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Invent. Math. 140 (2000), 143–170.
FINITE GENERATION OF
HOCHSCHILD HOMOLOGY ALGEBRAS
LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR
Abstract. We prove converses of the Hochschild-Kostant-Rosenberg Theo- rem, in particular: If a commutative algebra S is flat and essentially of finite type over a noetherian ring k, and the Hochschild homology HH∗(S |k) is a finitely generated S-algebra for shuffle products, then S is smooth over k.
Introduction Let S be a commutative algebra over a commutative noetherian ring k. Shuffle products on the Hochschild complex define the Hochschild homology alge- bra HH∗(S |k), which is graded-commutative and is natural in S and k, cf. [11], [23]. Since HH 0 (S |k) is S itself, and HH 1 (S |k) is the S-module of K¨ahler differentials Ω^1 S |k, there is a canonical homomorphism of graded algebras
ω∗ S |k :
S Ω
1 S |k →^ HH∗(S^ |k)
mapping differential forms to Hochschild homology. It provides a piece of the product: ωnS |k is injective if n! is invertible in S. Little more is known in general. In a special case the story is complete. Recall that S is regular over k if it is flat, and the ring S ⊗k k is regular for each homomorphism k → k to a field k. The algebra S is smooth if it is regular and essentially of finite type, cf. [16].
If S is smooth over k, then Ω^1 S | k is projective and ω S∗ |k is bijective.
This classical result is due to Hochschild, Kostant, and Rosenberg [19] when k is a perfect field, and can be extended, with some work, to cover noetherian rings. Using their homology theory of commutative algebras [1], [26], Andr´e and Quillen provide a generalization and a converse: a noetherian k-algebra S is regular if and only if it is flat, the S-module Ω^1 S | k is flat, and the map ω∗ S |k is bijective, cf. [2]. Our main result explains why shuffle product structures have remained elusive. It establishes a conjecture of Vigu´e-Poirrier [32], proved by her and Dupont [32], [13] when S is positively graded and S 0 = k is a field of characteristic zero.
Theorem on Finite Generation. If S is a flat k-algebra essentially of finite type and the S-algebra HH∗(S |k) is finitely generated, then S is smooth over k.
As a consequence, S is smooth if ω∗ S |k is surjective or, more generally, if the S-module HH∗(S |k) is finite. The last result also follows from an earlier
Theorem on Semi-Rigidity. If S is a flat k-algebra essentially of finite type, and HH 2 i− 1 (S |k) = 0 = HH 2 j (S |k) for some i, j > 0 , then S is smooth over k.
Date: March 22, 2000, 9 h 1 min. 1991 Mathematics Subject Classification. 13D99, 13C40, 18G15. L.L.A. was partly supported by a grant from the NSF. 1
2 L. L. AVRAMOV AND S. IYENGAR
When k is a field this is proved independently by Avramov and Vigu´e-Poirrier [8] in arbitrary characteristic, and by Campillo, Guccione, Guccione, Redondo, Solotar, and Villamayor [10] if char(k) = 0. Rodicio [28] conjectured that Hochschild homol- ogy over a field k is rigid : If HHm(S |k) = 0 for some m > 0, then HHn(S |k) = 0 for all n > m; he and Lago [20], [28] prove this when S is complete intersection, and Vigu´e-Poirrier [31] when S is positively graded and char(k) = 0. Over noetherian rings k semi-rigidity is established by Rodicio [29]. His crucial observation is that this property can be proved in the wider context of augmented commutative algebras, using a result of Avramov and Rahbar-Rochandel on large homomorphisms of local rings [22] to replace the specific constructions of resolutions over S ⊗k S, on which the approach in [8], [10] is based. On the other hand, Larsen and Lindenstrauss [21] show that HH 2 i− 1 (S |Z) 6 = 0 = HH 2 i(S |Z) for any ring of algebraic integers S 6 = Z and all i > 0, so Hochschild homology over Z is not rigid. In Sections 1 and 2 we use DG (=differential graded) homological algebra to study large homomorphisms, further developing results and ideas applied to Hoch- schild homology in [8], free resolutions in [7], [3], and Andr´e-Quillen homology in [6], [4]. As a bonus, we get a concise proof of a local semi-rigidity theorem. Sections 3 and 4 are at the heart of our argument, and go a long way towards determining the structure of large homomorphisms with finitely generated Tor al- gebras. In positive residual characteristic the local semi-rigidity theorem easily yields the desired finiteness result. In characteristic zero, besides DG homological algebra we use the finiteness results on Andr´e-Quillen homology from [6], [4]; the architecture of our proof mirrors, to some extent, the topological approach in [13], viewed trough the looking glass [5] between local algebra and rational homotopy. We return to Hochschild homology in the last two sections. In Section 5 we put together our local results to prove the theorems above. We also show by various examples that their hypotheses cannot be significantly relaxed. In Section 6 we study nilpotence properties of shuffle products in Hochschild ho- mology. When k is a field of characteristic 0 and S a locally complete intersection k-algebra essentially of finite type, we prove that Hochschild homology is nilpo- tent : There is an integer s ≥ 1 such that HH> 1 (S |k)s^ = 0. We provide examples that illustrate that this need not hold when k is a field of positive characteris- tic. On the other hand, the presence of divided powers on Hochschild homology entails that it is nil for any algebra of positive characteristic: If qS = 0, then wq^ = 0 for each w ∈ HH> 1 (S |k). In an earlier version of this paper we had asked whether Hochschild homology is also nil when k is a field of characteristic 0, and suggested S = k[x, y]/(x^2 , xy, y^2 ) as a test case; L¨ofwall and Sk¨oldberg, and in- dependently Larsen and Lindenstrauss, showed that if k is a field of characteristic 0 then HH∗(S |k) is not nil. Thus, the general form of the Theorem on Finite Generation is not a corollary of the Theorem on Semi-Rigidity.
- DG algebras Let (P, p, k) be a local ring P with maximal ideal p and residue field k = P/p. In this paper DG algebras over P are assumed to be graded commutative: if a is an element of degree i and b is one of degree j, then ab = (−1)ij^ ba, and a^2 = 0 when i is odd. A morphism that induces an isomorphism in homology is called a quasiisomorphism, and it is often marked by the appearance of the symbol ' next to its arrow. Details on DG algebra can be found in [25], [5, §1], [3, §1].
4 L. L. AVRAMOV AND S. IYENGAR
Proof. An obvious induction shows that we may restrict to A〈w|∂(w) = z〉. Filter- ing A〈w〉 by the internal degree of A, we get a spectral sequence with
(^0) Ep,q = A\〈w|∂(w) = z〉p,q =⇒ Hp+q(A〈w〉)
and differential defined by 0 d(A) = 0 and 0 d(w) = z. The regularity of z implies (^1) Ep,q = 0 if p 6 = 0 and 1 E 0 ,q = A/(z), hence Hn(A〈w〉) = 2 E 0 ,n = Hn(A/(z)).
Let C be a DG module over a DG algebra A. For r ∈ Z the r’th shift of C is the DG module Σr^ C having (Σr^ C)n = Cn−r for all n, differential ∂
Σr^ (c)
Σr^
(−1)r∂(c)
and action aΣr^ (c) = (−1)iΣr(ac) for a ∈ Ai, where Σr^ : C → ΣrC is the degree r map sending c ∈ Cn−r to c ∈ (Σr^ C)n. We use this construction to show that the adjunction of a finite package of exterior variables preserves finiteness properties.
1.4. Lemma. Let A be a DG algebra, Z a finite set of cycles of even degree, and
A〈W 〉 = A〈W |∂(W ) = Z〉. (1) If Hn(A) = 0 for n ≥ s, then Hn(A〈W 〉) = 0 for n ≥ s +
w∈W deg(w). (2) If the algebra H∗(A) is noetherian, then the graded H∗(A)-module H∗(A〈W 〉) is finite and annihilated by cls(z) for all z ∈ Z.
Proof. The inclusions of DG algebras A ⊂ A〈w 1 〉 ⊂ · · · ⊂ A〈W 〉 show that it suffices to treat the case A〈W 〉 = A〈w|∂(w) = z〉. We then have an exact sequence
0 → A ι −→ A〈w〉 θ −→ Σr^ A → 0
where ι is the inclusion and θ(a + wb) = Σr(b). The homology exact sequence
Σr−^1 H∗(A) ð −→ H∗(A)
H∗(ι) −−−→ H∗(A〈w〉)
H∗ (θ) −−−→ Σr^ H∗(A)
immediately implies (1). For (2) note that ð(Σr−^1 (h)) = cls(z)h and H∗(ι) is a homomorphism of algebras, hence cls(z) annihilates the graded H∗(A)-module H∗(A〈w〉). This module is finite because the H∗(A)-modules H∗(A) and Σr^ H∗(A) are noetherian and the maps H∗(ι) and H∗(θ) are H∗(A)-linear.
Using [17, (1.3.5)] and induction, or referring to [3, (7.2.10)], we have
1.5. Lemma. If φ : A → B is a quasiisomorphism, and Z ⊆ A is a set of cycles, then φ extends to a quasiisomorphism of DG algebras
φW : A〈W |∂(W ) = Z〉 → B〈W |∂(W ) = φ(Z)〉
such that φW (w) = w for each w ∈ W.
1.6. Lemma. Let α : A → B and β : B → C be surjective homomorphisms of (graded) algebras, and set I = Ker α, J = Ker(βα), K = Ker β. If the induced map Torφ 2 (C, C) : TorA 2 (C, C) → TorB 2 (C, C) is surjective, then the exact sequence 0 → I → J → K → 0 induces an exact sequence of (graded) C-modules
0 → I/IJ → J/J^2 → K/K^2 → 0.
Proof. The standard change of rings spectral sequence with
(^2) Ep,q = TorB p (Tor
A q (B, C), C) =⇒^ Tor
A p+q (C, C)
FINITE GENERATION OF HOCHSCHILD HOMOLOGY 5
yields an exact sequence of graded C-modules
TorA 2 (C, C) Torα 2 (C,C) −−−−−−−→ TorB 2 (C, C) −ð→^2
TorA 1 (B, C)
TorA 1 (β,C) −−−−−−→ TorA 1 (C, C)
Torα 1 (C,C) −−−−−−−→ TorB 1 (C, C) −→ 0.
Since Torα 2 (C, C) is surjective, we have ð 2 = 0. Canonical isomorphisms identify the tail of the exact sequence above with the desired exact sequence.
We also need a special case of Theorem 1.1, proved in [3, (7.2.9)].
1.7. For each minimal semifree extension P [X] of a regular local ring (P, p, k) the surjective homomorphism P → k can be factored as P [X] ↪→ P [X]〈X′〉 −→' k, where
card(X′ n) =
dim P for n = 1 ; card(Xn− 1 ) for n ≥ 2 ; ∂(P [X]〈X′〉) ⊆ (p + (X))P [X]〈X′〉.
Proof of Theorem 1.1. The argument is broken down into several steps.
Step 1. P [X] = P [ Y , Z˜ ] where Y˜ t Z is a set of variables over P , φ maps Y˜ bijectively to Y , and Ker φ = (p, Z)P [X], where p is a regular sequence in p that is linearly independent modulo p^2.
Since φ 0 : P → Q is a surjective homomorphism of regular local rings, Ker φ 0 is minimally generated by a set p that is linearly independent modulo p^2. Thus, the morphism φ factors as a composition of surjective morphisms
P [X] P [X]/(p) = Q[X]
α Q[Y ]. Since the graded Q-algebra Q[Y ]^ is free, the surjective homomorphism of graded Q-algebras α^ : Q[X]^ → Q[Y ]^ is split by a homomorphism of graded Q-algebras
σ : Q[Y ]^ → Q[X]. It follows that Torα
∗ (Q, Q)^ ◦^ Tor
σ ∗ (Q, Q) is the identity map of TorQ[Y^ ]
∗ (Q, Q), so in particular Tor
α
2 (Q, Q) is surjective. Lemma 1.6 applied to α^ and β : Q[Y ]^ → Q produces an exact sequence of graded Q-modules
0 → (I/(X)I)^ → (QX)^ → (QY )^ → 0
where I = Ker α. For each j ≥ 1, choose in P Xj a set Y˜j that φ maps bijectively onto Yj , and a set Zj ⊆ I whose image in I/(X)I is a basis of that Q-module.
Thus, Y˜ t Z generates the ideal of elements of positive degree of the graded Q- algebra Q[X], and hence is a generating set of the algebra. Nakayama’s Lemma
then implies that Y˜ t Z generates the P -algebra P [X]. We conclude from the equalities card( Y˜j ) + card(Zj ) = card(Xj ) that Y˜ t Z is a set of variables over P.
Step 2. φ factors as P [ Y , Z˜ ] ⊂^ ι → P [ Y , Z˜ ]〈U 〉
eφ −→ Q[Y ], where U 1 = {uz |z ∈ p} and ∂(uz ) = z for z ∈ p ; Uj+1 = {uz |z ∈ Zj } and ∂(uz) − z ∈ P [ Y˜<j , Z<j ]〈U 6 j 〉 for z ∈ Zj.
First, we factor φ as a composition of morphisms of DG algebras
P [ Y , Z˜ ] ⊂^ ι(1) → P [ Y , Z˜ ]〈U 1 |∂(U 1 ) = p〉 π(1) '^ Q[^ Y , Z˜ ] κ
(1) Q[Y ]
FINITE GENERATION OF HOCHSCHILD HOMOLOGY 7
As π = lim−→ π(j)^ stays a surjective quasiisomorphism and κ = lim−→ κ(j)^ becomes
an isomorphism, φ factors through the surjective quasiisomorphism φ˜ = κπ.
Step 3. Z> 1 (P [ Y , Z˜ ]〈U 〉) ⊆ (p + ( Y , Z˜ ))P [ Y , Z˜ ]〈U 〉.
For this argument it is convenient to revert to the notation P [X]. Since P [X] is minimal and φ is surjective, the DG algebra Q[Y ] is minimal. Choose by 1.7 a quasiisomorphism Q[Y ]〈Y ′〉 → k and extend it by Lemma 1. to a quasiisomorphism P [X]〈U, Y ′〉 → Q[Y ]〈Y ′〉. If P [X]〈X′〉 → k is a quasiiso- morphism given by 1.7, then P [X]〈X′〉 and P [X]〈U, Y ′〉 are quasiisomorphic DG modules over P [X], cf. [3, (1.3.1)]. By [3, (1.3.3)] we then get a quasiisomorphism
k〈X′〉 = k ⊗P [X] P [X]〈X′〉 ' k ⊗P [X] P [X]〈U, Y ′〉 = k〈U, Y ′〉.
As ∂(k〈X′〉) = 0, we obtain (in)equalities of formal power series
∏^ ∞
i=
(1 − (−t)i)(−1)
i− (^1) card(X i′ )
n
rankk k〈X′〉ntn
n
rankk Hn(k〈X′〉)tn
n
rankk Hn(k〈U, Y ′〉)tn
n
rankk k〈U, Y ′〉ntn
∏^ ∞
i=
(1 − (−t)i)(−1)
i− (^1) (card(Ui)+card(Y (^) i′ ))
Applying successively 1.7 for P [X], Step 2, and 1.7 for Q[Y ] we get
card X i′ =
dim P = card U 1 + dim Q = card U 1 + card Y 1 ′ for i = 1 ; card Xi− 1 = card Ui + card Yi− 1 = card Ui + card Y (^) i′ for i ≥ 2.
Thus, rankk Hn(k〈U, Y ′〉) = rankk Hn(k〈U, Y ′〉) for all n, so ∂(k〈U, Y ′〉) = 0. Put in other terms, we have ∂(P [X]〈U, Y ′〉) ⊆ (p + (X))P [X]〈U, Y ′〉. Since P [X]〈U 〉 is a DG subalgebra of P [X]〈U, Y ′〉 and the latter is acyclic, we have
Z> 1 (P [X]〈U 〉) = Z> 1 (P [X]〈U, Y ′〉) ∩ (P [X]〈U 〉> 1 ) = ∂(P [X]〈U, Y ′〉) ∩ (P [X]〈U 〉> 1 ) ⊆ (p + (X))P [X]〈U, Y ′〉 ∩(P [X]〈U 〉) = (p + (X))P [X]〈U 〉
where the last equality arises from the freeness of P [X]〈U, Y ′〉^ over P [X]〈U 〉.
Step 4. ∂(uz ) − z ∈ (p, Y˜<j , Z<j )P [ Y˜<j , Z<j ]〈U 6 j 〉 for z ∈ Zj.
Putting together the results of the last two steps, for z ∈ Zj we get
∂(uz) − z ∈ (p + ( Y , Z˜ ))P [ Y , Z˜ ]〈U 〉 ∩P [ Y˜<j , Z<j ]〈U 6 j 〉 = (p + ( Y˜<j− 1 , Z<j− 1 ))P [ Y˜<j , Z<j ]〈U 6 j 〉.
At this point, we have established all the assertions of the theorem.
8 L. L. AVRAMOV AND S. IYENGAR
Proof of Theorem 1.2. Choose a minimal set p of generators of p. It contains dim P elements, so Hn(P [X]〈W |∂(W ) = p〉) = 0 for n ≥ s + dim P by Lemma 1.4. The morphism π factors through P [X]〈W 〉 → P [X]〈W 〉 /(W, p) = k[X], the arrow is a quasiisomorphism by Lemma 1.3, and k[X] is minimal. Thus, after changing notation we may assume that P = k, and (hence) s + dim P = s. Setting Jn = 0 for n < s − 1, Js− 1 = ∂s(k[X]s), and Jn = k[X]n for n ≥ s we get a DG ideal J of k[X], with H∗(J) = 0. Let k[X] ↪→ k[X]〈U 〉 −→' k[Y ] be the factorization of φ given by Theorem 1.1. That theorem guarantees that the module of cycles Z> 1 (k[X]〈U 〉) is contained in (X)k[X]〈U 〉. As k[X]〈U 〉^ is free over k[X], it follows that H∗(Jk[X]〈U 〉) = 0, and so we obtain
(Z> 1 (k[X]〈U 〉))s^ ⊆ Z((X)sk[X]〈U 〉) ⊆ Z(Jk[X]〈U 〉) = ∂(Jk[X]〈U 〉).
Since k[X]〈U 〉 → k[Y ] is a surjective quasiisomorphism, we have Z> 1 (k[Y ]) = π(Z> 1 (k[X]〈U 〉), and so (H> 1 (k[Y ]))s^ = 0, as desired.
- Large homomorphisms Following Levin [22], we say that a surjective homomorphism ϕ : R → S of local rings with residue field k is large if for each n ∈ Z it induces a surjective map
Torϕn (k, k) : TorRn (k, k) → TorSn (k, k).
For instance, if ϕ is split by a ring homomorphism ψ : S → R such that ϕψ = idS , then Torϕn (k, k) ◦ Torψn (k, k) = idTorSn (k,k) by functoriality, and hence ϕ is large. The next result is the generalization [29] of the main theorems of [8], [10].
2.1. Theorem. If ϕ : (R, m, k) → (S, n, k) is a large homomorphism of local rings and TorRn (S, S) = 0 for some even positive n and some odd positive n, then Ker ϕ is generated by a regular sequence that extends to a minimal set of generators of m.
The theorem is proved at the end of this section. The major ingredient is the following result, which plays a fundamental role in the next section as well.
2.2. Theorem. Let ρ : P → R and ϕ : R → S be surjective homomorphisms of local rings such that (P, p, k) is regular, Ker ρ ⊆ p^2 , and ϕ is large. There exist a regular local ring (Q, q, k), a homomorphism σ : Q → S with Ker σ ⊆ q^2 , and a commutative diagram of morphisms of DG algebras
P [X] ⊂^ → P [X]〈U 〉
φe ' ^ Q[Y^ ]
R
eρ ' ↓↓ ⊂ (^) → R〈U 〉
eπ ' ↓↓ ϕe '
S
eσ ' ↓↓
where ˜ρ 0 = ρ, σ˜ 0 = σ, ϕ˜ 0 = ϕ, labeled maps are surjective quasiisomorphisms, and the following hold
∂(P [X]) ⊆ (p + (X))^2 P [X] ; ∂(P [X]〈U 〉) ⊆ (p + (X))P [X]〈U 〉 ; ∂(Q[Y ]) ⊆ (q + (Y ))^2 Q[Y ] ; ∂(R〈U 〉) ⊆ mR〈U 〉.
Furthermore, the DG algebra Q[Y ]〈U 〉 = Q[Y ] ⊗P [X] P [X]〈U 〉 satisfies
∂(U 1 ) = 0 and ∂(Uj+1) ⊆ (q + (Y<j ))Q[Y<j ]〈U 6 j 〉 for j ≥ 1. Before starting on the proof, we recall some properties of large homomorphisms.
10 L. L. AVRAMOV AND S. IYENGAR
- γn and δn are connecting maps in the exact sequences of Tor induced by (∗).
- The map Torϕn (k, k) is surjective because ϕ is large. Using a suitably bigraded version of 1.3, one readily sees that
TorP^ [X]
p (k, k)q^ ∼=^ k〈X ′′〉p,q
where X p,q′′ = X p′+1 if q = 1 and X p,q′′ = ∅ otherwise. On the other hand, we have
TorP ∗ [X](k, k) = H∗(k ⊗P [X] P [X]〈X′〉) = k〈X′〉
where the first equality holds by definition, and the second by 1.7. Thus, we get
∑^ ∞
n=
p+q=n
rankk 1 EP p,q^ [X]
tn^ =
∑^ ∞
n=
rankk TorP n^ [X](k, k)
tn^ ,
so the spectral sequence (∗∗) stops on the first page, and so αn is surjective. A similar argument establishes the surjectivity of βn. The diagram commutes because of the naturality of all the maps involved, so each φn is surjective, and thus by Nakayama’s Lemma φ is surjective, as desired.
2.4. A DG algebra A over R is said to be a DG Γ -algebra, if each a ∈ A of even positive degree has a sequence
a(j)
j> 1 of^ divided powers^ satisfying standard identities, cf. [17, (1.7.1), (1.8.1)], among them a(0)^ = 1, a(1)^ = a, as well as
a(i)a(j)^ =
(i + j)! i!j! a(i+j)^ and ∂
a(j)
= ∂(a)a(j−1)^ for all i, j ≥ 1.
Any semifree Γ-extension R〈U 〉 is a DG Γ-algebra in which the divided powers of the elements of U are the natural ones, cf. e.g. [17, (1.8.4)].
2.5. Let ϕ : (R, m, k) → (S, n, k) be a surjective homomorphism of local rings. A factorization of ϕ in the form R ↪→ R〈U 〉 −'→ S is called an acyclic closure of ϕ if ∂(U 1 ) minimally generates Ker ϕ, and {cls(∂(u))|u ∈ Un+1} is a minimal generating set of Hn(R〈U 6 n〉) for each n ≥ 1. By [17, (1.9.5)], acyclic closures are unique up to isomorphism as DG Γ-algebras.
Thus, there is a “smallest” resolutions of S with a structure of semifree Γ- extension of R, and in that class it is “as unique as” a minimal resolution is among free resolutions. Here is a simple relation between the two concepts.
2.6. If H∗(R〈U 〉) ∼= S and ∂(R〈U 〉) ⊆ mR〈U 〉, then R〈U 〉 is an acyclic closure of the homomorphism ϕ. Indeed, if that fails, then for some n ≥ 0 we have
u∈Un+1 ru∂(u) =^ ∂(v) with ru ∈ R, not all ru ∈ m, and v ∈ R〈U 6 n〉. It follows that z =
u∈Un+1 ruu^ −^ v^ is a cycle in Zn+1(R〈U 〉). Since H> 1 (R〈U 〉) = 0, there exists an element w ∈ R〈U 〉 such that ∂(w) = z /∈ mR〈U 〉, contradicting the minimality of R〈U 〉.
The converse of the last remark does not hold in general. One case when it does is for S = R/m, by a well known theorem of Gulliksen and Schoeller, cf. [17, (1.6.4)] or [3, (6.3.5)]. The homomorphism R → k is obviously large, so the next result constitutes a substantial extension. The proof here differs from those originally given, independently, by Avramov and by Rahbar-Rochandel, cf. [22, (2.5)].
2.7. Corollary. If ϕ : (R, m, k) → (S, n, k) is a large homomorphism, and R〈U 〉 is an acyclic closure of ϕ, then ∂(R〈U 〉) ⊆ mR〈U 〉.
FINITE GENERATION OF HOCHSCHILD HOMOLOGY 11
Proof. If R is complete, then by Cohen’s Structure Theorem there is a surjective homomorphism ρ : P → R, where (P, p, k) is a regular local ring, and Ker ρ ⊆ p^2. Theorem 2.2 now yields a DG algebra R〈U 〉 with ∂(R〈U 〉) ⊆ mR〈U 〉. By 2.6, it is an acyclic closure of ϕ, hence each acyclic closure has the desired property by 2.5. In general, ϕ̂ : R̂ → Ŝ is a large homomorphism by 2.3.2. If R〈U 〉 is an acyclic
closure ϕ, then it is easy to see that R̂ 〈U 〉 = R̂ ⊗R R〈U 〉 is one of ϕ̂ , hence
∂(R〈U 〉) ⊆ (R〈U 〉) ∩ ∂(R̂ 〈U 〉) ⊆ (R〈U 〉) ∩ m(R̂ 〈U 〉) = mR〈U 〉
where the second inclusion holds by the already established case.
The non-vanishing homology classes below are also used in [8], [10], [29].
2.8. Corollary. If x 1 ,... , xe minimally generate Ker ϕ and the Koszul complex K = R〈t 1 ,... , te |∂(ti) = xi〉 has H 1 (K) minimally generated by c elements, then
k〈t 1 ,... , te〉 ⊕ k〈u 1 ,... , uc〉 t 1 · · · te ⊆ TorR ∗ (S, S) ⊗S k
where deg(ti) = 1 for 1 ≤ i ≤ e and deg(uj ) = 2 for 1 ≤ j ≤ c.
Proof. By 2.5, ϕ has an acyclic closure R〈U 〉 such that U 1 = {t 1 ,... , te}, U 2 = {u 1 ,... , uc}, and cls(∂(u 1 )),... , cls(∂(u 2 )) minimally generate H 1 (K). In the DG algebra S〈U 〉 = S ⊗R R〈U 〉 we have ∂(U 1 ) = 0 and ∂(U 2 ) ⊆ SU 1 , hence
Z = S〈t 1 ,... , te〉 ⊕ S〈u 1 ,... , uc〉 t 1 · · · te ⊆ S〈U 〉
is a submodule of cycles. By Corollary 2.7, ∂(R〈U 〉) ⊆ mR〈U 〉, so the composition
Z ⊗S k → H∗(S〈U 〉) ⊗S k → H∗(S〈U 〉 ⊗S k) = k〈U 〉
is injective. As H∗(S〈U 〉) = TorR ∗ (S, S), this proves our assertion.
Proof of Theorem 2.1. By hypothesis, ϕ : R → S is a large homomorphism with TorRn (S, S) = 0 for some even positive n and some odd positive n. By the preceding corollary we then have c = 0, that is, H 1 (K) = 0. This implies that the sequence x 1 ,... , xe is regular; it is linearly independent modulo m^2 by 2.3.1.
- Finite generation Let S ← R → S′^ be homomorphisms of commutative rings. The t-product of Cartan and Eilenberg [11, §XI.4] provides TorR ∗ (S, S′) with a natural structure of graded-commutative algebra, which in degree 0 is the standard product on S ⊗R S′. The product may be computed from any flat resolution of S over R. In particular, if A is a DG algebra with An a flat R-module for each n, H 0 (A) ∼= S, and Hn(A) = 0 for n 6 = 0, then TorR ∗ (S, S′) ∼= H∗(A ⊗R S′) as graded algebras. In this section we focus on large homomorphisms of local rings with finitely generated Tor algebras. In non-zero characteristic we describe them completely.
3.1. Theorem. Let ϕ : R → S be a surjective homomorphism of local rings. If R has residual characteristic p > 0 , then the following are equivalent. (i) The S-algebra TorR ∗ (S, S) is finitely generated, and ϕ is large. (ii) Each S-algebra S′^ defines a natural isomorphism of graded S′-algebras TorR ∗ (S, S′) ∼=
S′^ (Σ^ S
′e)
with e = edim R − edim S, and ϕ is large. (iii) The ideal Ker ϕ is generated by an R-regular sequence that extends to a minimal system of generators of the maximal ideal of R.
FINITE GENERATION OF HOCHSCHILD HOMOLOGY 13
(ii) =⇒ (i) is clear. (i) =⇒ (iii). Under our hypothesis, the algebra TorR ∗ (S, S)/p TorR ∗ (S, S) is gen- erated over S by finitely many elements of positive degree. By 3.5 their p’th powers are equal to 0, so TorRn (S, S) = p TorRn (S, S) for n 0, and hence TorRn (S, S) = 0 by Nakayama’s Lemma. Theorem 2.1 yields the desired conclusion. (iii) =⇒ (ii) is well known, but we include an argument for completeness. By hypothesis, Ker ϕ is minimally generated by an R-regular sequence x that is linearly independent modulo m^2. It follows that x has length e = edim R − edim S, and the Koszul complex R〈T |∂(T ) = x〉 yields
TorR ∗ (S, S′) = H∗(R〈T 〉 ⊗RS′) = S′〈T 〉 =
S′^ ΣS
′e^.
Furthermore, R → k has an acyclic closure of the form R〈T, V 〉. By Lemma 1.3 the morphism R〈T, V 〉 → R〈T, V 〉 /(T, x) = S〈V 〉 is a quasiisomorphism. As R〈T, V 〉 is a minimal resolution of k by the theorem of Gulliksen and Schoeller, recalled before Corollary 2.7, we see that S〈V 〉 is a minimal resolution of k over S, hence Torϕ ∗ (k, k) : R〈T, V 〉 ⊗Rk → S〈V 〉 ⊗S k is surjective, that is, ϕ is large.
Proof of Theorem 3.2. In this proof ϕ is a large homomorphism, as in 3.3, and the S-algebra TorR ∗ (S, S) is finitely generated.
The homomorphism ϕ̂ : R̂ → Ŝ is large by 2.3.2. The Ŝ-algebra Tor Rb ∗ (S,̂^ Ŝ ) is isomorphic to TorR ∗ (S, S)⊗S Ŝ, and so finitely generated. Thus, we may assume that R is m-adically complete. Let ρ : P → R be a Cohen presentation with (P, p, k) regular and Ker ρ ⊆ p^2. Theorem 2.2 now applies and we adopt its notation, modified in accordance with 3.3.1. Since P [X, U ]^ is free over P [X], the quasiisomorphism Q[Y ] ' −→ S yields
Q[Y, U ] = Q[Y ] ⊗P [X] P [X, U ] ' −→ S ⊗P [X] P [X, U ] = S ⊗R R[U ].
By Theorem 2.2, the DG algebra R[U ] is a free resolution of S over R, so
H∗(S ⊗R R[U ]) = TorR ∗ (S, S).
We conclude that H∗(Q[Y, U ]) is finitely generated over S, say by the classes of z 1 ,... , zg. Let Z = {z^21 ,... , z g^2 }, pick a minimal generating set q of q, and set
A = Q[Y, U, V, W |∂(V ) = Z ; ∂(W ) = q].
By Lemma 1.4, H∗(A) is a finite module over the noetherian ring H∗(Q[Y, U ]), and is annihilated by the ideal generated by q and {cls(z 1 )^2 ,... , cls(zg)^2 }. This ideal has finite colength, so there is an integer s such that Hn(A) = 0 for all n ≥ s. Since Q is a regular local ring, the morphism Q[W |∂(W ) = q] → k is a quasiiso- morphism. As Q[Y, U, V ] is a bounded below complex of free Q-modules, it induces a quasiisomorphism of DG algebras
A = Q[Y, U, V, W ] = Q[Y, U, V ] ⊗Q Q[W ] ' −→ Q[Y, U, V ] ⊗Q k = k[Y, U, V ]
so, in particular, Hn(k[Y, U, V ]) = 0 for all n ≥ s. On the other hand,
∂(Y ) ⊆ (Y )^2 k[Y, U, V ] ; ∂(U ) ⊆ (Y )(Y, U )k[Y, U, V ] , ∂(V ) ⊆ (Y, U )^2 k[Y, U, V ] ,
14 L. L. AVRAMOV AND S. IYENGAR
where the first two relations are provided by Theorem 2.2, and the last one holds by construction. Thus, k[Y, U, V ] is a minimal semifree extension of the field k and Hn(k[Y, U, V ]) = 0 for n ≥ s. By Theorem 1.1.3, the surjective morphism
k[Y, U, V ] k[Y, U, V ]/(Y ) = k[U, V ]
shows that in H∗(k[U, V ]) the product of any s elements is equal to zero. Setting r = max{deg(v)|v ∈ V }, we get an isomorphism of DG algebras k[U, V ] ∼= k[U<r, V ] ⊗k k[U>r ]
where ∂(U ) = 0. In homology it induces an isomorphism of k-algebras
H∗(k[U, V ]) ∼= H∗(k[U<r , V ]) ⊗k k[U>r]. We conclude that
n=r card^ Un^ < s, hence^ U^ is finite, as desired. If R or S is a complete intersection, then 3.3.3.2 yields Un = ∅ for n 6 = 1, 2.
- Split homomorphisms The main result here is a structure theorem for certain split homomorphisms.
4.1. Theorem. Let S ψ −→ R ϕ −→ S be homomorphisms of local rings with ϕψ = idS. When R has residual characteristic 0 and fdS R < ∞ the following are equivalent. (i) The S-algebra TorR ∗ (S, S) is finitely generated. (ii) Each S-algebra S′^ defines a natural isomorphism of graded S′-algebras TorR ∗ (S, S′) ∼=
S′^ (Σ^ S
′e) ⊗S′ (^) Sym S′^ (Σ (^2) S′c)
with e = edim R − edim S and c = e − (dim R − dim S). (iii) The (Ker ϕ)-adic completion of the ring R is isomorphic as an S-algebra to S[[x 1 ,... , xe]]/(f ), where f is a length c regular sequence in (x 1 ,... , xe)^2. The proof shows that the finiteness of the flat dimension fdS R could be dropped if Quillen’s conjecture 3.3.3 holds in characteristic 0. The arguments use Tate complexes, whose construction we recall next.
4.2. Let x = x 1 ,... , xe and f = f 1 ,... , fc be regular sequences in a commutative ring P that satisfy fj =
∑e i=1 gij^ xi^ for^ j^ = 1,... , c, set^ R^ =^ P/(f^ ) and^ S^ =^ P/(x), and let π : P → R and ϕ : R → S be the canonical projections.
∑In the Koszul complex^ R〈T^ 〉^ =^ R〈t^1 ,... , te^ |∂(ti) =^ π(xi)〉^ the elements^ zj^ = e i=1 π(gij^ )ti^ satisfy^ ∂(zj^ ) =^
∑e i=1 π(gij^ xi) =^ π(fj^ ) = 0. Tate [30, Theorem 5], cf. also [17, (1.5.4)] or [3, (6.1.9)], proves that the DG algebra
R〈T, U 〉 = R〈T, U |∂(ti) = π(xi) ; ∂(uj ) = zj 〉
is a resolution of S over R. Thus, TorR ∗ (S, S) = H∗(A) for the DG algebra
A = S ⊗R R〈T, U 〉 = S
T, U |∂(T ) = 0 ; ∂(uj) =
∑^ e
i=
aij ti for 1 ≤ j ≤ c
with aij = ϕπ(gij ). When S contains a field of characteristic 0 the discussion works equally well with R[T, U ] in place of R〈T, U 〉, as noted in 3.3.1.
The preceding construction has strong implications for homology.
4.3. Proposition. If R, S, A are as in 4.2 then the S-algebra B = TorR ∗ (S, S)
has a bigrading with Bn = TorRn (S, S) =
⊕n `=0 B
(`) n , such that the following hold.
16 L. L. AVRAMOV AND S. IYENGAR
(i) =⇒ (iii). The maps P → R → S define a Jacobi-Zariski exact sequence Dn+1(S |P ; k) → Dn+1(S |R; k) → Dn(R|P ; k) → Dn(S |P ; k)
cf. [1, (5.1)]. By flat base change [1, (4.54)], we have
Dn(S |P ; k) ∼= Dn(k|(P ⊗S k); k) for all n ∈ Z.
The last module vanishes for n ≥ 2 because the ring P ⊗S k ∼= k[[x 1 ,... , xe]] is regular, cf. [1, (6.26)]. Putting these facts together, we get
Dn+1(S |R; k) ∼= Dn(R|P ; k) for n ≥ 2 ,
By hypothesis, TorR ∗ (S, S) is a finitely generated algebra over S, so Dn(S |R; k) = 0 for n 0 by Theorem 3.4, and thus Dn(R|P ; k) = 0 for n 0. By [6, (3.2)] the projective dimension pdP R is finite, hence f is a regular sequence by 3.3.3.2. We can now apply Proposition 4.3, whose notation we adopt. It yields a direct sum decomposition B = C ⊕ D of B = TorR ∗ (S, S), where C =
n< 2 ` B
(`) n is an ideal and D =
n B
(n) 2 n is a subalgebra.^ The same proposition shows that E =
n B
(n+e) 2 n+e is an ideal of the graded algebra^ B, and^ CE^ = 0. By hypothesis^ B is finitely generated as an algebra over the noetherian ring S, hence the ideal E of B is finitely generated, and thus E is finite as a module over the algebra B/C = D.
The vanishing lines of A (`) n yield exact sequences of graded^ S-modules
0 → D → S[U ] −→∂ S[U ] ⊗S ST ;
S[U ] ⊗S
∧e− 1 (ST ) ∂ −→ S[U ] ⊗S
∧e (ST ) → E → 0.
The map b ∈ S[U ] 7 → b · t 1 · · · te ∈ S[U ] ⊗S
∧e (ST ) is a degree e homomorphism τ : S[U ] → E of graded D-modules. As ∂(S[T, U ]) ⊆ nS[T, U ], we see that
τ ⊗D k : S[U ] ⊗D k → E ⊗D k
is bijective. For each n ∈ Z the degree n component of the D-module S[U ] is a finite S-module, and vanishes for n < 0, so by the appropriate version of Nakayama’s Lemma the D-module S[U ] is finite. In particular, each u ∈ U satisfies an equation
ur^ + zr− 1 ur−^1 + · · · + z 1 u + z 0 = 0 ∈ S[U ]
of integral dependence with zj ∈ D. Differentiating one with minimal r, we get ( rur−^1 + (r − 1)zr− 1 ur−^2 + · · · + z 1
∂(u) = 0 ∈ S[U ] ⊗S ST.
The minimality of r implies that the coefficient of ∂(u) is non-zero, hence it is not a zero-divisor on the free S[U ]-module S[U ] ⊗S ST , and so ∂(u) = 0. Thus, ∑e i=1aij^ ti^ =^ ∂(uj^ ) = 0^ for^ j^ = 1,... , c
so all aij vanish. Since aij = gij (0,... , 0) where gij ∈ S[[x 1 ,... , xe]] appear in equalities fj =
∑e i=1 gij^ xi, we get^ fj^ ∈^ (x^1 ,... , xe) (^2) for j = 1,... , c, as desired.
- Hochschild homology In this section we bring the local results of the preceding discussion to bear on the Hochschild homology of flat k-algebras essentially of finite type. We start by recalling the classical interpretation of Hochschild homology as a derived functor.
FINITE GENERATION OF HOCHSCHILD HOMOLOGY 17
5.1. Let S be a flat k-algebra, set R = S ⊗k S and let μ : R → S be the multipli- cation map μ(a′^ ⊗ a′′) = a′a′′. The flatness hypothesis yields an isomorphism
HH∗(S |k) ∼= TorR ∗ (S, S)
of graded S-algebras, cf. Cartan-Eilenberg [11, §XI.6] or Loday [23, §4.2]. If n is a prime ideal of S and m = μ−^1 (n), then μ induces a surjective local homomorphism ϕ : Rm → Sn, and there are canonical isomorphisms
TorR ∗ (S, S) ⊗S Sn ∼= TorS ∗ n^ ⊗kSn(Sn, Sn) ∼= TorR ∗ m(Sn, Sn). Next we prove the theorems announced in the introduction.
5.2. Theorem. If S is a flat commutative algebra essentially of finite type over a commutative noetherian ring k, and HHn(S |k) = 0 for an even positive n and an odd positive n, then S is smooth over k.
Proof. Due to the isomorphisms of 5.1, Theorem 2.1 shows that (Ker μ)m is gener- ated by an Rm-regular sequence, so S is smooth, cf. [23, (3.4.2)].
5.3. Theorem. If S is a flat commutative algebra essentially of finite type over a commutative noetherian ring k, and the algebra HH∗(S |k) is finitely generated over S, then S is smooth over k.
Proof. Let n be a prime ideal of S, and set k = Sn/nSn. When char(k) > 0 Theorem 3.1 and 5.1 show that Ker ϕ is generated by an Rm-regular sequence; as in the preceding proof, it follows that S is smooth. When char(k) = 0, consider the homomorphism S → S ⊗k S given by a 7 → a ⊗ 1. It localizes to a homomorphism ψ : Sn → Rm satisfying ϕψ = idSn. Thus, Theorem 4.1 applies, and shows that the Sn-module TorR 1 m(Sn, Sn) ∼= Ω^1 Sn|k is free, hence S
is smooth by the Jacobian criterion, cf. [16, (17.15.8)] or [1, (7.31)].
Proposition 5.6 and Example 5.7 show that the homological hypothesis in the statements of the preceding theorems cannot be significantly weakened. We briefly consider relaxing the finiteness hypothesis on the k-algebra S.
5.4. Remark. An attentive reader might have noticed that the preceding proofs show that S is regular over k even when the hypothesis that S is essentially of finite type over k is weakened to an assumption that (S ⊗k S)m is noetherian for each prime ideal m in S ⊗k S containing Ker μ. We have stated the results under the stronger hypothesis because it is easy to check, and because Ferrand [15, (3.6)] proves that it covers most cases: If Sn ⊗k Sn is noetherian for some prime ideal n of S, then Sn is essentially of finite type over k.
For complete intersections we can prove more by using a special resolution.
5.5. Let P = k[x 1 ,... , xe] be a polynomial ring over a noetherian ring k, and let f = f 1 ,... , fc be a P -regular sequence such that S = P/(f ) is flat over k.
5.5.1. If ∂i(fj ) is the image in S of the partial derivative ∂fj /∂xi, then
HH∗(S|k) ∼= H∗
S
t 1 ,... , te; u 1 ,... , uc
∣∂(ti) = 0 ; ∂(uj ) =
∑^ e
i=
∂i(fj )ti
When k contains a field of characteristic zero the isomorphism is implicit in a general theorem of Quillen [26, (8.6)]; explicitly, it appears in an argument of
FINITE GENERATION OF HOCHSCHILD HOMOLOGY 19
- Nilpotence In this section we study nilpotence properties of shuffle products in Hochschild homology. Our main result in this direction significantly generalizes [32, (2.7)], where it is assumed that S is graded and finite over S 0 = k.
6.1. Theorem. Let k be a field, and S a k-algebra essentially of finite type that is locally complete intersection. If char(k) = 0, or if S is reduced and for each minimal prime ideal q of S the field extension k ⊆ Sq is separable, then
(HH> 1 (S|k))s^ = 0 for some integer s ≥ 1.
Proof. First we treat a special case: S = P/(f ) satisfies the hypotheses of 5.5.1.
Proposition 4.3.1 then yields B = C ⊕ D with C =
n< 2 m B
(`) n and^ D^ = ⊕ n B
(n) 2 n , and shows that^ C e+1 (^) ⊆ ⊕ n+e< 2 ` B
(`) n.^ By Proposition 4.3.2 the last module is trivial, so it remains to prove that D> 1 is nilpotent. Proposition 4.3.
yields D> 1 ⊆
n> 1 (0 :^ j)A
(n) 2 n , where^ j^ is the ideal in^ S^ generated by the^ c^ ×^ c minors of the Jacobian matrix
∂fj /∂xi
, so we show that (0 : j) is nilpotent. If char(k) = 0, then Eisenbud, Huneke, and Vasconcelos [14, (2.2)] prove that (0 : j) is the nilradical of S. If S is reduced and generically smooth over k, then the linear map Sc^ → Se^ given by the Jacobian matrix is injective, hence (0 : j) = 0. Next we turn to the general case: S is a localization of a residue ring of a polynomial ring P over k. Fix n ∈ Spec(S), and let m be its inverse image in P ; by hypothesis Sn = Pm/(f ) where f is a Pm-regular sequence that we may take in P. The Koszul complex K = P 〈T |∂(T ) = f 〉 satisfies H 1 (K)m ∼= H 1 (Km) = 0, so we can find h ∈ P r m with h H 1 (K) = 0. The isomorphism P ′^ ∼= P [y]/(hy − 1) identifies Spec(P ′) with the open set D(h) = {p ∈ Spec P |h /∈ p} of Spec P. For each p ∈ D(h) we have H 1 (K ⊗R P (^) p′) = 0, so the sequence f is P (^) p′-regular, and hence the ideal (f )P ′^ can be generated by a P ′-regular sequence. If g is a lifting of such a sequence in the polynomial ring P [y], then Sn is a localization of the k- algebra Sh ∼= P [y]/(hy− 1 , g), where hy− 1 , g is a P [y]-regular sequence. Hochschild homology algebras commute with localization, so for each s we have
(HH> 1 (S|k)s)h ∼= (HH> 1 (Sh|k))s^.
The first part of the proof shows that the right hand side vanishes for s 0. The open sets D(h)∩Spec S cover Spec S, so by quasi-compactness we can find h 1 ,... , ht such that Spec S =
⋃t i=1 D(hi).^ For large enough^ s^ we have (HH>^1 (Shi |k)) s (^) = 0
for i = 1,... , t, hence we conclude that (HH> 1 (S|k))s^ = 0, as desired.
The next example shows that, in general, Hochschild homology is not nilpotent; it also serves to illustrate that the hypotheses of the theorem above are sharp.
6.2. Example. If k is a field of characteristic p > 0 and S = k[x]/(xp^ − a) with a /∈ kp, then for the purely inseparable field extension k ⊆ S we have
HH∗(S |k) = H∗
S〈t, u|∂(t) = 0; ∂(u) = 0〉
= S〈t, u〉
by 5.5.1, and the product rule in 2.4 yields u · u(p)^ · · · u(p
s) (^6) = 0 for all s ≥ 1.
Nevertheless, when k is a field of characteristic p > 0 the ideal HH> 1 (S |k) is nil of exponent p, due to the following immediate consequence of 5.1 and 3.5.
6.3. Remark. If there is a positive integer q such that qS = 0, then wq^ = 0 for each homology class w ∈ HH> 1 (S |k).
20 L. L. AVRAMOV AND S. IYENGAR
In view of the preceding theorem and remark, in an earlier version of this paper we raised the question whether Hochschild homology over a field of characteristic 0 is nil. L¨ofwall and Sk¨oldberg, and independently Larsen and Lindenstrauss, answered our question in the negative for the test algebra that we suggested. With their permission, we include the argument of Larsen and Lindenstrauss.
6.4. Proposition. If k is a field of characteristic 0 and S = k[x, y]/(x^2 , xy, y^2 ), then there is a class cls(z) ∈ HH 4 (S |k) such that cls(z)n^6 = 0 for each n ≥ 1.
Proof. We set n = (x, y) ⊆ S and denote ⊗ tensor products over k. The Hochschild complex C of the k-algeba S has degree n component Cn = S ⊗n⊗n^ and differential
∂(s ⊗ a 1 ⊗ · · · ⊗ an) = (sa 1 ) ⊗ a 2 ⊗ · · · ⊗ an + (−1)n(ans) ⊗ a 1 ⊗ · · · ⊗ an− 1 ,
due to the equality n^2 = 0, cf. [11, §IX.6] or [23, §1.1]. In particular, ∂(C) ⊆ nC, so if z is a cycle and zn^6 ∈ nC, then cls(z)n^6 = 0. A direct computation shows that
z = 1 ⊗ (x ⊗ x ⊗ y ⊗ y − y ⊗ x ⊗ x ⊗ y + y ⊗ y ⊗ x ⊗ x − x ⊗ y ⊗ y ⊗ x) ∈ C 4
is a cycle. Denote cn the coefficient with which the tensor monomial
vn = 1 ⊗ x ︸ ⊗ · · · ⊗︷︷ x︸ 2 n
⊗ y ⊗ · · · ⊗ y ︸ ︷︷ ︸ 2 n
∈ C 4 n = S ⊗ n⊗^4 n
appears in zn. By the definition of shuffle product, cf. [11, §XI.6] or [23, §4.2], any monomial occuring in a product involving one of the elements 1 ⊗ y ⊗ x ⊗ x ⊗ y, 1 ⊗ y ⊗ y ⊗ x ⊗ x, or 1 ⊗ x ⊗ y ⊗ y ⊗ x contains y ⊗ x as a submonomial, so cn is equal to the coefficient of vn in (1 ⊗ x ⊗ x ⊗ y ⊗ y)n. It is clear that c 1 = 1, so we assume that cn− 1 = ((n − 1)!)^2 for some integer n ≥ 2. Note that cn = bncn− 1 , where bn is the coefficient with which vn appears in v 1 · vn− 1 , and that each such appearance comes from a permutation ξ that shuffles the x’s separately from the y’s. Thus, if
Ξ denotes the set of (2, 2 n − 2)-shuffles, then bn =
ξ∈Ξ sign(ξ)
. The equalities
∑
ξ∈Ξ
sign(ξ) =
(^2) ∑n− 1
i=
∑^2 n
j=i+
(−1)i+j−^3 = 1 + 0 + 1 + 0 + · · · + 1 ︸ ︷︷ ︸ 2 n− 1
= n ,
yield cn = (n!)^2 6 = 0 ∈ k, hence zn^ ∈/ nC for each n ≥ 0, as desired.
Acknowledgement We thank Michael Larsen, Ayelet Lindenstrauss, Clas L¨ofwall, and Emil Sk¨oldberg for interesting correspondence in relation to this paper.
References [1] M. Andr´e, Homologie des algebres commutatives, Grundlehren Math. Wiss. 206 , Springer- Verlag, Berlin, 1974. [2] M. Andr´e, Algebres gradu´ees associ´ees et alg`ebres sym´etriques plates, Comment. Math. Helv. 49 (1974), 277–301. [3] L. L. Avramov, Infinite free resolutions, in: Six lectures in commutative algebra, Bellaterra, 1996, Progress in Math. 166 , Birkh¨auser, Boston, 1998; pp. 1–118. [4] L. L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Ann. of Math. (2) 150 (1999), 455-487. [5] L. L. Avramov, S. Halperin, Through the looking glass: A dictionary between rational ho- motopy theory and local algebra, in: Algebra, algebraic topology, and their interactions; Stockholm, 1983, Lecture Notes Math. 1183 , Springer-Verlag, Berlin, 1986; pp. 1–27.
