in dimension 1.
Next, we compare Ext In arbitrary dimension, we only consider this for local Cohen-Macaulay rings of minimal multiplicity.We first record some general preliminary lemmas.Lemma 5.1.17.Let (R, m, k) be a local ring such that m 2 = 0, m = 0. Let e := µ(m).Then µ Proof.Since m 2 = 0 and m = 0, we have m ⊆ (0 : R m) R. Hence m = (0 : R m).Then we have µ Since tensoring with S preserves exactness, so tensoring a minimal free resolution (F • , ∂ • ) of an R-module M with S and remembering S⊗∂ now have entries in mS, the maximal ideal of S, we see that S⊗ R Ω R M ∼ = Ω S (S ⊗ S M ).Also, if ω R exists, then owing to the fact that S/mS is a field, we see that ω S also exists and let, the claim be true for rings with dimension d − 1.Let x ∈ m be such that m 2 = (x, x 1 , ..., x d−1 )m (see [3, 4.6.14(c)]).So, R xR has minimal multiplicity.Now we have where the first isomorphism follows from [3, Proposition 3.3.3(a)],and the second isomorphism is by Lemma 5.1.18.So,  Proof.We note that, mod S = Mod S ∩ mod R. Since S is also 1-dimensional Cohen-Macaulay, we have CM(S) = collection of all finitely generated torsion-free S-modules = collection of all torsion-free S-modules ∩ mod R. Hence the conclusion follows from Proposition 5.2.10.
When (R, m) is a local Cohen-Macaulay ring of dimension 1 and I an m-primary ideal, then considering the finite birational extension R ⊆ B(I), where B(I) denotes blow-up of I ([11, Definition 4.3, Remark 4.4]), it is shown in [11,Theorem 4.6] that CM(B(I)) = Ul I (R).Thus, in this particular case, we get another proof of the fact that Ul I (R) is an exact subcategory of mod R, which is very different from Corollary 4.7.
For further applications, we first record a lemma connecting Ext 1 UlI (R) (−, −) to Ext 1 B(I) (−, −).Here, we will keep in mind that, when dim R = 1, we write Ul satisfies the axiom [E0 op ].Next, we will show that (A ′ , E| A ′ ) satisfies the axiom [E1 op ].Let B ′ B e and B C p be two admissible epics in E such that they are in E| A ′ , which means that there exist two kernel-cokernel pairs A B C that A, B, C, D, B ′ ∈ A ′ .Then we will show that B ′ C p•e is an admissible epic in E| A ′ .From Proposition 3.4 we get that, A ′ B ′ C i ′ p•e is a kernel-cokernel pair in E. By the hypothesis of Lemma 3.3 and the diagram of Proposition 3.4 we get that, by [6, Proposition 5.2] we get that γ is also the pushout of σ ⊕ β in E by the sum map A⊕A Σ − → A. Next, α is the pullback of γ in E ′ by the diagonal map C ∆ − → C ⊕C, so by [6, Proposition 5.2] we get that α is also the pullback of γ in E by the diagonal map C where γ is the pullback of σ by g in E and f * (g) = [α] E ′ , where α is the pullback of σ by g in E ′ .Since (A ′ , E ′ ) is an exact subcategory of (A, E), by [6, Proposition 5.2] we get that [γ] E = [α] E .Also, by definition φ A,M ([α] E ′ ) = [α] E .Thus f (g) = φ A,M (f * (g)), so the above square commutes.

Lemma 4 . 4 .
, Theorem A.11(b)]), so a similar argument as the previous one shows e R (I, M ) = e S (IS, S ⊗ R M ).Thus φ I(M ) = φ IS (S ⊗ R M ).The function φ I : CM s (R) → Z satisfies φ I (M ) ≤ 0, is constant on isomorphism classes of modules in CM s (R), additive on finite biproducts, and subadditive on short exact sequences of modules in CM s (R).Proof.It is obvious that φ I is constant on isomorphism classes of modules in CM s (R), and additive on finite biproducts (direct sums).Now, e R (I, −) is additive on short exact sequences of modules in CM s (R) by[3, Corollary 4.7.7].Since λ R ((−) ⊗ R R/I) is always subadditive on short exact sequences, this proves φ I is also subadditive on short exact sequences of modules in CM s (R).Now we finally prove that φ I (M ) ≤ 0 for all M ∈ CM s (R).If dim M = 0, then e R (I, M ) = λ R (M ) ≥ λ R (M/IM ), so φ I (M ) ≤ 0. Now assume that s = dim M > 0. We may assume that R has infinite residue field due to 4.3.By [3, Corollary 4.6.10],we have e R (I, M ) = e R ((x), M ) for some system of parameters x = x 1 , ..., x s on M , which is a reduction of I with respect to M .Then, x is a M -regular sequence, since M is Cohen-Macaulay ([3, Theorem 2.1.2(d)]).Let J = (x).Since x is a M -regular sequence, by[3, Theorem 1.1.8]we have

Definition 4 . 5 .
the last inequality follows by noticing that J ⊆ I).Hence φ I (M ) ≤ 0. Let (R, m) be a Noetherian local ring, and I be an m-primary ideal.Let s ≥ 0 be an integer.We denote by Ul s I (R) the full subcategory of all modules M ∈ CM s (R) such that φ I (M ) = 0, i.e., λ R (M/IM ) = e R (I, M ).When s = dim R, we will denote this subcategory simply by Ul I (R).When I = m, Ul s m (R) will be denoted by Ul s (R).We will also denote Ul dim R m (R) simply by Ul(R).Note that, when s = dim R = 1 and R is Cohen-Macaulay, Ul I (R) is exactly the collection of all I-Ulrich modules as defined in[11, Definition 4.1].Remark 4.6.In terms of [21, Definition 2.1], a non-zero R-module M is Ulrich if M ∈ Ul s (R) for some s ≥ 0 in our notation.When R is Cohen-Macaulay, then the modules in Ul(R) are simply the maximally generated modules as studied in[2].Corollary 4.7.For each integer s ≥ 0 and m-primary ideal I, (Ul s I 8) and we have corresponding commutative diagram of long exact sequences by Corollary 3.9.With the extension-closed subcategory X = mod R, where (R, m, k) is a Noetherian local ring, and the subadditive function φ(−) = µ(−) : mod R → Z being the number of generators function in Definition 4.10, we see that Ext 1 R (M, N ) µ = M, N in the notation of [35, Definition 1.35].So by our discussion, Theorem 4.8, Proposition 3.8 and Corollary 3.9 recovers [35, Corollary 1.36, Proposition 1.37, Proposition 1.38].

Corollary 4 . 13 .
Let (X , E) be an exact subcategory of Mod R. Let G : X → fl(R) be an additive halfexact functor, where fl(R) denotes the full subcategory of mod R consisting of finite length modules.Set E G := {σ ∈ E : G(σ) is a short exact sequence of modules}.Then, (X , E G ) is an exact subcategory of Mod R. Proof.From Lemma 4.12 it follows that, E G = E φ , where φ(−) := λ R (G(−)) : X → Z. Hence the claim now follows from Theorem 4.8.Now again, let X = mod R, where R is Noetherian.Basic examples of additive half-exact functors
15) and e R (I, −) is additive on short exact sequence of modules of the same dimension ([3, Corollary 4.7.7]),hence we get the following lemma by following the same proof as in [30, Proposition 17]: Lemma 4.16.Let (R, m) be an equidimensional local ring satisfying (S 1 ).Let M ∈ S 1 (R).Let d = dim R ≥ 1, and I be an m-primary ideal.Then, the function n → λ(Tor R 1 (M, R/I n+1 )) is given by a polynomial of degree ≤ d − 1 for n ≫ 0. So, in particular, the limit lim n→∞ λ(Tor R 1 (M, R/I n+1 )) n d−1 exists.If R and M are as in Lemma 4.16, then let us denote e T I (M ) := (d − 1)! lim n→∞ λ(Tor R 1 (M, R/I n+1 )) n d−1 .

5. 1 .
Some computations and applications of the subfunctor Ext 1 R (−, −) µ .We start this subsection with a characterisation of regularity among Cohen-Macaulay rings of positive dimension d in terms of vanishing of certain Ext 1 R (−, −) µ (see Definition 4.10 and the discussion after 4.11 for notation).

Theorem 5 . 1 . 1 .
Let R be a local Cohen-Macaulay ring of dimensiond ≥ 1.Then, R is regular if and only if Ext 1 R (Ω d−1 R k, N) µ = 0 for some finitely generated non-zero R-module N of finite injective dimension.

Lemma 5 . 1 . 7 .
Let (R, m, k) be a local ring, and I an ideal of R, which is not principal.Then, for every x ∈ I \ mI, we have (xm : m I) = (xm : R I) = ((x) : R I).Proof.Since trivially (xm : m I) ⊆ (xm : R I) ⊆ ((x) : R I) always holds, it is enough to prove the inclusion ((x) : R I) ⊆ (xm : m I).So, let y ∈ ((x) : R I), which means yI ⊆ (x).If y were a unit, then we would have I ⊆ y −1 (x) = (x) ⊆ I, implying I = (x), contradicting our assumption that I is not principal.Thus, we must have y ∈ m.Now pick an arbitrary element r ∈ I. Then yr ∈ yI ⊆ (x), so yr = xs for some s ∈ R. If s / ∈ m, then x = s −1 yr ∈ mI, which is a contradiction.Hence, s ∈ m.So, yr = xs ∈ xm.Since r ∈ I was arbitrary, we get yI ⊆ xm.Hence y ∈ (xm : m I).
where the last two isomorphisms follow from [3, Proposition 3.3.3(a),Theorem 3.3.5(a)]).Now Hom R/xR ((m/xm) † , k) ∼ = Hom R/xR (k † , m/xm) ∼ = Hom R/xR (k, m/xm) ∼ = (xm : m m)/xm = ((x) : R m)/xm (where the first isomorphism holds because R/xR is Artinian, so one can invoke [10, Lemma 3.14, Remark 3.15], and the last equality follows from Lemma 5.1. This concludes the d = 1 case.Now let, dim R = d > 1 and suppose the claim has been proved for all non-regular local Cohen-Macaulay rings of dimension 1, ..., d − 1 admitting a canonical module.Since Ω d−1 R k has co-depth 1, by [28, Proposition 11.21](see 5.1.5)we get that 0 xR k over R/xR (by 5.1.5,[28, Proposition 11.21]), their direct sum 0 which is what we wanted to prove.Proof. of Theorem 5.1.1:Follows by combining Corollary 5.1.4and Proposition 5.1.9.For an arbitrary local ring of positive depth, we give a characterization of the ring being regular in terms of vanishing of Ext 1 R (k, M ) µ for some M ∈ mod R of finite projective dimension.For this, we first recall the definitions of weakly m-full and Burch submodules of a module from [12, Definition 3.1, 4.1] and subsequently, we relate that property to the vanishing of certain Ext 1 R (k, −) µ .Definition 5.1.10.Let (R, m, k) be a local ring and let, N be an R-submodule of a finitely generated R-module M .Then N is called a weakly m-full submodule of M if (mN : M m) = N .Also, N is called a Burch submodule of M if m(N : M m) = mN .Proposition 5.1.11.Let (R, m, k) be a local ring and let, N be an R-submodule of a finitely generated R-module M such that Ext 1 R (k, N ) µ = 0. Then (mN : M m) = N + Soc(M ), i.e., N + Soc(M ) is a weakly m-full submodule of M .So, in particular, if we moreover have depth(M ) > 0, then N is a weakly m-full submodule of M .Proof.Clearly, N ⊆ (mN : M m) and Soc(M ) = (0 : M m) ⊆ (mN : M m), so N + Soc(M ) ⊆ (mN : M m).Now choose an element f ∈ (mN : M m), and we aim to show f ∈ N + Soc(M ).If f ∈ N , then we are done.Otherwise, say f / ∈ N .As f ∈ (mN : M m), so mf ⊆ mN ⊆ N , hence m(N + Rf ) = mN .Thus Rf + N N is a non-zero (as f ∈ N ) k-vector space (as m(N + Rf ) ⊆ N ).Moreover, Rf + N N is cyclically generated by the image of f in M/N .Hence Rf + N N ∼ = k, so we have a short exact sequence ) =⇒ (3) Obvious.(3)=⇒ (1) By Auslander-Buchsbaum formula, depth M = 1, so Soc(M ) = 0. Hence by prime avoidance, we can choose x ∈ m, which is both R and M -regular.ThenxM ∼ = M , so Ext 1 R (k, xM ) µ ∼ = Ext 1 R (k, M ) µ = 0.By Proposition 5.1.11we get that, xM is a weakly m-full submodule of M .Since depth(M/xM ) = 0, xM is a Burch submodule of M by [12, Lemma 4.3].Moreover, pd R/xR(M/xM ) = pd R M < ∞, and pd R R/xR < ∞, so pd R M/xM < ∞.Hence Tor R ≫0 (k, M/xM ) = 0. Thus pd R k < ∞ by [12, Theorem 1.2], so R is regular.Next, we try to relate Ext 1 R (M, N ) µ to Ext 1 R (M, N ) for all M, N ∈ mod R, when (R, m) is a regular local ring of dimension 1 (i.e., a local PID).For this, we first record a preliminary lemma.Lemma 5.1.13.Let (R, m) be a local ring and x ∈ m be a non-zero-divisor.Then, Ext 1 R (R/xR, R/I) µ = m Ext 1 R (R/xR, R/I) ∼ = m I + xR for every proper ideal I of R. Proof.Calculating Ext 1 R (R/xR, R/I) from the minimal free-resolution 0 → R •x − → R → R/xR → 0 of R/xR, we see that Ext 1 R (R/xR, R/I) ∼ = R/(xR + I) is a cyclic R-module.By Lemma 5.2.1 we have m Ext 1 R (R/xR, R/I) ⊆ Ext 1 R (R/xR, R/I) µ .We finally claim that Ext 1 R (R/xR, R/I) µ = Ext 1 R (R/xR, R/I).Indeed, we have an exact sequence σ : 0 → xR/xI → R/xI → R/xR → 0. Now, we have a natural surjection R r →rx+xI − −−−−− → xR/xI, whose kernel is {r ∈ R : rx ∈ xI}.Since x is a non-zero-divisor, rx ∈ xI if and only if r ∈ I. Hence, the kernel is I. Hence R/I ∼ = xR/xI.So, we get the exact sequence σ : 0 xR, R/I) by Lemma 5.1.2.Now using this lemma, we can compare the structure of Ext 1 R (M, N ) µ to Ext 1 R (M, N ) for every pair of finitely generated modules M, N over a DVR.Proposition 5.1.14.Let (R, m) be a regular local ring of dimension 1.Then, Ext 1 R (M, N ) µ = m Ext 1 R (M, N ) for all finitely generated R-modules M and N .So, in particular, Ext 1 R (k, N ) µ = 0 for all finitely generated R-modules N .

2
and a = b, we get σ is split exact.In the following, H 0 m (−) denotes the zero-th local cohomology module.For a finite length R-module M , ℓℓ(M ) will stand for the smallest integer n ≥ 0 such that m n M = 0. Proposition 5.1.16.Let (R, m) be a regular local ring of dimension 1.Let L ∈ mod(R).Let c be an integer such that c ≥ ℓℓ(H 0 m (L)).Consider the function φ L (−) := λ( R m c ⊗ −) : mod(R) → N ∪ {0}.Then for any free R-module F we have, Ext 1 1, where λ(ω R ) = λ(R) follows from Matlis duality.Lemma 5.1.18.Let (R, m, k) be a local ring and let, M be a finitely generated R-module.If x is M -regular, then ΩRM xΩRM ∼ = Ω R/xR M xM .Proof.Since x is M -regular, we have Tor R 1 (M, R xR ) = 0. Hence tensoring the short exact sequence 0 Finally, we also have r(R) = r(S) by [3, Proposition 1.2.16(b)].Proposition 5.1.20.Let (R, m, k) be a local Cohen-Macaulay ring of minimal multiplicity admitting a canonical module ω R and also let, m = 0. Then µ((Ω R ω R ) † ) = r(R) 2 − 1.Proof.Due to 5.1.19,we may pass to the faithfully flat extension S := R[X] m[X] and assume the residue field is infinite.We will prove the claim by induction on dim R = d.First let, d = 0. Since R has minimal multiplicity, we have m 2 = 0. Hence by Lemma 5.1.17we get that,

. 5 .
this sequence gives us Ω R Tr(R/I) ∼ = R/ ann R (I) and the first Betti number of Tr(R/I) is 1.Hence, Ext 1 R (Tr(R/I), R/ ann R (I)) µ = m Ext 1 R (Tr(R/I), R/ ann R (I)) by Lemma 5.1.3.Now we prove the equivalence of the three conditions as follows:(1) =⇒ (2): Let σ : 0 → F → X → M → 0 be an exact sequence, where F is a finitely generated free R-module.Since depth R = 0, we get that k embeds inside R, i.e., k is torsionless.Hence Ext 1 R (Tr k, R) = 0 by[28, Proposition 12.5].Hence, Ext 1 R (Tr k, F ) = 0. So, we get an exact sequence 0 → Hom R (Tr k, F ) → Hom R (Tr k, X) → Hom R (Tr k, M ) → 0. Then by[28, Exercise 13.36]  we get that, the sequence 0→ F ⊗ R k → X ⊗ R k → M ⊗ R k → 0 is also exact, which means σ is µ-additive.Thus Ext 1 R (M, F ) µ = Ext 1 R (M, F ). (2) =⇒ (3) Obvious.(3) =⇒ (1): Assume Ext 1 R (Tr k, R) µ = Ext 1 R (Tr k, R).Now if possible let, depth R > 0.Then m contains a non-zero-divisor, so ann R (m) = 0. Then by the first part of this proposition, we getExt 1 R (Tr k, R) µ = m Ext 1 R (Tr k, R).Hence Ext 1 R (Tr k, R) = m Ext 1 R (Tr k, R).Then by Nakayama's lemma, Ext 1 R (Tr k, R) = 0.Hence by[28, Proposition 12.5], we have an embedding k → k * * .Consequently, k * = 0, i.e., depth R = 0.For general local Cohen-Macaulay rings of dimension 1, we now show that if Ext 1 R (M, R) µ = 0 for some I-Ulrich module ([11, Definition 4.1]) M ⊆ Q(R) containing a non-zero-divisor of R, then I is principal.For this, we first need the following lemma.For the remainder of this section, given R-submodules M, N of Q(R), by (M : N ) we will mean {x ∈ Q(R) : xN ⊆ M }.Lemma 5.1.25.Let (R, m) be a local Cohen-Macaulay ring of dimension 1.Let I be an m-primary ideal of R admitting a principal reduction a ∈ I. Assume (I : I) = R.If M ⊆ Q(R) is an I-Ulrich module, and contains a non-zero-divisor of R, then the natural inclusion map Hom R (M, (a)) → Hom R (M, I), induced by the inclusion (a) → I, is an isomorphism.Proof.The natural inclusion 0 → (a) i − → I induces the following commutative diagram: 0 Hom R (M, (a)) Hom R (M, I) 0 ((a) : M ) (I : M ) α→{x →αx} α→{x →αx} where the rows are natural inclusion maps, and the vertical arrows are isomorphisms due to [26, Proposition 2.4(1)].So, it is enough to show that (I : M ) ⊆ ((a) : M ).Indeed, if x ∈ (I : M ), then xM ⊆ I. Since M is I-Ulrich, M is a B(I) = R I a -module ([11, Remark 4.4, Theorem 4.6]).Hence I a M ⊆ M , so I a xM ⊆ xM ⊆ I. Thus 1 a xM ⊆ (I : I) = R, so xM ⊆ (a).Therefore x ∈ ((a) : M ).Proposition 5.1.26.Let (R, m) be a local Cohen-Macaulay ring of dimension 1.Let I be an m-primary ideal of R admitting a principal reduction a ∈ I. Assume (I : I) = R.If there exists an I-Ulrich module M ⊆ Q(R), containing a non-zero-divisor of R such that Ext 1 R (M, R) µ = 0, then I ∼ = R. Proof.Consider the short exact sequence 0 → (a) i − → I → I/(a) → 0. Since I n+1 = aI n for all n ≫ 0 and I contains a non-zero-divisor, we have a is a non-zero-divisor.So, (a) ∼ = R.If a ∈ mI, then I n+1 ⊆ mII n = mI n+1 , which implies I n+1 = 0 by Nakayama's lemma.This contradicts the fact that I is m-primary.So, a / ∈ mI, hence µ(I/(a)) = µ(I) − 1.Thus the above short exact sequence is µ-additive.So, by Corollary 3.9 we get the following induced exact sequence: 0 → Hom R (M, (a)) → Hom R (M, I) → Hom R (M, I/(a)) → Ext 1 R (M, (a)) µ ∼ = Ext 1 R (M, R) µ = 0 where the induced map Hom R (M, (a)) → Hom R (M, I) is an isomorphism by Lemma 5.1.25.Hence we get Hom R (M, I/(a)) = 0. Now I/(a) has finite length.So, if I/(a) = 0, then Ass(I/(a)) = {m}.Hence Ass (Hom R (M, I/(a))) = Supp(M ) ∩ Ass(I/(a)) = Supp(M ) ∩ {m} = {m}, contradicting Hom R (M, I/(a)) = 0. Thus, we must have I/(a) = 0, i.e., I = (a) ∼ = R.Lemma 5.2.9.Let R be a commutative Noetherian ring and let R ⊆ S ⊆ Q(R) be a ring extension.Then, Hom R (M, N ) = Hom S (M, N ) for all M, N ∈ Mod(S) where N is torsion-free S-module.Proof.HomS (M, N ) ⊆ Hom R (M, N ) is clear.Hence it is enough to show Hom R (M, N ) ⊆ Hom S (M, N ).Let a b ∈ S ⊆ Q(R), where a, b ∈ R, so b ∈ R is a non-zero-divisor.Let m ∈ M and f ∈ Hom R (M, N ).af (m) = f b a b m − af (m) = f (am) − af (m) = 0. Since Nis torsion-free and b is a non-zerodivisor on R, it is a non-zerodivisor on N .So, m ∈ M and f ∈ Hom R (M, N ) were arbitrary, we conclude Hom R (M, N ) ⊆ Hom S (M, N ).Now as a consequence, we can deduce the following:Proposition 5.2.10.Let R ⊆ S be a birational extension (S ⊆ Q(R)) of commutative rings.Let S R be the standard exact structure on Mod R. Let X be the strictly full subcategory of Mod S consisting of all torsion-free S-modules.Then it holds that S R | X = S S | X , and (X , S R | X ) is a strictly full exact subcategory of Mod R, and (X ∩ mod R, S R | X ∩mod R ) is an exact subcategory of Mod R (hence, also of mod R).Proof.By definition, S R | mod R∩X = S R | mod R ∩ S R | X .Since (mod R, S R | mod R) is an exact subcategory of Mod R, the second part of the statement would readily follow from the first part of the statement and Lemma 3.2.First, we show that X is strictly full in Mod R. If M, N are two R-modules and f : M → N is an R-linear isomorphism and N is moreover a torsion-free S-module extending its R-module structure, then M has an S-module structure, extending its R-module structure, given by s • m := f −1 (sf (m)) for all s ∈ S, m ∈ M .Note that, with this structure, M is moreover a torsion-free S-module.Indeed, let s ∈ S be a non-zero-divisor such that s • m = 0. Then sf (m) = f (s • m) = 0, so f (m) = 0 as N is S-torsionfree.Thus m = 0 as f is an isomorphism.Moreover, Hom R (M, N ) = Hom S (M, N ) for all torsion-free S-modules M, N by Lemma 5.2.9.Consequently, we notice that S R | X = S S | X .So, to show (X , S R | X ) is an exact subcategory of Mod R, it is enough to show that (X , S S | X ) is an exact subcategory of Mod S (by Lemma 5.2.8).Now it is well known that for any ring S, the subcategory of all S-torsionfree modules is closed under extensions in Mod S. Hence (X , S S | X ) is an exact subcategory of Mod S by Proposition 3Corollary 5.2.11.Let R be a local Cohen-Macaulay ring of dimension 1.Let R ⊆ S be a finite birational extension.Then, (CM(S), S R | CM(S) ) is a strictly full exact subcategory of mod R.
satisfies the axiom [E1 op ].Let B ′ B A ′ be an arbitrary morphism in ∩ λ A λ .Now A B i is an admissible monic in E λ for all λ.Since (A λ , E λ ) is an exact category for all λ, by [6, Proposition 2.12(iv)] we have the following pushout commutative diagram with rows being kernel-cokernel pairs in E satisfies the axiom [E1 op ].Now we will show that (∩ λ A λ , ∩ λ E λ ) satisfies the axiom [E2].Let A B i be an admissible monic in ∩ λ E λ and A f − → λ : . By [6, Definition 2.1] and [6, Remark 2.4], it is enough to show that E φ is closed under isomorphisms and (A, E φ ) satisfies the axioms [E0], [E0 op ], [E1 op ], [E2] and [E2 op ].First, we will show that E φ is closed under isomorphisms.Let M N L be a kernel-cokernel pair in E φ .Also let, M ′ → N ′ → L ′ be a kernel-cokernel pair in A such that it is isomorphic to M N L , which implies [39,preserves minimal MCM approximation by[39, Theorem 1.4]).By uniqueness of minimal MCM approximation, we have