HOMOLOGICAL PROPERTIES OF POWERS OF THE MAXIMAL IDEAL OF A LOCAL RING

LIANA M. ŞEGA

Abstract. It is known that the powers 𝔪^{n} of the maximal ideal of a local Noetherian ring share certain homological properties for all sufficiently large integers n. For example, the natural homomorphisms R → R/𝔪^{n} are Golod, respectively, small, for all large n. We give effective bounds on the smallest integers n for which such properties begin to hold.

Introduction

Let R be a local commutative Noetherian ring with maximal ideal 𝔪 and residue field k = R/𝔪. We study homological properties of the powers 𝔪^{n} which hold for all large values of n.

One such property is established by Levin [19]; he proves that the natural homomorphism R → R/𝔪^{n} is Golod for all large n. Lee [17] defines the Golod invariant of R to be the smallest number s such that R → R/𝔪^{n} is Golod for all n ≥ s. The results of this paper are better stated in terms of a Golod index G(R), defined to be one less than the invariant introduced by Lee.

In order to study the Golod property, we consider two related homological properties. One is based on the notion of small homomorphism introduced by Avramov [3], the other arises from Levin’s proof of his theorem; we refer to the corresponding sections for precise definitions. We define indices A(R) and L(R), in analogy to the Golod index. Results of Avramov and Levin show that these are natural numbers that provide bounds for G(R) as follows:

A(R) ≤ G(R) ≤ L(R) .

We obtain bounds for A(R) and L(R) in terms of numerical invariants of the associated graded ring gr_{𝔪}(R) with respect to the 𝔪-adic filtration. Recall that the 𝔪-adic completion R̂ has a minimal Cohen presentation R̂ ≅ Q/𝔞, with (Q, 𝔫) a regular local ring and 𝔞 ⊆ 𝔫^{2}. We summarize below our results: Theorem 1. Let (R, 𝔪) be a local Noetherian ring with 𝔪 ≠ 0. Let R̂ ≅ Q/𝔞 be a minimal Cohen presentation and let pol reg(R) denote the Castelnuovo-Mumford regularity of gr_{𝔪}(R) over gr_{𝔫}(Q). The following then hold:

(1) inf{i ≥ 1 | 𝔞 ∩ 𝔫^{i+2} ⊆ 𝔫𝔞} ≤ A(R) ≤ L(R) ≤ max{1, pol reg(R)}.

(2) If R is a complete intersection, then A(R) = inf{i ≥ 1 | 𝔞 ∩ 𝔫^{i+2} ⊆ 𝔫𝔞}.

(3) If R is a hypersurface or a Golod Artinian ring, then

A(R) = G(R) = L(R) = max{1, pol reg(R)} .

(4) If edim R − dim R ≤ 1, or if edim R ≤ 2, or if edim R = 3 and R is a complete intersection, then A(R) = G(R).

(5) The graded k-algebra gr_{𝔪}(R) is Koszul if and only if L(R) = 1.

We study the behavior of the indices under factorization of regular sequences. Part (2) of the next theorem answers partially a question of Roos [29]. Theorem 2. For each local Noetherian ring (R, 𝔪) the following hold:

(1) If x is a regular element such that (x) ≠ 𝔪, then A(R) ≤ A(R/(x)).

(2) If x ∉ 𝔪^{2} and the initial form of x is gr_{𝔪}(R)-regular, then L(R) = L(R/(x)). The index L(R) is a particular case of an index defined in Section 3 for any R-

module M, and denoted L_{R}(M). If pol reg(M) denotes the Castelnuovo-Mumford regularity of gr_{𝔪}(M) over a certain polynomial ring, then we obtain an inequality L_{R}(M) ≤ pol reg(M) + 1, which gives the corresponding inequality of Theorem 1(1). We obtain an application to delta invariants δ_{R}^{i}(M), defined by Auslander [2] when R is Gorenstein, and by Martsinkovsky [24] in general. Theorem 3. If M is a finite module over a non-regular local ring (R, 𝔪, k), then δ_{R}^{i}(𝔪^{n}M) = 0 for all i ≥ 0 and all n > pol reg(M).

This generalizes a theorem of Yoshino [39], which shows that if R is Gorenstein, then δ_{R}^{i}(𝔪^{n}) = 0 for all large n.

In Section 1 we discuss notions of regularity; the definitions involve Castelnuovo-Mumford regularity over different graded rings of the associated graded module gr_{𝔪}(M). In Section 2 we construct canonical homomorphisms

τ_{i}^{M}: Tor_{i}^{R}(M, k) → Tor_{i}^{gr_{𝔪}(R)} ( gr_{𝔪}(M), k) ,

which can be computed by means of free resolutions of either the first or the second module argument. We show that M has a linear resolution, that is, reg_{R}(M) = 0, if and only if the maps τ_{i}^{M} are bijective for all i.

In Section 3 we introduce for each finite R-module M the index L_{R}(M) and we bound it by pol reg(M)+1. As consequences, we obtain effective versions of results of Levin and Avramov on Poincaré series, and of similar results of Roos for Bass series. Theorem 3 is proved in Section 4. In Section 5 we define the index A(R) and use results of Avramov [3] to establish the relevant parts of Theorems 1 and 2. In Section 6 we discuss the invariant G(R) and prove Theorem 1(4). We also prove that if edim R − dim R ≤ 1, then the ring R/𝔪^{n} is Golod for all integers n ≥ 2. Since the equality edim R − dim R = 0 characterizes regularity, this generalizes Golod’s classical example [12].

In Section 7 we introduce the index L(R) by the formula L(R) = L_{R}(𝔪) and derive the equalities of Theorem 1(3) from results proved earlier in the paper. We also prove Theorem 1(5), which extends a characterization of graded Koszul algebras noted by Roos [29]. In particular, this shows that if R is Koszul, that is, reg_{R}(k) = 0, then Ext_{R}^{∗}(k, k) is finitely generated as a graded algebra under Yoneda product. While these two conditions are equivalent for graded algebras, we give an example of a local ring for which they are not.

In Section 8 we prove Theorem 2(2). In Section 9 we consider graded rings. Adapting our definitions to this case, we prove that if R is a graded Golod ring which is not a field, then A(R) = G(R) = L(R) = max{1, pol reg(R)}. We end the paper with various examples.

1. Regularity of modules

In this paper all rings are commutative Noetherian and all modules are assumed finitely generated.

1.1. If A is a ring and a = a_{1}, …, a_{e} is a sequence of elements of A, then K(a; A) denotes the Koszul complex on a. For an A-module N we set K(a; N) = K(a; A)⊗_{A}N and H_{∗}(a; N) = H_{∗} (K(a; N)). If φ: A → B is a ring homomorphism, then clearly K(a; A) ⊗_{A} B = K(φ(a); B). If the A-module structure of N is induced through the homomorphism φ, then we systematically identify K(a; N) and K(φ(a); N).

1.2. Let k be a field, A = ⨁_{n=0}^{∞} A_{n} a commutative graded algebra with A_{0} = k, and N = ⨁_{n=0}^{∞} N_{n} a graded A-module. For each d ∈ ℤ we denote N(d) the graded A-module with N(d)_{p} = N_{d+p}. We denote A_{+} the maximal homogeneous ideal of A and write k for the residue field A/A_{⩾ 1} modulo the maximal homogeneous ideal of A. The module N has a minimal graded free resolution

G = ⋯ → G_{i} ⟶^{∂_{i}} G_{i−1} → ⋯ → G_{1} ⟶^{∂_{1}} G_{0} ,

where for each i the module G_{i} is isomorphic to a direct sum of copies of A(−j) and ∂_{i}(G_{i}) ⊆ (A_{⩾ 1})G_{i−1}. Any two minimal graded free resolutions are isomorphic as complexes of graded A-modules, so the number of direct summands of G_{i} isomorphic to A(−j) is an invariant of N, called the ij ’th graded Betti number β_{ij}^{A}(N). The Castelnuovo-Mumford regularity of N is the number

reg_{A}(N) = sup{s ∈ ℤ | β_{i,i+s}^{A}(N) ≠ 0 for some i ∈ ℕ} .

We note that β_{i,j}^{A}(N) = rank_{k} Tor_{i}^{A}(N, k)_{j}; these numbers can be calculated from any free resolution of k or N over A.

1.3. Assume further that A = A_{0}[A_{1}]. We can present A as k[u]/I, where k[u] is a polynomial ring over k with variables u = u_{1}, …, u_{r} in degree 1 and I a homogeneous ideal. We define the polynomial regularity of N by the formula

pol reg(N) = reg_{k[u]}(N) .

The next lemma shows that the right-hand side does not depend on the choice of the presentation.

1.4. Lemma. If A = k[u]/I and A = k[v]/J are two presentations as above, then reg_{k[u]}(N) = reg_{k[v]}(N).

Proof. The canonical maps k[u] ⟶^{α} A ⟵^{β} k[v] define a surjective homomorphism

k[u, v] = k[u] ⊗_{k} k[v] → A

through which both α and β factor, so it suffices to prove that reg_{k[u]}(N) = reg_{k[u,v]}(N). Using induction on the number of variables in v, it suffices to show that reg_{P} (N) = reg_{P [v]}(N), where P = k[u] and v is a single variable which acts on N through a surjective homomorphism γ: P [v] → A such that γ|_{P} = α. Replacing v with the linear form v − ∑_{j=1}^{r} a_{j}u_{j}, where a_{j} ∈ k are such that γ(v) = α(∑_{j=1}^{r} a_{j}u_{j}), we can assume vN = 0. By the graded graded version of a well-known result (see [10, 1.6.13]) we have an exact sequence

⋯ → H_{i}(u; N) → H_{i}(u, v; N) → H_{i−1}(u; N(−1)) ⟶^{±v} H_{i−1}(u; N) → ⋯

of Koszul homology. Since vN = 0, it splits into short exact sequences

0 → H_{i}(u; N) → H_{i}(u, v; N) → H_{i−1}(u; N(−1)) → 0 .

The complex K(u; P), respectively K(u, v; P [v]), is a minimal graded free resolution of k over P, respectively P [v], hence we have

β_{i,j}^{P}(N) = rank_{k} ( H_{i}(u; N))_{j} and β_{i,j}^{P [v]}(N) = rank_{k} ( H_{i}(u, v; N))_{j} .

Computing ranks from the exact sequences above, we obtain

β_{i,j}^{P}(N) + β_{i−1,j−1}^{P}(N) = β_{i,j}^{P [v]}(N)

for all i, j ∈ ℤ. These equalities show that the regularities coincide. □

1.5. If a graded A-module N has finite length, then pol reg(N) can be expressed as the largest integer s for which N_{s} ≠ 0. Indeed, let A = k[u]/I be a presentation as in 1.3, with variables u = u_{1}, …, u_{e}. Since K(u; N) is a minimal free resolution of k over k[u], we have Tor_{i}^{k[u]}(N, k)_{i+n} ≅ H_{i}(u; N)_{i+n} = 0 for all i ≥ 0 and all n > s, and Tor_{e}^{k[u]}(N, k)_{e+s} ≅ H_{e}(u, N)_{e+s} = {y ∈ N_{s}|uy = 0} = N_{s} ≠ 0.

We extend next the notion of regularity to local rings. The notation for a local ring is (R, 𝔪, k), where 𝔪 is the maximal ideal and k is the residue field. 1.6. For a local ring (R, 𝔪, k) and an R-module M we denote gr(M) the associated graded module ⨁_{n=0}^{∞} 𝔪^{n}M/𝔪^{n+1}M with respect to the 𝔪-adic filtration. Thus, gr(R) is a graded ring and gr(M) is a graded gr(R)-module. Note that gr(R) is a polynomial ring if and only if R is regular.

We define the regularity of M over R by the formula

reg_{R}(M) = reg_{gr(R)} ( gr(M)) .

and the polynomial regularity of M by the formula

pol reg(M) = pol reg ( gr(M)) .

1.7. As usual, R̂ denotes the 𝔪-adic completion of R. We say that Q/𝔞 is a Cohen presentation of R if R̂ ≅ Q/𝔞, where (Q, 𝔫, k) is a regular local ring and 𝔞 an ideal of Q. If 𝔞 is contained in 𝔫^{2}, then we say that the presentation is minimal. Cohen’s Structure Theorem guarantees that such presentations always exist.

The ring R̂ is R-flat, with maximal ideal 𝔪R̂, hence there are equalities gr(R̂) = gr(R) and gr(M̂) = gr(M). Thus, the regularity of M over R, as well as the polynomial regularity of M, do not change under 𝔪-adic completion.

Let R̂ = Q/𝔞 be a Cohen presentation. Since gr(R) is then a homomorphic image of gr(Q), which is a polynomial ring, we have:

pol reg(M) = pol reg(gr(M)) = reg_{gr(Q)}(gr(M)) = reg_{Q}(M) .

1.8. Example. The Loewy length of an R-module M, denoted ℓℓ_{R}(M), is the smallest positive integer n for which 𝔪^{n}M = 0. Recall by 1.5 that if gr(M) has finite length, then pol reg ( gr(M)) is equal to the largest integer n such that gr(M)_{n} ≠ 0; this is also the largest integer n such that 𝔪^{n}M ≠ 0. For an Artinian R-module M we have thus pol reg(M) = ℓℓ_{R}(M) − 1.

1.9. Example. Assume that R is a hypersurface, that is, R̂ has a Cohen presentation R̂ = Q/(𝔞), with 𝔞 = (a) for some 𝔞 ∈ 𝔫. In this case, pol reg(R) = mult(R)−1. Indeed, it is well-known that mult(R) is the smallest integer s such that a ∈ 𝔫^{s}. By 1.7 the invariant pol reg(R) is equal to the Castelnuovo-Mumford regularity of gr(R) over the polynomial ring gr(Q). If a^{∗} denotes the initial form of a in gr(R), then gr(R) = gr(Q)/(a^{∗}), hence

0 ⟶ gr(Q)(−s) ⟶^{a^{∗}} gr(Q) ⟶ 0

is a minimal graded free resolution of gr(R) over gr(Q), so pol reg(R) = s − 1.

2. Linear resolutions

Let (R, 𝔪, k) be a local ring and set G = gr(R). An R-module M is said to have a linear resolution if reg_{R}(M) = 0. We describe next constructions that will be used to characterize such modules. 2.1. Let X be a complex of R-modules filtered by subcomplexes

X = F^{0} ⊇ F^{1} ⊇ ⋯ ⊇ F^{p} ⊇ · · · .

The associated graded complex (gr_{F} (X), d) has

( gr_{F} (X))_{i,p} = F_{i}^{p}/F_{i}^{p+1}

and the differential d is induced by ∂.

If the complex X is minimal, that is ∂(X) ⊆ 𝔪X, then it has a filtration defined by F_{i}^{p} = 𝔪^{p−i}X_{i} for each i, that is:

F^{p} = ⋯ → X_{p+1} → X_{p} → 𝔪X_{p−1} → ⋯ → 𝔪^{p−1}X_{1} → 𝔪^{p}X_{0} → 0 .

In this case we set E(X) = gr_{F} (X) and note that E(X) is a complex of graded Gmodules, with E(X)_{i} = gr(X_{i})(−i) for each i, and the differential d is homogeneous of degree 0. 2.2. For an R-module M we choose a minimal free resolution X of M over R. It induces a map E(X) → gr(M). Choose a graded free resolution U of gr(M) over G and choose a lifting of the identity map on gr(M) to a morphism of complexes of graded G-modules ι^{M}: E(X) → U; this lifting is unique up to homotopy. For each z ∈ X_{i} we denote \overline{z} the class of z in E(X)_{i,i} = X_{i}/𝔪X_{i}. Calculating Tor_{i}^{R}(M, k) from the resolution X and Tor_{i}^{G} ( gr(M), k) from the resolution U, we define homomorphisms of R-modules λ_{i}^{M} as follows:

λ_{i}^{M}: Tor_{i}^{R}(M, k) = X_{i} ⊗_{R} k ⟶ H_{i}(U ⊗_{G} k)_{i} = Tor_{i}^{G} ( gr(M), k)_{i}
λ_{i}^{M}(z ⊗_{R} 1) = cls (ι^{M}(\overline{z}) ⊗_{G} 1) .

We then define τ_{i}^{M}: Tor_{i}^{R}(M, k) → Tor_{i}^{G} ( gr(M), k) to be the composition of λ_{i}^{M} with the inclusion Tor_{i}^{G} ( gr(M), k)_{i} ↪ Tor_{i}^{G} ( gr(M), k). 2.3. Proposition. Let (R, 𝔪, k) be a local ring, M a finite R-module, and X a minimal free resolution of M over R. The following properties are equivalent:

(1) The module M has a linear resolution.

(2) E(X) is a minimal free resolution of gr(M) over G.

(3) E(X) is a free resolution of gr(M) over G.

(4) For each i the maps τ_{i}^{M} and λ_{i}^{M} are bijective.

(5) For each i the map τ_{i}^{M} is surjective.

The content of the proposition is more or less known; see also [16]. Due to lack of a proper reference, we provide a proof, based on the next lemma.

For each integer j we denote R{j} the R-module R, filtered by F^{p} = 𝔪^{p−j}.

2.4. Lemma. Let (R, 𝔪, k) be a local ring and M a finite R-module. There exists a filtered resolution X′ of M with X′_{i} = ⨁_{j=1}^{s_{i}} R{a_{ij}} for each i ≥ 0, such that the associated graded complex is a minimal graded free resolution of gr(M) over G.

Proof. For an element m ∈ M we denote m^{∗} its initial form in gr_{𝔪}(M) and ord(m) the order of m^{∗}. Let m_{1}^{∗}, …, m_{s}^{∗} be a minimal system of homogeneous generators of gr_{𝔪}(M) over G and set a_{0j} = ord(m_{j}). By [34, II-6, Cor. 2], the elements m_{1}, …, m_{s} generate M. Set X′_{0} = ⊕_{j=1}^{s}R{a_{0j}} and let ∂_{0} : X′_{0} → M be the surjection that sends to m_{j} the basis element corresponding to R{a_{0j}}. The associated graded module of X′_{0} is U_{0} = ⊕_{j=1}^{s}G( − a_{0j}), and the map d_{0} : U_{0} → gr(M) induced by ∂_{0} is homogeneous and surjective. We set M_{0} = Ker(∂_{0}) and consider the filtration F_{0} induced by X′_{0}. Note that gr_{F_{0}}(X′_{0}) = Ker(d_{0}). The resolution X′ is obtained by iterating this procedure. □

Proof of Proposition 2.3. The implications (2) ⇒ (3) and (4) ⇒ (5) are clear.

(1) ⇒ (2) Let X′ be the filtered resolution of M from Lemma 2.4. The associated graded complex has U_{i} = ⨁_{j=1}^{s_{i}} G(−a_{ij}). Since reg_{R}(M) = 0, we get a_{ij} = i for all i and j, hence the filtration of X′ is given by F_{i}^{p} = 𝔪^{p−i}X′_{i}. The compatibility of this filtration with the differential implies that X′ is a minimal free resolution. Any two minimal free resolutions of M over R are isomorphic, and such an isomorphism preserves the filtration described above, hence U = E(X′) ≅ E(X).

(3) ⇒ (4) Both E(X) and U are graded free resolutions of gr(M) over G, so H_{∗}(ι^{M} ⊗_{G}k) is bijective. Since the complex E(X) is minimal, we have H_{∗} ( E(X)⊗_{G} k) = E(X) ⊗_{G} k. The map ν_{i}: X_{i} ⊗_{R} k → ( E(X) ⊗_{G} k)_{i,i} defined by ν_{i}(z ⊗_{R} 1) = \overline{z} ⊗_{G} 1 is clearly bijective. Since λ_{i}^{M} = H_{i}(ι^{M} ⊗_{G} k)_{i} ∘ ν_{i} for each i, the map λ_{i}^{M} is bijective, and so is τ_{i}^{M}, because Tor_{i}^{G}(gr(M), k)_{p} ≅ ( E(X) ⊗_{G} k)_{i,p} = 0 for i ≠ p.

(5) ⇒ (1). The image of the map τ_{i}^{M} is in Tor_{i}^{G}(gr(M), k)_{i}. If τ_{i}^{M} is surjective for each i, then Tor_{i}^{G}(gr(M), k)_{p} = 0 for p ≠ i, hence reg_{R}(M) = 0. □

We defined the homomorphisms λ_{i}^{M}: Tor_{i}^{R}(M, k) → Tor_{i}^{G} ( gr(M), k)_{i} in terms of free resolutions of M and respectively gr(M). We show next that one can also compute these maps using free resolutions of k. To do so, we take a more general point of view. 2.5. Let X be a minimal complex of free R-modules. Since E(X)_{i,p} = 0 for p < i and E_{i,i} = X_{i}/𝔪X_{i}, we have

H_{i} ( E(X))_{i} = Z_{i} ( E(X))_{i} ⊆ X_{i}/𝔪X_{i} .

Recall that for each z ∈ X_{i} we denote \overline{z} its class in E(X)_{i,i}. We then define canonical homomorphisms

ν_{i}^{X}: H_{i}(X) → H_{i} ( E(X))_{i} , by setting ν_{i}^{X}( cls(z)) = \overline{z} .

If X and Y are two minimal complexes of R-modules, then an easy calculation shows that E(X ⊗_{R} Y) = E(X) ⊗_{G} E(Y) as complexes of graded G-modules, hence for each i we have a map

ν_{i}^{X⊗_{R}Y} : H_{i}(X ⊗_{R} Y) → H_{i} ( E(X) ⊗_{G} E(Y))_{i} .

Let M and N be two R-modules. We consider them as complexes concentrated in degree 0 and note that E(M) = gr(M) and E(N) = gr(N). Let κ^{X}: X → M be a minimal free resolution of M over R and κ^{Y} : Y → N a minimal free resolution of N over R. They induce morphisms E(κ^{X}): E(X) → gr(M) and E(κ^{Y}): E(Y) → gr(N). For each graded free resolution κ^{U}: U → gr(M) of gr(M) over G and each graded free resolution κ^{V} : V → gr(N) of gr(N) over G we consider morphisms of graded complexes ι^{M}: E(X) → U and ι^{N}: E(Y) → V as in 2.2; these morphisms are unique up to homotopy. For each integer i we set

α_{i,i} = H_{i} ( E(X) ⊗_{G} E(κ^{Y}))_{i} and β_{i,i} = H_{i} ( E(κ^{X}) ⊗_{G} E(Y))_{i} .

The commutative diagram below defines maps λ_{i}(M, N) as indicated:

[figure: Tor_{i}^{R}(M, N); λ_{i}(M,N); Tor_{i}^{G} ( gr(M), gr(N))_{i}; H_{i}(X ⊗_{R} N); ν_{i}^{X⊗_{R}N}; H_{i}(E(X) ⊗_{G} gr(N))_{i}; H_{i}(ι^{M} ⊗_{G}gr(N))_{i}; H_{i}(U ⊗_{G} gr(N))_{i}; H_{i}(X ⊗_{R} Y); ν_{i}^{X⊗_{R}Y}; ≅; H_{i}(X⊗_{G}κ^{Y} ); ≅; H_{i}(κ^{X}⊗_{R}Y); H_{i}(E(X) ⊗_{G} E(Y))_{i}; H_{i}(ι^{M} ⊗_{R}ι^{N})_{i}; β_{i,i}; α_{i,i}; H_{i}(U ⊗_{G} V)_{i}; ≅; H_{i}(U⊗_{G}κ^{V} )_{i}; ≅; H_{i}(κ^{U} ⊗_{G}V)_{i}; H_{i}(M ⊗_{R} Y); ν_{i}^{M⊗_{R}Y}; H_{i}(gr(M) ⊗_{G} E(Y))_{i}; H_{i}(gr(M)⊗_{G}ι^{N})_{i}; H_{i}(gr(M) ⊗_{G} V)_{i}]

We note that λ_{i}(M, k) coincides with the map λ_{i}^{M} defined at the beginning of the section. Thus, these maps can be calculated from any horizontal line of the diagram above, with N = k.

3. Indices of modules

Each homomorphism of R-modules ψ: M → N induces homomorphisms of graded vector spaces

Tor_{∗}^{R}(ψ, k): Tor_{∗}^{R}(M, k) → Tor_{∗}^{R}(N, k) ,
Ext_{R}^{∗}(ψ, k): Ext_{R}^{∗}(N, k) → Ext_{R}^{∗}(M, k) ,
Ext_{R}^{∗}(k, ψ): Ext_{R}^{∗}(k, M) → Ext_{R}^{∗}(k, N) .

Let n be an integer. For each R-module M consider the canonical inclusion

μ_{M}^{(n)} : 𝔪^{n}M → 𝔪^{n−1}M .

If φ: (Q, 𝔫) → (R, 𝔪) is a surjective homomorphism of local rings, then 𝔪^{n}M = 𝔫^{n}M, so the notation μ_{M}^{(n)} will be used without reference to a specific ring. 3.1. We define the Levin index of M over R by the formula:

L_{R}(M) = inf{s ≥ 1 | Tor_{∗}^{R}(μ_{M}^{(n)}, k) = 0 for all n ≥ s} .

The isomorphism of functors Ext_{R}^{∗}(−, k) ≅ Hom_{k} ( Tor_{∗}^{R}(−, k), k) shows that

L_{R}(M) = inf{s ≥ 1 | Ext_{R}^{∗}(μ_{M}^{(n)}, k) = 0 for all n ≥ s} .

Results of Levin [21] show that L_{R}(M) < ∞, cf. 3.6 for more details. We define the Roos index of M over R by the formula

R_{R}(M) = inf{s ≥ 1 | Ext_{R}^{∗}(k, μ_{M}^{(n)}) = 0 for all n ≥ s} .

Roos [27] noted that Levin’s arguments can be adapted to show that R_{R}(M) < ∞, cf. 3.7 for details.

Following [16], we say that a ring R is Koszul if its residue field has a linear resolution, that is, reg_{R}(k) = 0; recall that the notions of regularity and polynomial regularity of local rings were discussed in the first section.

3.2. Theorem. If (R, 𝔪) is a Koszul local ring and M a finite R-module, then

(1) reg_{R}(𝔪^{i}M) = max{reg_{R}(M) − i, 0} for all i ≥ 0.

(2) L_{R}(M) = reg_{R}(M) + 1.

3.3. Theorem. For a finite module M over an arbitrary local ring R there is an inequality

max{L_{R}(M), R_{R}(M)} ≤ pol reg(M) + 1 .

We postpone the proofs for the moment, in order to give an application of Theorem 3.3.

The Poincaré series of a finite R-module M is the formal power series

P_{M}^{R}(t) = ∑_{i=0}^{∞} rank_{k} Tor_{i}^{R}(M, k)t^{i} .

The Bass series of M is the formal power series

I_{R}^{M}(t) = ∑_{i=0}^{∞} rank_{k} Ext_{R}^{i}(k, M)t^{i} .

The Hilbert series of M is the formal power series

Hilb_{M}^{R}(t) = ∑_{i=0}^{∞} rank_{k}(𝔪^{i}M/𝔪^{i+1}M)t^{i}

The following corollary contains effective versions of [6, 4.1.8], [21, Theorem 2] (see also [6, 6.3.6]); we also include versions for Bass series.

3.4. Corollary. Set p = pol reg(M). For each submodule M′ contained in 𝔪^{p+1}M and for each integer n > p the following hold:

P_{M/M′}^{R}(t) = P_{M}^{R}(t) + t P_{M′}^{R}(t)  I_{R}^{M/M′}(t) = I_{R}^{M}(t) + t I_{R}^{M′}(t) .
P_{𝔪^{n}M}^{R}(t) = Hilb_{𝔪^{n}M}^{R}(−t) P_{k}^{R}(t)  I_{R}^{𝔪^{n}M}(t) = Hilb_{𝔪^{n}M}^{R}(−t) I_{R}^{k}(t) .

Proof. Let α be the inclusion M′ ↪ M, let β be the inclusion M′ ↪ 𝔪^{p+1}M, and γ the inclusion 𝔪^{p+1}M ↪ M. We have then α = γ ∘ μ^{(p+1)} ∘ β, hence

Tor_{∗}^{R}(α, k) = Tor_{∗}^{R}(γ, k) ∘ Tor_{∗}^{R}(μ^{(p+1)}, k) ∘ Tor_{∗}^{R}(β, k) .

Theorem 3.3 gives Tor_{∗}^{R}(μ_{M}^{(p+1)}, k) = 0, hence Tor_{∗}^{R}(α, k) = 0. Consider the short exact sequence

0 → M′ ⟶^{α} M ⟶^{π} M/M′ → 0 .

The long exact sequence obtained by applying Tor_{∗}^{R}(−, k) splits into short exact sequences. Computing ranks, we obtain the equality P_{M/M′}^{R}(t) = P_{M}^{R}(t) + t P_{M′}^{R}(t). The corresponding equality for Bass series is obtained similarly.

The expression for P_{𝔪^{n}M}^{R}(t) can be deduced using the calculations in [21, Theorem 2] or [6, 6.3.6]. Similar computations apply to Bass series. □

Proof of theorem 3.2. We choose a minimal free resolution Y of k over R. By Proposition 2.3 the complex E(Y) is a minimal free resolution of k over G. For all integers i and n we have

Tor_{i}^{G}(gr(M), k) ≅ H_{i} ( gr(M) ⊗_{G} E(Y)) = H_{i} ( E(M ⊗_{R} Y)) ,
Tor_{i}^{G} ( gr(𝔪^{n}M), k) ≅ H_{i} ( gr(𝔪^{n}M) ⊗_{G} E(Y)) = H_{i} ( E(𝔪^{n}M ⊗_{R} Y)) .

Since E(𝔪^{n}M ⊗_{R} Y)_{i,p} = E(M ⊗_{R} Y)_{i,p+n} we further have

H_{i} ( E(𝔪^{n}M ⊗_{R} Y))_{p} ≅ \begin{cases}H_{i} ( E(M ⊗_{R} Y))_{p+n} & if p > i \\ Z_{i} ( E(M ⊗_{R} Y))_{p+n} & if p = i . \end{cases}

These isomorphisms prove (1). In particular, we have reg_{R}(𝔪^{n}M) = 0 for n ≥ reg_{R}(M), hence the maps λ_{i}^{𝔪^{n}M} are bijective by Proposition 2.3(4). As shown in 2.5, we have λ_{i}^{𝔪^{n}M} = λ_{i}(𝔪^{n}M, k), and this map can be computed from the last row of the diagram considered there, as the composition H_{i} ( gr(𝔪^{n}M) ⊗_{G} ι^{k})_{i} ∘ ν_{i}^{𝔪^{n}M⊗_{R}Y}. Since both V and E(Y) are graded free resolutions of k over G, the first map is an isomorphism. It follows that ν_{i}^{𝔪^{n}M⊗_{R}Y} is bijective for all i and all n ≥ reg_{R}(M). To simplify notation, we denote these maps ν_{i}^{(n)} and we recall their definition:

ν_{i}^{(n)} : H_{i}(𝔪^{n}M ⊗_{R} Y) → H_{i}( E(𝔪^{n}M ⊗_{R} Y))_{i} ⊆ \frac{𝔪^{n}M ⊗_{R} Y_{i}}{𝔪^{n+1}M ⊗_{R} Y_{i}} ,
ν_{i}^{(n)}(cls(z)) = z + (𝔪^{n+1}M ⊗_{R} Y_{i}) .

We set l = L_{R}(M) and r = reg_{R}(M). To prove the inequality r +1 ≥ l, we show that the map

H(μ_{M}^{(n)} ⊗_{R} Y): H(𝔪^{n}M ⊗_{R} Y) → H(𝔪^{n−1}M ⊗_{R} Y)

is zero for all n ≥ r + 1. Let z ∈ 𝔪^{n}M ⊗_{R} Y be a cycle of degree i, with n ≥ r + 1. We regard z as a cycle of the subcomplex 𝔪^{n−1}M ⊗_{R} Y. Since ν_{i}^{(n−1)} is bijective and ν_{i}^{(n−1)}(cls(z)) = 0, it follows that z is a boundary in 𝔪^{n−1}M ⊗_{R} Y.

We assume next that r + 1 > l. We want to prove that Tor_{i}^{G} ( gr(M), k)_{i+r} = 0 for all i, which contradicts r = reg_{R}(M). By the above we have

Tor_{i}^{G} ( gr(M), k)_{i+r} ≅ H_{i} ( E(M ⊗_{R} Y))_{i+r} .

Let z be an element of 𝔪^{r}M ⊗_{R} Y_{i} whose image \overline{z} in E(M ⊗_{R} Y)_{i,i+r} is a cycle. We need to show that \overline{z} is a boundary of E(M ⊗_{R} Y). We regard \overline{z} as a cycle of E(𝔪^{r}M ⊗_{R} Y). Since the map ν_{i}^{(r)} is bijective, there exists a cycle z′ ∈ 𝔪^{r}M ⊗_{R} Y_{i} such that \overline{z} = \overline{z}′ in E(𝔪^{r}M ⊗_{R} Y)_{i,i} = E(M ⊗_{R} Y)_{i,i+r}. The assumption that r ≥ l and the definition of l imply that the map H(μ_{M}^{(r)} ⊗_{R} Y): H(𝔪^{r}M ⊗_{R} Y) → H(𝔪^{r−1}M ⊗_{R} Y) is zero, hence z′ is a boundary of the complex 𝔪^{r−1}M ⊗_{R} Y. It follows that \overline{z}′, and hence \overline{z}, is a boundary of E(M ⊗_{R} Y). □

3.5. For an arbitrary local ring (R, 𝔪) we choose a minimal system of generators g of 𝔪 and set K^{R} = K(g; R). If M is an R-module, then we set K(M) = K(g; M) and H_{∗}(M) = H_{∗} (K(M)). For an R-homomorphism ψ: M → M′ we denote H_{∗}(ψ) the induced map H_{∗}(M) → H_{∗}(M′). If φ: R → S is a surjective homomorphism of local rings with Ker(φ) ⊆ 𝔪^{2}, then φ(g) is a minimal system of generators of the maximal ideal of S. Thus, we can identify K^{S} and K^{R} ⊗_{R} S. If the R-module structure of M is induced through the homomorphism φ, then, as in 1.1, we identify K(φ(g); M) and K(g, M) and let the notation K(M) refer to either complex.

The complex Hom_{R}(K^{R}, M) is known to be isomorphic to K(M), cf. [10, 1.6.10]. We denote H^{∗}(M) its homology. Also, we denote H^{∗}(φ): H^{∗}(M′) → H^{∗}(M) the map induced by the homomorphism φ: M → M′, which is the k-dual of H_{∗}(φ), and we make identifications similar to those above. 3.6. Levin [21, Lemma 1] proves that H_{∗} (μ_{M}^{(n)}) for all large n. Also, [21, Lemma 2] shows that H_{∗} (μ_{M}^{(n)}) = 0 implies Tor_{∗}^{R} (μ_{M}^{(n)}, k) = 0. In particular, L_{R}(M) < ∞.

3.7. Since the map H^{∗} (μ_{M}^{(n)}) is dual to H_{∗} (μ_{M}^{(n)}), it is also zero for all large n. As noted by Roos [27], techniques similar to those of Levin show that H^{∗} (μ_{M}^{(n)}) = 0 implies Ext_{R}^{∗} (k, μ_{M}^{(n)}) = 0. In particular, R_{R}(M) < ∞.

Proof of Theorem 3.3. Since R̂ is R-flat, with maximal ideal 𝔪R̂, there are equalities L_{R}(M) = LR̂(M̂) and R_{R}(M) = RR̂(M̂). Thus, we can assume that R has a minimal Cohen presentation R = Q/𝔞. Since Q is regular, the Koszul complex K^{Q} is a minimal free resolution of k over Q, so Tor_{∗}^{Q}(−, k) = H_{∗}(−) and Ext_{Q}^{∗}(k, −) = H^{∗}(−), in the notation of 3.5. For any Q-module there are isomorphisms H_{i}(N) ≅ H^{d−i}(N), where d = dim(Q), cf. [10, 1.6.10]. We have thus:

L_{Q}(M) = inf{s ≥ 1 | H_{∗} (μ_{M}^{(n)}) = 0 for all n ≥ s}
= inf{s ≥ 1 | H^{∗} (μ_{M}^{(n)}) = 0 for all n ≥ s} = R_{Q}(M) .

By 3.5 the maps H_{∗} (μ_{M}^{(n)}) and H^{∗} (μ_{M}^{(n)}) do not depend on whether M is viewed as a module over R or over Q. In our notation, 3.6 and 3.7 translate as follows:

max{L_{R}(M), R_{R}(M)} ≤ L_{Q}(M) = R_{Q}(M) < ∞ .

Note that Q is a Koszul local ring, since gr(Q) is a polynomial ring over k, hence a minimal graded free resolution of k over this ring is given by the Koszul complex on the variables. By 3.2 we have then L_{Q}(M) = reg_{Q}(M) + 1. To finish the proof, recall from 1.7 that reg_{Q}(M) = pol reg(M). □

4. Higher delta invariants

For a finite module M over a Gorenstein local complete ring (R, 𝔪) Auslander defined the delta invariant δ_{R}(M) to be the smallest integer n such that there exists an epimorphism X ⊕ R^{n} → M, where X is a maximal Cohen Macaulay module with no free summand. For an integer i ≥ 0 he defined an i-th higher delta invariant δ_{R}^{i}(M) by the formula δ_{R}^{i}(M) = δ_{R}(Ω_{R}^{i}(M)), where Ω_{R}^{i}(M) denotes the i-th syzygy module in a minimal free resolution of M over R, cf. [2, §5].

If R is not regular, then Auslander proved δ_{R}^{i}(k) = 0 for all i > 0, cf. [2, 5.7]. Yoshino [39] studied the vanishing of the numbers δ_{R}^{i}(R/𝔪^{n}) for positive integers i and n. He conjectured that if R is not regular, then they all vanish. One of his main results [39, (2.1)] shows that there exists an integer s such that δ_{R}^{i}(R/𝔪^{n}) = 0 for all n ≥ s and all i > 0, or, equivalently, δ_{R}^{i}(𝔪^{n}) = 0 for all n ≥ s and all i ≥ 0.

Buchweitz has noted, cf. [26, 2.3], that δ_{R}^{i}(M) equals the rank of the kernel of

ε_{R}^{i}(M, k): Ext_{R}^{i}(M, k) → Ex^{∨}t_{R}^{i}(M, k) ,

where ε_{R}^{∗}(−, −) is the natural transformation from absolute cohomology to Tate cohomology. The latter cohomology theory is defined for all modules over a Gorenstein ring, but not in general. Over an arbitrary ring, Vogel has defined a cohomology theory, Ex^{∨}t_{R}^{∗}(−, −), which comes with a natural transformation ε_{R}^{∗}(−, −) as above, and coincides with Tate cohomology when R is Gorenstein. Vogel’s theory is described in [11], cf. also [24].

Martsinkovsky [24] sets ξ_{R}^{i}(M) = rank_{k} Ker (ε_{R}^{i}(M, k)) and proves:

4.1. ([25, Theorem 6]). If R is not regular, then ξ_{R}^{i}(k) = 0 for all i.

We use the cohomological interpretation of delta invariants to extend Yoshino’s result to arbitrary local rings, to generalize it to submodules 𝔪^{n}M of any finite R-module, and to obtain bounds for the vanishing of δ_{R}^{i}(𝔪^{n}M). 4.2. Theorem. If M is a finite module over a non-regular local ring (R, 𝔪, k), then the homomorphism ε_{R}^{i}(𝔪^{n}M, k) is injective for all i ≥ 0 and all n ≥ L_{R}(M). In particular, if R is Gorenstein, then δ_{R}^{i}(𝔪^{n}M) = 0 for all n > pol reg(M).

Proof. For each integer n we set M_{n} = 𝔪^{n−1}M/𝔪^{n}M and form the exact sequence

0 → 𝔪^{n}M ⟶^{μ_{M}^{(n)}} 𝔪^{n−1}M → M_{n} → 0 .

It induces long exact sequences in cohomology, both for Ext and Ex^{∨}t. The naturality of ε_{R}^{∗}(−, k) implies that for each i there is a commutative diagram

\xymatrix{Ext_{R}^{i}(𝔪^{n−1}M, k) \ar[r]^{Ext_{R}^{i}(μ_{M}^{(n)},k)} \ar[d]_{ε_{R}^{i}(𝔪^{n−1}M,k)} & Ext_{R}^{i}(𝔪^{n}M, k) \ar[r]^{ð^{i}} \ar[d]_{ε_{R}^{i}(𝔪^{n}M,k)} & Ext_{R}^{i+1}(M_{n}, k) \ar[d]_{ε_{R}^{i+1}(M_{n},k)} \\ Ex^{∨}t_{R}^{i}(𝔪^{n−1}M, k) \ar[r]^{Ex^{∨}t_{R}^{i}(μ_{M}^{(n)},k)} & Ex^{∨}t_{R}^{i}(𝔪^{n}M, k) \ar[r]^{_{ð^{∨}_{i}}} & Ex^{∨}t_{R}^{i+1}(M_{n}, k)}

where ð^{i} and ð^{∨}^{i} denote connecting homomorphisms. If n ≥ L_{R}(M), then we have Ext_{R}^{i}(μ_{M}^{(n)}, k) = 0 by the definition in 3.1, hence ð^{i} is injective. Since 𝔪M_{n} = 0, the map ε_{R}^{i+1}(M_{n}, k) can be identified with the natural map ε_{R}^{i+1}(k, k)⊗_{R}Hom_{k}(M_{n}, k). The map ε_{R}^{i+1}(k, k) is injective by 4.1, hence so is ε_{R}^{i+1}(M_{n}, k). The commutativity of the right-hand square implies that ε_{R}^{i}(𝔪^{n}M, k) is injective, hence δ_{R}^{i}(𝔪^{n}M) = 0. The last statement of the theorem follows by 3.3. □

5. Small homomorphisms

Let (R, 𝔪, k) be a local ring and let φ: R → S be a surjective homomorphism of rings. Due to the functoriality of Tor and Ext in the ring variable there are homomorphisms of graded vector spaces

Tor_{∗}^{φ}(k, k): Tor_{∗}^{R}(k, k) → Tor_{∗}^{S}(k, k)
Ext_{φ}^{∗}(k, k): Ext_{S}^{∗}(k, k) → Ext_{R}^{∗}(k, k) .

Recall that Ext_{φ}^{∗}(k, k) is a homomorphism of k-graded algebras, where multiplication on the Ext’s is given by the Yoneda products.

Following Avramov [3], we say that a surjective homomorphism φ: R → S is small if Tor_{∗}^{φ}(k, k) is injective, or, equivalently, if the algebra homomorphism Ext_{φ}^{∗}(k, k) is surjective.

For each integer n we consider the canonical homomorphism

ρ^{(n)} : R → R/𝔪^{n} .

By [3, 4.1] the homomomorphism ρ^{(n)} is small for all large n. 5.1. We define the Avramov index by the formula

A(R) = inf{s ≥ 0 | ρ^{(s+1)} is small} .

We note that if ρ^{(s)} is small for some integer s, then ρ^{(n)} is small for all n ≥ s. Indeed, if υ: R/𝔪^{n} → R/𝔪^{s} is the induced map, then the functoriality of Tor gives

Tor_{∗}^{ρ^{(s)}}(k, k) = Tor_{∗}^{υ}(k, k) ∘ Tor_{∗}^{ρ^{(n)}}(k, k) .

Thus, the definition of A(R) can be reformulated in terms similar to those of the other indices:

A(R) = inf{s ≥ 0 | ρ^{(n)} is small for all n > s} .

5.2. By [3, 3.9] a homomorphism φ is small if and only if the induced homomorphism φ̂: R̂ → Ŝ is small. Thus, A(R) = A(R̂) and we will assume whenever necessary that R is complete, with Cohen presentation R = Q/𝔞 as in 1.7.

For completeness, we include a proof of the following known result. 5.3. Proposition. For a local ring (R, 𝔪, k) an inequality A(R) ≤ 1 holds if and only if Ext_{R}^{∗}(k, k) is generated as an algebra by its elements of degree 1. Moreover, A(R) = 0 holds if and only if 𝔪 = 0.

Proof. The inequality A(R) ≤ 1 implies the surjectivity of the algebra homomorphism Ext_{ρ^{(2)}}^{∗}(k, k). Since Ext_{R/𝔪^{2}}^{∗}(k, k) is the tensor algebra over k generated by the elements of degree 1, cf. [28, §1, Remark 3], the conclusion follows.

If R = k, then it is clear that A(R) = 0. Conversely, if A(R) = 0, then we have an injection Tor_{1}^{ρ^{(1)}}(k, k): Tor_{1}^{R}(k, k) = 𝔪/𝔪^{2} → Tor_{1}^{k}(k, k) = 0, hence 𝔪 = 0 by Nakayama’s Lemma. □

Recall that a ring R is said to be a complete intersection if the ideal 𝔞 in some Cohen presentation is generated by a Q-regular sequence. It is known that this notion does not depend on the choice of the presentation, cf. [3, 7.3.3], for example. If the ideal 𝔞 is principal, then R is a hypersurface. 5.4. Proposition. Let (R, 𝔪) be a local ring with 𝔪 ≠ 0 and minimal Cohen presentation R̂ = Q/𝔞. The following then hold:

(1) A(R) ≥ inf{i ≥ 1 | 𝔞 ∩ 𝔫^{i+2} ⊆ 𝔫𝔞}.

(2) If R a complete intersection, then equality holds in (1).

(3) If R is a hypersurface, then A(R) = max{1, mult(R) − 1}.

(4) If x is a regular sequence in R and 𝔪 ≠ (x), then A(R) ≤ A(R/(x)).

The proof is based on results about small homomorphisms from [3]. 5.5. A DG algebra is a complex (Λ, ∂) with an unitary associative product such that the differential satisfies the Leibnitz rule: ∂(ab) = ∂(a)b + (−1)^{|a|}a∂(b), where |a| denotes the homological degree of a. In addition, we assume DG algebras to be graded commutative, that is ab = (−1)^{|a||b|}ba for all a, b ∈ Λ, and a^{2} = 0 when |a| is odd. We refer to [6] and [19] for details.

A system of divided powers on a DG algebra Λ is an operation that associates to every element a ∈ Λ of even positive degree a sequence of elements a^{(i)} ∈ Λ with i = 0, 1, 2, … satisfying certain axioms, cf. [14, 1.7.1]. A DG Γ-algebra is a DG algebra with divided powers which are compatible with the differential, in the sense that ∂(a^{(i)}) = ∂(a)a^{(i−1)} for every a ∈ Λ of positive even degree and every i ≥ 1.

By Gulliksen [13] and Schoeller [32] there exists a minimal free resolution of k over R which has a structure of DG Γ-algebra; it is obtained by Tate’s procedure of adjoining divided powers variables (cf. [38]). We call it a minimal Tate resolution of k over R. Note that Tor_{∗}^{R}(k, k) inherits a structure of DG Γ-algebra.

Let Λ be a DG Γ-algebra and denote Λ_{>0} the ideal of elements of positive degree. The module of Γ-indecomposables of Λ is the quotient of Λ_{>0} by the submodule generated by all elements of the form uv with u, v ∈ Λ_{>0} and w^{(n)} with w ∈ Λ_{2i}, n ≥ 2. We denote π_{∗}(R) the module of Γ-indecomposables of Tor_{∗}^{R}(k, k).

The next result is our main tool in the study of small homomorphisms. 5.6. ([3, 3.1]) A surjective homomorphism of local rings φ: R → S is small if and only if the induced homomorphism π_{∗}(φ): π_{∗}(R) → π_{∗}(S) is injective.

We proceed to describe the maps π_{1}(φ) and π_{2}(φ). 5.7. By 5.2 we can consider a minimal Cohen presentation R = Q/𝔞. We then have S = Q/𝔟 for an ideal 𝔟 ⊇ 𝔞. Since π_{1}(R) = Tor_{1}^{R}(k, k), the map π_{1}(φ) = Tor_{1}^{φ}(k, k) is canonically identified with the natural map 𝔫/𝔫^{2} → 𝔫/(𝔟, 𝔫^{2}). It is injective if and only if 𝔟 ⊆ 𝔫^{2}. When this happens, π_{1}(φ) is bijective.

Assume that 𝔟 ⊆ 𝔫^{2}. The proof of [14, Proposition 3.3.4] canonically identifies π_{2}(R) with 𝔞/𝔫𝔞 and π_{2}(S) with 𝔟/𝔫𝔟; under these identifications, π_{2}(φ) is the natural homomorphism 𝔞/𝔫𝔞 → 𝔟/𝔫𝔟. This map is injective if and only if 𝔞 ∩ 𝔫𝔟 ⊆ 𝔫𝔞, or, equivalently, if a minimal set of generators of 𝔞 can be completed to a minimal set of generators of 𝔟.

The behavior of smallness under factorization of a regular sequence is described by the following result of Tate [38, Theorem 4] and Scheja [33, Satz 1], in the form given by Gulliksen [14, 3.4.1]. 5.8. Let x be a regular sequence in R and set \overline{R} = R/(x). The canonical homomorphism ψ: R → \overline{R} induces isomorphisms

π_{j}(ψ): π_{j}(R) ≅ π_{j}(\overline{R}) for j ≥ 3

and an exact sequence

0 → π_{2}(R) ⟶^{π_{2}(ψ)} π_{2}(\overline{R}) → (x)/𝔪(x) → π_{1}(R) ⟶^{π_{1}(ψ)} π_{1}(\overline{R}) → 0 .

In particular, one sees from here: if R is regular, then π_{i}(R) = 0 for i ≠ 1; if R is a complete intersection, then π_{i}(R) = 0 for i ≠ 0, 1. 5.9. Lemma. Let (R, 𝔪, k) be a local ring, 𝔠 an ideal contained in 𝔪^{2}, and x a regular sequence. Set S = R/𝔠 and denote φ the canonical homomorphism R → S. Also, set \overline{R} = R/(x) and \overline{S} = R/(𝔠, x). If the induced homomorphism \overline{φ}: \overline{R} → \overline{S} is small, then φ is small.

Proof. The naturality of the module of indecomposables yields a commutative diagram

\xymatrix{π_{∗}(R) \ar[r]^{π_{∗}(φ)} \ar[d]_{π_{∗}(ψ)} & π_{∗}(S) \ar[d] \\ π_{∗}(\overline{R}) \ar[r]^{π_{∗}(\overline{φ})} & π_{∗}(\overline{S})}

Since \overline{φ} is small, the map π_{∗}(\overline{φ}) is injective. Also, π_{⩾ 2}(ψ) is injective by 5.8, so the commutativity of the diagram implies that π_{⩾ 2}(φ) injective. Since 𝔠 ⊆ 𝔪^{2}, the map π_{1}(φ) is bijective by 5.7. Thus, π_{∗}(φ) is injective, hence φ is small by 5.6. □

Proof of Proposition 5.4. By 5.2 we can consider a minimal Cohen presentation R = Q/𝔞. Also, the hypothesis 𝔪 ≠ 0 implies A(R) ≥ 1.

(1) By 5.7 we have

inf{i ≥ 1 | 𝔞 ∩ 𝔫^{i+2} ⊆ 𝔫𝔞} = inf{i ≥ 1 | π_{2}(ρ^{(i+1)}) is injective} .

By 5.6 the index A(R) is an upper bound for the number on the right hand side.

(2) Let n ≥ 1 be an integer such that 𝔞 ∩ 𝔫^{n+2} ⊆ 𝔫𝔞. Set S = R/𝔪^{n+1} and φ = ρ^{(n+1)}. Note that S = Q/𝔟 with 𝔟 = (𝔞, 𝔫^{n+1}). We have 𝔟 ⊆ 𝔫^{2} and 𝔞∩𝔫𝔟 ⊆ 𝔫𝔞, so π_{1}(φ) and π_{2}(φ) are injective by 5.7. Also, by 5.9 we have π_{i}(R) = 0 for i > 2, so π_{∗}(φ) is injective, and then φ is small by 5.6.

(3) We have a = (a) and the multiplicity of R is equal to the smallest integer n such that a ∈ 𝔫^{n}. The equality A(R) = max{1, mult(R) − 1} then follows from (2).

(4) The inequality A(R) ≤ A(R/(x)) follows by applying Lemma 5.9 to 𝔠 = 𝔪^{n+1}, where n = A(R/(x)). Note that n ≥ 1, cf. 5.3 □

5.10. Remark. The proof above shows that the number inf{i ≥ 1 | 𝔞 ∩ 𝔫^{i+2} ⊆ 𝔫𝔞} does not depend on the minimal Cohen presentation R = Q/𝔞 and is an invariant of the ring R; we denote it s(R). 5.11. Remark. If n > A(R), then Proposition 5.4(1) implies 𝔞 ∩ 𝔫^{n+1} ⊆ 𝔫𝔞, that is, any minimal system of generators of 𝔞 is part of a minimal system of generators of (𝔞, 𝔫^{n}).

6. Golod rings and Golod homomorphisms

A surjective homomorphism φ: R → S is called Golod if

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

Levin [19, 3.15] proves that ρ^{(n)} is Golod for all large n. 6.1. We define the Golod index of R by the formula

G(R) = inf{s ≥ 0 | ρ^{(n)} is Golod for all n > s} .

Note that G(R) is one less than the Golod invariant G(R) introduced in [17]. 6.2. By [3, 3.5] a Golod homomorphism is small. In particular, one has

A(R) ≤ G(R) .

6.3. The condition G(R) = 0 holds if and only if R is a field. Indeed, if G(R) = 0, then 6.2 implies A(R) = 0, hence R is a field by 5.3. The converse is clear.

Golod [12] studied local rings (R, 𝔪, k) satisfying

P_{k}^{R}(t) = \frac{(1 + t)^{edim R}}{1 − ∑_{j=1}^{∞} rank H_{j}(K^{R})t^{j+1}} ,

Rings with this property are now called Golod rings. If R = Q/𝔞 is a minimal Cohen presentation as in 1.7, then H(K^{R}) ≅ Tor_{∗}^{Q}(R, k), hence the projection Q → Q/𝔞 is a Golod homomorphism if and only if R is a Golod ring.

6.4. Poincaré series are invariant under completion, so a surjective homomorphism φ: R → S is Golod if and only if the induced homomorphism φ̂: R̂ → Ŝ is Golod. Thus, G(R) = G(R̂) and we may assume whenever necessary that R is complete.

Next we show that in some cases the indices A(R) and G(R) are equal. 6.5. Theorem. An equality A(R) = G(R) holds if one of the following conditions is satisfied:

(1) R is Golod and Artinian. In this case both indices are equal to pol reg(R).

(2) edim R ≤ dim R + 1.

(3) edim R = 2.

(4) edim R = 3 and R is a complete intersection.

The proof of Theorem 6.5 requires some preparation. One of the ideas involved is to connect Golod rings to Golod homomorphisms. To do this we use their cohomological characterizations. As in Section 5, π_{∗}(R) denotes the module of indecomposables of Tor. Its vector space dual π^{∗}(R) = Hom_{k}(π_{∗}(R), k) is a graded Lie algebra, called the homotopy Lie algebra of R; we refer to [4, §10] for details. 6.6. Avramov and Löfwall prove that a local ring S is Golod if and only if π^{⩾ 2}(S) is a free Lie algebra and that a homomorphism φ: R → S is Golod if and only if the kernel of the induced map π^{∗}(φ): π^{∗}(S) → π^{∗}(R) is a free Lie algebra, cf. [3, 3.5], [5, 3.4], [22, Corollary 2.4]. 6.7. Proposition. Let φ: R → S be a surjective homomorphism of local rings. If φ is small and the ring S is Golod, then the homomorphism φ is Golod.

Proof. The map π^{∗}(φ) is surjective by 5.6, and π^{1}(φ) is bijective by 5.7. Denoting L the kernel of π^{∗}(φ), we have L^{1} = 0. If S is Golod, then π^{⩾ 2}(S) is a free Lie algebra, cf. 6.6. Subalgebras of free Lie algebras are free by [18, A.1.10], hence L = L^{⩾ 2} is free and φ is Golod by 6.6. □

We recall two facts on Koszul complexes, using the notation of 3.5. 6.8. ([20, 1.6]) If there is an R-submodule V of 𝔪K^{R} with V^{2} = 0 and such that

Z_{≥1}(K^{R}) ⊆ V + B(K^{R}) ,

then the ring R is Golod. 6.9. ([36, §2, Lemma 1]). If (Q, 𝔫, k) is a regular local ring and p is a positive integer, then

∂(𝔫^{p−1}K_{⩾ 1}^{Q}) = ∂(K^{Q}) ∩ 𝔫^{p}K^{Q} .

In view of Proposition 6.7, we plan to prove most of Theorem 6.5 by showing that, under the given assumption, the ring R/𝔪^{n} is Golod for all n > A(R). If edim R = dim R, that is, if R is regular, then by Golod’s example [12] the ring R/𝔪^{n} is Golod for all n ≥ 2. The next proposition generalizes this result; the proof uses some ideas from of [37, §2, Lemma 2] and [35, §5, Lemma 2]. 6.10. Proposition. If (R, 𝔪, k) is a local ring with edim R ≤ dim R + 1, then the ring R/𝔪^{n} is Golod for each integer n ≥ 2.

Proof. By 6.4 we may assume that R has a minimal Cohen presentation R = Q/𝔞. Since edim R ≤ dim R + 1 and Q is catenary, it follows that ht 𝔞 ≤ 1. Since Q is factorial, there exist an element x ∈ 𝔫 and an ideal 𝔟 such that 𝔞 = x𝔟. Let s be the largest integer for which x ∈ 𝔫^{s}. We denote \overline{R} the ring R/𝔪^{n} = Q/(𝔫^{n}, x𝔟) and \overline{𝔪} its maximal ideal. Set K = K^{Q} and \overline{K} = K^{R}. By 3.5 we have

\overline{K} = K ⊗_{Q} \overline{R} = K/(𝔫^{n}, x𝔟)K .

If \overline{y} is a cycle in \overline{K} of degree j ≥ 1, and y its preimage in K, then

∂(y) = a + xb with a ∈ 𝔫^{n}K_{j−1} and b ∈ 𝔟K_{j−1} .

Differentiating, we obtain x∂(b) = −∂(a) ∈ 𝔫^{n+1}K. Now K is a complex of free Qmodules. If c_{1}, …, c_{r} are the coefficients of ∂(b) in a basis of K_{j−2}, then xc_{i} ∈ 𝔫^{n+1} for all i. Since x is not contained in 𝔫^{s+1} and Q is a regular ring, we conclude that c_{i} ∈ 𝔫^{n+1−s}, hence ∂(b) ∈ 𝔫^{n+1−s}K.

Let T_{1}, … T_{e} be a basis of K_{1}, with ∂(T_{i}) = g_{i} for each i. Note that g_{1}, …, g_{e} minimally generate 𝔫, hence x = a_{1}g_{1} + … a_{e}g_{e} with a_{i} ∈ 𝔫^{s−1}. For t = a_{1}T_{1} + ⋯ + a_{e}T_{e} we have t ∈ 𝔫^{s−1}K and ∂(t) = x. Setting u = y − tb, we then obtain

∂(u) = a + xb − ∂(t)b + t∂(b) = a + t∂(b) ∈ 𝔫^{n}K .

By 6.9 there is an element v ∈ 𝔫^{n−1}K such that ∂(u) = ∂(v). Since u is a cycle of positive degree in K, we have u − v = ∂(w) for some w ∈ K. In conclusion, any \overline{y} ∈ Z_{⩾ 1}(\overline{K}) can be written as

\overline{y} = \overline{t} \overline{b} + \overline{v} + ∂(w) with \overline{b} ∈ \overline{𝔟} and \overline{v} ∈ \overline{𝔪}^{n−1}\overline{K}

The submodule V = (\overline{t} \overline{𝔟}, \overline{𝔪}^{n−1}\overline{K}) is contained in \overline{𝔪}\overline{K}. Indeed, if the ideal \overline{𝔟} is not contained in \overline{𝔪}, then s ≥ 2 and hence \overline{t} ∈ 𝔪. The product of any two cycles of the form \overline{t} \overline{b} + \overline{v}, with \overline{b} ∈ \overline{𝔟} and \overline{v} ∈ \overline{𝔪}^{n−1}\overline{K} is equal to zero, hence the ring R is Golod by 6.8. □

To continue, we need two more results of Avramov, Kustin and Miller [8]. 6.11. ([8, 6.1]) Let S be homomorphic image of a regular local ring Q. If pd_{Q}(S) ≤ 3, then there is a Golod homomorphism from a complete intersection (of codimension less than 2) onto S. 6.12. ([8, 5.13]) Let (R, 𝔪, k) be a local ring, 𝔠 an ideal contained in 𝔪^{2}, and set \overline{R} = R/𝔠. If the natural homomorphism R → \overline{R} is Golod and x = x_{1}, …, x_{r} is a regular sequence that can be extended to a minimal generating set for 𝔠, then the induced homomorphism R/(x) → \overline{R} is Golod.

Proof of Theorem 6.5. Let n be an integer such that n > A(R), that is, the map ρ^{(n)} is small. We may assume A(R) > 0, hence n ≥ 2 (otherwise, R is a field and both indices are zero). We have to prove that ρ^{(n)} is Golod.

(1) For an ideal 𝔠 of R the canonical map R → R/𝔠 is small if and only if 𝔠 = (0), cf. [3, 4.7]. We have thus 𝔪^{n} = 0, hence ρ^{(n)} is Golod for trivial reasons. By 1.5, pol reg(R) is the largest integer s for which 𝔪^{s} ≠ 0, hence A(R) = G(R) = pol reg(R).

For the rest of the proof we assume R = Q/𝔞 with (Q, 𝔫, k) regular and 𝔞 ⊆ 𝔫^{2}, cf. 1.7.

(2) By Proposition 6.10, the ring \overline{R} = R/𝔪^{s} is Golod for each s ≥ 2, so Proposition 6.7 implies that ρ^{(n)} is Golod.

(3)By Scheja [33, Satz 9] the ring \overline{R} = R/𝔪^{n} = Q/(𝔞, 𝔫^{n}) is either Golod or a complete intersection. If it is Golod, then ρ^{(n)} is Golod by Proposition 6.7. Assume now that \overline{R} is a complete intersection. By Remark 5.11, a minimal system of generators of 𝔞 can be completed to a minimal system of generators of (𝔞, 𝔫^{n}), hence 𝔞 is generated by a regular sequence. If dim R > 0, then codim R ≤ 1, so the problem is settled by (2). If dim R = 0, then 𝔞 is generated by a maximal regular sequence and thus (𝔞, 𝔫^{n}) = 𝔞. Therefore 𝔪^{n} = 0, and ρ^{(n)} is the identity map.

(4) We have 𝔞 = (x) for a regular sequence x = x_{1}, …, x_{c}. By Remark 5.11, x is part of a minimal system of generators of (x, 𝔫^{n}). Set S = R/𝔪^{n} = Q/(𝔫^{n}, x). If c ≤ 1, then the assertion follows from (2). If c ≥ 2, then by 6.11 there exists a regular sequence x′ in (x′) ⊆ (x, 𝔪^{n}), of length at most two, such that the map Q/(x′) → S is Golod. Examining the proof of [8, 2.17], we see that we can modify x′ to be either x_{1}, x_{2} or x_{1}. The conclusion then follows from 6.12. □

Comments. We proved that A(R) = G(R) for all complete intersections R with edim R ≤ 3. It would be interesting to see whether there are complete intersections of higher embedding dimension for which the equality does not hold. So far I did not find any example of a local ring with A(R) < G(R).

7. The Levin index of a ring

7.1. We define the Levin index of the ring R by the formula

L(R) = L_{R}(𝔪) .

We set μ^{(n)} = μ_{R}^{(n)} : 𝔪^{n} → 𝔪^{n−1} and note that μ^{(n)} = μ_{𝔪}^{(n−1)}, hence we have

L(R) = inf{s ≥ 1 | Tor_{∗}^{R}(μ^{(n)}, k) = 0 for all n > s} = max{1, L_{R}(R) − 1} .

7.2. Theorem. For a local ring R there are inequalities

G(R) ≤ L(R) ≤ max{1, pol reg(R)} .

Proof. The proof of [19, 3.15] shows that if Tor_{∗}^{R}(μ^{(n)}, k) = 0 for some n ≥ 2, then ρ^{(n)} is Golod; this proves the first inequality. The second one follows from Theorem 3.3. □

7.3. The proofs of [21, Theorem 2] and [20, 2.8] contain calculations of Poincaré series based on a choice of an integer s ≥ 2 such that Tor_{∗}^{R}(μ^{(n)}, k) = 0 for all n ≥ s, hence Theorem 7.2 yields

P_{k}^{R/𝔪^{n}}(t) = \frac{P_{k}^{R}(t)}{1 − t^{2} Hilb_{𝔪^{n}}^{R}(−t) P_{k}^{R}(t)}

for each n > max{1, pol reg(R)}.

The next proposition follows by combining 5.4(3) and 6.5(1) with 7.2.

7.4. Proposition. Let (R, 𝔪) be a local ring which is not a field. If R is a hypersurface or a Golod Artinian ring, then

A(R) = G(R) = L(R) = max{1, pol reg(R)}.  □

We characterize next rings that satisfy L(R) = 1. A similar result for graded algebras is mentioned by Roos [29, Remark 3.4].

7.5. Proposition. A local ring R is Koszul if and only if L(R) = 1. If R is Koszul, then Ext_{R}^{∗}(k, k) is generated by its elements of degree 1.

Proof. If R is Koszul, then 3.2 gives L(R) = max{1, reg_{R}(R)} = 1. Conversely, assume that L(R) = 1. We denote (X, ∂) a minimal free resolution of k over R and set U = E(X); this is the associated graded complex of X with respect to the natural filtration, as defined in 2.1 The fact that L(R) = 1 means that the natural map H_{∗}(𝔪^{n}X) → H_{∗}(𝔪^{n−1}X) is zero for all n ≥ 1. To prove that R is Koszul, we show that the complex U is acyclic. For all integers i and n the module H_{i}(U)_{i+n} is the homology of the complex

𝔪^{n−1}X_{i+1}/𝔪^{n}X_{i+1} → 𝔪^{n}X_{i}/𝔪^{n+1}X_{i} → 𝔪^{n+1}X_{i−1}/𝔪^{n+2}X_{i−1} .

Let \overline{x} be an element in Z_{i}(U)_{i+n} for some i ≥ 1, that is, x ∈ 𝔪^{n}X_{i} and ∂(x) ∈ 𝔪^{n+2}X_{i+1}. Since the map H_{∗}(𝔪^{n+2}X) → H_{∗}(𝔪^{n+1}X) is zero, it follows that ∂(x) = ∂(a) for some a ∈ 𝔪^{n+1}X_{i}. Thus x − a is a cycle of X of positive degree, hence x−a = ∂(b) for some b ∈ X_{i+1}. Since ∂(b) ∈ 𝔪^{n}X and the map H_{∗}(𝔪^{n}X) → H_{∗}(𝔪^{n−1}X) is zero, it follows that ∂(b) = ∂(c) for some c ∈ 𝔪^{n−1}X_{i+1}. We conclude that \overline{x} = ∂(\overline{c}) in U, hence U is acyclic and thus reg_{R}(k) = 0 by 2.3.

The last assertion follows by 5.3. □

For graded k-algebras, the converse of the last assertion of the proposition holds by [22, Theorem 1.2]. However, the converse does not hold for local rings, as can be seen from the following example: 7.6. Let (Q, 𝔫, k) be a 2-dimensional regular local ring and u, v a system of parameters. We set 𝔞 = (u^{2} + v^{3}, uv) and R = Q/𝔞 = k[u, v]/(u^{2} + v^{3}, uv). This is a local complete intersection such that Ext_{R}^{∗}(k, k) is generated in degree 1, cf. [37, Theorem 5]. Still, gr(R) ≅ k[u, v]/(u^{2}, uv, v^{4}) and β_{2,4}^{gr(R)} ≠ 0, hence gr(R), and thus R is not Koszul.

8. Reduction by a regular sequence

In this section we study the behavior of the Levin index under factorization of a regular sequence, in connection with a question of Roos. In [29] and [30] he introduces the graded vector spaces

\overline{S}_{𝔪^{n}} = Im ( Ext_{R}^{⩾ 1}(ψ^{(n)}, k)) ⊆ Ext_{R}^{⩾ 1}(R/𝔪^{n+1}, k) ,

where ψ^{(n)} : R/𝔪^{n+1} → R/𝔪^{n} is the canonical map, and considers the following properties of the ring R: (M_{s}) \overline{S}_{𝔪^{n}} = 0 for all n > s. ( L_{s} ) R satisfies M_{s} and the induced homomorphisms \overline{S}_{𝔪^{n}} → \overline{S}_{𝔪^{n+1}} are bijective

for n = 1, …, s − 3. As pointed out in [29, Remark 3.4], a graded k-algebra R is Koszul if and only if it satisfies M_{2}. In general, these properties measure how far the algebra is from being Koszul. 8.1. Roos [29, 7(iv)] asks the following question: Let R be a Cohen-Macaulay ring and \overline{R} = R/(an R-sequence). Is any of the conditions L_{s} and/or M_{s} true for R if and only if it is true for \overline{R}? Although in [29] the ring R is graded, the question makes sense for local rings as well.

The isomorphism of functors Hom_{k} ( Tor_{⩾ 1}^{R}(−, k), k) ≅ Ext_{R}^{⩾ 1}(−, k) shows that (M_{s}) is equivalent to L(R) < s. Thus, the question of Roos can be partly reformulated as whether the Levin index is invariant under factorization of a regular sequence. We show next that the answer is negative, unless certain assumptions are made on the regular sequence. 8.2. Let (R, 𝔪) be a regular local ring with dim(R) > 0. If x is an element in 𝔪^{n} ∖ 𝔪^{n+1} for some integer n ≥ 1, then L(R) = 1 and L(R/(x)) = n − 1 by 7.4 and 1.9.

A sequence x = x_{1}, …, x_{m} of elements in R is called strictly regular if the initial forms x_{1}^{∗}, …, x_{m}^{∗} form a regular sequence. It is known and easy to see that a strictly regular sequence is an R-sequence.

We give next a partial answer to Roos’s question. 8.3. Theorem. If (R, 𝔪) is a local ring and x = x_{1}, …, x_{m} is a strictly regular sequence in 𝔪 ∖ 𝔪^{2}, then the following hold:

(1) L(R) = L (R/(x)).

(2) R satisfies M_{s} if and only if R/(x) satisfies M_{s}.

(3) R satisfies L_{s} if and only if R/(x) satisfies L_{s}.

The proof of the theorem follows from Proposition 8.7 below. Here are some preliminaries: 8.4. ([6, 3.1]) Let X be a minimal Tate resolution of k over R (see 5.5) and x a regular element in 𝔪 ∖ 𝔪^{2}. If T ∈ X_{1} satisfies ∂(T) = x, then X/(x, T)X is a minimal free resolution of k over R/(x). 8.5. Lemma. Let (R, 𝔪, k) be a local ring, x be an element of R, and T be an element of X_{1} with ∂(T) = x. If (𝔪^{n+1} : x) ⊆ 𝔪^{n} and ∂(T c) ∈ 𝔪^{n+1}X for some integer n and some c ∈ X, then T c ∈ 𝔪^{n}X.

Proof. The Leibnitz rule gives ∂(T c) = xc − T ∂(c). Multiplying by T, we obtain T ∂(T c) = T (xc) = x(T c). This implies x(T c) ∈ 𝔪^{n+1}X. We set b = T c ∈ X_{s} (where s is the homological degree). Since X_{s} is a free R-module, we consider b_{1}, …, b_{r} to be the components of b in a basis. Then b_{i}x ∈ 𝔪^{n+1} for any i. The assumption on x implies b_{i} ∈ 𝔪^{n}, hence T c ∈ 𝔪^{n}X. □

8.6. If k is infinite, then each superficial element x which is regular satisfies the condition (𝔪^{n+1} : x) ⊆ 𝔪^{n} for all large integers n, cf. [31, Remarks, I-9]; such elements exist by [31, 3.2, I-8]. Also, if x ∈ 𝔪 ∖ 𝔪^{2} is strictly regular, then (𝔪^{n+1} : x) ⊆ 𝔪^{n} for all n ≥ 0. 8.7. Proposition. Let (R, 𝔪, k) be a graded Noetherian ring, x a regular element not contained in 𝔪^{2}, and set \overline{R} = R/(x) and \overline{𝔪} = 𝔪/(x). For any positive integer n such that (𝔪^{n+1} : x) ⊆ 𝔪^{n} the following hold:

(1) The induced homomorphism \overline{S}_{\overline{𝔪}}^{n} → \overline{S}_{𝔪^{n}} is surjective.

(2) The induced homomorphism \overline{S}_{\overline{𝔪}}^{n−1} → \overline{S}_{𝔪^{n−1}} is injective.

Proof. We first reformulate the statement in terms of homology. For each s we denote U_{𝔪^{s}} the image of ( Tor_{⩾ 1}^{R}(ψ^{(s)}, k)) in Tor_{⩾ 1}^{R}(R/𝔪^{s}, k). In view of the isomorphisms Hom_{k} ( Tor_{⩾ 1}^{R}(−, k), k) ≅ Ext_{R}^{⩾ 1}(−, k), we conclude that U_{𝔪^{s}} is canonically isomorphic to the vector-space dual of \overline{S}_{𝔪^{s}}. Thus, for (1) we have to prove that the induced map U_{𝔪^{n}} → U\overline{𝔪}^{n} is injective and for (2) we have to prove that the induced map U_{𝔪^{n−1}} → U\overline{𝔪}^{n−1} is surjective.

Let X be a minimal Tate resolution of k over R. By 8.4 the complex X/(x, T)X is a minimal free resolution of k over R/x, where T ∈ X_{1} satisfies ∂(T) = x. For all integers i we identify Tor_{⩾ 1}^{R}(R/𝔪^{i}, k) with H_{⩾ 1}(X/𝔪^{i}X) and Tor \overline{R} ; ⩾ 1( \overline{R}/ \overline{𝔪}^{i}, k) with H_{⩾ 1} (X/(𝔪^{i}, x, T)X). Overbars denote residue classes, as appropriate to the context.

(1) Let cls(\overline{y}) be an element of H_{⩾ 1}(X/𝔪^{n+1}X) whose image in U\overline{𝔪}^{n} zero. Thus ∂(y) ∈ 𝔪^{n+1}X and

y = ∂(a) + b + xc + T g with b ∈ 𝔪^{n}X and a, c, g ∈ X .

The Leibnitz rule gives xc = ∂(T c) + T ∂(c) and we obtain:

y = ∂(a) + b + ∂(T c) + T ∂(c) + T g = ∂(a′) + b + T g′ with a′, g′ ∈ X .

Differentiating, we get ∂(T g′) = ∂(y) − ∂(b). Since ∂(y) ∈ 𝔪^{n+1}X and b ∈ 𝔪^{n}X, we obtain ∂(T g′) ∈ 𝔪^{n+1}X. By Lemma 8.5 we have T g′ ∈ 𝔪^{n}X and thus \overline{y} is a boundary in X/𝔪^{n}X. We thus have cls(\overline{y}) = 0 in H_{⩾ 1}(X/𝔪^{n}X), hence the map U_{𝔪^{n}} → U\overline{𝔪}^{n} is injective

(2) For cls(\overline{y}) ∈ H_{⩾ 1} (X/(𝔪^{n}, x, T)X) we have

∂(y) = a + xb + T c with a ∈ 𝔪^{n}X and b, c ∈ X .

The Leibnitz rule gives xb = ∂(T b) + T ∂(b) and thus

∂(y − T b) = a + T c′ with c′ ∈ X .

Differentiating, we obtain ∂(T c′) = −∂(a); since a ∈ 𝔪^{n}X, we have ∂(a) ∈ 𝔪^{n+1}X. Lemma 8.5 yields T c′ ∈ 𝔪^{n}X, hence ∂(y − T b) ∈ 𝔪^{n}X. We conclude that \overline{y} − \overline{T} \overline{b} is a cycle in X/𝔪^{n}X and thus cls(\overline{y}) is the image of cls(\overline{y} − \overline{T} \overline{b}) ∈ H_{⩾ 1}(X/𝔪^{n}X). Thus, the induced map H_{⩾ 1}(X/𝔪^{n}X) → H_{⩾ 1} (X/(𝔪^{n}, x, T)X) is surjective. This implies the surjectivity of the map U_{𝔪^{n−1}} → U\overline{𝔪}^{n−1}. □

9. Graded rings

In this section we consider graded Noetherian rings. Let k be a field. Adapting the notation of a local ring, we denote (R, 𝔪, k) a graded Noetherian ring R = ⊕_{i=0}^{∞}R_{i} satisfying R = R_{0}[R_{1}], with maximal irrelevant ideal 𝔪 = ⊕_{i=1}^{∞}R_{i} and R_{0} = k. We use the notation k also for the residue field R/𝔪. All R-homomorphisms are assumed homogeneous. The notions and results used so far have analogous graded versions. We only mention that the notion corresponding to a minimal Cohen presentation is a presentation of the form R = k[u_{1}, …, u_{r}]/𝔞, with a homogeneous ideal 𝔞 ⊆ 𝔫^{2}, where 𝔫 = (u_{1}, …, u_{r}), and that the Koszul complexes K^{R} are understood as K^{R} = K(g, R) for a chosen basis g of R_{1}. Also, Tate resolutions become graded resolutions in a natural way. 9.1. Let R = k[u_{1}, …, u_{r}]/𝔞 be a minimal presentation as above. We noted in Remark 5.10 that the number s(R) = inf{i ≥ 1 | 𝔞 ∩ 𝔫^{i+2} ⊆ 𝔫𝔞} does not depend on the choice of the presentation. It is easy to see that s(R) is equal to 1 if 𝔞 = (0) and is one less the maximum of the degrees of a minimal system of generators of 𝔞, otherwise.

All the results of this paper have analogous versions for graded rings. There are also some improvements of the statements, which are collected in the next theorem. 9.2. Theorem. Let R be a graded ring as above, which is not a field.

(1) The ring R is Koszul if and only if A(R) = 1, if and only if G(R) = 1, if and only if L(R) = 1.

(2) s(R) ≤ A(R) ≤ L(R) ≤ max{1, pol reg(R)}.

(3) If R is a complete intersection, then A(R) = s(R).

(4) If a linear form y is a non-zero divisor, then L(R) = L(R/(y)).

(5) If R is Golod, then A(R) = G(R) = L(R) = max{1, pol reg(R)}.

The proofs of (1)–(4) are mainly contained in the previous sections. The only part that needs a proof is (5).

9.3. We denote MH(K^{R}) the set of all matric Massey products of H_{⩾ 1}(K^{R}), as defined in ([23, §1]. It is a submodule of H_{⩾ 1}(K^{R}), containing the usual products. The map H(K^{φ}): K^{R} → K^{S} induced by a homomorphism of local rings φ: R → S satisfies H(K^{φ})(MH(K^{R})) ⊆ MH(K^{S}) (see [23, 3.10]). By [3, 4.6], if φ is small then the induced homomorphism

φ̃: H_{⩾ 1}(K^{R})/ MH(K^{R}) → H_{⩾ 1}(K^{S})/ MH(K^{S})

is injective. Also, Golod [12] shows that the ring R is Golod if and only if MH(K^{R}) = 0, cf. [5, (2.3)]. For our purposes, we will use the graded version of these results.

Proof of Proposition 9.2(5). Set n = A(R) + 1 and S = R/𝔪^{n}. In particular, the map ρ^{(n)} : R → S is small. Note that n ≥ 2 by 5.3. Since R is a Golod ring, we have MH(K^{R}) = 0 by 9.3 and then the induced map H_{⩾ 1}(K^{R}) → H_{⩾ 1}(K^{S}) is injective by 9.3. Set s = pol reg(R). There exists then an integer i such that H_{i}(K^{R})_{i+s} ≠ 0, hence H_{i}(K^{S})_{i+s} ≠ 0. Since K^{S} = K^{R}/𝔪^{n}K^{R} by 3.5, we observe that H_{i}(K^{S})_{j} = 0 for all j ≥ n + i. We conclude that i + s < n + i, that is n > s, hence A(R) ≥ s. From Theorem 9.2(2) we also know that L(R) ≤ max{1, s}, so the inequalities between the indices give the desired equalities. □

Comments. By 6.5 and 9.2(5) all Golod rings R which are either hypersurfaces, or local and Artinian, or graded satisfy the equality A(R) = G(R). I do not know whether Golod local rings of positive dimension satisfy this equality.

9.4. If R is a complete intersection on quadrics, then R is a Koszul algebra. Thus s(R) = A(R) = G(R) = L(R) = 1, while pol reg(R) = codim R − 1. We note that all Koszul algebras which are not regular rings satisfy s(R) = A(R) = 1; thus, the equality between these two invariants is not specific to complete intersections.

9.5. Consider the ring R = k[u, v]/(u^{3}, uv^{2}). Since codim(R) = 1, this ring is Golod by a result of Shamash [35, §5, Corollary (2)]. Note that s(R) = 2 and pol reg(R) = 3. By Theorem 9.2(5) we have A(R) = G(R) = L(R) = 3. Thus, we have A(R) > s(R) in this case.

9.6. Consider the ring R = k[u, v]/(u^{3}, v^{3}). Since edim(R) ≤ 2, we know that 2 = s(R) = A(R) = G(R) by 6.5. Also, pol reg(R) = 4 and we can see that L(R) = 4. Indeed, by Tate [38, Theorem 4] a minimal free resolution of k over R has the form

X = R⟨S, Y, U, V |∂(U) = u, ∂(V) = v, ∂(S) = u^{2}U, ∂(T) = v^{2}V ⟩ .

Express Tor_{⩾ 1}^{R}(R/𝔪^{4}, k) as H_{⩾ 1}(X/𝔪^{4}X) and Tor_{⩾ 1}^{R}(R/𝔪^{3}, k) as H_{⩾ 1}(X/𝔪^{3}X). For degree reasons, the image of u^{2}UT is a cycle in X/𝔪^{4}X, which is not a boundary in X/𝔪^{3}X, hence the map Tor_{⩾ 1}^{R}(R/𝔪^{4}, k) → Tor_{⩾ 1}^{R}(R/𝔪^{3}, k) is not zero.

Acknowledgments

I wish to thank my thesis advisor Luchezar Avramov for advice, encouragement, and many inspiring discussions on the subject of this paper.

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

   E-mail address: lmsega@math.purdue.edu