J. Pure Appl. Algebra (to appear)

ACYCLICITY OF TATE CONSTRUCTIONS

SRIKANTH IYENGAR

Abstract. We prove that a Tate construction A⟨u_{1}, …, u_{n} | ∂(u_{i}) = z_{i}⟩ over a DG algebra A, on cycles z_{1}, …, z_{n} in A_{⩾ 1}, is acyclic if and only the map of graded-commutative algebras H_{0}(A)[y_{1}, …, y_{n}] → H(A), with y_{i} ↦ cls(z_{i}), is an isomorphism. This is used to establish that if a large homomorphism R → S has an acyclic closure R⟨U⟩ with sup{i | U_{i} ≠ ∅} = s < ∞, then s is either 1 or an even integer.

Let A be a Differential Graded algebra (henceforth abbreviated to DG algebra), and let w_{1}, …, w_{n} be classes in H(A). Choose cycles z_{1}, …, z_{n} in A with cls(z_{i}) = w_{i} for each i, and consider the DG algebra A⟨U⟩ = A⟨u_{1}, …, u_{n} | ∂(u_{i}) = z_{i}⟩; the Tate construction on A over the cycles w_{1}, … w_{n}, cf. [1, (6.1)].

Our main result, Theorem 2.3, asserts when |w_{i}| ≥ 1 for each i, the Tate construction A⟨U⟩ is acyclic if and only if the canonical map of graded-commutative algebras:

H_{0}(A)[y_{1}, …, y_{n}] → H(A) where y_{i} ↦ w_{i} ,

is an isomorphism; here H_{0}(A)[y_{1}, …, y_{n}] denotes the free graded-commutative algebra over H_{0}(A) on the graded set {y_{1}, …, y_{n}}, cf. 1.1. Our theorem generalizes a result of Blanco, Majadas, and Rodicio [4, Theorem 1] who consider the case where A is the Koszul complex on a set of elements in a ring and each class w_{i} has degree one.

The crux of the matter is the interplay between regularity, in the sense of Tate [10], of a sequence of elements in a graded-commutative algebra and quasiregularity, a concept introduced here by extrapolating from commutative rings, which are viewed as graded-commutative algebras concentrated in degree 0. In this context, we prove that, as in the classical case, a regular sequence is quasiregular, while the converse holds under suitable separatedness assumptions. This is the content of Section 1.

Section 2 combines the tools developed in Section 1 with those of Tate, to arrive at the following: Let A⟨U⟩ be the Tate construction over A on a classes w = w_{1}, …, w_{n} in H(A); we make no assumptions on the degrees of the elements. Then, under separatedness assumptions, the following are equivalent.

(a) the sequence w_{1}, …, w_{n} is regular;

(b) the sequence w_{1}, …, w_{n} is quasi-regular;

(c) the canonical map H(A) → H(A⟨U⟩) is surjective;

(d) the canonical map \frac{H(A)}{(w) H(A)} → H(A⟨U⟩) is an isomorphism; This is the essence of Theorem 2.2. The equivalence of (a), (c), and (d) is well known; thus the only new input is (a) ⟺ (b), which is subject of Section 1. Now, when A is a commutative ring (concentrated in degree 0), A⟨U⟩ is exactly the Koszul complex on the z_{i}’s, so we recover a fact well known in the theory of commutative rings. In the special case where |z_{i}| ≥ 1 for all i, we build on the equivalence above to arrive at Theorem 2.3 announced at the beginning of this introduction.

Date: May 30, 2000.

1991 Mathematics Subject Classification. 13C99, 13D99.

Key words and phrases. Tate constructions, regular sequences, large homomorphisms.

Research supported by a grant from the E.P.S.R.C.

From our point of view these results are handy tools to possess when studying DG algebras. One justification for this claim is that they may be used to prove the following theorem:

Let R → S be a large homomorphism of local rings and R⟨U⟩ the acyclic closure of S over R. If s = {i | U_{i} ≠ ∅} < ∞, then s is either 1 or an even integer.

Section 3 is dedicated to the proof of this result. The reader will find large homomorphisms defined in loc. cit.; for the moment it suffices to note that for a local ring R with residue field k, the canonical surjection R → k is large, as is any surjective local homomorphism R → S which admits a section. In fact, in these cases one has a stronger conclusion: s ≤ 2. This is well known for the map R → k, cf. [1, (7.3)] and the references therein, and for split homomorphisms it follows from work of Avramov and the author [3]. Since these two cases do not exhaust the class of large homomorphisms it maybe worthwhile to report the result above. Now, the special cases mentioned do raise the question: Is s ≤ 2 in general? When the characteristic of the residue field of R is 0, this question is equivalent to a conjecture of Quillen [9, (5.6)] on the vanishing of cotangent homology, for the particular case of a large homomorphism.

1. Regularity vs. Quasi-regularity

In this note, it is expedient to adopt the convention that all rings and modules are defined over an ubiquitous commutative ring 𝕜. For an element x in a graded set X = {X_{i}}_{⩾ 0}, we write |x| = i to indicate that x ∈ X_{i}; this is the degree of x. An algebra R = {R_{i}}_{i ⩾ 0} consists of 𝕜-modules R_{i} with a product compatible with the degrees and satisfying the usual properties.

A graded-commutative algebra is an algebra R = {R_{i}}_{i ⩾ 0} such that

r · s = (−1)^{|r||s|}s · r for r, s ∈ R and r^{2} = 0 when |r| is odd .

A commutative ring, in the usual sense of the word, will be viewed as a gradedcommutative algebra over ℤ, concentrated in degree 0.

1.1. Let R be a graded-commutative algebra. The free graded-commutative algebra over R on a set of variables X = {X_{i}}_{i ⩾ 0} is the tensor product over R of the polynomial algebra on X_{even} and the exterior algebra on X_{odd}; it is denoted R[X]. It is characterized by the following universal property:

If ϕ: R → S is a morphism of graded-commutative algebras and α: X → S a degree 0 map of graded sets, then there is a unique morphism of graded-commutative algebras ϕ̃: R[X] → S extending ϕ and such that ϕ̃(x) = α(x) for each x ∈ X.

An element in R[X] is called a polynomial, although it may involve both polynomial and exterior variables. It is useful to keep in mind that R[X] has a second grading, the external grading denoted deg(·), with deg(r) = 0 for r ∈ R and deg(x) = 1 for each x ∈ X. In general, R[X] is not a graded-commutative algebra under the external grading.

1.2. An element r ∈ R is regular if r is not invertible and

(0: _{R} r) = \begin{cases}0 & if |r| is even \\ (r) & if |r| is odd \end{cases}

Thus, a regular element has the smallest possible annihilator. A sequence r_{1}, …, r_{n} in R is a regular sequence if r_{i+1} is regular on R/(r_{1}, …, r_{i}) for 0 ≤ i ≤ n − 1.

For example, for X = {x_{1}, …, x_{n}} a set of variables over R, the sequence x_{1}, …, x_{n} is regular on R[X].

1.3. Let R be a graded-commutative algebra and I an ideal in R. Consider the bi-graded 𝕜-module gr_{I}(R) with

gr_{I}(R)_{i,n} = (\frac{I^{n}}{I^{n+1}} )_{i} for i, n ≥ 0 ;

the index i is the internal degree, denoted | · |, and n is the external degree, denoted deg(·). The product on R induces one on gr_{I}(R) which is compatible with both the degrees: |ab| = |a| + |b| and deg(ab) = deg(a) + deg(b) for a, b ∈ gr_{I}(R).

The commutativity of R ensures that for a, b ∈ gr_{I}(R), one has ab = (−1)^{|a||b|}ba, so that gr_{I}(R), when graded by the internal degree alone, is a graded-commutative algebra. This is the associated graded algebra of R with respect to I. Note that the canonical inclusion R/I = (gr_{I}(R))_{∗,0} ↪ gr_{I}(R) is a morphism of gradedcommutative algebras.

1.4. Let r = r_{1}, …, r_{n} be elements in a graded-commutative algebra R, let I = (r), and let x_{1}, …, x_{n} a set of variables with |x_{i}| = |r_{i}| for each i. By the universal property of the free graded-commutative algebra, cf. 1.1, the canonical inclusion R/I ↪ gr_{I}(R) extends to a map of graded-commutative algebras

φ_{r}: \frac{R}{I}[x_{1}, …, x_{n}] → gr_{I}(R) with φ_{r}(x_{i}) = r_{i}

This map is surjective; taking a cue from the classical case, the sequence r_{1}, …, r_{n} is said to be quasi-regular if the map φ_{r} is an isomorphism. The issue is only whether the map in question is injective, so in analyzing this map, we can exploit the fact that it respects the external grading as well. Observe that if r_{1}, …, r_{n} is quasi-regular, then so is any permutation of the r_{i}’s.

Unraveling the definition leads to the following test for quasi-regularity.

1.5. Let r_{1}, …, r_{n} be elements in a graded-commutative algebra R, let I be the ideal generated by the r_{i}’s, and let R[X] = R[x_{1}, …, x_{n}], with |x_{i}| = |r_{i}| for all i. The following conditions are equivalent:

(i) The sequence r_{1}, …, r_{n} is quasi-regular; (ii) For a polynomial F ∈ R[X] with deg(F) = v (the external degree !), if

F (r_{1}, …, r_{n}) ∈ I^{v+1}, then F ∈ IR[X]; (iii) For an F ∈ R[X] with deg(F) = v if F (r_{1}, …, r_{n}) = 0, then F ∈ IR[X].

The proof is as in [8, §16].

One may verify with no difficulty that the sequence x_{1}, …, x_{n} in the free gradedcommutative algebra R[x_{1}, …, x_{n}] is quasi-regular. The following result is a partial converse.

1.6. Proposition. Let R be a graded-commutative algebra, let r_{1}, …, r_{n} ∈ R be a quasi-regular sequence with |r_{i}| ≥ 1 for each i, and let y_{1}, …, y_{n} be variables with |y_{i}| = |r_{i}|. The morphism of graded-commutative algebras R_{0}[y_{1}, …, y_{n}] → R, with y_{i} ↦ r_{i} is injective.

Proof. To begin with, any polynomial in R_{0}[Y] = R_{0}[y_{1}, …, y_{n}] may be viewed as a polynomial in R[Y] via the canonical inclusion R_{0}[Y] ↪ R[Y]. Moreover, R_{0}[Y] ∩ IR[Y] = {0}, for the ideal I = (r_{1}, …, r_{n}).

Suppose that F ∈ R_{0}[Y] is a non-zero polynomial in the kernel of the map R_{0}[Y] → R, that is to say, F (r_{1}, …, r_{n}) = 0. We may write F = ∑_{i=a}^{b} F_{i} with 0 ≤ a ≤ b, deg(F_{i}) = i, and F_{a} ≠ 0. Then

F_{a}(r_{1}, …, r_{n}) = − ∑_{i=a+1}^{b} F_{i}(r_{1}, …, r_{n}) ∈ I^{a+1} ,

so 1.5 yields F_{a} ∈ IR[Y]; hence, F_{a} ∈ R_{0}[Y] ∩ IR[Y] = {0}, a contradiction. □

The next two propositions assert that, as for commutative rings, quasi-regularity is closely related to regularity.

1.7. Proposition. If a sequence r_{1}, …, r_{n} in a graded-commutative algebra R is regular, then it is quasi-regular.

The converse holds under appropriate separatedness assumptions. In what follows, the Jacobson radical for a commutative ring Q is denoted Jac(Q).

1.8. Proposition. Let r_{1}, …, r_{n} be elements in a graded-commutative algebra R such that either |r_{i}| ≥ 1 for all i, or each R_{j} is noetherian over R_{0} and if |r_{i}| = 0, then r_{i} is contained Jac(R_{0}).

If the sequence r_{1}, …, r_{n} is quasi-regular, then it is regular.

The proofs of these results are similar to those for commutative rings as found in Matsumura [8, §16], although twice as long, for the even and odd degrees require separate arguments. The details are spelled out, in the odd parity case at any rate, to convince the skeptical reader, as well as the writer.

We begin with a few observations akin to those of [8, §16].

1.9. Let r_{1}, …, r_{n} be a quasi-regular sequence, I = (r_{1}, …, r_{n}), and s an element in R.

(a) If |s| is odd and (I: _{R} s) = I + (s), then (I^{v}: _{R} s) = I^{v} + (s) for all v ≥ 1.

(b) If |s| is even and (I: _{R} s) = I, then (I^{v}: _{R} s) = I^{v} for all v ≥ 1.

(a) We induce on v, the basis v = 1 being the given. Assume that the desired equality holds for some integer v ≥ 1. If rs ∈ I^{v+1} ⊂ I^{v}, then by the induction hypothesis r ∈ I^{v} +(s). Thus, there is a polynomial F in R[X] = R[x_{1}, …, x_{n}] with deg(F) = v such that r = F (r_{1}, …, r_{n}) + as for some a ∈ R. Now, sF ∈ R[X] is such that sF (r_{1}, …, r_{n}) = s(r − as) = sr = ±rs ∈ I^{v+1}; since deg(F) = v and the sequence r_{1}, …, r_{n} is quasi-regular, according to 1.5 one has sF ∈ IR[X], that is to say, sc ∈ I for each coefficient c of the polynomial F. Thus, c ∈ (I: _{R} s) = I + (s), so that F ∈ (I +(s))R[X] and r = F (r_{1}, …, r_{n})+as ∈ (I +(s))I^{v} +(s) = I^{v+1}+(s), which is the desideratum.

The proof of (b) is analogous.

1.10. Proof of Proposition 1.7. We induce on n, the cardinality of the sequence r_{1}, …, r_{n}; the basis step n = 1 is a straightforward verification. The induction hypothesis is that any regular sequence of length n is quasi-regular.

Let r_{1}, …, r_{n+1} be a regular sequence; since the sequence r_{1}, …, r_{n} is regular as well, it must be quasi-regular. Set I = (r_{1}, …, r_{n}), J = (r_{1}, …, r_{n+1}), and set R[X] = R[x_{1}, …, x_{n}]. By 1.5, it suffices to establish that if a polynomial F ∈ R[x_{1}, …, x_{n+1}] = R[X, x_{n+1}], with deg(F) = v, is such that F (r_{1}, …, r_{n+1}) = 0, then F ∈ JR[X, x_{n+1}]. We may assume that F ∉ R[X].

Suppose that |x_{n+1}| = |r_{n+1}| is odd; the regularity of r_{n+1} on R/(r_{1}, …, r_{n}) translates to the equality (I: _{R} r_{n+1}) = I + (r_{n+1}). Now we induce on v = deg(F); the basis step v = 0 is tautological. Assume that the desired conclusion holds for all polynomials of external degree v or less. If deg(F) = v + 1, then F = G + x_{n+1}H, with G, H ∈ R[X], deg(G) = v + 1, and deg(H) = v. Now

G(r_{1}, …, r_{n}) + r_{n+1}H(r_{1}, …, r_{n}) = F (r_{1}, …, r_{n+1}) = 0 ,

so that by 1.9.a, one has H(r_{1}, …, r_{n}) ∈ (I^{v+1} : _{R} r_{n+1}) = I^{v+1}+(r_{n+1}). Therefore, there is a polynomial H̃ ∈ R[X] with deg(H̃) = v + 1 such that H(r_{1}, …, r_{n}) = H̃(r_{1}, …, r_{n}) + ar_{n+1} for some element a ∈ R. For G̃ = G + r_{n+1}H̃ ∈ R[X] we find

G̃(r_{1}, …, r_{n}) = G(r_{,} …, r_{n}) + r_{n+1}H̃(r_{1}, …, r_{n})
= G(r_{,} …, r_{n}) + r_{n+1}(H(r_{1}, …, r_{n}) − ar_{n+1})
= G(r_{,} …, r_{n}) + r_{n+1}H(r_{1}, …, r_{n})
= F (r_{1}, …, r_{n+1}) = 0 .

Since r_{1}, …, r_{n} is quasi-regular, G̃ ∈ IR[X] ⊂ JR[X, x_{n+1}], so that G = G̃ − r_{n+1}H̃ ∈ JR[X, x_{n+1}]. Now we attend to H: The polynomial r_{n+1}H ∈ R[X] is such that

r_{n+1}H(r_{1}, …, r_{n}) ∈ r_{n+1}(I^{v+1} + (r_{n+1})) = r_{n+1}I^{v+1} ⊂ I^{v+1} ;

the quasi-regularity of r_{1}, …, r_{n} yields r_{n+1}H ∈ IR[X], that is to say, r_{n+1}c ∈ I for each coefficient c of H; hence c ∈ (I: _{R} r_{n+1}) = I +(r_{n+1}) = J, and H ∈ JR[X]. Therefore, F = G + x_{n+1}H ∈ JR[X, x_{n+1}], which is what we seek.

The case where |x_{n+1}| is even is handled analogously. □

Now we prove Proposition 1.8.

1.11. Proof of Proposition 1.8. Set I = (r_{1}, …, r_{n}). The hypothesis ensures that

( ⋂_{v ⩾ 1} I^{v})_{i} ⊆ ⋂_{v ⩾ 1} Jac(R_{0})R_{i} = 0 for all i ≥ 0 ,

where the equality on the right is due to Krull’s intersection theorem, cf. [8, (8.10)]; thus ⋂_{v ⩾ 1} I^{v} = 0. We claim that r_{1} is regular.

Indeed, suppose that |r_{1}| is odd, and that sr_{1} = 0 with s ∈ R. It suffices to prove the following statement: For each integer v ≥ 1, there is an element a_{n} ∈ R and a polynomial F_{v} ∈ R[x_{2}, …, x_{n}] with

deg(F_{v}) = v and s = a_{v}r_{1} + F_{v}(r_{2}, …, r_{n}) .

For then, modulo r_{1}, one has s ∈ ⋂_{v ⩾ 1} I^{v} = 0; thus s ∈ (r_{1}) as desired.

We resort to an induction on v to find the a_{v}’s and F_{v}’s. Since sr_{1} = 0, 1.5 applied to the polynomial sx_{1} ∈ R[X] = R[x_{1}, …, x_{n}] yields s ∈ I; say s = a_{1}r_{1} + ∑_{i ⩾ 2}^{n} a_{i}r_{i}. Then, a_{1} and F_{1} = ∑_{i ⩾ 2}^{n} a_{i}x_{i} form the basis of the induction.

Assume that a_{v} and F_{v} with the desired properties have been found, for some integer v ≥ 1. Then

0 = r_{1}s = r_{1}(a_{v}r_{1} + F_{v}(r_{2}, …, r_{n})) = r_{1}F_{v}(r_{2}, …, r_{n}) .

By 1.5, the coefficients of the polynomial x_{1}F_{v} ∈ R[X] are in I. Observe that since F_{v} ∈ R[x_{2}, …, x_{n}] the set of coefficients of the polynomials F_{v} and x_{1}F_{v} coincide, which allows us to deduce that the coefficients of F_{v} are all in I. Thus, after a spot of rearrangement, we can find a polynomial F_{v+1} ∈ R[x_{2}, …, x_{n}] and an element a ∈ R such that F_{v}(r_{2}, …, r_{n}) = ar_{1} + F_{v+1}(r_{2}, …, r_{n}), so that

s = a_{v}r_{1} + F_{v}(r_{2}, …, r_{n}) = (a_{v} + a)r_{1} + F_{v+1}(r_{2}, …, r_{n}) .

Setting a_{v+1} = a completes the induction step.

The proof in the case when |r_{1}| is even is similar, and a tad simpler.

Now, set S = R/(r_{1}). To complete the proof the proposition, it suffices to prove that the (images of the) elements r_{2}, …, r_{n} are a quasi-regular sequence in S. Once again we are faced with two cases; we content ourselves by considering the situation when |r_{1}| is odd.

Suppose that F (r_{2}, …, r_{n}) = 0 for an F ∈ S[x_{2}, …, x_{n}] with deg(F) = v. Pick an element F̃ ∈ R[x_{2}, …, x_{n}], with deg(F̃) = v, such that it maps to F under the canonical surjection R[x_{2}, …, x_{n}] → S[x_{2}, …, x_{n}]. Therefore, F (r_{2}, …, r_{n}) = 0 translates to F̃(r_{2}, …, r_{n}) = ar_{1} for some a ∈ R; in particular ar_{1} ∈ I^{v}. Since r_{1}, …, r_{n} is quasi-regular, a repeated application of 1.5.ii yields a ∈ (r_{1}) + I^{v−1}, so we may find a polynomial G ∈ R[X] = R[x_{1}, …, x_{n}] with deg(G) = v − 1 so that a = a′r_{1} + G(r_{1}, …, r_{n}) for some a′ ∈ R. Hence, F̃(r_{2}, …, r_{n}) = ar_{1} = G(r_{1}, …, r_{n})r_{1}, that is to say the polynomial F̃ − Gx_{1} evaluated at r_{1}, …, r_{n} is zero. Thus, by 1.5, we obtain F̃ − Gx_{1} ∈ IR[X]. Remains to note that F̃ does not involve any non-zero terms with the variable x_{1} so that F̃ ∈ IR[X] and hence its image F ∈ JS[X], where J is the ideal in S generated by r_{2}, …, r_{n}. □

2. Tate extensions

In the theory of commutative rings, regularity of a sequence is related to the acylicity of the associated Koszul complex. The analogous fact, in the context of graded-commutative algebras, was established by Tate [10]. This result is recalled below, as formulated by Avramov [1, (6.1.7)]; we refer the reader to loc. cit. and [5], for unexplained notation, definitions, and basic constructions concerning DG algebras.

2.1. Let A be a DG algebra, let w = w_{1}, …, w_{n} be classes in H(A), let z_{1}, …, z_{n} be cycles such that cls(z_{i}) = w_{i} for each i, and let A⟨U⟩ = A⟨u_{1}, …, u_{n} | ∂(u_{i}) = z_{i}⟩.

Implications (i) ⟹ (ii) ⟹ (iii) hold among the following conditions:

(i) w_{1}, …, w_{n} is a regular sequence in H(A);

(ii) the canonical map \frac{H(A)}{(w) H(A)} → H(A⟨U⟩) is an isomorphism;

(iii) the canonical map H(A) → H(A⟨U⟩) is surjective. The conditions are equivalent if |w_{i}| ≥ 1 for all i, or each H_{j}(A) is noetherian over H_{0}(A) and if |w_{i}| = 0, then w_{i} ∈ Jac(H_{0}(A)).

Combining the result above with Proposition 1.8, we arrive at 2.2. Theorem. Let A be a DG algebra, let w = w_{1}, …, w_{n} be classes in H(A), such that |w_{i}| ≥ 1 for all i, or each H_{j}(A) is noetherian over H_{0}(A) and if |w_{i}| = 0, then w_{i} ∈ Jac(H_{0}(A)). Let z_{1}, …, z_{n} be cycles such that cls(z_{i}) = w_{i} for each i, and let A⟨U⟩ = A⟨u_{1}, …, u_{n} | ∂(u_{i}) = z_{i}⟩.

The following conditions are equivalent.

(i) the canonical morphism of graded-commutative algebras is an isomorphism:

\frac{H(A)}{(w) H(A)}[x_{1}, …, x_{n}] → gr_{(w)}(H(A)) with x_{i} ↦ w_{i};

(ii) the canonical map \frac{H(A)}{(w) H(A)} → H(A⟨U⟩) is an isomorphism;

(iii) the canonical map H(A) → H(A⟨U⟩) is surjective. □

With additional hypotheses, one arrives at a more gratifying result; it generalizes a theorem of Blanco, Majadas, and Rodicio, [4, Theorem 1] who prove the special case where A is the Koszul complex on a set of elements over a commutative ring and each class w_{i} has degree 1.

2.3. Theorem. Let A be a DG algebra, let w_{1}, …, w_{n} be classes in H(A), such that |w_{i}| ≥ 1 for all i. Let z_{1}, …, z_{n} be cycles such that cls(z_{i}) = w_{i} for each i, and let A⟨U⟩ = A⟨u_{1}, …, u_{n} | ∂(u_{i}) = z_{i}⟩.

The following are equivalent:

(i) The morphism of H_{0}(A)-algebras H_{0}(A)[y_{1}, …, y_{n}] → H(A) with y_{i} ↦ w_{i}, is an isomorphism;

(ii) the DG algebra A⟨U⟩ is acyclic.

Proof. (i) ⟹ (ii) The sequence w_{1}, …, w_{n} is regular on H(A); now invoke 2.1.

(ii) ⟹ (i) Since |z_{i}| ≥ 1, the map H(A) → H(A⟨U⟩) is an isomorphism in degree 0, hence surjective, and so, thanks to 2.1, we deduce that the sequence w_{1}, …, w_{n} is regular on H(A), and that (w_{1}, …, w_{n}) is the ideal of elements of positive degree in H(A). In particular, the map in question H_{0}(A)[y_{1}, …, y_{n}] → H(A) is surjective; injectivity is asserted by Proposition 1.6, for, according to Proposition 1.7, a regular sequence is quasi-regular. □

The following corollary was suggested to me by J.P.C. Greenlees; it may be viewed as the graded-commutative analogue of the Auslander-Buchsbaum test for detecting regular local rings: A noetherian local ring (R, 𝔪, k) is regular if and only if, for elements r_{1}, …, r_{e} ∈ 𝔪 that map to a basis for the k-vector space 𝔪/𝔪^{2}, the Koszul complex R⟨u_{1}, …, u_{e} | ∂(u_{i}) = r_{i}⟩ is acyclic.

2.4. Corollary. Let R be a graded-commutative algebra with R_{0} a field k, let 𝔪 be the kernel of the canonical surjection R → k, and assume that the k-vector space 𝔪/𝔪^{2} is finitely dimensional. Let r_{1}, …, r_{n} ∈ R map to a basis for 𝔪/𝔪^{2}, and set R⟨U⟩ = R⟨u_{1}, …, u_{n} | ∂(u_{i}) = r_{i}⟩. The following conditions are equivalent.

(i) R is isomorphic to a free graded-commutative algebra over k;

(ii) The morphism of graded-commutative algebras k[y_{1}, …, y_{n}] → R, where

y_{j} ↦ r_{j}, is an isomorphism; (iii) The DG algebra R⟨U⟩ is acyclic.

Proof. (ii) ⟹ (i) is tautological; (ii) ⟺ (iii) is a special case of Theorem 2.3.

(i) ⟹ (ii) Let R ⟶^{≅} k[X], and let α be the composed k[Y] = k[y_{1}, …, y_{n}] → R ⟶^{≅} k[X]. The choice of the r_{i} ensures that the induced map of graded kvector spaces kY → kX is an isomorphism, therefore, by the (graded version of) Nakayama’s lemma, α is surjective; let J be its kernel. The universal property of free algebras provides a morphism of k-algebras σ: k[X] → k[Y] such that α∘σ = id. In particular, Tor^{α} (k, k) ∘ Tor^{σ} (k, k) is the identity map of Tor^{k[X]} (k, k) , and hence Tor^{α} (k, k) is surjective. Thus, [2, (1.6)] applied to α and β: k[X] → k, yields an exact graded k-vector spaces 0 → J/(Y)J → kY → kX → 0. Since the map kY → kX is an isomorphism, J = (Y)J, so that another application of Nakayama’s lemma yields J = 0, that is to say, α is an isomorphism. □

3. Large homomorphisms

A surjective local homomorphism ϕ: (R, 𝔪, k) → (S, 𝔫, k) is large if the induced map Tor^{ϕ} (k, k) : Tor^{R} (k, k) → Tor^{S} (k, k) is surjective. For example, the surjection R → k is large, as is a split local homomorphism ϕ: R → S, that is to one, one which admits a section ρ: S → R such that ϕ ∘ ρ = id^{S}.

This paper originates in an attempt to deal with the following question; confer [1, §6.3] for information concerning acyclic closures.

Problem. Let R → S be a large homomorphism and let R⟨U⟩ be the acyclic closure of S over R. If U_{i} = ∅ for all i ≫ 0, then is it true that U_{i} = 0 for i ≥ 3?

This problem may be viewed as a DG algebra version of a conjecture of Quillen on the vanishing of cotangent homology [9, (5.6)], specialized to large homomorphisms. When the characteristic of the field k is zero, the question above coincides with Quillen’s conjecture for the particular case of large homomorphisms; cf. [9, (9.5)] or [2, (3.3.2)].

As mentioned in the introduction, the question above has an affirmative answer for the large homomorphism R → k; confer [1, §7.3]. In fact, in this case, it suffices to assume that U_{i} = 0 for some integer i ≥ 1. Avramov and the author [3], have settled the question, once again in the affirmative, in the case when the homomorphism R → S is split. Note that there are large homomorphisms for which sup{i | U_{i} ≠ ∅} is either 1 or 2, cf. [10, Theorem 4], or [2, (4.2)].

Not all large homomorphisms are of the preceding types, cf. [7], so it seems worth while to report the following

3.1. Theorem. Let R → S be a large homomorphism and let R⟨U⟩ be the acyclic closure of S over R. If sup{i | U_{i} ≠ ∅} = s < ∞, then either s = 1 or s is even.

The proof requires some preparation, and is deferred to the end of this section. In the sequel, for a DG algebra A we write sup(A) for sup{i | H_{i}(A) ≠ 0}; in

particular, sup(A) = −∞ if H(A) = 0.

Let A↣A⟨x | ∂(x) = z⟩ be the Tate extension with |x| = d and w = cls(z).

3.2. When d is odd, there is an exact sequence of chain maps

0 → A → A⟨x⟩ → A ⟶^{υ} 0 where υ(a + xb) = b

and the corresponding homology long exact sequence is

⋯ → H_{i−d}(A) ⟶^{w} H_{i}(A) → H_{i}(A⟨x⟩) → H_{i−d−1}(A) → ⋯

In particular, if sup(A) < ∞, then sup(A⟨x⟩) < ∞; the converse holds if z ∈ 𝔪Z(A).

Iterating this, we arrive at: Let z_{1}, …, z_{n} be cycles in a DG algebra A and let A⟨X⟩ = A⟨x_{1}, …, x_{n} | ∂(x_{i}) = z_{i}⟩. If |x_{i}| is odd for each i and sup(A) < ∞, then sup(A⟨X⟩) < ∞. The converse holds if if z_{i} ∈ 𝔪Z(A) for each i.

3.3. When d is even, there is an exact sequence of chain maps

0 → A → A⟨x⟩ ⟶^{υ} A⟨x⟩ → 0 where υ( ∑_{j} a_{j}x^{(j)}) = ∑_{j} a_{j}x^{(j−1)}

and this results in the homology long exact sequence

⋯ → H_{i−d}(A⟨x⟩) → H_{i}(A) → H_{i}(A⟨x⟩) → H_{i−d−1}(A⟨x⟩) → ⋯

Therefore, if sup(A⟨x⟩) < ∞, then sup(A) < ∞.

An iteration leads to: Let z_{1}, …, z_{n} be cycles in a DG algebra A and let A⟨X⟩ = A⟨x_{1}, …, x_{n} | ∂(x_{i}) = z_{i}⟩. If |x_{i}| is even for each i and sup(A⟨X⟩) < ∞, then sup(A) < ∞.

A semi-free Γ -extension A⟨X⟩ of a DG algebra X is a DG algebra obtained by a successive adjunction of exterior and divided powers variables, cf. [1, (6.1)]. We shall require the following facts concerning acyclic closures of large homomomorphisms.

3.4. Let (R, 𝔪, k) → (S, 𝔫, k) be a large homomorphism, and let R⟨U⟩ be its acyclic closure. Then

(a) R⟨U⟩ is minimal, that is to say, ∂(R⟨U⟩) ⊆ 𝔪R⟨U⟩;

(b) there is a semi-free Γ-extension R⟨U⟩ ↣R⟨U, Ũ⟩, with R⟨U, Ũ⟩ the acyclic

closure of k over R. The first statement is a result of Avramov and Rahbar-Rochandel [7, (2.5)], also cf. [2, (2.7)]. Given this fact, the second assertion follows from [7, (1.1.2)].

For the rest of this paper, it is convenient to adopt the following terminology: A partial acyclic closure of S over R is a DG Γ-algebra R⟨U⟩ which admits a semi-free Γ-extension R⟨U⟩⟨W ⟩ which is an acyclic closure of S over R.

3.5. Proposition. Let R → S be a large homomorphism, and let R⟨V ⟩⟨X⟩ be a partial acyclic closure of S over R with card(X) < ∞. If sup(R⟨V, X⟩) < ∞, then sup(R⟨V ⟩) < ∞.

Proof. It is enough to prove the result when X = {x}. The case when the degree of x is even is settled by 3.3; when |x| = 1, since R⟨V, x⟩ is a partial acyclic closure of S over R it must be that ∂(x) ∈ 𝔪, so 3.2 yields the desired result. Thus, it remains to consider the case when |x| ≥ 3 and odd.

Since R → S is large, the acyclic closure of S over R extends to an acyclic closure of k over R, 3.4, so that there is a semi-free Γ-extension R⟨V, x⟩ ↣R⟨Y ⟩ with R⟨Y ⟩ the acyclic closure of k over R. Adjoining the set of variables W = Y_{1} \ V_{1}, we obtain a diagram of semi-free Γ-extensions

R⟨V ⟩⟨W ⟩ ↣R⟨V, x⟩⟨W ⟩ ↣R⟨Y ⟩ .

Set A = R⟨V, W ⟩, and observe that the DG Γ-algebra R⟨V, x, W ⟩ may be obtained by adjoining x to A, that is to say, R⟨V, x, W ⟩ = A⟨x⟩. Thanks to 3.2, it suffice to prove that sup(A) < ∞; by the same token, since sup(R⟨V, x⟩) < ∞ we obtain that sup(A⟨x⟩) < ∞. By construction, one has H_{0}(A) = k, and

Z_{⩾ 1}(A) ⊆ Z_{⩾ 1}(R⟨Y ⟩) ∩ A ⊆ ∂(R⟨Y ⟩) ∩ A ⊆ 𝔪R⟨Y ⟩ ∩A = 𝔪A

where, reading from the left, the second inclusion holds since R⟨Y ⟩ is acyclic, the third since it is minimal 3.4, and the last since R⟨Y ⟩ is a free A-module. Now we have all the prerequisites to invoke [6, Lemma 2], which yields sup(A) < ∞. □

Proof of Theorem 3.1. We may assume that s ≥ 3. Set V = ⋃_{i=1}^{s−1} U_{i}. Since R⟨U⟩ = R⟨V ⟩⟨U_{s}⟩ is acyclic, with homology S, Theorem 2.3 yields H(R⟨V ⟩) is isomorphic to S[w_{1}, …, w_{n}], where |w_{i}| = s − 1.

If s is odd, then |w_{i}| is even for each i, so that sup(R⟨V ⟩) = ∞. On the other hand, thanks to Proposition 3.5, the acyclicity of R⟨V ⟩⟨U_{s}⟩ entails sup(R⟨V ⟩) < ∞; a contradiction. □

References

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Pure Maths, Hicks Building, University of Sheffield, Sheffield, S7 3RH, U.K.

  E-mail address: s.iyengar@shef.ac.uk