KOSZUL MODULES

JÜRGEN HERZOG AND SRIKANTH IYENGAR

To Wolmer Vasconcelos on his 65th birthday

Abstract. This paper introduces a new class of modules over noetherian local rings, called Koszul modules. It is proved that when a local ring R is either a complete intersection or Golod, all high syzygies of finitely generated R-modules are Koszul. A stronger result is obtained when R is Golod: the (2 emb dim R + 1)st syzygy of every R-module is Koszul. In addition, results are established that demonstrate that Koszul modules possess good homological properties; for instance, their Poincaré series is a rational function.

Introduction

Consider a noetherian local ring R with maximal ideal 𝔪 and residue field k. Given a finitely generated R-module M and F its minimal free resolution, denote lin^{R}(F) the associated graded complex of F arising from the standard 𝔪-adic filtration; the details are given in Section 1. This is a complex of free modules over the associated graded ring gr_{𝔪}(R) of R and its differentials depend only on the linear part - in a sense made precise in Section 4 - of the corresponding differentials on F; whence the notation.

We say that M is Koszul if lin^{R}(F) is acyclic. Such modules have appeared previously in literature under the name: modules with linear resolution; confer [21]. We prefer to call them ‘Koszul modules’ for they are module theoretic analogues of Koszul algebras, introduced by Löfwall[15] and Priddy [17]: many of the results in this paper support this claim. As is usual, R is said to be a Koszul ring if the R-module k is Koszul. This is tantamount to requiring that the graded k-algebra gr_{𝔪}(R) be Koszul in the classical sense of the word; see (1.7).

In studying Koszul modules, it is profitable to consider the larger class of modules M for which the number lpd_{R} (M) = sup{i | H_{i}(lin^{R}(F)) ≠ 0} is finite. These are the modules in whose minimal free resolutions the linear part predominates. Observe that lpd_{R} (M) ≤ d is finite if and only if Ω^{d} (M), the dth syzygy of M, is Koszul.

A basic issue in this context is that of existence of modules of finite linear dominance. Here one encounters a rather surprising phenomenon: if lpd_{R} (k) < ∞, then R is Koszul, at least when R is a standard graded k-algebra. Nevertheless, even when R is Koszul there can be modules M with lpd_{R} (M) = ∞. Indeed, if lpd_{R} (M) is finite, then by Proposition 1.6 the Poincaré series of M is a rational function, and its denominator term depends only on the Hilbert series of gr_{𝔪}(R). However, even over the seemingly innocuous Koszul algebra k[x_{1}, x_{2}, x_{3}]/(x_{1}, x_{2}, x_{3})^{2} ⊗_{k} k[y_{1}, y_{2}, y_{3}]/(y_{1}, y_{2}, y_{3})^{2} there are finitely generated modules with transcendental Poincaré series; see Jacobsson [12]. Furthermore, Roos [19] has constructed a Koszul algebra R and a family of R-modules {M_{n}}_{n∈ℕ} such that the Poincaré series of each M_{n} is rational, but these do not share a common denominator. This leads one to consider the problem of classifying the rings for which each module is of finite (or bounded) linear dominance.

We prove if a Koszul local ring R is either a complete intersection or Golod, then lpd_{R} (M) < ∞ for every finitely generated R-module M; this is the content of Corollary 5.8. There is a dichotomy that emerges here: when R is a complete intersection, there is no global upper bound on lpd_{R} (M), unless R is a hypersurface. This follows from Proposition 6.7 which asserts, more generally, that for a Gorenstein standard graded k-algebra R, if there is a global bound on linear dominance, then R is a hypersurface. On the other hand, when R is Golod, Corollary 6.2 yields that lpd_{R} (M) ≤ 2 emb dim R + 1 for every finitely generated R-module M.

Date: August 6, 2002.

Modules of finite linear dominance are particularly well behaved with regards to their homological properties. One example of this phenomenon is the rationality of Poincaré series mentioned above. More sophisticated manifestations concern the left module structure of Ext_{R}(M, k) over the k-subalgebra R^{!} of Ext_{R}(k, k) generated by Ext_{R}^{1}(k, k). For instance, we prove in Theorem 2.1 that if lpd_{R} (M) is finite, then Ext_{R}(M, k) is finitely generated over R^{!} by elements with degrees ≤ lpd_{R} (M). In particular, when the ring R is Koszul, R^{!} = Ext_{R}(k, k); this last result is well known when R is a standard graded k-algebra.

Over Koszul local rings, Theorem 2.1 admits a vast generalization, best captured by the formula

lpd_{R} (M) = reg_{R^{!}} Ext_{R}(M, k) .

It is this equality, in conjunction with work of Backelin and Roos [6], that allows us to deduce the finiteness of linear dominance of modules over complete intersections and Golod rings, discussed previously. It implies also that when lpd_{R} (M) is finite, the R^{!}-module Ext_{R}(M, k) and all its higher syzygies are finitely generated. The term on the left in the formula above is the regularity of Ext_{R}(M, k) as an R^{!}-module; the relevant definition and the proofs of the preceding results are the contents of Section 5.

Now for an overview of the paper: Section 1 contains the basic definitions and first results about Koszul modules, and more generally, about modules with finite linear dominance. While the bulk of the results in this paper are stated and proved for modules over local rings, they are applicable also to graded modules over graded k-algebras. Now, for a graded module M over a standard graded ring R, the invariant lpd_{R} (M) is closely linked to the regularity of M over R. Section 1 explores also some of these issues.

The main goal in Section 2 is the proof of the result concerning the finite generation of Ext_{R}(M, k) over R^{!}. This requires detailed calculations involving products on Ext_{R}(k, k) and its action on Ext_{R}(M, k). These play a key role also in the subsequent development, as do the results in the short Section 3 that is devoted to dealing with certain delicate issues pertaining to left and right module structures on Ext_{R}(k, k).

Section 4 is the heart of the paper. It is devoted entirely to the proof of Theorem 4.1 that expresses the homology of the linear part of the minimal resolution F of M, over a Koszul ring R, in terms of the (co)homology of Tor^{R} (k, M) when viewed as module over R^{!}. A crucial ingredient in its proof is Theorem 4.5 that provides an explicit description of lin^{R}(F) in terms of the action of Ext_{R}^{1}(k, k) on Tor^{R} (k, M). The material in Section 5 has already been described. One of the main contributions of Section 6 is Theorem 6.1 which states that given a surjective homomorphism of Koszul local rings φ: S → R and M a finitely generated R-module, one has

lpd_{S} (M) = lpd_{R} (M) whenever M is φ-Golod .

Its proof utilizes a new characterization of φ-Golod modules whose applicability extends beyond the task on hand. As corollary, we deduce the global bound for lpd_{R} (M), when R is a Golod ring.

This research is inspired by the paper of Eisenbud, Floystad, and Schreyer [8] concerning modules over the exterior algebra: many of the results presented here are either analogous to, or extensions of, those in loc. cit. For instance, the part of Corollary 5.8 concerning complete intersections is akin to [8, (3.1)] which asserts that, for finitely generated graded modules over the exterior algebra, the linear parts of minimal resolutions predominate. This is no coincidence since homological algebra over complete intersections is similar to that over exterior algebras. Furthermore, Theorem 5.4, relating linear dominance to regularity over the Ext algebra, also has its counterpart in [8, (2.4)]. Having said this, it should be noted that [8] makes critical use of the Bernstein-Gelfand-Gelfand correspondence. This is not available in our framework, which encompasses both local and graded rings, and, more to the point, a good number of our results do not require the ring to be Koszul.

1. Koszul modules

Let (R, 𝔪, k) be a local ring. A complex of R-modules

C: ⋯ ⟶ C_{n+1} ⟶^{∂_{n+1}} C_{n} ⟶^{∂_{n}} C_{n−1} ⟶ ⋯

is said to be minimal if ∂(C_{n}) ⊆ 𝔪C_{n−1}. The standard filtration F of a minimal complex C is defined by the subcomplexes {F^{i}C}_{i≥0}, where

(F^{i}C)_{n} = 𝔪^{i−n}C_{n}  for all  n ∈ ℤ ,

with the convention that 𝔪^{j} = R for j ≤ 0. The minimality of the complex C ensures that ∂(F^{i}C) ⊆ F^{i}C for each integer i so that F is a filtration of C. The associated graded complex with respect to this filtration is denoted lin^{R}(C), and called the linear part of C. By construction, lin^{R}(C) is a complex of graded modules over the graded ring gr_{𝔪}(R) and has the property that

lin_{n}^{R}(C) = gr_{𝔪}(C_{n})(−n) .

Here, and everywhere else, for any graded object V = {V_{i}}_{i∈ℤ}, we denote V (n) the graded object with degree ith component V_{i+n}. In particular, when C is a complex of free modules the complex of gr_{𝔪}(R)-modules lin^{R}(C) is free and, by construction, the differentials on it can be described by matrices of linear forms; this partly explains our terminology.

Now we introduce the main objects of interest. Recall that over local rings, each finitely generated module has a minimal free resolution, and that this is unique (up to isomorphism). Thus, one may speak of the minimal free resolution of such a module. Definition 1.1. A finitely generated R-module M is said to be Koszul if lin^{R}(F) is acyclic, where F is the minimal free resolution of M.

Koszul modules are precisely those modules that have linear resolutions, in the terminology of Şega [21, §2]; this is justified by Proposition 1.3. In order to state this result, we introduce certain families of maps akin to those in [loc. cit., (2.2)]. 1.2. Let M be finitely generated R-module and F its minimal R-free resolution. Set A = gr_{𝔪}(R). Let U a graded A-free resolution of gr_{𝔪}(M). The augmentation F → M gives rise to an A-linear map lin^{R}(F) → gr_{𝔪}(M). Since the complex of A-modules lin^{R}(F) is free, this map lifts to a morphism of complexes of A-modules κ: lin^{R}(F) → U; such a lift is unique up to homotopy.

For each z ∈ F_{n}, denote \overline{z} its residue class in lin_{n}^{R}(F)_{n} = F_{n}/𝔪F_{n}. For each integer n, set

χ_{M}^{n}: Ext_{A}^{n}(gr_{𝔪}(M), k) = H_{−n}(Hom_{A}(U, k)) ⟶ Hom_{R}(F_{n}, k) = Ext_{R}^{n}(M, k)
χ_{M}^{n} (cls (α)) (z) = α(κ(\overline{z})) .

A routine calculation reveals that χ_{M}^{n} is well defined, and k-linear.

Now, since A and gr_{𝔪}(M) are graded, so is Ext_{A}^{n}(gr_{𝔪}(M), k). Note that each χ_{M}^{n} factors through Hom_{A}(lin_{n}^{R}(F), k), which is concentrated in (internal) degree −n, so χ_{M}^{n} decomposes as

Ext_{A}^{n}(gr_{𝔪}(M), k) ↠^{π_{M}^{n}} Ext_{A}^{n}(gr_{𝔪}(M), k)_{−n} ⟶^{ν_{M}^{n}} Ext_{R}^{n}(M, k) .

We remark that χ_{M}^{n} and ν_{M}^{n} are the k-vector space duals of the maps τ_{n}^{M} and λ_{n}^{M} respectively, defined in [21, (2.2)]; this is immediate from the corresponding definitions.

It turns that that χ_{k} is a homomorphism of k-algebras, and χ_{M} is χ_{k}-equivariant; see Proposition 2.13. These properties are not used in this section.

The next item provides diverse characterizations of Koszul modules. Mostly, it is a repackaging of well known results; cf. [21, (2.3)]. The implication (1) ⇒ (2) uses an elementary lemma on filtered exact sequences; a detailed proof of this latter result, given for completeness, is relegated to the end of this section. Proposition 1.3. Let M be a finitely generated R-module and F its minimal free resolution. The following conditions are equivalent.

(1) M is Koszul.

(2) lin^{R}(F) is a minimal free resolution of gr_{𝔪}(M) over gr_{𝔪}(R).

(3) The gr_{𝔪}(R)-module gr_{𝔪}(M) has a linear resolution.

(4) The maps π_{M}, ν_{M} and χ_{M} are bijective.

(5) The map χ_{M} is injective.

Proof. (1) ⇒ (2) is a tautology. As to the reverse implication: lin^{R}(F) is minimal, by construction, and acyclic, since M is Koszul. Hence Lemma 1.12 yields that H_{0}(lin^{R}(F)) = gr_{𝔪}(M).

The remaining implications can be read off [21, (2.3)], keeping in mind that χ^{M} and ν_{M}^{n} are the k-vector space duals of the maps τ_{n}^{M} and λ_{n}^{M} respectively. □

Over any local ring (R, 𝔪, k), there are two canonical ‘test’ modules: R itself and k. The former is always Koszul; however, requiring that k be Koszul is a severe restriction on R. Definition 1.4. The local ring (R, 𝔪, k) is Koszul if the R-module k is Koszul.

This notion of a Koszul ring coincides with the one due to Herzog, Reiner, and Welker [11] who defined it by the property that the associated graded algebra is Koszul, in the usual sense of the word; see Remark 1.7.

Koszul modules over Koszul rings are the focus of Sections 5 and 6. For the moment though, we concentrate on Koszul modules in general. To begin with, we introduce an invariant that is one measure of how far a module is from being Koszul. Definition 1.5. Let M be a finitely generated R-module and F its minimal free resolution. Define

lpd_{R} (M) = sup{i | H_{i}(lin^{R}(F)) ≠ 0} .

When lpd_{R} (M) is finite, we say that for the complex F the linear part predominates; this is keeping in line with the terminology of Eisenbud, Floystad, and Schreyer [8]. Note that lpd_{R} (M) = 0 if and only if M is Koszul. More generally, for any integer d, one has lpd_{R} (M) ≤ d if and only if the dth syzygy module Ω^{d} (M) of M is Koszul. 1.6. Poincaré series. Let M be a finitely generated R-module. Recall that the nth Betti number of M, denoted b_{n}(M), is the integer: rank_{R} F_{n}, where F is the minimal free resolution of M. As is usual, we set P_{M}^{R}(t) = ∑_{n ⩾ 0} b_{n}(M)t^{n}; this is the Poincaré series of M.

It is known that P_{M}^{R}(t) need not be a rational; not even for M = k; see the discussion surrounding [1, (4.3.10)]. Moreover, in [19], Roos constructs a ring R and a family of finitely generated modules {M_{n}}_{n∈ℕ} such that P_{M_{n}}^{R}(t) is rational for each n, but these do not have a common denominator. These comments may serve to highlight the relevance of the following result. In what follows, for any graded k-vector space V, its Hilbert series ∑_{n∈ℤ} rank_{k}(V_{n})t^{n} is denoted Hilb_{V} (t). Proposition. Let (R, 𝔪, k) be a local ring. If lpd_{R} (M) < ∞, then there exists a polynomial q(t) with integer coefficients such that

P_{M}^{R}(t) = \frac{q(t)}{D(t)} where D(t) = Hilb_{gr_{𝔪}(R)}(−t)(1 + t)^{dim R} .

In particular, the denominator of P_{M}^{R}(t) depends only on the Hilbert series of gr_{𝔪}(R).

Proof. Set d = lpd_{R} (M). Then P_{M}^{R}(t) = t^{d}P_{Ω^{d}(M)}^{R}(t) + p(t), where p(t) is a polynomial of degree d − 1. Thus, P_{M}^{R}(t) has the desired form if and only if P_{Ω^{d}(M)}^{R}(t) does. Furthermore Ω^{d} (M) is Koszul. Hence, by passing to Ω^{d} (M) one may assume that M is Koszul.

Let F be the minimal free resolution of M. By Proposition 1.3, the complex of free gr_{𝔪}(R)- modules lin^{R}(F) is a free resolution of gr_{𝔪}(M); this yields the first of the following identities. The second holds because this resolution is, by construction, linear and Hilbert series are additive on short exact sequences.

P_{M}^{R}(t) = P_{gr_{𝔪}(M)}^{gr_{𝔪}(R)}(t)  and  Hilb_{gr_{𝔪}(M)}(t) = Hilb_{gr_{𝔪}(R)}(t)P_{gr_{𝔪}(M)}^{gr_{𝔪}(R)}(−t) .

Hence P_{M}^{R}(t) = Hilb_{gr_{𝔪}(M)}(−t)Hilb_{gr_{𝔪}(R)}(−t)^{−1}. It remains to note that the Hilbert series of gr_{𝔪}(M) and gr_{𝔪}(R) are both rational functions with denominator (1 − t)^{dim R}. □

The graded case. Let k be a field. By a standard graded k-algebra, we mean a graded k-algebra R = {R_{i}}_{i ⩾ 0} such that R_{0} = k, rank_{k}(R_{1}) is finite, and R_{1} generates R as a k-algebra. Such a k-algebra has a unique homogeneous maximal ideal: 𝔪 = R_{i ⩾ 1}.

Let R be a standard graded k-algebra.

Let C be a minimal complex of graded R-modules; this includes the assumption that the differentials are homogeneous of degree 0. As in the local case one can define its standard filtration with respect to 𝔪 and the associated graded complex lin^{R}(C). Observe that since R is standard graded, the k-algebras R and gr_{𝔪}(R) are naturally isomorphic. Moreover, for each integer n, the R-modules R(n) and gr_{𝔪}(R(n)) are isomorphic. In particular, if C is a complex of free R-modules, then so is lin^{R}(C).

Given these considerations, one can speak of Koszul modules, and, more generally, the invariant lpd_{R} (M), associated to any graded finitely generated R-module M. In keeping with tradition, when the R-module k is Koszul, we say that R is a Koszul algebra, rather than a Koszul ring. Remark 1.7. The k-algebra R is Koszul if and only if k has linear resolution; this is contained in (the graded analogue of) Proposition 1.3, since gr_{𝔪}(k) = k and gr_{𝔪}(R) = R. Thus, the class of Koszul algebras as introduced above coincides with the classical one. This observation along with (1.3) implies also that a local ring is Koszul if and only if its associated graded algebra is Koszul.

When one speaks of the Koszul property, the invariant that springs to mind is regularity; the next item compares it to linear dominance. 1.8. Let R be a standard graded k-algebra and M a finitely generated graded R-module. Let F be the minimal graded free resolution of M; in particular, F_{n} = ⨁_{i∈ℤ} R(−i)^{b_{n,i}} for each integer n. The integer b_{n,i} is the nith graded Betti number of M.

The regularity of M is the integer

reg_{R} M = sup{r ∈ ℤ | b_{n,n+r} ≠ 0 for some integer n} .

The standard filtration of F is compatible with internal gradings so F^{i}F is a graded subcomplex of F for each integer i. Thus, the associated graded lin^{R}(F) is a complex of bigraded R-modules, where R is viewed as a bigraded k-algebra with components

R_{p,q} = \begin{cases}0 & if p ≠ q \\ R_{p} & if p = q. \end{cases}

Observe that lin_{n}^{R}(F) ≅ ⨁_{i∈ℤ} R(−n, −i)^{b_{n,i}}. In detail: an element f ∈ lin_{n}^{R}(F) is bihomogenous of degree (p, q) if f ≡ g mod 𝔪^{p+1}F_{n}, where g is homogeneous of degree q and lies in 𝔪^{p}F_{n} \ 𝔪^{p+1}F_{n}. The differential in lin^{R}(F) can be described by matrices of bihomogenous forms of degree (1, 1), so it decomposes as a direct sum of subcomplexes of graded R-modules

lin^{R}(F) = ⨁_{r∈ℤ} F^{r} with F_{n}^{r} = R(−n, −n − r)^{b_{n,n+r}} .

The subcomplex F^{r} is called the rth linear strand of F. Keeping track of the degrees leads us to the realization that Lemma. reg_{R} M = sup{r ∈ ℤ | F^{r} ≠ 0}. □

This completes the preparation for the result below. It can been deduced also from Theorem 2.1, which throws a different light on the matter on hand. We provide a direct, and elementary, proof because justifying the latter calls for some fairly sophisticated technology.

It should be noted that its converse need not hold. Indeed, Eisenbud and Avramov [2, (1)] have proved that every finitely generated module over a Koszul algebra has finite regularity. However, not every such module has finite linear dominance; see (1.6). Proposition 1.9. If lpd_{R} (M) is finite, then so is reg_{R} M.

Proof. Suppose that reg_{R} M is infinite. Let S = {r ∈ ℤ | F^{r} ≠ 0}; here we are using the notation from (1.8). The preceding Lemma implies that the set S is infinite. For each r ∈ S, let n(r) be the least integer such that F_{n(r)}^{r} ≠ 0. For every integer n there are only finitely many r ∈ S with n(r) = n, since F_{n} is finitely generated, so that in S there is an infinite strictly increasing sequence r_{1} < r_{2} < ⋯ with n(r_{1}) < n(r_{2}) < · · ·. By definition F_{n}^{r_{k}} = 0 for n < n(r_{k}); moreover, F^{r_{k}} is minimal. Thus, H_{n(r_{k})}(F^{r_{k}}) ≠ 0 for each r_{k}, hence H_{n(r_{k})}(lin^{R}(F)) ≠ 0, since F^{r_{k}} is a direct summand of lin^{R}(F). In particular, lpd_{R} (M) = ∞, as desired. □

Here is one corollary of the preceding result. Proposition 1.10. Let R be a standard graded k-algebra. If lpd_{R} (k) < ∞, then R is Koszul.

Proof. If lpd_{R} (k) is finite, then Proposition 1.9 yields that reg_{R} k is finite, and then, according to Avramov and Peeva [3, (2)], R is Koszul. □

The proposition can be rephrased as: In the graded case, lpd_{R} (k) < ∞ implies lpd_{R} (k) = 0. Question 1.11. Let (R, 𝔪, k) be a local ring. If lpd_{R} (k) < ∞, then is lpd_{R} (k) = 0?

Theorem 2.1 provides some information on this problem - see (2.3) - but not enough to solve it. The precise relation between regularity and linear dominance remains mysterious; however,

see the remarks after Theorem 2.1. For the special case of Koszul modules though a result of Römer provides a complete and elegant description. To state it one requires a notion introduced in [10]: A graded finitely generated R-module M is componentwise linear if for each integer i the R-submodule of M generated by M_{i} has a linear resolution. Theorem (Römer). Let M be a finitely generated graded R-module. Then M is Koszul if and only if M is componentwise linear.

The last item in this section is a simple lemma on filtered modules; it was used in proving (1.3). 1.12. Filtered modules. A filtered module is an R-module U with a filtration {U^{i}}_{i ⩾ 0} such that U^{0} = U and U^{n+1} ⊆ U^{n} for each n ≥ 0. The associated graded module of a filtered module U is the graded R-module gr(U) with degree n component U^{n}/U^{n+1}.

A homomorphism of filtered modules is an R-module homomorphism φ: U → V such that φ(U^{n}) ⊆ V^{n}. In this case, Ker φ is filtered with (Ker φ)^{n} = (Ker φ) ∩ U^{n}. Similarly, Coker φ is filtered with (Coker φ)^{n} = (V^{n} + φ(U))/φ(U).

Lemma. Let U ⟶^{φ} V ⟶^{ψ} W be homomorphisms of filtered R-modules. If the associated graded sequence is exact, then the canonical homomorphism Coker(gr(ψ)) → gr(Coker ψ) is bijective.

Proof. Fix an integer n ≥ 0. The degree n part of the homomorphism in question is the surjection

Coker(gr(ψ))_{n} = \frac{W^{n}}{W^{n+1} + ψ(V^{n})} ⟶ \frac{W^{n} + ψ(V)}{W^{n+1} + ψ(V)} = gr(Coker ψ)_{n} .

This map is injective if and only if W^{n} ∩ (W^{n+1} + ψ(V)) ⊆ W^{n+1} + ψ(V^{n}). Thus, it suffices to prove that W^{n} ∩ ψ(V) ⊆ ψ(V^{n}). To this end we establish by a descending induction on s that

W^{n} ∩ ψ(V^{s}) ⊆ ψ(V^{n})  for all  0 ≤ s ≤ n .

The basis step s = n is tautological. Suppose that the inclusion above holds for s + 1, for some integer s where 0 ≤ s ≤ n − 1. The exactness of the sequence

\frac{U^{s}}{U^{s+1}} ⟶^{gr(φ)_{s}} \frac{V^{s}}{V^{s+1}} ⟶^{gr(ψ)_{s}} \frac{W^{s}}{W^{s+1}} ,

yields: ψ^{−1}(W^{s+1}) ∩ V^{s} ⊆ φ(U^{s}) + V^{s+1}. Thus, W^{s+1} ∩ ψ(V^{s}) ⊆ ψ(V^{s+1}). This explains the inclusion on the right in the computation below, while the one of the left holds because n ≥ s + 1.

W^{n} ∩ ψ(V^{s}) ⊆ W^{s+1} ∩ ψ(V^{s}) ⊆ ψ(V^{s+1}) .

Therefore, W^{n} ∩ ψ(V^{s}) ⊆ W^{n} ∩ ψ(V^{s+1}). It remains to invoke the induction hypothesis. □

2. Cohomology of modules of finite linear dominance

The highlight of this section is the result below which reveals a striking property of modules of finite linear dominance. A far stronger statement can be established when R is Koszul; cf. (5.5). Theorem 2.1. Let (R, 𝔪, k) be a local ring and M a finitely generated R-module. If lpd_{R} (M) is finite, then as a left ⟨ Ext_{R}^{1}(k, k) ⟩-module Ext_{R}(M, k) has a finite generating set consisting of elements of degree ≤ lpd_{R} (M). Remark 2.2. There is a precise analogue of this result for standard graded k-algebras. Now, for such a k-algebra R, the bigraded k-algebra ⟨ Ext_{R}^{1}(k, k) ⟩ lies on the ‘diagonal’: each (homogeneous) element has degree (n, −n), for some integer n. Consequently, when Ext_{R}(M, k) is finitely generated, reg_{R} M is finite. In this way, one recovers Proposition 1.9 from the theorem above. Remark 2.3. The preceding result might be a first step to solving Question 1.11. Indeed, it implies that when lpd_{R} (k) < ∞ the inclusion of k-algebras ⟨Ext_{R}^{1}(k, k)⟩ ⊆ Ext_{R}(k, k) is a module finite extension. This is even a free extension as the inclusion is one of Hopf algebras. However, these properties in themselves do not imply that R is Koszul; see Example 2.5.

The proof of Theorem 2.1 is given in (2.14). It is worth spelling out a special case: Corollary 2.4. If the local ring R is Koszul, then ⟨ Ext_{R}^{1}(k, k) ⟩ = Ext_{R}(k, k). □

This result is classical for standard graded k-algebras. Moreover, for such algebras the converse is also true: the identity ⟨Ext_{R}^{1}(k, k)⟩ = Ext_{R}(k, k) captures the Koszul property; cf. [15, Theorem 1.2]. This is not so in general, as the example below from loc. cit. illustrates. Example 2.5. Let R = k[x, y]/(x^{2}+y^{3}, xy). Then, since R is a complete intersection, a computation due to Sjödin [22, Theorem 5] gives:

Ext_{R}(k, k) = T (u, v) /(v^{2}, [u^{2}, v]) where |u| = 1 = |v| .

Here T (u, v) denotes the tensor algebra on variables u, v. In particular Ext_{R}(k, k) is generated by Ext_{R}^{1}(k, k). However, R is not Koszul since the associated graded algebra is k[x, y]/(x^{2}, xy, y^{4}) and this is not Koszul.

As is clear by now, one of the protagonists in this work is Ext_{R}(k, k). We begin by a review of basic computations concerning its algebra structure and its action on Tor^{R} (k, M) and Ext_{R}(M, k); these play a key role in the sequel. No doubt all this is old hat to the experts, but suitable references are difficult to find.

The following notation that will be operative throughout Sections 2 – 4. Notation 2.6. Let (R, 𝔪, k) be a local ring and M a complex of R-modules. For any (graded) k-vector space V denote V^{∨} the (graded) dual vector space Hom_{k}(V, k). Moreover

(a) let r_{1}, …, r_{e} be a minimal generating set of 𝔪;

(b) let x_{i} be the residue class of r_{i} in 𝔪/𝔪^{2}. Thus, x_{1}, …, x_{e} is a basis for 𝔪/𝔪^{2};

(c) let y_{1}, …, y_{e} be the basis of (𝔪/𝔪^{2})^{∨} dual to x_{1}, …, x_{e};

(d) let P be an R-free resolution of k such that P_{0} = R and P_{1} = ⨁_{i=1}^{e} Re_{i} with ∂(e_{i}) = r_{i},

and let ε: P → k be the augmentation. 2.7. The Ext-algebra. Since P is an R-free resolution of k, one may identify the k-vector space Ext_{R}^{n}(k, k) with H_{−n}(Hom_{R}(P, P)). Composition endows the complex Hom_{R}(P, P) with a structure of a DG (=Differential Graded) algebra. This induces on its homology, that is to say, on Ext_{R}(k, k), a structure of a graded k-algebra. In the sequel, this is taken to be the product on Ext_{R}(k, k). It differs from the classical Yoneda product only by sign; cf. [22, Lemma 1].

Let F be a free resolution of M. Then Ext_{R}^{n}(M, k) = H_{−n}(Hom_{R}(F, P)). Akin to the situation described above, composition makes Hom_{R}(F, P) a left DG Hom_{R}(P, P)-module, and passing to homology gives a left Ext_{R}(k, k)-module structure on Ext_{R}(M, k).

Similarly, Tor_{n}^{R} (k, M) = H_{n}(P ⊗_{R} F) and Tor^{R} (k, M) is a left Ext_{R}(k, k)-module: Hom_{R}(P, P) acts on P ⊗_{R} M by evaluation on the left-hand factor of the tensor product. This makes P ⊗_{R} M a DG left module over the DG algebra Hom_{R}(P, P). Passing to homology yields the desired structure. Now, Hom_{R}(P, P) acts also on P ⊗_{R} M and the quasiisomorphism P ⊗_{R} F ↠ P ⊗_{R} M is a homomorphism of DG Hom_{R}(P, P) modules. Thus, one can compute the Ext_{R}(k, k) action on Tor^{R} (k, M) via the complex P ⊗_{R} M.

Next we delve deeper into Ext_{R}^{1}(k, k). As is well known, Ext_{R}^{1}(k, k) = (𝔪/𝔪^{2})^{∨}. Its basis y_{1}, …, y_{e} can be described quite explicitly in terms of cycles in Hom_{R}(P, P): 2.8. There are degree −1 cycles θ_{1}, …, θ_{e} in Hom_{R}(P, P) with y_{i} = cls (θ_{i}) and such that

θ_{i}(e_{j}) = \begin{cases}1 & if i = j \\ 0 & if i ≠ j . \end{cases}  for all  1 ≤ i, j ≤ e .

A proof of this result is contained in [22, Theorem 1]; see also [1, (10.2.1)]. 2.9. Products in ⟨Ext_{R}^{1}(k, k)⟩. Suppose that the local ring R is 𝔪-adically complete. Cohen’s structure theorem provides a surjective local homomorphism π: (Q, 𝔫, k) → R with Q regular; one can ensure that Ker(π) ⊆ 𝔫^{2}. Let r̃_{1}, …, r̃_{e} be elements in Q such that π(r̃_{i}) = r_{i} for each integer i. Then (r̃_{1}, …, r̃_{e}) = 𝔫 since Ker(π) ⊆ 𝔫^{2}. Let s_{1}, …, s_{c} be a minimal generating set for Ker(π). For each integer k, write

s_{k} = ∑_{1 ⩽ i ⩽ j ⩽ e} a_{ijk} r̃_{i}r̃_{j} with a_{ijk} ∈ Q .

In what follows, given an element a ∈ Q, its residue class in k = Q/𝔫 is denoted \overline{a}.

It is established in [22, §3], see also [1, (10.2.2)], that there is a basis u_{1}, …, u_{c} of Ext_{R}^{2}(k, k) such that for all integers i, j, p with 1 ≤ i < j ≤ e and 1 ≤ p ≤ e, one has

[y_{i}, y_{j}] = − ∑_{k=1}^{c} \overline{a}_{ijk}u_{k}  and  y_{p}^{2} = − ∑_{k=1}^{c} \overline{a}_{ppk}u_{k} .

Here [y_{i}, y_{j}] denotes the (graded) commutator y_{i}y_{j} + y_{j}y_{i} of the elements y_{i} and y_{j}.

A complex M is bounded below if H_{i}(M) = 0 for i ≪ 0, and degreewise finite if the R-module H_{i}(M) is finitely generated for each i. Such complexes admit minimal free resolutions. 2.10. The action of Ext_{R}^{1}(k, k) on Tor^{R} (k, M). Let M be a complex of R modules that is bounded below and degree finite, and let F be its minimal free resolution. Then

Tor^{R} (k, M) = H(k ⊗_{R} F) = k ⊗_{R} F .

As has been described in (2.7), the k-algebra Ext_{R}(k, k) acts on Tor^{R} (k, M); hence, via the identification above, it acts on k ⊗_{R} F. In turns out that the action of Ext_{R}^{1}(k, k) on k ⊗_{R} F is determined completely by the linear part of the differential on F. This is the gist of the following result. A piece of notation: for any element z ∈ F, we write \overline{z} for its residue class in k ⊗_{R} F. Lemma. Let z be an element in F_{n} and write ∂(z) = ∑_{j=1}^{e} r_{j}f_{j} with f_{j} ∈ F_{n−1}. Then y_{i} ·\overline{z} = −\overline{f_{i}} for each integer 1 ≤ i ≤ e.

Proof. The complex of R-modules F is free so the canonical surjection ε ⊗_{R} F: P ⊗_{R} F ↠ k ⊗_{R} F is a quasiisomorphism. Since (P ⊗_{R} F)_{n} = F_{n} ⊕ (P_{1} ⊗ F_{n−1}) ⊕ (P_{⩾ 2} ⊗_{R} F)_{n} the cycle \overline{z} lifts to a cycle in P ⊗_{R} F of the form

z̃ = z − ∑_{j=1}^{e}(e_{j} ⊗_{R} f_{j}) + w with w ∈ (P_{⩾ 2} ⊗_{R} F) .

Let θ_{i} ∈ Hom_{R}(P, P) be the degree −1 cycle corresponding to y_{i} that is provided by (2.8). Then θ_{i}(z̃) = −f_{i} + θ_{i}(w) so that y_{i} · cls \binom{}{z} = cls (−f_{i} + θ_{i}(w)) in H(P ⊗_{R} F). Therefore, in k ⊗_{R} F one has that

y_{i} · \overline{z} = ε(y_{i} · cls \binom{}{z}) = ε(cls (−f_{i} + θ_{i}(w))) = cls (ε(−f_{i} + θ_{i}(w))) = −\overline{f_{i}} ,

where the last equality holds because θ_{i}(w) ∈ (P_{⩾ 1} ⊗_{R} F) = Ker(ε). □

Here is the cohomological counterpart of (2.10).

2.11. The action of Ext_{R}^{1}(k, k) on Ext_{R}(M, k). As before, let M be a bounded below, degreewise finite, complex of R-modules, and let F be its minimal free resolution. By definition, Ext_{R}(M, k) = H(Hom_{R}(F, P)); see (2.7). However, the complex of R-modules F is free, so the canonical surjection Hom_{R}(F, ε): Hom_{R}(F, P) → Hom_{R}(F, k) is a quasiisomorphism. The minimality of F ensures that the differential on Hom_{R}(F, k) is trivial, so that

Ext_{R}(M, k) = Hom_{R}(F, k) .

Via this identification, Hom_{R}(F, k) acquires a left Ext_{R}(k, k)-module structure. Lemma. Let α be a degree 1 − n homomorphism in Hom_{R}(F, k), and let z ∈ F_{n}. Write ∂(z) = ∑_{j=1}^{e} r_{j}f_{j} with f_{j} ∈ F_{n−1}. Then (y_{i} · α)(z) = (−1)^{1−n}α(f_{i}) for each integer 1 ≤ i ≤ e.

Proof. Under the quasiisomorphism Hom_{R}(F, ε), the element α lifts to a cycle α̃ in Hom_{R}(F, P), of degree 1 − n. Now α̃(z) ∈ P_{1}, since its degree is 1. Say α̃(z) = ∑_{j=1}^{e} c_{j}e_{j}, with c_{j} ∈ R. Then

∂α̃(z) = ∑_{j=1}^{e} c_{j}r_{j}  and  α̃∂(z) = ∑_{j=1}^{e} r_{j}α̃(f_{j}) .

Since ∂α̃ = (−1)^{1−n}α̃∂, we obtain that ∑_{j=}^{e} (c_{j} − (−1)^{1−n}α̃(f_{j})) r_{j} = 0 in P_{0}; recall that P_{0} = R. As r_{1}, …, r_{e} is a minimal generating set for 𝔪, the element c_{j} − (−1)^{1−n}α̃(f_{j}) is in 𝔪, for each integer j. Now εα̃ = α and 𝔪 ⊂ Ker(ε), consequently ε(c_{j}) = (−1)^{1−n}α(f_{j}) for each integer j. Therefore, with θ_{i} the cycle in Hom_{R}(P, P) corresponding to y_{i}, given by (2.8), one obtains:

(y_{i} · α)(z) = ε(θ_{i}(α̃(z))) = ε(θ_{i}( ∑_{j=1}^{e} c_{j}e_{j})) = ε(c_{i}) = (−1)^{1−n}α(f_{i}) .

This is the equality we seek. □

Two more ingredients are required before one can deal with Theorem 2.1 Here is the first. 2.12. Let A be a standard graded k-algebra and G a graded degreewise finite A-module with G_{i} = 0 for i ≪ 0. Set s = inf{i | G_{i} ≠ 0}, and

D = ⨁_{n∈ℕ} Ext_{A}^{n}(k, k)_{−n}  and  U = ⨁_{n∈ℕ} Ext_{A}^{n}(G, k)_{−n−s} .

Then D is a k-subalgebra, called the diagonal subalgebra, of Ext_{A}(k, k). In particular, D acts on Ext_{A}(G, k) giving it a structure of a left D-module. Since D is situated on the diagonal, the subspace U of Ext_{A}(G, k) inherits this structure; we call it the diagonal submodule of Ext_{A}(G, k). It should be emphasized that this is an D-submodule, but not necessarily an Ext_{A}(k, k)-submodule.

It turns out that D coincides with ⟨ Ext_{A}^{1}(k, k) ⟩, the k-subalgebra of Ext_{A}(k, k) generated by its degree 1 elements. Moreover, U is generated as a left D-module by U^{0}, which is to say, by Hom_{k}(G_{s}, k). These statements, and more, are contained in the next result; in it T (A_{−1}^{∨}) denotes the tensor algebra on the k-vector space A_{−1}^{∨} concentrated in degree −1, and (Im(μ_{R}^{∨})) denotes its 2-sided ideal generated by Im(μ_{R}^{∨}). Proposition. Let μ_{R}: A_{1} ⊗_{k} A_{1} → A_{2} and μ_{M}: A_{1} ⊗_{k} G_{s} → G_{s+1} be the respective product maps. Then ⟨ Ext_{A}^{1}(k, k) ⟩ = D, and there are canonical isomorphisms:

D ≅ \frac{T (A_{−1}^{∨})}{(Im(μ_{R}^{∨}))}  and  U ≅ \frac{D ⊗_{k} G_{−s}^{∨}}{D · Im (μ_{M}^{∨})} ,

where the one on the left is of k-algebras, and the one on the right is of left D-modules.

Proof. The assertions concerning D are special cases of a result of Löfwall [15, Corollary 1.1]. His arguments can be extended without any ado to obtain the desired formula for U. □

The next item establishes some structural properties of the maps defined in (1.2).

2.13. Let (R, 𝔪, k) be a local ring and M a finitely generated R-module. Set A = gr_{𝔪}(R). The construction in (1.2) yields diagrams

Ext_{A}(k, k) ⟶^{π_{k}} ⨁_{n∈ℕ} Ext_{A}^{n}(k, k)_{−n} ⟶^{ν_{k}} Ext_{R}(k, k)

Ext_{A}(gr_{𝔪}(M), k) ⟶^{π_{M}} ⨁_{n∈ℕ} Ext_{A}^{n}(gr_{𝔪}(M), k)_{−n} ⟶^{ν_{M}} Ext_{R}(M, k) ,

such that the composed map in the first diagram is χ_{k}, and in the second it is χ_{M}. Proposition. The maps π_{k}, ν_{k}, and χ_{k} are homomorphisms of k-algebras. Moreover, π_{M} is π_{k}-equivariant, ν_{M} is ν_{k}-equivariant, and χ_{M} is χ_{k}-equivariant.

Proof. First we tackle χ_{k} and χ_{M}. Since these maps are k-linear, by construction, it suffices to prove that for ε ∈ Ext_{A}(k, k) and α ∈ Ext_{A}(gr_{𝔪}(M), k) one has

(†)  χ_{M}(ε · α) = χ_{k}(ε) · χ_{M}(α) .

This would settle the equivariance of χ_{M}, and specializing M to k would yield also that χ_{k} is an algebra homomorphism. There are further reductions: note that

(‡)  Ker(π_{k}^{n}) = Ext_{A}^{n}(k, k)_{i<−n}  and  Ker(π_{M}^{n}) = Ext_{A}^{n}(gr_{𝔪}(M), k)_{i<−n} .

In particular, when checking (†), one may assume that ε lies in ⨁_{n∈ℕ} Ext_{A}^{n}(k, k)_{−n}, the diagonal subalgebra of Ext_{A}(k, k). Then, it suffices to verify (†) for ε = y_{i}, where y_{1}, …, y_{e} are the elements defined in (2.6). This is because, by Proposition 2.12, the diagonal subalgebra is generated by Ext_{A}^{1}(k, k), that is to say, by (𝔪/𝔪^{2})^{∨}. Now Ext_{R}^{1}(k, k) = (𝔪/𝔪^{2})^{∨} as well; a direct computation establishes that χ_{k}^{1} is the identity. The long and short of this discussion is that the desired results on χ_{k} and χ_{M} follow once we check (†) for ε = y_{i} for each integer 1 ≤ i ≤ e.

Let F be the minimal R-free resolution of M, let U be the minimal graded A-free resolution of gr_{𝔪}(M), and κ: lin^{R}(F) → U the comparison map; see (1.2). Then Ext_{R}^{n}(M, k) = Hom_{R}(F_{n}, k) and Ext_{A}^{n}(gr_{𝔪}(M), k) = Hom_{A}(U_{n}, k). Fix a homomorphism α in Hom_{A}(U, k) of degree 1 − n and an element z ∈ F_{n}. By definition

χ_{M}(α)(z) = α(κ(\overline{z})) ,

where \overline{z} denotes the residue class of z in F/𝔪F. Write ∂(z) = ∑_{j=1}^{e} r_{j}f_{j}. Then, in lin^{R}(F) one has ∂(\overline{z}) = ∑_{j=1}^{e} x_{j}\overline{f}_{j}, where x_{j} is the residue class in 𝔪/𝔪^{2} of the element r_{j}. Since κ is a morphism of complexes one obtains: ∂(κ(\overline{z})) = κ(∂(\overline{z})) = ∑_{j=1}^{e} x_{j}κ(\overline{f}_{j}). The formula above and (2.11) yield

χ_{M}(y_{i} · α)(z) = (y_{i} · α)(κ(\overline{z})) = (−1)^{1−n}α(κ(\overline{f}_{i})) ;
(y_{i} · χ_{M}(α))(z) = (−1)^{1−n}χ_{M}(α)(f_{i}) = (−1)^{1−n}α(κ(\overline{f}_{i})) .

Since z was chosen arbitrarily, this yields: χ_{M}(y_{i} · α) = y_{i} · χ_{M}(α), for each integer i, as desired. This completes the proof that χ_{k} is a homomorphism of k-algebras and χ_{M} is χ_{k}-equivariant. As to the remaining assertions: Since each graded vector space Ext_{A}^{n}(k, k) is concentrated in

degrees ≤ −n; one obtains from (‡) that Ker(π_{k}) is a 2-sided ideal in Ext_{A}(k, k) and that Ker(π_{M}) is a (left) Ext_{A}(k, k)-submodule of Ext_{A}(gr_{𝔪}(M), k). Therefore, π_{k} is a k-algebra homomorphism and π_{M} is π_{k}-equivariant. The surjectivity of these homomorphisms and the identities χ_{k} = ν_{k} ∘π_{k} and χ_{M} = ν_{M} ∘ π_{k} entail the stated properties of ν_{k} and ν_{M}. □

2.14. Proof of Theorem 2.1. As usual, it is convenient to write A for gr_{𝔪}(R).

Suppose lpd_{R} (M) = 0, so that M is Koszul. Then Proposition 1.3 provides the isomorphism

χ_{M}: ⨁_{n∈ℕ} Ext_{A}^{n}(gr_{𝔪}(M), k)_{−n} ⟶^{≅} Ext_{R}(M, k) .

Proposition 2.12 yields that the module on the left is generated as an ⟨ Ext_{A}^{1}(k, k) ⟩-module by Ext_{A}^{0}(gr_{𝔪}(M), k)_{0}. Therefore, one obtains that the ⟨ Ext_{R}^{1}(k, k) ⟩-module Ext_{R}(M, k) is generated by Ext_{R}^{0}(M, k), since χ_{M} is χ_{k}-equivariant; see Proposition 2.13. This is the result we seek.

Suppose d = lpd_{R} (M) ≥ 1. The first syzygy Ω (M) of M is defined by the exact sequence 0 → Ω (M) → F → M → 0, where F is a free module and Ω (M) ⊆ 𝔪F. When Hom_{R}(−, k) is applied to it, it unwraps into the long exact sequence

⋯ → Ext_{R}^{i}(Ω (M) , k) ⟶^{ð^{i}} Ext_{R}^{i+1}(M, k) → Ext_{R}^{i+1}(F, k) → Ext_{R}^{i+1}(Ω (M) , k) → · · · .

Due to the choice of F, one obtains that ð^{i} is an isomorphism for each i ≥ 0. Furthermore, it is well known, and not difficult to prove, that the ð^{i} are compatible with the corresponding Ext_{R}(k, k)-module structures. Thus, the long exact sequence above can be packaged into the short exact sequence of graded left Ext_{R}(k, k)-modules

0 → Ext_{R}(Ω (M) , k)(1) → Ext_{R}(M, k) → Ext_{R}^{0}(M, k) → 0 .

Here Ext_{R}^{0}(M, k) has the trivial left Ext_{R}(k, k)-action. The exact sequence above is also one of left ⟨ Ext_{R}^{1}(k, k) ⟩-modules, since the latter is a subalgebra of Ext_{R}(k, k). Now, lpd_{R} (Ω (M)) = lpd_{R} (M) − 1; thus, by an iteration, one can assume that Ext_{R}(Ω (M) , k) is generated over ⟨ Ext_{R}^{1}(k, k) ⟩ by elements in degrees ≤ d − 1. Therefore, in the exact sequence above, the module on the left, and hence also the one in the middle, is generated by elements in degrees ≤ d. □

3. Conjugation

Let (R, 𝔪, k) be a local ring. The first item in this section describes a natural procedure for converting each left Ext_{R}(k, k)-module into a right Ext_{R}(k, k)-module with the same underlying k-vector space, and vice versa. Its relevance to the topic on hand is explained by (3.2). 3.1. Conjugation. Set E = Ext_{R}(k, k). This graded k-algebra can be enriched to a graded Hopf algebra, and part of that structure is an anti-automorphism σ: E → E, called the conjugation; see [9, Chapter II, §2], and the paper of Milnor and Moore [16, (8.4)]. For each graded left E-module U, the graded k-vector space underlying it has the structure of right E-module, with product:

u ∗ ε = (−1)^{|ε||u|}σ(ε)u for u ∈ U and ε ∈ E .

Denote U^{σ} the right E-module defined thus. This procedure is functorial: Each homomorphism f: U → V of graded left E-modules transforms into a homomorphism of graded right E-modules: f: U^{σ} → V^{σ}. These remarks may be summed up as: conjugation is an equivalence between the categories of left E-modules and right E-modules.

The remark below is used in Section 4; it holds, more generally, for modules over Hopf algebras. Lemma. Let U and V be graded left E-modules. There is an natural isomorphism of bigraded k-vector spaces

Tor^{E} (U^{σ}, V) ≅ Tor^{E} (V^{σ}, U) .

Proof. One has the natural ‘twisting’ homomorphism of graded k-vector spaces

U^{σ} ⊗_{k} V → V^{σ} ⊗_{k} U where u ⊗_{k} v ↦ (−1)^{|u||v|}v ⊗_{k} u .

An simple, and gratifying, computation establishes that this map descends to a homomorphism U^{σ} ⊗_{E} V → V^{σ} ⊗_{E} U; it is bijective for it has an obvious inverse. The derived avatar of this isomorphism is the one we seek. □

3.2. Set E = Ext_{R}(k, k). Let M be a bounded below, degreewise finite, complex of R-modules. Then Hom-Tensor adjunction yields a natural isomorphism of graded k-vector spaces:

(⋆)  Ext_{R}(M, k) ≅ Hom_{k}(Tor^{R} (k, M) , k) .

Via this isomorphism, the left action of E on Tor^{R} (k, M) passes to a right action on Ext_{R}(M, k). Now Ext_{R}(M, k) has a second right E-module structure arising via conjugation - see (3.1) - from its usual left E-module structure.

The following proposition tells us that these two structures on Ext_{R}(M, k) coincide. This coincidence is exploited in the proofs of theorems 4.1 and 6.1.

Proposition. Suppose that R is Koszul. Then the isomorphism in (⋆) is compatible with the corresponding right E-module structures:

Ext_{R}(M, k)^{σ} ≅ Hom_{k}(Tor^{R} (k, M) , k) .

Dually: The isomorphism Hom_{k}(Ext_{R}(M, k)^{σ}, k) ≅ Tor^{R} (k, M) is one of left E-modules. Remark. This result is basic, but there seems to be no reference to it in the literature, except for the special case M = k; see [9, (2.3.3)]. In fact, (we believe that) we can do away with the hypotheses on R: that it is Koszul, and on M: that it is bounded below and degreewise finite). However, the only (completely satisfactory) proof we could come up with for this, more general, statement is rather technical, and presenting it would be too great a digression.

Proof. The ring R is Koszul, so E = ⟨ Ext_{R}^{1}(k, k) ⟩; see (2.4). This means that the elements y_{1}, …, y_{e}, from (2.6), generate the k-algebra E, since Ext_{R}^{1}(k, k) = (𝔪/𝔪^{2})^{∨}. Consequently, it suffices to verify the required compatibility for the action of each of the y_{i}. The effect of conjugating the y_{i} is easy to describe:

(⋆⋆)  σ(y_{i}) = −y_{i}  for all  1 ≤ i ≤ e .

This is immediate from the proof of [16, (8.2)].

Let F be a minimal free resolution of M. Then Tor^{R} (k, M) = k ⊗_{R} F and Ext_{R}(M, k) = Hom_{R}(F, k); the induced left action of Ext_{R}^{1}(k, k) on k ⊗_{R} F and Hom_{R}(F, k) is described in (2.10) and (2.11) respectively. The isomorphism in question is then the usual adjunction:

Hom_{R}(F, k) ≅ Hom_{k}(k ⊗_{R} F, k) .

Let α ∈ Hom_{R}(F, k) be an element of degree 1 − n. Let z be an element in F_{n}; write ∂(z) = ∑_{j=1}^{e} r_{j}f_{j}. By (2.10), the right action of y_{i} on α induced by the isomorphism above is given by:

(α · y_{i})(z) = −α(f_{i}) .

On the other hand, the right action on Hom_{R}(F, k) induced by conjugation is given by

(α∗y_{i})(z) = (−1)^{1−n}(σ(y_{i}) · α)(z) = −(−1)^{1−n}(σ · α)(z) = −α(f_{i}) ,

where second equality is by (⋆⋆) above^{1} and the last is by (2.11). Thus α · y_{i} = α ∗ y_{i} on elements of degree n. Thus α · y_{i} = α ∗ y_{i}, since both are of degree −n. This settles the issue. □

4. The (co)homology of the linear part of a complex

This section is dedicated entirely to the proof of the theorem below. It generalizes, and is inspired by, the article of Eisenbud, Floystad, and Schreyer [8, (3.4)], who have dealt with the case where R is either a polynomial ring or an exterior algebra. Some of the key ideas in the proof are contained already in work of Buchweitz [5]. 4.1. Let (R, 𝔪, k) be a local ring and M a complex of R-modules that is bounded below and degreewise finite. Set R^{!} = ⟨ Ext_{R}^{1}(k, k) ⟩. Let T = {T^{i}}_{i ⩽ 0} be the graded k-vector space with T^{−i} = Tor_{i}^{R} (k, M) for each integer i; it acquires from Tor^{R} (k, M) the structure of a left R^{!}-module. Theorem. Let F be the minimal free resolution of M. If R is Koszul, then for all n, i ∈ ℤ one has isomorphisms of k-vector spaces

H_{n}(lin^{R}(F))_{i} = Ext_{R^{!}}^{i−n}(k, T)^{−i}
Hom_{k}(H_{n}(lin^{R}(F))_{i}, k) = Tor_{i−n}^{R^{!}} (k, Ext_{R}(M, k))^{i} .

Remark. We have introduced T so that all R^{!}-modules that appear are upper graded.

1In other words: by the stars above....

In the sequel this result is invoked only for the special case where M is a module (concentrated in degree 0). The more general result is established with a view towards future applications. At any rate, nothing is gained by restricting oneself to modules.

The proof of Theorem 4.1 requires considerable preparation and is given at the end of the section. An important intermediate step is Theorem 4.5 which describes lin^{R}(F) in terms of the action of Ext_{R}^{1}(k, k) on Tor^{R} (k, M); this result does not require R to be Koszul.

In the remainder of this section, one has to deal with Tor^{R} (k, M) and Ext_{R}(k, k) simultaneously. The grading on the former is homological whilst on the latter it is cohomological. Keeping it this way leads to utter confusion during calculations, at least for us. So we adopt the following notation for these and revert to the more conventional one when required.

Notation 4.2. Set A = gr_{𝔪}(R) and let E be the k-algebra with E_{i} = Ext_{R}^{−i}(k, k) for each i ∈ ℤ. Thus, A is concentrated in non-negative degrees whilst E is concentrated in non-positive degrees. The product on E is taken over from Ext_{R}(k, k).

In addition, the notation introduced in (2.6) will be used without specific comment.

4.3. Consider the k-algebra A ⊗_{k} E with product defined by

(x′ ⊗_{k} y′) · (x ⊗_{k} y) = x′x ⊗_{k} yy′ for all x′, x ∈ A and y′, y ∈ E .

Observe that (𝔪/𝔪^{2}) ⊗_{k} (𝔪/𝔪^{2})^{∨} is the subspace A_{1} ⊗_{k} E_{−1} of A ⊗_{k} E. The k-vector space 𝔪/𝔪^{2} is finite dimensional, so the canonical map

(𝔪/𝔪^{2}) ⊗_{k} (𝔪/𝔪^{2})^{∨} ⟶ Hom_{k}(𝔪/𝔪^{2}, 𝔪/𝔪^{2})

is an isomorphism. Denote Δ the element in (𝔪/𝔪^{2}) ⊗_{k} (𝔪/𝔪^{2})^{∨} corresponding to the identity map on 𝔪/𝔪^{2}. Observe that

Δ = ∑_{i=1}^{e}(x_{i} ⊗_{k} y_{i}) ,

in terms of the chosen basis x_{1}, …, x_{e} of 𝔪/𝔪^{2} and its dual y_{1}, …, y_{e}.

Lemma. Δ^{2} = 0, when viewed as an element in A ⊗_{k} E.

Proof. Let R̂ denote the completion of R at its maximal ideal 𝔪. The canonical map R → R̂ is flat and the maximal ideal 𝔪̂ of the local ring R̂ equals 𝔪R̂. Therefore, gr_{𝔪}(R) ≅ gr𝔪̂(R̂) and Ext_{R}(k, k) ≅ ExtR̂(k, k). So by passing to R̂ one can assume that R is complete. Now one is in a position to utilize the results in (2.9) and (2.10); we adopt their notation.

The elements x_{1}, …, x_{e} form a basis for 𝔪/𝔪^{2}, so they generate the k-algebra A. By construction, each x_{i} is the residue class of the element r̃_{i}. Thus, for any integer k, one has that

∑_{1 ⩽ i ⩽ j ⩽ e} \overline{a}_{ijk}x_{i}x_{j} = ∑_{1 ⩽ i ⩽ j ⩽ e} a_{ijk}r̃_{i}r̃_{j} mod (𝔫^{2} + Ker(π)) = 0 , (⋆)

where the equality on the right holds because ∑_{i,j} a_{ijk}r̃_{i}r̃_{j} = s_{k}, which is an element in Ker(π).

Recall that Δ = ∑_{i=1}^{e} x_{i} ⊗_{k} y_{i}. This leads to the following computation, where the second equality holds as A is commutative; the third is by (2.9), while the last is due to (⋆) above.

Δ^{2} = ∑_{1 ⩽ i<j ⩽ e} (x_{i}x_{j} ⊗_{k} y_{j}y_{i} + x_{j}x_{i} ⊗_{k} y_{i}y_{j}) + ∑_{i=1}^{e} (x_{i}^{2} ⊗_{k} y_{i}^{2})

= ∑_{1 ⩽ i<j ⩽ e} (x_{i}x_{j} ⊗_{k} [y_{i}, y_{j}]) + ∑_{i=1}^{e} (x_{i}^{2} ⊗_{k} y_{i}^{2})

= ∑_{1 ⩽ i<j ⩽ e_{1 ⩽ p ⩽ c}} −(x_{i}x_{j} ⊗_{k} \overline{a}_{ijp}u_{p}) + ∑_{1 ⩽ i ⩽ e_{1 ⩽ p ⩽ c}} −(x_{i}^{2} ⊗_{k} \overline{a}_{iip}u_{p})

= ∑_{1 ⩽ i ⩽ j ⩽ e_{1 ⩽ p ⩽ c}} −(x_{i}x_{j} ⊗_{k} \overline{a}_{ijp}u_{p})

= ∑_{p=1}^{c} −( ∑_{1 ⩽ i ⩽ j ⩽ e} \overline{a}_{ijp}x_{i}x_{j}) ⊗_{k} u_{p}

= 0 .

Thus Δ^{2} = 0; this is the result we seek. □

4.4. Construction I. Let U = {U_{i}}_{i∈ℤ} be a graded left E-module. For each integer n, let

β_{n}(U) = A(−n) ⊗_{k} U_{n}

be the graded free A-module, with A acting via the left-hand factor of the tensor product. Let δ_{n}: β_{n}(U) → β_{n−1}(U) be the k-linear map with

δ_{n}(s ⊗_{k} u) = ∑_{i=1}^{d}(sx_{i} ⊗_{k} y_{i}u)  for all  (s ⊗_{k} u) ∈ β_{n}(U) .

Claim. For each integer n, the map δ_{n} is degree 0 homomorphism of A-modules and δ_{n−1} ∘δ_{n} = 0. In other words, (β_{∗}(U), δ_{∗}) is a complex of graded free A-modules.

Indeed, it is elementary to verify that δ_{n} is A-linear and that it preserves (internal) degrees. Thus, the only moot point is whether δ_{n} is a differential. Note that the k-vector space β_{∗}(U) is a right module over A ⊗_{k} E with product defined by (s ⊗_{k} u) · (x ⊗_{k} y) = (sx ⊗_{k} yu). In particular, δ(s ⊗_{k} u) = (s ⊗_{k} u) · Δ, where Δ is the element studied in (4.3). It remains to invoke Lemma 4.3.

This brings us to one of the major landmarks on the road to Theorem 4.1.

4.5. Let F be a minimal free resolution of M; remember that M is assumed to be a bounded below, degreewise finite, complex of R-modules.

Recall that E = Ext_{R}(k, k). Thus Tor^{R} (k, M) is a left module over E-module - see (2.7) - and one may construct the complex of graded A-modules β(Tor^{R} (k, k)), as in (4.4). The crux of the theorem below is that this complex is naturally isomorphic to lin^{R}(F). First, a description of the candidate for the isomorphism: Since F is minimal Tor_{n}^{R} (k, M) = k ⊗_{R} F_{n} = F_{n}/𝔪F_{n}. This latter vector space is exactly the degree n component of lin_{n}^{R}(F), so multiplication by (−1)^{n} yields an inclusion of k-vector spaces: Tor_{n}^{R} (k, M) ↪ lin_{n}^{R}(F). This extends to a degree 0 homomorphism of graded A-modules

κ_{n}: A(−n) ⊗_{k} Tor_{n}^{R} (k, M) ⟶ lin_{n}^{R}(F) .

Combining these yields a homomorphism of graded A-modules: β(Tor^{R} (k, M)) → lin^{R}(F).

Theorem. The homomorphism κ: β(Tor^{R} (k, M)) → lin^{R}(F) is an isomorphism of complexes.

Remark. The definition of κ utilizes signed multiplication on k ⊗_{R} F rather than the identity because the multiplication on Ext is the composition, and not the Yoneda, product.

Proof. Since E acts on Tor^{R} (k, M) via inclusion into Ext_{R}(k, k), the results of (2.10) apply; in the proof we adopt also its notation. Also, we identify (k ⊗_{R} F) with Tor^{R} (k, M).

Fix an integer n. The A-module lin_{n}^{R}(F) is free and generated by the subspace of elements of degree n. Since this latter subspace is isomorphic to (k ⊗_{R} F)_{n} via κ_{n}, one obtains that κ_{n} is an isomorphism. It remains to prove that κ is compatible with the differentials.

Let z be an element in F_{n} \ 𝔪F_{n}. Write ∂(z) = ∑_{j=1}^{d} r_{j}f_{j} ∈ F_{n−1}. Then ∂(\overline{z}) = ∑_{j=1}^{d} x_{j}\overline{f}_{j} in lin^{R}(F); this explains the first row in the diagram below. As to the second: the equality is due to Lemma 2.10, while the rest are by definition.

(1 ⊗_{k} \overline{z}) ⟶^{κ} (−1)^{n} \overline{z} ⟶^{∂} (−1)^{n} ∑_{j=1}^{d} x_{j}\overline{f}_{j} ;
(1 ⊗_{k} \overline{z}) ⟶^{δ} ∑_{j=1}^{d}(x_{j} ⊗_{k} y_{j}\overline{z})  − ∑_{j=1}^{d}(x_{j} ⊗_{k} \overline{f}_{j}) ⟶^{κ} (−1)^{n} ∑_{j=1}^{d} x_{j}\overline{f}_{j}

Thus ∂κ = κδ on elements of the form 1 ⊗_{k} \overline{z} with z ∈ F_{n} \ 𝔪F_{n}. The A-module β_{n}(k ⊗_{R} F) is spanned by such elements and all the maps in question are A-linear, hence one deduces that κ commutes with the differentials. This completes the proof of the theorem. □

By definition, when (R, 𝔪, k) is Koszul, the linear part of the minimal resolution of k is acyclic, with homology equal to k, and also minimal. Then the resolution of k over gr_{𝔪}(R), provided by the theorem above, coincides with one given by results of Priddy and Löwall; [17], [15]. Corollary 4.6. If R is Koszul, then β(Tor^{R} (k, k)) is the minimal A-free resolution of k. □

The next item is a formula for the homology of the complex β(U) constructed in (4.4). This calls for an alternate description of that complex. 4.7. Construction II. Let U = {U_{i}}_{i∈ℤ} be a graded degreewise finite left E-module. As per our convention, A^{∨} denotes the graded k-dual of A, so A_{−n}^{∨} = Hom_{k}(A_{n}, k), for each n. Let

X_{n} = E(n) ⊗_{k} (A^{∨})_{−n}

be the graded free left E-module with multiplication induced from the left action of E on itself. Let ∂_{n}: X_{n} → X_{n−1} be the k-linear map with

∂_{n}(e ⊗_{k} a) = ∑_{i=1}^{d}(ey_{i} ⊗_{k} x_{i}a)  for all  (e ⊗_{k} a) ∈ X_{n} .

It is immediate that ∂_{n} is E-linear and that it has degree 0. Arguments akin to those used in (4.4) establish that ∂_{n−1} ∘ ∂_{n} for each integer n. Thus, X is a complex of graded free E-modules. In particular, one can construct Hom_{E}(X, U); a priori, this is a complex of k-vector spaces. It turns out that this complex coincides with β(U), up to shift in degrees.

To see this, note that the bigraded k-vector spaces underlying β(U) and X are A ⊗_{k} U and E ⊗_{k} A^{∨} respectively, up to a reordering of degrees. More precisely

β_{n}(U)_{i} = A_{i−n} ⊗_{k} U_{n}  and  X_{n,i} = E_{i+n} ⊗_{k} (A^{∨})_{−n} .

Therefore, the canonical evaluation map: A^{∨} ⊗_{k} A → k gives rise to the k-linear map

Φ: β(U) → Hom_{E}(X, U) with Φ(s ⊗_{k} u)(e ⊗_{k} f) = ef(s)u

for all (s ⊗_{k} u) ∈ β(U) and (e ⊗_{k} f) ∈ X. This is a bijection. Indeed, classical Hom-Tensor adjointness yields

Hom_{E}(X_{n}, U) = Hom_{E}(E(n) ⊗_{k} (A^{∨})_{−n}, U)
≅ Hom_{k}((A^{∨})_{−n}, Hom_{E}(E(n), U))
≅ Hom_{k}((A^{∨})_{−n}, U(−n)) .

Under this identification Φ coincides, at the level of bigraded vector spaces, with the canonical map A_{i−n} ⊗_{k} U_{n} → Hom_{k}((A^{∨})_{n−i}, U_{n}), so Φ is bijective. Moreover, while it is not a morphism of complexes, it is compatible with differentials; this can be verified easily by a direct computation. This discussion justifies the following Lemma. As a k-vector space, H_{n}(β(U))_{i} ≅ H_{n−i}(Hom_{E}(X, U))_{i} for all integers n, i ∈ ℤ. □

The lemma above and Theorem 4.5 lead to the following counterpart of Corollary 4.6. It describes the resolution of k over the Ext_{R}(k, k) when R is Koszul, since the Ext-algebra is E, except for the grading convention.

Corollary 4.8. If R is Koszul, then X is the minimal free resolution of k over E.

Proof. Let G be a minimal free resolution of k over R. Since R is Koszul, lin^{R}(G) is acyclic and H_{0}(lin^{R}(G)) ≅ k; see (1.3). Hence, by Theorem 4.5, the same holds for β(Tor^{R} (k, k)). Thus, Lemma 4.7 yields

H(Hom_{E}(X, Tor^{R} (k, k))) ≅ k .

Now, Tor^{R} (k, k) = Hom_{k}(E, k) as left E-modules, by the special case M = k of (3.2), so

Hom_{E}(X, Tor^{R} (k, k)) ≅ Hom_{E}(X, Hom_{k}(E, k)) ≅ Hom_{k}(X, k) .

Thus, H(Hom_{k}(X, k)) ≅ k. Since k is a field, this is tantamount to: H(X) ≅ k, as desired. □

As a direct application of the calculation above and Lemma 4.7 one obtains a formula for the homology of the complex β(U). It should be noted that it is contained in the work of Buchweitz [5], who establishes a similar result for non-necessarily-commutative Koszul k-algebras.

Proposition 4.9. Let U be a graded degreewise finite left E-module. If R is Koszul, then

H_{n}(β(U))_{i} = Ext_{E}^{i−n}(k, U)_{i}  for all  n, i ∈ ℤ . □

The preceding results all culminate in the

Proof of Theorem 4.1. The first isomorphism follows from Theorem 4.5 and the special case U = Tor^{R} (k, M) of Proposition 4.9; only, one has to translate the lower grading in the latter result to upper grading.

By the procedure described in (3.1), the left R^{!}-module Ext_{R}(M, k) gives rise to the right module Ext_{R}(M, k)^{σ}, with the same underlying graded k-vector space. Since Ext_{R}(M, k) is degreewise finite, the following natural homomorphism (of graded k-vector spaces) is bijective:

Tor^{R} (k, M) ⟶^{≅} Hom_{k}(Ext_{R}(M, k)^{σ}, k) .

According to Proposition 3.2, this isomorphism is compatible with the corresponding left R^{!}-module structures. This explains the second of the following isomorphisms of graded k-vector spaces; the first is the usual one.

Hom_{k}(Tor_{n}^{R^{!}} (Ext_{R}(M, k)^{σ}, k) , k) ≅ Ext_{R^{!}}^{n}(k, Hom_{k}(Ext_{R}(M, k)^{σ}, k))
≅ Ext_{R^{!}}^{n}(k, Tor^{R} (k, M)) .

The k-vector space Tor_{n}^{R^{!}} (Ext_{R}(M, k)^{σ}, k) is degreewise finite. This is because (R^{!})^{0} = k, and both R^{!} and Ext_{R}(M, k) are degreewise finite and concentrated in non-negative degrees; see 5.2. Thus, the composed isomorphism above dualizes to the first isomorphism below, while Lemma 3.1 provides the next one.

Hom_{k}(Ext_{R^{!}}^{n}(k, Tor^{R} (k, M)), k) ≅ Tor_{n}^{R^{!}} (Ext_{R}(M, k)^{σ}, k)
≅ Tor_{n}^{R^{!}} (k, Ext_{R}(M, k)) .

Combining this with the first isomorphism in the theorem completes the proof. □

5. Linear dominance as regularity

In this section we establish that finitely generated modules over local rings that are either complete intersection or Golod have finite linear dominance. We deduce these results by interpreting linear dominance of a module as the “regularity” of an appropriate module over the Ext-algebra of the local ring.

To this end, it is expedient to introduce a notion of regularity that is applicable to rings that may not be commutative or even noetherian. Our aim is to develop this notion only to a point which allows us to state (and prove!) the result we seek. Also, since the theory is intended to be applied to Ext-algebras of local rings, the algebras and modules are assumed to be upper graded. 5.1. Regularity. Let E = {E^{i}}_{i ⩾ 0} be a graded degreewise finite k-algebra with E^{0} = k, and let U = {U^{i}}_{i∈ℤ} be a graded degreewise finite left E-module.

We define the regularity of U to be the integer

reg_{E} U = sup{r | Ext_{E}^{n}(U, k)^{−n−r} ≠ 0 for some integer n} .

When E is commutative (and noetherian), this notion of regularity coincides with the classical one; see (⋆) below. As usual, the isomorphism of graded k-vector spaces: Hom_{k}(Tor_{n}^{E} (k, U) , k) ≅ Ext_{E}^{n}(U, k) implies that regularity may be read off from appropriate Tor functors:

(⋆)  reg_{E} U = sup{r | Tor_{n}^{E} (k, U)^{n+r} ≠ 0 for some integer n} .

For our purposes, the Ext-definition of regularity is more convenient. One reason for this is the presence of the following test to detect modules of finite regularity. Lemma. If reg_{E} k is finite and the Ext_{E}(k, k)-module Ext_{E}(U, k) is finitely generated, then reg_{E} U is finite as well.

Proof. The hypothesis yields a surjective degree 0 homomorphism

⨁_{i∈ℤ} Ext_{E}(k, k)(−i)^{b_{i}} ↠ Ext_{E}(U, k) ,

where all but finitely many of the b_{i} are zero. Then reg_{E} U ≤ reg_{E} k + sup{i ∈ ℤ | b_{i} ≠ 0}. □

One nice thing about E is that many modules over it have minimal resolutions. Here is a brief resume of the basic facts concerning them. 5.2. Minimal resolutions. Suppose the E-module U, in addition to being degreewise finite, is bounded below; in other words, U^{i} = 0 for i ≪ 0. Then U admits a graded free resolution X such that for each integer n, one has

X_{n} ≅ ⨁_{i∈ℤ} E(−i)^{b_{n,i}} with b_{n,i} = 0 for i ≪ 0 ,

and k ⊗_{E} X has trivial differentials. Such a minimal free resolution exists and is unique (up to isomorphism) because Nakayama’s lemma applies to all bounded below E-modules. At any rate, it is guaranteed by results of Eilenberg [7]. Taking a cue from the local case, we call the left E-module Coker(∂_{n+1}) the nth syzygy of U.

The existence of such minimal resolution implies: Lemma 5.3. Let U be a bounded below, degreewise finite, left E-module. If reg_{E} U is finite, then U and all its higher syzygies are finitely generated.

Proof. Let X be the minimal free resolution of U. Fix an integer n. We adopt the notation from (5.2). Since Tor_{n}^{E} (k, U) = k ⊗_{E} X_{n} and reg_{E} U is finite, one obtains from the equality (⋆) in (5.1) that b_{n,i} = 0 for all but finitely many i. Hence X_{n} is finitely generated. It remains to note that X_{n} maps surjectively onto the nth syzygy of U. □

Here is the advertised relation between linear dominance and regularity; that such a theorem might exist was suggested by [8] which contains an analogues statement in the case where R is an exterior algebra. Its proof is short only because most of the groundwork has already been laid. Observe that one cannot to do away with the hypothesis that R is Koszul as reg_{R^{!}} Ext_{R}(k, k) = 0. Theorem 5.4. Let (R, 𝔪, k) be a Koszul local ring and let M be a finitely generated R-module. Set R^{!} = Ext_{R}(k, k). Then

lpd_{R} (M) = reg_{R^{!}} Ext_{R}(M, k) .

In particular, lpd_{R} (M) < ∞ if the Ext_{R^{!}}(k, k)-module Ext_{R^{!}}(Ext_{R}(M, k), k) is finitely generated.

Proof. The formula for lpd_{R} (M) is justified by the following equalities, in which Theorem 4.1 implies the first and the second is due to (⋆) in (5.1) above.

lpd_{R} (M) = sup{n | Tor_{i−n}^{E} (k, Ext_{R}(M, k))^{i} ≠ 0 for some integer i}
= reg_{R^{!}} Ext_{R}(M, k) .

Specializing this equality to M = R one obtains: reg_{R^{!}} k = lpd_{R} (R) = 0. Thus, the finiteness of lpd_{R} (M) is a consequence of Lemma 5.1. □

Combining the preceding result with Lemma 5.3 yields this next result. Over Koszul rings, it constitutes a vast generalization of Theorem 2.1. Corollary 5.5. If lpd_{R} (M) is finite, then the left R^{!}-module Ext_{R}(M, k) and all its higher syzygies are finitely generated. □

We apply the preceding results to modules over complete intersections and Golod rings. The latter class may be less familiar than the former to many readers, so the next item recapitulates the relevant definitions and also some elementary results concerning those; see [9], [13], or [1, §5]. 5.6. Let (R, 𝔪, k) be a local ring and M a finitely generated R-module.

Let K^{R} be the Koszul complex for R and set K^{M} = K^{R} ⊗_{R} M. There is a coefficientwise inequality of formal power series

P_{M}^{R}(t) ≼ \frac{∑_{i} rank_{k} H_{i}(K^{M})t^{i}}{1 − ∑_{j} rank_{k} H_{j}(K^{R})t^{j+1}} .

If equality holds, then the module M is Golod. One says that R is a Golod ring if k is Golod.

Here is a relative version of the Golod property: Let φ: (S, 𝔫, k) ↠ (R, 𝔪, k) be a surjective local homomorphism. There is a coefficientwise inequality of formal power series:

P_{M}^{R}(t) ≼ \frac{P_{M}^{S}(t)}{1 − t (P_{R}^{S}(t) − 1)} .

If equality holds, then M is φ-Golod. If k is φ-Golod, then φ is a Golod homomorphism. Remark. Let S be regular, with emb dim S = emb dim R, and let K^{S} be the Koszul complex for S. Then H_{0}(K^{S}) = k and the regularity of S ensures that H_{i}(K^{S}) = 0 for i ≥ 1. Since emb dim S = emb dim R, one has that K^{R} = K^{S} ⊗_{S} R and K^{M} = K^{S} ⊗_{S} M; thus ∑_{i} rank_{k} H_{i}(K^{M})t^{i} = P_{M}^{S}(t) and ∑_{j} rank_{k} H_{j}(K^{M})t^{j} = P_{R}^{S}(t). Hence the R-module M is Golod if and only if it is φ-Golod.

Now for the principal result of this section. Theorem 5.7. Let R be a Koszul local ring. If there is a Golod homomorphism φ: S → R such that S is a complete intersection, then lpd_{R} (M) < ∞ for every finitely generated R-module M.

Proof. Set R^{!} = Ext_{R}(k, k). A result of Backelin and Roos [6, Theorem. 2.a] provides that the Ext_{R^{!}}(k, k)-module Ext_{R^{!}}(Ext_{R}(M, k), k) is finitely generated. It remains to invoke (5.4). □

Here is a digestible special case of the preceding result. Corollary 5.8. Let R be a Koszul local ring. If R is either a complete intersection or Golod, then lpd_{R} (M) < ∞ for every finitely generated R-module M.

Proof. If R is a complete intersection, then Theorem 5.7 yields the desired result, for the identity homomorphism R → R is Golod.

Suppose R is Golod. Let R̂ denote the completion of R at its maximal ideal. Then R̂ is Golod and lpd_{R} (M) = lpdR̂(M̂). Thus, by passing to R̂ one may assume that R is complete. Then Cohen’s structure theorem provides a surjective local homomorphism φ: S → R with S a regular local ring; one can ensure also that emb dim S = emb dim R. By Remark (5.6), the homomorphism φ is Golod, so the result we seek is once again contained in Theorem 5.7. □

Remark 5.9. Let R be a multigraded ring, and M a finitely generated multigraded R-module. Set R^{!} = Ext_{R}(k, k). In [6, §4], the authors surmise that the Ext_{R^{!}}(k, k)-module Ext_{R^{!}}(Ext_{R}(M, k), k) is finitely generated. If this turns out to be the case, then when R is Koszul, Theorem 5.4 would imply that lpd_{R} (M) < ∞.

6. Modules over Golod rings

The main contribution of this section is a ‘change of base’ result for computing linear dominance: Theorem 6.1. Let φ: S ↠ R be a surjective local homomorphism such that R and S are Koszul. Let M be a finitely generated R-module. If M is φ-Golod, then

lpd_{R} (M) = lpd_{S} (M) .

A couple of remarks are in order. First, in the graded case it is not necessary to assume a priori that S is Koszul: when R is Koszul and φ is Golod, the bigraded version of the equality of Poincaré series in (5.6) yields that reg_{S} k = 0, so that S is Koszul. Second, it is not crucial that φ be surjective; the theorem and its proof extend to general Golod homomorphisms.

The theorem above allows for a quantitative strengthening of the part of Corollary (5.8) dealing with Golod rings. It is best described in terms of the following invariant, modeled on the global dimension of rings: For each local ring R, we define

gl lpd (R) = sup{lpd_{R} (M) : M is a finitely generated R-module} .

Here is the advertized result. Corollary 6.2. If a local ring R is Koszul and Golod, then gl lpd (R) ≤ 2 emb dim R.

Proof. By completing R at its maximal ideal, one may assume that R is complete. Then Cohen’s structure theorem provides a surjective homomorphism φ: S → R where S is a regular local ring with emb dim S = emb dim R. Since S is regular φ is Golod; see the remark in (5.6). Let M be a finitely generated R-module, and set d = proj dim_{S} M. Then Ω^{d} (M), the dth syzygy of M over R, is φ-Golod; this is by [1, (5.3.2)]. Thus

lpd_{R} (M) ≤ lpd_{R} (Ω^{d} (M)) + d = lpd_{S} (Ω^{d} (M)) + d ≤ d + d = 2d ,

where the equality is by Theorem 6.1. It remains to note that d ≤ emb dim S. □

Remark. Suppose that R is a standard graded k-algebra. Assume that R, in addition to being Koszul and Golod, is Cohen-Macaulay. Fairly elementary arguments, involving reduction to the case where dim R = 0, allow one to deduce that gl lpd (R) ≤ dim R. This leads us to believe that the upper bound in the corollary above can be improved considerably.

The proof of Theorem 6.1, given in (6.6), uses some ideas from [18], [6], and [1] and the following elementary remark concerning modules over tensor algebras. 6.3. Let Λ = {Λ^{i}}_{i ⩾ 1} be a graded k-vector space. We denote T (Λ) the tensor algebra on Λ. Lemma. Let U be a graded, bounded below, degreewise finite, left T (Λ)-module. Let μ: Λ⊗_{k}U → U be the map defined by μ(λ⊗_{k}u) = λu for λ ∈ Λ and u ∈ U. The following conditions are equivalent.

(1) μ is injective.

(2) U is free.

(3) for some (respectively, any) homogeneous k-linear section of the surjection U ↠ Coker(μ), the induced homomorphism T (Λ) ⊗_{k} Coker(μ) → U of left T (Λ)-modules is bijective.

Proof. The tensor algebra being what it is - a free algebra - one has the short exact sequence of right T (Λ)-modules: 0 → Λ ⊗_{k} T (Λ) ⟶^{ι} T (Λ) → k → 0, where ι(λ ⊗_{k} t) = λt. Applying − ⊗_{T(Λ)} U to it leads to the exact sequence

0 → Tor_{1}^{T(Λ)} (k, U) → (Λ ⊗_{k} U) ⟶^{μ} U → k ⊗_{T(Λ)} U → 0 .

In particular, μ is injective if and only if Tor_{1}^{T(Λ)} (k, U) = 0. Since U admits a minimal free resolution - see (5.2) - this last condition is equivalent to the freeness of U; this settles (1) ⟺ (2). The remaining non-trivial implication (2) ⇒ (3) follows by a standard argument based on Nakayama’s lemma. □

6.4. Let φ: (S, 𝔫, k) ↠ (R, 𝔪, k) be a Golod homomorphism. By [13, Theorem 1.5], there is an exact sequence of complexes of R-modules:

(†)  0 → X ⊗_{S} R → Y → Y ⊗_{R} F → 0 ,

where X is the minimal resolution of k as an S-module, Y is the minimal resolution of k as an R-module, and F is a complex of free R-modules with trivial differentials. Since the complexes are minimal, tensoring with k leads to the exact sequence of graded k-vector spaces

0 → Tor^{S} (k, k) → Tor^{R} (k, k) → Tor^{R} (k, k) ⊗_{k} V → 0 ,

where V = k ⊗_{R} F. Dualizing this yields the exact sequence

(‡)  0 → Λ ⊗_{k} Ext_{R}(k, k) → Ext_{R}(k, k) → Ext_{S}(k, k) → 0 ,

where we write Λ for Hom_{k}(V, k). In the sequel, we view Λ as a subset of Ext_{R}(k, k) by identifying it with (the image of) Λ ⊗_{k} 1. Via this inclusion, Λ acts on Ext_{R}(M, k) for each R-module M. Lemma. Suppose that the local ring R is Koszul. Let M be a finitely generated R-module. Then there is an exact sequence of graded k-vector spaces

Λ ⊗_{k} Ext_{R}(M, k) ⟶^{μ} Ext_{R}(M, k) ⟶^{π} Ext_{S}(M, k) .

where μ(λ ⊗_{k} w) = λw and the map π is the canonical one. Moreover, M is φ-Golod if and only if μ is injective if and only if π is surjective.

Proof. Applying − ⊗_{R} M to the exact sequence (†) above leads to the exact sequence

0 → X ⊗_{S} M → Y ⊗_{R} M → (Y ⊗_{R} M) ⊗_{R} F → 0 .

Taking homology gives the exact sequence: Tor^{S} (k, M) → Tor^{R} (k, M) → Tor^{R} (k, M) ⊗_{k} V. Dualizing this leads to the exact sequence of graded k-vector spaces:

Λ ⊗_{k} Ext_{R}(M, k) ⟶^{μ′} Ext_{R}(M, k) ⟶^{π} Ext_{S}(M, k) .

There is a subtlety here: Since Ext_{R}(M, k) appears as the dual of Tor^{R} (k, M), one has μ′(λ⊗_{k}w) = λ ∗ w, where the product ∗ is the conjugate - in the sense of (3.1) - of the right action of Ext_{R}(k, k) on Ext_{R}(M, k) that is induced from Tor^{R} (k, M). However, since R is Koszul and M is finitely generated, this coincides with the usual left action of Ext_{R}(k, k) on Ext_{R}(M, k); see Proposition 3.2. In other words, μ′ = μ, and the exact sequence above is the one we seek.

As to the remaining assertion: μ and π arise from exact sequences of complexes, so μ is injective if and only if π is surjective. The exactness of the sequence above yields the coefficientwise inequality of formal power series:

P_{M}^{R}(t)(1 − Hilb_{Λ}(t)) ≼ P_{M}^{S}(t) .

Moreover, equality holds if and only if μ is injective and π is surjective. It remains to note that Hilb_{Λ}(t) = t(P_{R}^{S}(t) − 1), by (†), since the homomorphism φ is Golod. □

6.5. Let φ: (S, 𝔫, k) ↠ (R, 𝔪, k) be a Golod homomorphism. The inclusion Λ ↪ Ext_{R}(k, k), from (6.4), gives rise to a homomorphism of k-algebras T (Λ) → Ext_{R}(k, k). This map is injective: this is because Ext_{R}(k, k) is free as a left T (Λ)-module, by the exact sequence (‡) in (6.4) and Lemma 6.3. Consequently, one can view T (Λ) as a k-subalgebra of Ext_{R}(k, k).

At any rate, Ext_{R}(M, k) acquires a structure of a left T (Λ)-module, for any R-module M. The result below characterizes φ-Golod modules in terms of this structure; it appears to be new, although the basic ingredients in the proof are not; cf. [18], [6], [13]. Only the implication (1) ⇒ (5) is invoked in the sequel; however, the argument is (to us) more transparent with all five conditions on hand. Theorem. Let φ: (S, 𝔫, k) ↠ (R, 𝔪, k) be a Golod homomorphism, and let M be a finitely generated R-module. Suppose that R is Koszul. Then the following conditions are equivalent.

(1) M is φ-Golod.

(2) The left T (Λ)-module Ext_{R}(M, k) is free.

(3) There is an isomorphism of left T (Λ)-modules:

Ext_{R}(M, k) ≅ T (Λ) ⊗_{k} Ext_{S}(M, k) ,

and the map Ext_{R}(M, k) → Ext_{S}(M, k) induced by T (Λ) → k is the canonical one.

(4) The following canonical map of bigraded k-algebras is bijective:

Ext_{Ext_{S}(k,k)}(Ext_{S}(M, k), k) → Ext_{Ext_{R}(k,k)}(Ext_{R}(M, k), k) .

(5) There is an isomorphism of bigraded vector spaces

Ext_{Ext_{S}(k,k)}(Ext_{S}(M, k), k) ≅ Ext_{Ext_{R}(k,k)}(Ext_{R}(M, k), k) .

Remark. We ask that R be Koszul only because of Lemma 6.4. In its turn, the latter result required that hypothesis because its proof invokes Proposition 3.2. Thus, in the light of the remarks following loc. cit., it is to be expected that the equivalences above hold for all local rings.

Proof. Set E_{R} = Ext_{R}(k, k) and E_{S} = Ext_{S}(k, k).

(1) ⟺ (2) ⟺ (3): This is immediate from (6.4) and (6.3).

(3) ⇒ (4): By definition, the Golod property for φ translates to the R-module k being φ-Golod. Therefore, the special case M = k of the already established implication (1) ⇒ (3) yields that

(a) E_{R} is free when viewed as a left T (Λ)-module, and

(b) the canonical map k ⊗_{T(Λ)} E_{R} → E_{S} is bijective. Observe that the map in (b) is one of graded k-algebras and also of right E_{R}-modules. In the isomorphisms below (b) gives the one on the left and (a) the other.

Tor^{E_{R}} (E_{S}, Ext_{R}(M, k)) ≅ Tor^{E_{R}} (k ⊗_{T(Λ)} E_{R}, Ext_{R}(M, k)) ≅ Tor^{T(Λ)} (k, Ext_{R}(M, k)) .

These isomorphisms are compatible with the left E_{S}-module structures. Thus, the hypotheses implies that

(⋆⋆)  Tor_{q}^{E_{R}} (E_{S}, Ext_{R}(M, k)) = \begin{cases}Ext_{S}(M, k) & for q = 0 \\ 0 & otherwise. \end{cases}

A standard change of rings spectral sequence arising from the ring homomorphism E_{R} → E_{S} reads

E_{2}^{p,q} = Ext_{E_{S}}^{p}(Tor_{q}^{E_{R}} (E_{S}, Ext_{R}(M, k)) , k) ⟹ Ext_{E_{R}}^{p+q}(Ext_{R}(M, k), k) .

Due to the vanishing of Tor in (⋆⋆), this spectral sequence collapses, so the edge homomorphism

Ext_{E_{S}}^{p}(Tor_{0}^{E_{R}} (E_{S}, Ext_{R}(M, k)) , k) ⟶^{≅} Ext_{E_{R}}^{p}(Ext_{R}(M, k), k)

is bijective. Another invocation of (⋆⋆) settles the claim.

(4) ⇒ (5): This is a triviality.

(5) ⇒ (1): Since Ext_{S}(M, k) is bounded below and degreewise finite, the rank of the k-vector space Ext_{E_{S}}^{n}(Ext_{S}(M, k), k)^{i} is finite for each pair of integers n, i; see (5.2). The same remark applies to the k-vector space Ext_{E_{R}}^{n}(Ext_{R}(M, k), k)^{i}. Thus, the given isomorphism of bigraded vector spaces results in the coefficientwise equality of bigraded Poincaré series:

P_{Ext_{S}(M,k)}^{E_{S}}(t, s) = P_{Ext_{R}(M,k)}^{E_{R}}(t, s) .

On the other hand, one has always that

Hilb_{E_{S}}(s) P_{Ext_{S}(M,k)}^{E_{S}}(−t, s) = Hilb_{Ext_{S}(M,k)}(s)
Hilb_{E_{R}}(s) P_{Ext_{R}(M,k)}^{E_{R}}(−t, s) = Hilb_{Ext_{R}(M,k)}(s) .

From the preceding equalities one deduces that P_{k}^{R}(s) P_{M}^{S}(s) = P_{k}^{S}(s) P_{M}^{R}(s) . Therefore

P_{M}^{R}(s) = \frac{P_{k}^{R}(s)P_{M}^{S}(s)}{P_{k}^{S}(s)} = \frac{P_{M}^{S}(s)}{1 − t(P_{R}^{S}(s) − 1)}

where the last equality is due to φ being a Golod homomorphism. Thus, M is φ-Golod. □

Theorem 6.5 prepares us - admirably - for the: 6.6. Proof of Theorem 6.1. One may assume that M ≠ 0. Then, φ is Golod, by [14, (1.1)], so the implication (1) ⇒ (5) of Theorem 6.5 yields: reg_{S^{!}} Ext_{S}(M, k) = reg_{R^{!}} Ext_{R}(M, k). It remains to invoke Theorem 5.4. □

The analogue of Corollary 6.2 for standard graded k-algebras has a partial converse: Theorem 6.7. Let R be a standard graded k-algebra. If R is Gorenstein, then the following conditions are equivalent:

(1) gl lpd (R) < ∞;

(2) R is a hypersurface defined by a quadratic form.

Proof. To begin with, note that a hypersurface is Koszul if and only if it is defined by a quadratic form. Since a hypersurface is always Golod, Corollary 6.2 yields the implication (2) ⇒ (1).

Conversely, gl lpd (R) < ∞ implies that lpd_{R} (k) < ∞, so that R is Koszul, by Proposition 1.10. Suppose that R is not a hypersurface. Let d = dim R. One may assume that k is infinite, and choose an R-sequence x_{1}, …, x_{d} with x_{i} ∈ R_{1} for i = 1, …, d. Set R̄ = R/(x_{1}, … x_{d}). Then Ext_{R}^{d}(k, R)(−d) ≅ Hom_{R}(k, R̄) ≅ k(−s) with s ≥ 2, since R is not a hypersurface ring. Let F be a minimal graded free R-resolution of k, and set F^{∗} = Hom_{R}(F, R). Then

H_{i}(F^{∗}) ≅ \begin{cases}0 & if i ≠ −d, \\ k(d − s) & if i = −d. \end{cases}

We denote G the complex

F (d − s): ⋯ → F_{i+1}(d − s) → F_{i}(d − s) → F_{i−1}(d − s) → ⋯

shifted d steps to the right. Then

H_{i}(G) ≅ \begin{cases}0 & if i ≠ −d, \\ k(d − s) & if i = −d. \end{cases}

The isomorphism H_{−d}(G) ≅ k(d − s) ≅ H_{−d}(F) lifts to a morphism α: G → F^{∗} of graded complexes. Now, the differentials in F can be expressed by matrices of linear forms, since R is Koszul, so an elementary induction argument allows one to deduce that α(G) ⊆ 𝔪F^{∗}. In other words, the mapping cone C of α is a minimal complex.

Choose any integer n ≥ d. Let D^{n} be the truncated complex

⋯ → C_{i} → … → C_{−n+1} → C_{−n} → 0

shifted n steps to the left. The complex C is exact, by construction, so D^{n} is a graded minimal free R-resolution of M_{n} = Coker(C_{−n+1} → C_{−n}). Since s ≥ 2, the complex D^{n} has two linear strands, namely the complexes F_{i≥−n}^{∗} and G, both shifted n steps to the left. Thus the linear strands of M_{n} have non-trivial homology in homological degree n − d, and are exact for i > n − d.

This implies that lpd_{R} (M_{n}) = n − d. Since n can be chosen to be arbitrarily large, one obtains that gl lpd (R) = ∞. This contradicts the hypothesis. Therefore, R is a hypersurface. □

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Jürgen Herzog, Fachbereich Mathematik und Informatik, Universität-GHS Essen, 45117 Essen, Germany

E-mail address: juergen.herzog@uni-essen.de

Srikanth Iyengar, Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

E-mail address: iyengar@math.missouri.edu