Hochschild homology of finite dimensional algebras

Micheline VIGUÉ-POIRRIER Université Paris-Nord Institut Galilée Département de Mathématiques F-93430 Villetaneuse e-mail : vigue@math.univ-paris13.fr

September 4, 2003

1 Introduction

Let A be an augmented algebra over a field k. By definition, the Hochschild homology of A with coefficients in A (called here Hochschild homology) is the homology of the Hochschild complex (C_{∗}(A), b) and is denoted HH_{∗}(A) = ⊕_{n≥0} HH_{n}(A), [Lo]. We denote A = k ⊕ Ā where Ā is

the augmentation ideal and we assume that Ā is a finite dimensional k-vector space.Then each homology group HH_{n}(A)is finite dimensional. Recall the following formulas :

C_{n}(A) = A ⊗ Ā^{⊗n}

b(a_{0} ⊗ a_{1} … ⊗ a_{n}) = ∑_{i=0}^{n−1}(−1)^{i}a_{0} ⊗ … ⊗ a_{i}a_{i+1} ⊗ … ⊗ a_{n} + (−1)^{n}a_{n}a_{0} ⊗ a_{1} … ⊗ a_{n−1}

where a_{0} ∈ A, a_{i} ∈ Ā if i ≥ 1.

All the tensor products are over k.

We introduce the cyclic permutation t_{n} : Ā^{⊗n} → Ā^{⊗n} defined by t_{n}(a_{1} ⊗ … ⊗ a_{n}) = (−1)^{n−1} a_{n} ⊗ a_{1} … ⊗ a_{n−1}.

¿From Loday, [Lo], Proposition 2.2.14, the reduced cyclic homology groups HC̃_{n}(A) := HC_{n}(A)/HC_{n}(k) can be computed as the homology groups of the complex Ā^{⊗(n+1)}/(Id − t_{n+1}) endowed with the differential induced by b, when chark = 0, or chark = p and n < p − 1.

2 Characterization of the trivial algebra structure

Proposition 2.1 [Ro] Let A be an augmented algebra, where the augmentation ideal Ā has finite dimension d and satisfies Ā · Ā = 0, then

1. HH_{n}(A) = Coker(Id − t_{n+1}) ⊕ Ker(Id − t_{n}) as k-vector spaces for all n > 0 2. if char k = 0, HC̃_{n}(A) = Coker(Id − t_{n+1}) for all n > 0.

3. if char k = 0, for all n ≥ 2,

a_{n−1} = dimHC_{n−1}(A) = (1/n) ∑_{i=1}^{n}(−1)^{(n−1)i}d^{q(i,n)}

where q(i, n) = g.c.d.(i, n).

4. if char k = 0, then dimHH_{n}(A) = a_{n} + a_{n−1} for all n ≥ 1 and a_{0} = d.

Corollary 2.2 Let A be an augmented algebra, where the augmentation ideal Ā has finite dimension d, d ≥ 1 and satisfies Ā · Ā = 0, then

1. HH_{n}(A) ≠ 0 for all n > 0

2. if char k = 0, lim_{n→∞} \sqrt[n]{dim HH_{n}(A)} = d.

Theorem 2.3 Let A be an augmented algebra over a field k. Let Ā be its augmentation ideal, with dim_{k} Ā finite. Let A_{t} = k ⊕ Ā_{t} be the augmented algebra with trivial multiplication on Ā_{t} and Ā = Ā_{t} as k-vector space. We assume that char k = 0, or char k ≠ 0 and there exists N ≥ 2 such that Ā^{N} = 0 in A ; then we have

1. dim HH_{n}(A) ≤ dim HH_{n}(A_{t}) for all n ≥ 0

2. dim HC_{n}(A) ≤ dim HC_{n}(A_{t}) for all n ≥ 0

Proof: We define an increasing filtration A_{k} on A

A_{k} = A if k ≥ 0, A_{−p} = Ā^{p} if p ≥ 1

We filter Ā by

Ā_{k} = Ā if k ≥ 0, Ā_{−p} = Ā^{p} if p ≥ 1

This allows us to filter the Hochschild complex as follows :

\begin{matrix}F_{k}(C_{n}(A)) = ∑_{k_{0}+k_{1}+...+k_{n}≤k} A_{k_{0}} ⊗ Ā_{k_{1}} ⊗ … ⊗ Ā_{k_{n}} & for & k < 0 \\ F_{k}(C_{n}(A)) = C_{n}(A) & for & k ≥ 0 \end{matrix}

With the additional hypothesis that there exists N ≥ 2 such that Ā^{N} = 0, we get F_{k}(C_{n}(A)) = 0 for k < −(n + 1)N. This filtration gives rise to a spectral sequence (E_{∗∗}^{r}, d^{r}) converging to HH_{∗}(A) with

⊕_{p} E_{−p,n+p}^{0} = C_{n}(B)

B = k ⊕ \frac{Ā}{Ā^{2}} ⊕ \frac{Ā^{2}}{Ā^{3}} ⊕ …

We check that d^{0}((λ + ā_{0}) ⊗ ā_{1} … ⊗ ā_{n}) = λ(Id − t_{n}) (ā_{1} … ⊗ ā_{n}) where ā_{i} ∈ ⊕Ā^{p}/Ā^{p+1} for i ≥ 0.

So we have ⊕_{p} E_{−p,n+p}^{1} = HH_{n}(B) with B isomorphic to A_{t} as algebras.

A general fact about convergent spectral sequences implies that

dim HH_{n}(A) ≤ dim HH_{n}(A_{t}).

To prove 2), we use the reduced bicomplex B̄(A) to compute cyclic homology ([Lo], page 58), and we define on it an increasing filtration as above. ■

Now, we are interested in algebras A for which inequality 1 or 2 of theorem 2.3 becomes an equality.

Theorem 2.4 Let A be an augmented algebra over a characteristic zero field k. Let Ā be its augmentation ideal. We assume that Ā is a finite dimensional k-vector space. Let A_{t} = k⊕Ā_{t} be the augmented algebra with trivial multiplication on Ā_{t}, and Ā_{t} = Ā as k-vector space. Suppose that there exists n ≥ 1 such that

dim HC_{n}(A) = dim HC_{n}(A_{t})

then A is commutative and is isomorphic to A = S/I where S is the polynomial algebra k[X_{1}, …, X_{m}] and I is generated by

f_{i} = X_{i}^{2} − λ_{i}X_{i}, 1 ≤ i ≤ m,  g_{ij} = X_{i}X_{j} − \frac{λ_{j}}{2} X_{i} − \frac{λ_{i}}{2} X_{j}, 1 ≤ i < j ≤ m,  and λ_{j} ∈ k.

Proof: It is a refinement of the proof of theorem 1.4 of [Vi]. ■

Corollary 2.5 Let A be an augmented algebra over a characteristic zero field k. Let Ā be its augmentation ideal. We assume that Ā is a finite dimensional k-vector space and there exists N ≥ 2 such that Ā^{N} = 0, and Ā ≠ 0. Let A_{t} = k ⊕ Ā_{t} be the augmented algebra with trivial multiplication on Ā_{t}, and Ā_{t} = Ā as k-vector space. Suppose that there exists n ≥ 1 such that

dim HC_{n}(A) = dim HC_{n}(A_{t})

then the multiplication is trivial in the augmented algebra A, (namely, A is isomorphic to A_{t}, as augmented algebras).

Proof:

The hypothesis Ā^{N} = 0 implies λ_{i} = 0, 1 ≤ i ≤ m so that x^{2} = 0, for any x ∈ Ā. ■

Remark Theorem 2.4 and Corollary 2.5 remain valid if chark = p, p > 0, and n < p − 2.

Example 2.6 Let A = k[X]/(X^{2} − X) and A_{t} = k[X]/X^{2}. We check,[B-V], that

HC̃_{2n}(A_{t}) = HC̃_{2n}(A) = k
HC_{2n+1}(A_{t}) = HC_{2n+1}(A) = 0
HH_{n}(A_{t}) = k for all n > 0
HH_{n}(A) = 0 for all n > 0

This shows that the hypothesis Ā^{N} = 0 cannot be omitted in corollary 2.5. On the other hand, the Hochschild homology groups of A_{t} and A are quite distinct.

This observation leads us to hope that the equality of the dimensions of one Hochschild homology group of A andA_{t} characterizes the trivial product.

Theorem 2.7 Let A be an augmented algebra over a characteristic zero field k. Let Ā be its augmentation ideal and we assume that Ā has finite dimension. Let A_{t} = k⊕Ā_{t} be the augmented algebra with trivial multiplication on Ā_{t} and Ā_{t} = Ā as k-vector space. Suppose that there exists n ≥ 1 such that

HH_{n}(A) = HH_{n}(A_{t})

Then the multiplication is trivial in the augmented algebra A(namely A is isomorphic to A_{t} as augmented algebras).

Proof: It is analogous to the proof of theorem 1.6 in [Vi] but here we do not assume that A is commutative. ■

Remark Theorem 2.7 remains valid if chark = p > 3, and 1 ≤ n < p − 1.

3 Examples and remarks

Let A be an augmented algebra over a field k of characteristic zero. We assume that the augmentation ideal Ā has finite dimension d, d ≥ 2.

We have seen, in §§2, that if Ā · Ā = 0, then

lim_{n→∞} \sqrt[n]{dim HH_{n}(A)} = d.

Proposition 3.1 Let A be the quotient of a polynomial algebra k[X_{1}, …, X_{r}] by an ideal generated by a regular sequence (f_{1}, …, f_{r}) where f_{i} ∈ 𝔪^{2}, for all i, and 𝔪 = (X_{1}, …, X_{r}). Then there exist constants K_{1} and K_{2}, such that

K_{2} · n^{r} ≤ ∑_{0≤p≤n} dim HH_{p}(A) ≤ K_{1} · n^{r}

Proof: It relies on results proved in [B-V]. ■

Definition An algebra satisfying the hypothesis of proposition 3.1 is called a complete intersection.

Proposition 3.2 Let A be a finite dimensional smooth commutative algebra, then we have

dim HH_{n}(A) = 0 for all n > 0.

Proof: It is a direct consequence of the Hochschild-Kostant-Rosenberg, [H-K-R]. ■

Conjecture 3.3 Let A be an augmented commutative algebra over a field, where the augmentation ideal has finite dimension d, d ≥ 2. If A is neither smooth nor a complete intersection, then there exist real numbers C_{1}, C_{2}, 1 < C_{2} ≤ C_{1} ≤ d such that

C_{2}^{n} ≤ ∑_{0≤p≤n} dim HH_{p}(A) ≤ C_{1}^{n}

Example 3.4 A = k[x]/x^{2} ×_{k} B

A is the fiber product over k of k[x]/x^{2} and B, where B is a finite dimensional augmented commutative algebra which is not smooth.

The fact that B is not smooth implies that there exists y ∈ B̄ and y ∉ B̄^{2}. Consider X = (x, 0) ∈ Ā and Y = (0, y) ∈ Ā; we have X^{2} = XY = 0.

Proposition 9 of [La] implies that for n = 4m, m ≥ 1, dim HH_{n}(A) ≥ 2^{m−1}. So we have:

C_{2}^{n} ≤ ∑_{0≤p≤n} dim HH_{p}(A) ≤ C_{1}^{n}

where C_{2} = \sqrt[4]{2} and C_{1} ≈ 1 + dim B̄.

REFERENCES

[B-V] D.Burghelea and M.Vigué-Poirrier, Cyclic homology of commutative algebras, Lecture Notes in Mathematics, 1318,(1988),51-72.

[H-K-R] G. Hochschild, B. Kostant, A. Rosenberg, Differential forms on regular affine algebras, Trans A.M.S., 102, (1962), 383-408.

[La] P.Lambrechts, On the Betti numbers of the free loop space of a connected sum, J. London Math. Soc., 64, (2001), 205-228.

[Lo] J.L. Loday, Cyclic homology, Springer-Verlag, Berlin, (1992).

[Ro] J.E. Roos, Homology of free loop spaces, cyclic homology, and nonrational Poincaré-Betti series in commutative algebra, Lecture Note in Math., 1352, (1988), 173-189.

[Vi] M.Vigué-Poirrier,Hochschild homology criteria for trivial algebra structures, Trans. A.M.S., 354, (2002), 3869-3882.

[AMA - Algebra Montpellier Announcements - 01-2003] [September 2003]