ON COMMUTATIVE ALGEBRA HOMOLOGY IN PRIME CHARACTERISTICS

Sarah Whitehouse

Mathematics Institute University of Warwick Coventry CV4 7AL England E-mail address: sarah@uk.ac.warwick.maths

Abstract

We give decompositions of the Hochschild and cyclic homology of a commutative algebra in characteristic p into p−1 parts, and show these decompositions are compatible with the shuffle product structures. We also give a counterexample to a conjecture attributed to Barr, which asserts that a modified version of Harrison cohomology coincides with André/Quillen cohomology.

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Introduction

In this paper we consider some aspects of the homology of commutative algebras in prime characteristics. The theories which we shall consider are Hochschild, Harrison and cyclic homology. We give decompositions of the Hochschild homology and cyclic homology of a commutative algebra over a field of characteristic p into p − 1 parts.

In the first section we cover the decomposition of Hochschild homology. This result was announced by Gerstenhaber and Schack, but it seems to us that it is really necessary to modify the argument they sketch in the final section of [3] and to use generalised eigenspaces. In particular, it is not generally the case that one part of the decomposition is Harrison homology. The second section covers the analogous decomposition for cyclic homology. We note that in the case p = 3, the decomposition of cyclic homology into two parts coincides with Loday’s splitting of cyclic homology in terms of dihedral homology, where the algebra has the trivial involution [4]. The decompositions are closely related to work of Patras in the general Hopf algebra setting [5].

In the final section we discuss a conjecture attributed to Barr in [2]. The Harrison cohomology of a commutative algebra is known to coincide with the André/Quillen cohomology over a field of characteristic zero but not in prime characteristics. The conjecture is that a modified version of Harrison cohomology, taking into account torsion, always agrees with André/Quillen cohomology. We give a counterexample.

These results were announced in [7].

1. The decomposition of Hochschild homology

We briefly recall the definition of Hochschild homology. Let A be an associative algebra (with 1) over a commutative ring k and let M be an A-bimodule. We denote ⊗_{k} simply by ⊗. The Hochschild complex is C_{n}(A; M) = M ⊗ A^{⊗n}, with boundary b : C_{n}(A; M) → C_{n−1}(A; M) given by

b(m ⊗ a_{1} ⊗ ⋯ ⊗ a_{n}) = (ma_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n})

+∑_{i=1}^{n−1} (−1)^{i}(m⊗a_{1}⊗⋯⊗a_{i}a_{i+1}⊗⋯⊗a_{n})+(−1)^{n}(a_{n}m⊗a_{1}⊗⋯⊗a_{n−1}).

The Hochschild homology of A with coefficients in M, denoted HH_{n}(A; M), is the homology of this chain complex. If M = A, we write HH_{n}(A). We let the symmetric group Σ_{n} act on the left on C_{n}(A; M) by

σ(m ⊗ a_{1} ⊗ ⋯ ⊗ a_{n}) = (m ⊗ a_{σ^{−1}1} ⊗ ⋯ ⊗ a_{σ^{−1}n}),

and extend this linearly to an action of the group algebra kΣ_{n}.

The Eulerian idempotents e_{n}^{(j)} for j = 1, …, n, lying in the rational group algebra of the symmetric group ℚΣ_{n}, were defined by Gerstenhaber and Schack [2]. If A is a commutative algebra over a ground ring k containing ℚ, then they commute with the Hochschild boundary map b, be_{n}^{(j)} = e_{n−1}^{(j)}b, so that they yield a decomposition of Hochschild homology [2]. We wish to give a modified version of this result valid in prime characteristics. We fix a prime p.

1.1 Definition. The lambda operations λ_{n}^{q} in the integral group algebra of the symmetric group ℤΣ_{n} are defined as follows. We let T^{c}(A) denote the cotensor Hopf algebra over A. That is, (T^{c}(A))_{n}= A^{⊗n}, the comultiplication Δ is deconcatenation:

Δ(a_{1} ⊗ ⋯ ⊗ a_{n}) = ∑_{i=1}^{n}(a_{1} ⊗ ⋯ ⊗ a_{i}) ⊗ (a_{i+1} ⊗ ⋯ ⊗ a_{n}),

and the multiplication μ is the signed shuffle:

μ((a_{1} ⊗ ⋯ ⊗ a_{p}) ⊗ (a_{p+1} ⊗ ⋯ ⊗ a_{p+q})) = ∑(sgnσ)σ(a_{1} ⊗ ⋯ ⊗ a_{p+q}),

where the sum is over (p, q)-shuffles in Σ_{p+q}. The shuffle product is graded commutative.

Then we define λ^{q} = μ^{q−1} ∘ Δ^{q−1}, a degree zero algebra map on T^{c}(A). Its restriction to A^{⊗n} is denoted λ_{n}^{q} and this may be thought of as the action of a well-defined element of the group algebra ℤΣ_{n}, also denoted λ_{n}^{q}.

The lambda operations are related to the Eulerian idempotents by the following result of Loday [4].

1.2 Proposition.

λ_{n}^{q} = ∑_{i=1}^{n} q^{i}e_{n}^{(i)} for all q ≥ 1. □

So e_{n}^{(i)}λ_{n}^{q} = q^{i}e_{n}^{(i)}. We adopt the conventions that e_{n}^{(i)} = 0 for i > n, e_{0}^{(0)} = 1 and e_{n}^{(0)} = 0 for n > 0.

1.3 Proposition. The idempotents ē_{n}^{(i)} = ∑_{m≥0} e_{n}^{(i+(p−1)m)} for i = 1, …, p − 1, are defined in kΣ_{n}, where k contains 𝔽_{p}.

Proof. The proof is a modified version of the argument appearing in [3]. In the rational group algebra ℚΣ_{n} the following equation holds for i, q ∈ {1, …, p − 1} and m ≥ 0.

e_{n}^{(i+(p−1)m)} (λ_{n}^{q} − q^{i}) = (q^{i+(p−1)m} − q^{i}) e_{n}^{(i+(p−1)m)}.

Since q^{i+(p−1)m} − q^{i} is divisible by p, by choosing a sufficiently large r, we can remove multiples of p in the denominator, so that we have in 𝔽_{p}Σ_{n},

e_{n}^{(i+(p−1)m)} (λ_{n}^{q} − q^{i})^{r} = (q^{i+(p−1)m} − q^{i})^{r} e_{n}^{(i+(p−1)m)} = 0.

Thus ē_{n}^{(i)} (λ_{n}^{q} − q^{i})^{r} = 0. Now choosing r a power of p, we have (λ_{n}^{q} − q^{i})^{r} = (λ_{n}^{q})^{r}−q^{ir} = λ_{n}^{q^{r}}−q^{ir} = λ_{n}^{q^{r}}−q^{i}, in 𝔽_{p}Σ_{n}. Using the fact that ∑_{i=1}^{p−1} ē_{n}^{(i)} = 1, we see that the idempotents ē_{n}^{(i)} satisfy the equations:

λ_{n}^{q^{r}} = ∑_{i=1}^{p−1} q^{i}ē_{n}^{(i)}.

The matrix of coefficients (q^{i}) is Vandermonde and invertible over 𝔽_{p}, so the idempotents ē_{n}^{(i)} are defined in 𝔽_{p}Σ_{n}. □

We set \overline{HH}_{n}^{(i)}(A; M) = ē_{n}^{(i)}HH_{n}(A; M). It follows from the above that this is the generalised eigenspace of the lambda operation λ_{n}^{q} with respect to eigenvalue q^{i}, where q is a primitive root modulo p.

1.4 Theorem. Let A be a commutative algebra over a ring k containing 𝔽_{p}, and let M be a symmetric A-bimodule. Then we have a direct sum decomposition of Hochschild homology into p − 1 parts:

HH_{n}(A; M) = \overline{HH}_{n}^{(1)}(A; M) ⊕ ⋯ ⊕ \overline{HH}_{n}^{(p−1)}(A; M) for n > 0,
HH_{0}(A; M) = \overline{HH}_{0}^{(0)}(A; M).

Proof. We have seen that the idempotents ē_{n}^{(i)} are defined in kΣ_{n}. These idempotents commute with the Hochschild boundary, b, that is bē_{n}^{(i)} = ē_{n−1}^{(i)}b. So they give the stated decomposition of homology. □

2. The decomposition of cyclic homology

We recall the definition of cyclic homology. First, letting Ā = A/k, we replace the Hochschild complex by its normalised version, C̄_{n}(A; M) = M ⊗ Ā^{⊗n}, with boundary b as before. When M = A we denote this C̄_{n}(A). Then the cyclic homology of A over k, denoted HC_{∗}(A), may be defined as the homology of the total complex corresponding to the normalised (b − B) bicomplex, B(A):

⋮  ⋮  ⋮
b↓  b↓  b↓
C̄_{n}(A) ⟵^{B} C̄_{n−1}(A) ⟵^{B} C̄_{n−2}(A) ⟵^{B} ⋯
b↓  b↓  b↓
C̄_{n−1}(A) ⟵^{B} C̄_{n−2}(A) ⟵^{B} C̄_{n−3}(A) ⟵^{B} ⋯
b↓  b↓  b↓
⋮  ⋮  ⋮
b↓  b↓  b↓
C̄_{2}(A) ⟵^{B} C̄_{1}(A) ⟵^{B} C̄_{0}(A)
b↓  b↓
C̄_{1}(A) ⟵^{B} C̄_{0}(A)
b↓
C̄_{0}(A)

where B : C̄_{n−1}(A) → C̄_{n}(A) is defined by

B(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n−1}) =

∑_{j=0}^{n−1} (−1)^{(n−1)j}(1 ⊗ a_{j} ⊗ a_{j+1} ⊗ ⋯ ⊗ a_{n−1} ⊗ a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{j−1}).

Now, for a commutative algebra A, over a ground ring k containing ℚ, the Eulerian idempotents are well-behaved with respect to B as well as b, Be_{n−1}^{(j−1)} = e_{n}^{(j)}B, so that they decompose cyclic homology [4].

We show how to give a decomposition of cyclic homology over k containing 𝔽_{p}.

2.1 Proposition. Let A be a commutative algebra over k containing 𝔽_{p}. Then

\begin{aligned}Bē_{n−1}^{(i−1)} & = ē_{n}^{(i)}B, for i = 2, …, p − 1, \\ Bē_{n−1}^{(p−1)} & = ē_{n}^{(1)}B, \end{aligned}

where B is Connes’ boundary operator in the normalised context.

Proof. We have λ_{n}^{q}B = Bqλ_{n−1}^{q} as in [4]. Thus λ_{n}^{q^{r}}B = Bq^{r}λ_{n−1}^{q^{r}}. Then, in 𝔽_{p}Σ_{n}, choosing r as in 1.3,

∑_{i=1}^{p−1} q^{i}ē_{n}^{(i)}B = Bq^{r} ∑_{i=1}^{p−1} q^{i}ē_{n−1}^{(i)}

= Bq ∑_{i=1}^{p−1} q^{i}ē_{n−1}^{(i)}

= ∑_{i=2}^{p} q^{i}Bē_{n−1}^{(i−1)}

= qBē_{n−1}^{(p−1)} + ∑_{i=2}^{p−1} q^{i}Bē_{n−1}^{(i−1)}

The result follows by inverting the matrix (q^{i}). □

We set C̄_{n}^{(i)}(A) = ē_{n}^{(i)}C̄_{n}(A).

2.2 Theorem. If A is a commutative algebra over k containing 𝔽_{p} there is a decomposition of cyclic homology into p − 1 parts:

HC_{n}(A) = \overline{HC}_{n}^{(1)}(A) ⊕ ⋯ ⊕ \overline{HC}_{n}^{(p−1)}(A) for n > 0,

HC_{0}(A) = \overline{HC}_{0}^{(0)}(A) = A.

Proof. It follows from the above that the (b − B) bicomplex for cyclic homology is a direct sum of the sub-bicomplexes B̄(A)^{(i)}:

⋮  ⋮  ⋮  ⋮
b↓  b↓  b↓  b↓
C̄_{n+1}^{(i)} ⟵^{B} C̄_{n}^{(i−1)} ⟵ ⋯ C̄_{n−i+2}^{(1)} ⟵^{B} C̄_{n−i+1}^{(p−1)} ⟵^{B} ⋯
b↓  b↓  b↓  b↓
C̄_{n}^{(i)} ⟵^{B} C̄_{n−1}^{(i−1)} ⟵ ⋯ C̄_{n−i+1}^{(1)} ⟵^{B} C̄_{n−i}^{(p−1)} ⟵^{B} ⋯
b↓  b↓  b↓  b↓
⋮  ⋮  ⋮  ⋮

Thus, denoting by \overline{HC}_{∗}^{(i)}(A) the homology of the total complex associated to the above, we have the stated decomposition. □

For p = 2, the decomposition is trivial. For p = 3, the idempotents are ē_{n}^{(1)} = \frac{1}{2}(1 + op_{n}) and ē_{n}^{(2)} = \frac{1}{2}(1 − op_{n}), where p_{n} is the permutation (1 n)(2 n − 1)(3 n − 2) ⋯ ∈ Σ_{n} and op_{n} = (−1)^{n(n+1)/2}p_{n}. Thus the splitting of cyclic homology into two parts given here is Loday’s splitting in terms of dihedral homology, HC_{n}(A) = HD_{n}(A) ⊕ HD′_{n}(A), where A has the trivial involution [4].

The above also shows that the periodicity sequence linking Hochschild and cyclic homology respects their decompositions.

2.3 Proposition. For i = 2, …, p − 1 there are long exact sequences:

⟶ \overline{HH}_{n}^{(i)} ⟶^{I} \overline{HC}_{n}^{(i)} ⟶^{S} \overline{HC}_{n−2}^{(i−1)} ⟶^{B} \overline{HH}_{n−1}^{(i)} ⟶

and for i = 1 there is a long exact sequence:

⟶ \overline{HH}_{n}^{(1)} ⟶^{I} \overline{HC}_{n}^{(1)} ⟶^{S} \overline{HC}_{n−2}^{(p−1)} ⟶^{B} \overline{HH}_{n−1}^{(1)} ⟶

Proof. The exact sequences above are the long exact sequences in homology corresponding to the following short exact sequences of complexes.

For i = 2, …, p − 1 we have

0 ⟶ C̄_{∗}^{(i)}(A) ⟶ T ot(B̄(A)^{(i)}) ⟶ T ot(B̄(A)^{(i−1)})⟨2⟩ ⟶ 0,

where ⟨2⟩ indicates a shift in degree.

For i = 1 we have

0 ⟶ C̄_{∗}^{(1)}(A) ⟶ T ot(B̄(A)^{(1)}) ⟶ T ot^{+}(B̄(A)^{(p−1)})⟨2⟩ ⟶ 0,

where + indicates the presence of an extra C̄_{0}^{(0)}(A) in the complex. □

2.4 Corollary. \overline{HH}_{n}^{(1)}(A) ≅ \overline{HC}_{n}^{(1)}(A) for 2 < n < p.

If n = 2 < p we have a short exact sequence,

0 ⟶ \overline{HH}_{2}^{(1)}(A) ⟶ \overline{HC}_{2}^{(1)}(A) ⟶ H_{DR}^{0}(A) ⟶ 0.

Proof. For i = 1, the bicomplex is B̄(A)^{(1)}:

⋮  ⋮  ⋮
b↓  b↓  b↓
p  C̄_{p}^{(1)} ⟵^{B} C̄_{p−1}^{(p−1)} ⟵^{B} C̄_{p−2}^{(p−2)} ⟵^{B} ⋯
b↓  b↓  b↓
p − 1  C̄_{p−1}^{(1)} ⟵^{B}  0  ⟵^{B}  0  ⟵^{B} ⋯
b↓

⋮

C̄_{2}^{(1)}

b↓
C̄_{1}^{(1)} ⟵^{B} C̄_{0}^{(0)}
b↓

0

Since the homology of the first column is \overline{HH}_{∗}^{(1)}(A) and the homology of the total complex associated to B̄(A)^{(1)} is \overline{HC}_{∗}^{(1)}(A) the result for 2 < n < p follows. The presence of C̄_{0}^{(0)} requires a modification for n ≤ 2. The required exact sequence follows from the fact that B induces d : A → Ω_{A/k}^{1}

on C̄_{0}^{(0)}. □

2.5 Corollary. When k contains 𝔽_{p}, Harrison homology, Harr_{∗}(A), is a direct summand of Hochschild homology in dimensions n < p and of cyclic homology in dimensions 2 < n < p. If n = 2 < p we have an exact sequence,

0 ⟶ Harr_{2}^{(1)}(A) ⟶ \overline{HC}_{2}^{(1)}(A) ⟶ H_{DR}^{0}(A) ⟶ 0.

Proof. Note that for n < p, ē_{n}^{(1)} = e_{n}^{(1)}, and so \overline{HH}_{n}^{(1)}(A) = HH_{n}^{(1)}(A) which is known to coincide with Harr_{n}(A). □

3. The decompositions and the shuffle product

We explain how the decompositions of Hochschild and cyclic homology behave with respect to the shuffle products.

3.1 Proposition. The decomposition of the Hochschild homology of a commutative algebra A in characteristic p is compatible with the shuffle product, μ:

μ : \overline{HH}_{r}^{(i)}(A) × \overline{HH}_{s}^{(j)}(A) → \overline{HH}_{r+s}^{([i+j])}(A),

where [i + j] is the class of i + j modulo p − 1.

Proof. Let q be a primitive root modulo p. Then the i^{th} part of the decomposition \overline{HH}_{∗}^{(i)}(A) corresponds to the generalised eigenspace of the lambda operation λ^{q} with respect to eigenvalue q^{i}. Now λ^{q} is an algebra map for the shuffle product so that λ^{q}(μ(a ⊗ b)) = μ((λ^{q}a) ⊗ (λ^{q}b)). The result follows easily. □

3.2 Proposition. The decomposition of the cyclic homology of a commutative algebra A in characteristic p is compatible with the internal product, ∗:

∗ : \overline{HC}_{r}^{(i)}(A) × \overline{HC}_{s}^{(j)}(A) → \overline{HC}_{r+s+1}^{([i+j+1])}(A),

where [i + j + 1] is the class of i + j + 1 modulo p − 1.

Proof. We recall the definition of the product ∗. It is induced by the map of chain complexes

∗ : T ot B(A)_{r} ⊗ T ot B(A)_{s} → T ot B(A)_{r+s+1}

given by

x ∗ y = (μ(Bx_{r} ⊗ y_{s}), μ(Bx_{r} ⊗ y_{s−2}), …),

for x = (x_{r}, x_{r−2}, …), y = (y_{s}, y_{s−2}, …) and where μ is again the shuffle product. The result can be easily deduced using

1). λ^{q}(μ(a ⊗ b)) = μ((λ^{q}a) ⊗ (λ^{q}b)), 2). λ^{q}B = Bqλ^{q} and the description of \overline{HC}_{∗}^{(i)}(A) as the generalised eigenspace of λ^{q} cor-

responding to eigenvalue q^{i}, for q a primitive root modulo p. □

4. A counterexample to a conjecture of Barr

Now let A be a commutative algebra and consider the Hochschild cochain complex (Hom_{A}(C_{∗}(A), M), b). The Harrison cochain complex is the subcomplex consisting of those cochains which vanish on shuffles, giving rise to Harrison cohomology, Harr^{∗}(A; M). This commutative algebra cohomology theory is known to coincide with the André/Quillen theory when A is projective over k and k contains ℚ [6]. An example of Barr shows that the theories do not coincide in prime characteristics [1].

Note that the indecomposables of the cotensor algebra T^{c}(A) may have torsion. For example, μ((a⊗b)⊗(a⊗b)) = 2(a⊗b⊗a⊗b), so that when k has characteristic zero a Harrison 4-cochain must vanish on (a ⊗ b ⊗ a ⊗ b), but not when k has characteristic 2. Let us denote by HB^{∗}(A; M) the homology of the subcomplex of all Hochschild cochains vanishing not only on shuffles but also on those elements some multiple of which is a shuffle. According to [2], Barr conjectures that this gives André/Quillen cohomology. We give a counterexample.

4.1 Proposition. HB^{5}(𝔽_{2}[x]; 𝔽_{2}) ≠ 0, where 𝔽_{2} is a 𝔽_{2}[x]-module via the trivial x-action.

Proof. The idea is to give a Harrison 5-cocycle f as the boundary of an explicitly defined Harrison 4-cochain g. So of course as a Hochschild or Harrison cycle f is cohomologous to zero. We then show that f satisfies Barr’s condition, but is not the boundary of any Barr 4-cochain.

The Hochschild 4-cochain g is defined as follows,

g(1 ⊗ x ⊗ 1 ⊗ x) = g(x ⊗ 1 ⊗ x ⊗ 1) = g(x ⊗ 1 ⊗ 1 ⊗ x) = g(1 ⊗ x ⊗ x ⊗ 1) = 1,

g is zero on any other chain of the form (x^{i_{1}} ⊗ x^{i_{2}} ⊗ x^{i_{3}} ⊗ x^{i_{4}}), and we extend linearly.

Let f = δg. Now f ≠ 0, since for example f(1 ⊗ 1 ⊗ x ⊗ 1 ⊗ x) = 1. Next we check that g vanishes on shuffles. It is clearly sufficient to check

that g vanishes on those shuffles which involve

(1 ⊗ x ⊗ 1 ⊗ x), (x ⊗ 1 ⊗ x ⊗ 1), (x ⊗ 1 ⊗ 1 ⊗ x) or (1 ⊗ x ⊗ x ⊗ 1).

For example,

\begin{matrix}g(μ((1) ⊗ (x ⊗ 1 ⊗ x))) = & g(1 ⊗ x ⊗ 1 ⊗ x) − g(x ⊗ 1 ⊗ 1 ⊗ x) \\ + g(x ⊗ 1 ⊗ 1 ⊗ x) − g(x ⊗ 1 ⊗ x ⊗ 1) = 0. \end{matrix}

Similar calculations show that g does vanish on all such shuffles. Hence g is a Harrison cochain and so f = δg is a Harrison 5-cocycle.

Notice that as above (1 ⊗ x ⊗ 1 ⊗ x) has a multiple which is a shuffle and yet g(1 ⊗ x ⊗ 1 ⊗ x) ≠ 0, so g does not satisfy Barr’s condition. In fact, if f = δh then 1 = f(1⊗1⊗x⊗1⊗x) = δh(1⊗1⊗x⊗1⊗x) = h(1⊗x⊗1⊗x). So f is not the coboundary of any cochain satisfying Barr’s condition.

It remains to check that f itself does satisfy Barr’s condition. Now one easily checks that the only chains of the form (x^{i_{1}} ⊗ x^{i_{2}} ⊗ x^{i_{3}} ⊗ x^{i_{4}} ⊗ x^{i_{5}}) on which f is non-zero are:

\begin{matrix}a_{1} = (1 ⊗ 1 ⊗ x ⊗ 1 ⊗ x), & a_{2} = (1 ⊗ x ⊗ 1 ⊗ 1 ⊗ x), \\ a_{3} = (1 ⊗ 1 ⊗ x ⊗ x ⊗ 1), & a_{4} = (x ⊗ 1 ⊗ x ⊗ 1 ⊗ 1), \\ a_{5} = (x ⊗ 1 ⊗ 1 ⊗ x ⊗ 1), & a_{6} = (1 ⊗ x ⊗ x ⊗ 1 ⊗ 1). \end{matrix}

We have f(a_{i}) = 1 for i = 1, …, 6. We need only check for shuffle multiples involving these. We introduce the following notation.

\begin{matrix}b_{1} = (1 ⊗ 1 ⊗ 1 ⊗ x ⊗ x), & b_{2} = (x ⊗ 1 ⊗ 1 ⊗ 1 ⊗ x), \\ b_{3} = (x ⊗ x ⊗ 1 ⊗ 1 ⊗ 1), & b_{4} = (1 ⊗ x ⊗ 1 ⊗ x ⊗ 1). \end{matrix}

We want to consider all possible shuffles involving three 1’s and two x’s. Routine calculations yield that these have the following form.

z_{1}(a_{1} + a_{2}) + z_{2}(a_{1} + a_{5}) + z_{3}(a_{2} + a_{4}) + z_{4}(b_{1} + a_{2} + a_{3}) + z_{5}(b_{1} + a_{2} + a_{6}) + z_{6}(b_{2} + a_{1} + a_{4}) + z_{7}(b_{3} + a_{3} + a_{5}) + z_{8}(b_{3} + a_{2} + a_{6}) + z_{9}b_{4},

where z_{i} ∈ ℤ.

For example, a_{1} + a_{2} = μ((1 ⊗ x) ⊗ (1 ⊗ 1 ⊗ x)).

Now consider when the above expression has form q(∑ k_{i}a_{i} + ∑ l_{j}b_{j}), for q, k_{i}, l_{j} ∈ ℤ. Firstly, the coefficient of each a_{i} and b_{j} is a multiple of q, and adding the coefficients of a_{2}, a_{5} and b_{2} gives ∑_{i=1}^{8} z_{i}. So ∑_{i=1}^{8} z_{i} ∈ qℤ. Secondly, the sum of the coefficients of the a_{i}’s is q ∑ k_{i} = 2 ∑_{i=1}^{8} z_{i}. Hence, q ∑ k_{i} ∈ 2qℤ and ∑ k_{i} ≡ 0 (mod 2).

But then f(∑ k_{i}a_{i} + ∑ l_{j}b_{j}) = ∑ k_{i} ≡ 0 (mod 2). Hence, f does vanish on any element some multiple of which is a shuffle. So 0 ≠ [f] ∈ HB^{5}(𝔽_{2}[x]; 𝔽_{2}). □

Another version of the conjecture is given by considering the Hochschild chain complex with the shuffle product as a divided power algebra, and factoring out the divided powers. The above is also a counterexample to the conjecture that the cohomology of this complex then gives André/Quillen cohomology. Firstly, since a ⊗ b ⊗ a ⊗ b is a divided power we see that f is not the coboundary of any cochain vanishing on divided powers. Secondly, the arguments above show that f itself does vanish on all divided powers.

Acknowledgement

Thanks to M. Barr for helpful comments about the counterexample.

References

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[3] M. Gerstenhaber and S.D. Schack, “The shuffle bialgebra and the cohomology of commutative algebras,” J. Pure Appl. Alg. 70 (1991), 263–272.

[4] J-L. Loday, “Cyclic Homology,” Springer-Verlag 1992.

[5] F. Patras, “La décomposition en poids des algebrès de Hopf,” Ann. Inst. Fourier, Grenoble 43 (1993), 1067–1087.

[6] D. Quillen, “On the (co)-homology of commutative rings,” Proc. Symp. Pure Math. 17 (1970), 65–87.

[7] S.A. Whitehouse, “Γ-(co)homology of commutative algebras and some related representations of the symmetric group,” Ph.D. thesis, University of Warwick (1994).