Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 1, 291-301.

Hochschild Cohomology Rings of Algebras k[X]/(f)

Thorsten Holm

Institut für Algebra und Geometrie, Otto-von-Guericke-Universität Magdeburg Postfach 4120, D-39016 Magdeburg, Germany e-mail: thorsten.holm@mathematik.uni-magdeburg.d400.de

Abstract. Let k be a commutative ring with unity and let f ∈ k[X] a monic polynomial. We determine the ring structure of the Hochschild cohomology for the k-algebra k[X]/(f). This generalizes results of [4] on the Hochschild cohomology rings of modular group algebras of cyclic groups over fields.

1. Introduction

The multiplicative structure of the Hochschild cohomology of an algebra is not known in general and is difficult to calculate, even in examples. In [4] we determined the structure of the Hochschild cohomology for commutative modular group algebras over a field. With a different and more direct method this was recently generalized by Cibils and Solotar to commutative group algebras over arbitrary commutative rings k, showing that there exists an algebra isomorphism HH^{∗}(kG) ≅ kG ⊗_{k} H^{∗}(G, k) for any finite abelian group G, [2].

In this paper we deal with the Hochschild cohomology of algebras of the form k[X]/(f) where k is a commutative ring and f ∈ k[X] is a monic polynomial. This includes the case of group algebras of cyclic groups and the results of [4] are obtained as special cases. But apart from group algebras we also obtain results on the Hochschild cohomology rings for a much larger class of algebras.

Our approach relies on the construction of projective resolutions of algebras k[X]/(f) as bimodules over itself given in [3]. The authors in [3] were mainly interested in cyclic homology and therefore studied only the additive structure of Hochschild cohomology. So the main topic of this article is to introduce the multiplicative structure in this context.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

292 Th. Holm: Hochschild Cohomology Rings of Algebras k[X]/(f)

For the convenience of the reader we review in this section the basic definitions on Hochschild cohomology. Let k be a commutative ring with unity and let A be an associative k-algebra. In the sequel any commutative ring is assumed to have a unit element. By A^{en} = A ⊗_{k} A^{op} we denote the enveloping algebra of A. Recall that every A-bimodule is a (left-)A^{en}-module, and vice versa. For all i ≥ 0 set A^{⊗i} = A ⊗_{k} … ⊗_{k} A, the i-fold tensor product, and C^{i} = C^{i}(A) = Hom_{k}(A^{⊗i}, A). Define a differential operator d^{i} : C^{i} → C^{i+1} by setting

(d^{i}f)(a_{1} ⊗ … ⊗ a_{i+1}) = a_{1}f(a_{2} ⊗ … ⊗ a_{i+1})+

+ ∑_{j=1}^{i}(−1)^{j}f(a_{1} ⊗ … ⊗ a_{j}a_{j+1} ⊗ … ⊗ a_{i+1}) + (−1)^{i+1}f(a_{1} ⊗ … ⊗ a_{i})a_{i+1}

for all a_{1}, …, a_{i+1} ∈ A and f ∈ C^{i}. Then (C^{i}, d^{i})_{i≥0} is easily seen to be a complex of A^{en}-modules. The additive group HH^{i}(A) = ker(d^{i})/im(d^{i−1}) is called the i-th Hochschild cohomology group of A.

There is an equivalent definition of these cohomology groups via certain Ext-functors. For all i ≥ −1 set C̃_{i} = C̃_{i}(A) = A^{⊗i+2} with the obvious A^{en}-action and define a differential by setting

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

for all a_{0}, …, a_{i+1} ∈ A. Then (C̃_{i}, d̃_{i})_{i≥0} is a projective resolution of A as an A^{en}-module ([1], 2.11). It is usually called the standard resolution. Applying the contravariant functor Hom_{A^{en}}(−, A) to this resolution yields Hom_{A^{en}}(C̃_{∗}(A), A) ≅ C^{∗}(A). Thus by passing to cohomology one gets HH^{i}(A) ≅ Ext_{A^{en}}^{i}(A, A) for all i ≥ 0.

In addition to the additive structure one can define a multiplication for the cochains. The cup product of two cochains f ∈ C^{i}(A) and g ∈ C^{j}(A) is by definition the cochain f ∪ g ∈ HH^{i+j}(A) given by

(f ∪ g)(a_{1} ⊗ … ⊗ a_{i+j}) = f(a_{1} ⊗ … ⊗ a_{i}) · g(a_{i+1} ⊗ … ⊗ a_{i+j})

for all a_{1}, …, a_{i+j} ∈ A. By a straightforward calculation one verifies d^{i+j}(f ∪ g) = d^{i}f ∪ g + (−1)^{i}f ∪ d^{j}g for all f ∈ C^{i}(A), g ∈ C^{j}(A). Thus the cup product of two cocycles is again a cocycle, which implies that the cohomology class of f ∪ g only depends on the cohomology classes of f and g. So the cup product induces a well-defined product on Hochschild cohomology ∪ : HH^{i}(A) × HH^{j}(A) → HH^{i+j}(A). This product turns the graded k-vector space HH^{∗}(A) = ⨁_{i≥0} HH^{i}(A) into a graded k-algebra.

2. A periodic resolution

Let f = X^{n} + f_{n−1}X^{n−1} + … + f_{1}X + f_{0} ∈ k[X] be a monic polynomial. We consider the corresponding k-algebra A_{f} = k[X]/(f) where (f) denotes the ideal of k[X] generated by f. Without further notice we often set in the sequel A = A_{f} for abbreviation.

In [3], the authors construct periodic resolutions for the algebras A_{f} as bimodules over itself and they give homotopy equivalences between these resolutions and the standard resolution defined in Section 1.

We first fix some notation. By P, Q, P_{i}, … we denote polynomials in k[X], but also their cosets in A if there are no ambiguities possible. The following maps will appear in the sequel:

T : A → A^{en} , T (P) = 1 ⊗ P − P ⊗ 1 ,

ϵ_{0} : A^{⊗s+2} → A^{⊗s+3} , ϵ_{0}(a_{0} ⊗ … ⊗ a_{s+1}) = 1 ⊗ a_{0} ⊗ … ⊗ a_{s+1} (for s ≥ 0) , μ : A^{en} → A , μ(P ⊗ Q) = P Q .

As f is monic, for any P ∈ k[X] there exist polynomials P̄, P̃ ∈ k[X] with P = P̄f + P̃ and deg P̃ < deg f.

Proposition 2.1. ([3], 1.3) Let A = A_{f} = k[X]/(f) where f = ∑_{i=0}^{n} f_{i}X^{i} is a monic polynomial. Consider the sequence

C̄_{∗} = C̄_{∗}(A) : … ⟶^{d_{3}} A^{⊗2} ⟶^{d_{2}} A^{⊗2} ⟶^{d_{1}} A^{⊗2} ⟶^{μ} A ⟶ 0

where d_{2r+1}(P ⊗ Q) = (P ⊗ Q)T (X) , d_{2r}(P ⊗ Q) = (P ⊗ Q)(∑_{i=1}^{n} f_{i} ∑_{j=0}^{i−1} X^{j} ⊗ X^{i−j−1}) for all r ≥ 0. Then C̄_{∗}(A) is a projective resolution of A as an A^{en}-module.

Thus the Hochschild cohomology of A_{f} is the cohomology of the complex Hom_{A^{en}}(C̄_{∗}(A_{f}), A_{f}).

For f ∈ k[X] denote by f′ ∈ k[X] the (formal) derivative of f; for g ∈ k[X] let Ann(g) = Ann_{A}(g) = {a ∈ A | ga = 0} be the annihilator of g.

Proposition 2.2. Let A = k[X]/(f) as above.

1. Using the identification Hom_{A^{en}}(A^{⊗2}, A) ≅ A, the complex Hom_{A^{en}}(C̄_{∗}(A), A) takes the

form

A ⟶^{0} A ⟶^{f′} A ⟶^{0} A ⟶^{f′} A ⟶^{0} A ⟶^{f′} A ⟶ …

where A ⟶^{f′} A is the map P ↦ P f′ given by multiplication with the derivative f′.

2. The Hochschild cohomology groups of A are

HH^{i}(A) = \{ \begin{matrix}A & if i = 0 \\ Ann_{A}(f′) & if i odd \\ A/(f′) & if i even \end{matrix}

Proof. We identify Hom_{A^{en}}(A^{⊗2}, A) ≅ A by g ↦ g(1 ⊗ 1). Then for all i ≥ 0 and g ∈ Hom_{A^{en}}(A^{⊗2}, A) one has

d_{2i+1}^{∗}(g)(1 ⊗ 1) = g ∘ d_{2i+1}(1 ⊗ 1) = g(1 ⊗ X − X ⊗ 1) = g(1 ⊗ 1) · X − X · g(1 ⊗ 1) = 0

as g is an A^{en}-homomorphism and A is commutative. Similarly,

d_{2i}^{∗}(g)(1 ⊗ 1) = g(∑_{i>0} f_{i} ∑_{j=0}^{i−1} X^{j} ⊗ X^{i−j−1})

= ∑_{i>0} f_{i} ∑_{j=0}^{i−1} g(X^{j} ⊗ X^{i−j−1})

= g(1 ⊗ 1) ∑_{i>0} f_{i} ∑_{j=0}^{i−1} X^{i−1} = g(1 ⊗ 1) · f′

for all i ≥ 0. This proves the first part; the second assertion is an easy consequence. □

We now turn our attention to the multiplicative structure of HH^{∗}(A_{f}). In order to be able to compute the cup product (which is given in terms of the standard resolution) we consider homotopy equivalences between the standard resolution and the resolution of Proposition 2.1. These were given in [3] in terms of Hochschild homology; we sketch the construction and transfer their results to cohomology.

Define A^{en}-homomorphisms g_{∗} : C̃_{∗} → C̄_{∗} inductively as follows g_{0} = Id : A^{⊗2} → A^{⊗2} the identity map;

g_{1} : A^{⊗3} → A^{⊗2} , g_{1}(1 ⊗ P ⊗ 1) = −\frac{T (P)}{T (X)} ;
g_{2} : A^{⊗4} → A^{⊗2} , g_{2}(1 ⊗ P_{1} ⊗ P_{2} ⊗ 1) = −1 ⊗ \overline{P_{1}P_{2}} ;

g_{s} : A^{⊗s+2} → A^{⊗2} , g_{s}(1⊗P_{1}⊗…⊗P_{s}⊗1) = g_{s−2}(1⊗P_{1}⊗…⊗P_{s−2}⊗1)·g_{2}(1⊗P_{s−1}⊗P_{s}⊗1)

for s > 2.

The next result describes the induced homomorphisms g_{i}^{∗} = Hom_{A^{en}}(g_{i}, A). For homology this was shown in [3]; a proof of the cohomological version needed here can be found in [4].

Lemma 2.3. Identifying Hom_{A^{en}}(A^{⊗2}, A) ≅ A and Hom_{A^{en}}(A^{⊗s+2}, A) ≅ C^{s}(A) for s > 0 the following formulas hold for all r ≥ 0 and all P, P_{1}, …, P_{2r+1} ∈ A:

g_{2r}^{∗}(P)(P_{1} ⊗ … ⊗ P_{2r}) = P (−1)^{r} ∏_{i=1}^{r} \overline{P_{2i−1}P_{2i}} ,

g_{2r+1}^{∗}(P)(P_{1} ⊗ … ⊗ P_{2r+1}) = P (−1)^{r+1}P′_{1} ∏_{i=1}^{r} \overline{P_{2i}P_{2i+1}}.

Conversely, define A^{en}-homomorphisms h_{∗} : C̄_{∗} → C̃_{∗} by setting for all r ≥ 0

h_{2r}(1 ⊗ 1) = (−1)^{r} ∑_{i_{1},...,i_{r}=1}^{n}(f_{i_{1}} … f_{i_{r}}) ∑_{k_{1},...,k_{r}=1}^{i_{1}−1,...,i_{r}−1} 1 ⊗ X^{k_{1}} ⊗ X ⊗ … ⊗ X^{k_{r}} ⊗ X ⊗ X^{c};

h_{2r+1}(1 ⊗ 1) = (−1)^{r+1} ∑_{i_{1},...,i_{r}=1}^{n}(f_{i_{1}} … f_{i_{r}}) ∑_{k_{−}1,...,k_{r}=1}^{i_{1}−1,...,i_{r}−1} 1 ⊗ X ⊗ X^{k_{1}} ⊗ … ⊗ X^{k_{r}} ⊗ X ⊗ X^{c},

where we have set c = ∑_{j=1}^{r} i_{j} − ∑_{j=1}^{r} k_{j} − r for abbreviation. Again we are mainly interested in the induced maps h_{i}^{∗} = Hom_{A^{en}}(h_{i}, A).

Lemma 2.4. Setting c = ∑_{j=1}^{r} i_{j} − ∑_{j=1}^{r} k_{j} − r for abbreviation the following formulas hold for all r ≥ 0:

h_{2r}^{∗}(α)(1 ⊗ 1) = (−1)^{r} ∑_{i_{1},...,i_{r}=1}^{n}(f_{i_{1}} … f_{i_{r}}) ∑_{k_{1},...,k_{r}=1}^{i_{1}−1,...,i_{r}−1} X^{c} · α(1 ⊗ X^{k_{1}} ⊗ X ⊗ … ⊗ X^{k_{r}} ⊗ X ⊗ 1)

for α ∈ Hom_{A^{en}}(A^{⊗2r+2}, A), and

h_{2r+1}^{∗}(α)(1⊗1) = (−1)^{r+1} ∑_{i_{1},...,i_{r}=1}^{n}(f_{i_{1}} … f_{i_{r}}) ∑_{k_{1},...k_{r}=1}^{i_{1}−1,...,i_{r}−1} X^{c} ·α(1⊗X ⊗X^{k_{1}} ⊗…⊗X^{k_{r}} ⊗X ⊗1)

for α ∈ Hom_{A^{en}}(A^{⊗2r+3}, A).

Proof. By the above formula for h_{2r} one has for all r ≥ 0 and α ∈ Hom_{A^{en}}(A^{⊗2r+2}, A):

h_{2r}^{∗}(α)(1 ⊗ 1) = α((−1)^{r} ∑_{i_{1},...,i_{r}=1}^{n}(f_{i_{1}} … f_{i_{r}}) ∑_{k_{1},...,k_{r}=1}^{i_{1}−1,...,i_{r}−1}1 ⊗ X^{k_{1}} ⊗ X ⊗ … ⊗ X^{k_{r}} ⊗ X ⊗ X^{c})

= (−1)^{r} ∑_{i_{1}−1,...,i_{r}−1}^{n}(f_{i_{1}} … f_{i_{r}}) ∑_{k_{1},...,k_{r}=1}^{i_{1}−1,...,i_{r}−1}X^{c}· α(1 ⊗ X^{k_{1}} ⊗ X ⊗ … ⊗ X^{k_{r}} ⊗ X ⊗ 1)

because α is an A^{en}-homomorphism and A is commutative.

The second assertion is shown similarly. □

3. The even cohomology ring

In this section we determine the structure of the subring HH^{ev}(A) = ⨁_{i≥0} HH^{2i}(A) of elements of HH^{∗}(A_{f}) of even degree. Therefore we have to develop formulas for the cup product of even elements.

The A^{en}-homomorphisms g_{∗} and h_{∗} of complexes are homotopy equivalences [3], so they induce isomorphisms in cohomology. By Proposition 2.2 we know HH^{0}(A) ≅ A and HH^{2i}(A) ≅ A/(f′) for all i > 0, i.e. the elements of even degree in HH^{∗}(A) are represented by polynomials in k[X]. The strategy for computing the cup product of two elements is as follows. Map the elements with g_{∗} to the level of the standard resolution, compute the cup product according to the definition of Section 1 and then map the result back with h_{∗}.

Lemma 3.1. Let A = A_{f} = k[X]/(f) as above. Then the cup product on the even Hochschild cohomology ring of A is induced by multiplication in A.

Proof. Assume Q_{1} ∈ HH^{2r}(A), Q_{2} ∈ HH^{2s}(A) for some r, s ≥ 0. For abbreviation, Q_{1} ∪ Q_{2} denotes the element g^{∗}(Q_{1}) ∪ g^{∗}(Q_{2}) ∈ C^{∗}(A). Then by definition of the cup product and by Lemma 2.3

(Q_{1} ∪ Q_{2})(P_{1} ⊗ … ⊗ P_{2(r+s)}) = g_{2r}^{∗}(Q_{1})(P_{1} ⊗ … ⊗ P_{2r}) · g_{2s}^{∗}(Q_{2})(P_{2r+1} ⊗ … ⊗ P_{2(r+s)})
= (−1)^{r}Q_{1} ∏_{i=1}^{r} \overline{P_{2i−1}P_{2i}} · (−1)^{s}Q_{2} ∏_{i=r+1}^{r+s} \overline{P_{2i−1}P_{2i}}

= (−1)^{r+s}Q_{1}Q_{2} ∏_{i=1}^{r+s} \overline{P_{2i−1}P_{2i}}

for all P_{1}, …, P_{2(r+s)} ∈ A. Setting b = ∑_{j=1}^{r+s} i_{j} − ∑_{j=1}^{r+s} k_{j} − r − s it follows by Lemma 2.4

h_{2(r+s)}^{∗}(Q_{1} ∪ Q_{2})(1 ⊗ 1) =
= (−1)^{r+s} ∑_{i_{1},...,i_{r+s}=1}^{n}(f_{i_{1}} … f_{i_{r+s}}) ∑_{k_{1},...,k_{r+s}=1}^{i_{1}−1,...,i_{r+s}−1} X^{b}(Q_{1} ∪ Q_{2})(X^{k_{1}} ⊗ X ⊗ … ⊗ X^{k_{r+s}} ⊗ X)

= (−1)^{r+s} ∑_{i_{1},...,i_{r+s}=1}^{n}(f_{i_{1}} … f_{i_{r+s}}) ∑_{k_{1},...,k_{r+s}=1}^{i_{1}−1,...,i_{r+s}−1} X^{b}(−1)^{r+s}Q_{1}Q_{2} ∏_{i=1}^{r+s} \overline{X^{k_{i}+1}}
= Q_{1}Q_{2} ,

because \overline{X^{k_{i}+1}} = δ_{k_{i},n−1}. □

Now we are in the position to give a presentation for the even Hochschild cohomology ring of k[X]/(f) by generators and relations.

Theorem 3.2. Let k be a commutative ring and let f ∈ k[X] be a monic polynomial with corresponding k-algebra A_{f} = k[X]/(f). Then

HH^{ev}(A_{f}) ≅ k[x, z]/(f(x), f′(x)z)

where deg x = 0 and deg z = 2.

Proof. As HH^{0}(A_{f}) ≅ A_{f} and cup product in degree zero is multiplication in A_{f}, the coset x = X + (f) generates HH^{0}(A_{f}); the only relation is f(x) = 0.

Take z ∈ HH^{2}(A_{f}) ≅ A_{f}/(f′) to be the coset of the constant polynomial 1 ∈ k[X]. Then HH^{2}(A_{f}) is generated by z and HH^{0}(A_{f}), again because the cup product is induced by multiplication. As HH^{2}(A_{f}) ≅ A_{f}/(f′) one gets the relation f′(x)z = 0.

In higher degrees HH^{2i}(A_{f}) is generated by the i-fold cup product z^{i} = z ∪ … ∪ z and HH^{0}(A_{f}). The corresponding relation f′(x)z^{i} = 0 is a consequence of the above relation in degree two. □

Putting some restrictions on k we already obtain the complete Hochschild cohomology ring of kC_{n} from Theorem 3.2. Note that the following result applies in particular to integral group algebras.

Corollary 3.3. Let k be an integral domain and let C_{n} be a cyclic group of order n. If the characteristic of k does not divide n then

HH^{∗}(kC_{n}) ≅ k[x, z]/(x^{n} − 1, nx^{n−1}z)

where deg x = 0 and deg z = 2.

Proof. Under the given assumptions the group cohomology of C_{n} with trivial coefficient module k vanishes in all odd degrees. This implies HH^{2i+1}(kC_{n}) = 0 for all i ≥ 0 ([1], 2.11.2). Thus HH^{∗}(kC_{n}) = HH^{ev}(kC_{n}) and the assertion follows from Theorem 3.2. □

4. The cup product in odd degrees

We have already seen that for arbitrary f the cup product in HH^{ev}(A_{f}) is given by multiplication in A_{f}. The situation in odd degrees is more difficult as we will see in this section. The aim is to prove the following result. Note that by Proposition 2.2 the elements of the Hochschild cohomology groups are represented by polynomials.

Lemma 4.1. Let f = ∑_{i=0}^{n} f_{i}X^{i} ∈ k[X] be monic and set A = A_{f} = k[X]/(f). If i or j is even then the cup product ∪ : HH^{i}(A) × HH^{j}(A) → HH^{i+j}(A) is induced by multiplication in A. If i and j are odd then the cup product is given by the formula

Q_{1} ∪ Q_{2} = −Q_{1}Q_{2} ∑_{i_{1}=1}^{n}( ∑_{k_{1}=1}^{i_{1}−1} k_{1})f_{i_{1}}X^{i_{1}−2}.

Th. Holm: Hochschild Cohomology Rings of Algebras k[X]/(f) 297

Proof. If i and j are even this was proved in Lemma 3.1. So let Q_{1} ∈ HH^{2r+1}(A), HH^{2s}(A) for some r, s ≥ 0. Then by Lemma 2.3

(Q_{1} ∪ Q_{2})(P_{1} ⊗ … ⊗ P_{2(r+s)+1}) =

= g_{2r+1}^{∗}(Q_{1})(P_{1} ⊗ … ⊗ P_{2r+1}) · g_{2s}^{∗}(Q_{2})(P_{2r+2} ⊗ … ⊗ P_{2(r+s)+1})
= (−1)^{r+1}Q_{1}P′_{1} ∏_{i=1}^{r} \overline{P_{2i}P_{2i+1}} · (−1)^{s}Q_{2} ∏_{i=r+1}^{r+s} \overline{P_{2i}P_{2i+1}}
= (−1)^{r+s+1}Q_{1}Q_{2}P′_{1} ∏_{i=1}^{r+s} \overline{P_{2i}P_{2i+1}}.

Applying the induced map h^{∗} and setting b = ∑_{j=1}^{r+s} i_{j} −∑_{j=1}^{r+s} k_{j} −r−s it follows by Lemma 2.4 h_{2(r+s)+1}^{∗}(Q_{1} ∪ Q_{2})(1 ⊗ 1) =

= (−1)^{r+s+1} ∑_{i_{1},...,i_{r+s}=1}^{n}(f_{i_{1}} … f_{i_{r+s}}) ∑_{k_{1},...,k_{r+s}}^{i_{1}−1,...,i_{r+s}−1}X^{b}(Q_{1} ∪ Q_{2})(X ⊗ X^{k_{1}} ⊗ X ⊗ … ⊗ X^{k_{r+s}} ⊗ X)
= (−1)^{r+s+1} ∑_{i_{1},...,i_{r+s}=1}^{n}(f_{i_{1}} … f_{i_{r+s}}) ∑_{k_{1},...,k_{r+s}=1}^{i_{1}−1,...,i_{r+s}−1}X^{b} · (−1)^{r+s+1}Q_{1}Q_{2}X′ ∏_{i=1}^{r+s} \overline{X^{k_{i}+1}}
= Q_{1}Q_{2}

as \overline{X^{k_{i}+1}} = δ_{k_{i},n−1}.

Assume now Q_{1} ∈ HH^{2r+1}(A), Q_{2} ∈ HH^{2s+1}(A) for some r, s ≥ 0. Then by Lemma 2.3

(Q_{1} ∪ Q_{2})(P_{1} ⊗ … ⊗ P_{2(r+s+1)}) =
= g_{2r+1}^{∗}(Q_{1})(P_{1} ⊗ … ⊗ P_{2r+1}) · g_{2s+1}^{∗}(Q_{2})(P_{2r+2} ⊗ … ⊗ P_{2(r+s+1)})
= (−1)^{r+1}Q_{1}P′_{1} ∏_{i=1}^{r} \overline{P_{2i}P_{2i+1}} · (−1)^{s+1}Q_{2}P′_{2r+2} ∏_{i=r+1}^{r+s} \overline{P_{2i+1}P_{2i+2}}.

Setting a = ∑_{j=1}^{r+s+1} i_{j} − ∑_{j=1}^{r+s+1} k_{j} − r − s − 1 it follows by Lemma 2.4 h_{2(r+s+1)}^{∗}(Q_{1} ∪ Q_{2})(1 ⊗ 1) =

= (−1)^{r+s+1} ∑_{i_{1},...,i_{r+s+1}=1}^{n}(f_{i_{1}} … f_{i_{r+s+1}})  ∑_{k_{1},...,k_{r+s+1}=1}^{i_{1}−1,...,i_{r+s+1}−1}X^{a}(Q_{1} ∪ Q_{2})(X^{k_{1}} ⊗ X ⊗ … ⊗ X^{k_{r+s+1}} ⊗X)
= (−1)^{r+s+1} ∑_{i_{1},...,i_{r+s+1}=1}^{n}(f_{i_{1}} … f_{i_{r+s+1}})  ∑_{k_{1},...,k_{r+s+1}=1}^{i_{1}−1,...,i_{r+s+1}−1}X^{a}(−1)^{r+s+2}Q_{1}Q_{2}(X^{k_{1}})′X′ ∏_{i=2}^{r+s+1} \overline{X^{k_{i}+1}}
= −Q_{1}Q_{2} ∑_{i_{1}=1}^{n} f_{i_{1}} ∑_{k_{1}=1}^{i_{1}−1} X^{i_{1}−k_{1}−1}k_{1}X^{k_{1}−1}
= −Q_{1}Q_{2} ∑_{i_{1}=1}^{n}( ∑_{k_{1}=1}^{i_{1}−1})f_{i_{1}}X^{i_{1}−2}

which completes the proof of the lemma. □

5. Polynomials with zero derivative

A consequence of Theorem 3.2 is that if f′ = 0 then the even Hochschild cohomology ring of A_{f} is a polynomial ring over HH^{0}(A_{f}). In this section we determine the structure of the whole cohomology ring of A_{f} for certain polynomials with derivative f′ = 0 using the explicit formulas for the cup product given in Lemma 4.1. Note that non-zero polynomials with zero derivative can only occur in positive characteristic.

Lemma 5.1. Let k be a commutative ring of characteristic p > 0. Assume that f ∈ k[X] is a monic polynomial with f′ = 0, i.e., f has the form f = ∑_{j=0}^{m} f_{jp}X^{jp} ∈ k[X]. For A_{f} = k[X]/(f) we then have HH^{i}(A_{f}) = A_{f} for all i ≥ 0. The cup product ∪ : HH^{i}(A_{f}) × HH^{j}(A_{f}) → HH^{i+j}(A_{f}) is induced by multiplication in A_{f} if i or j is even; if i and j are odd then ∪ is given by

Q_{1} ∪ Q_{2} = \{ \begin{matrix}Q_{1}Q_{2}(∑_{j odd} f_{2j}X^{2j−2}) & if p = 2 \\ 0 & if p ≠ 2 \end{matrix}

Proof. As f′ = 0 we get HH^{i}(A_{f}) = A_{f} for all i ≥ 0 by Proposition 2.2. If i or j is even then ∪ : HH^{i}(A_{f}) × HH^{j}(A_{f}) → HH^{i+j}(A_{f}) is multiplication in A_{f} by Lemma 4.1. So it remains to consider the case where i and j are odd. If Q_{1} ∈ HH^{i}(A_{f}), Q_{2} ∈ HH^{j}(A_{f}) then

Q_{1} ∪ Q_{2} = −Q_{1}Q_{2} ∑_{i_{1}=1}^{mp}( ∑_{k_{1}=1}^{i_{1}−1} k_{1})f_{i_{1}}X^{i_{1}−2}

= −Q_{1}Q_{2} ∑_{j=1}^{m}( ∑_{k_{1}=1}^{jp−1} k_{1})f_{jp}X^{jp−2}

If p ≠ 2 then ∑_{k_{1}=1}^{jp−1} k_{1} ≡ 0 (mod p) for all j ≥ 1. Therefore Q_{1} ∪ Q_{2} = 0 in this case. We finally consider the case p = 2. Then

∑_{k_{1}=1}^{2j−1} k_{1} ≡ \{ \begin{matrix}0 (mod 2) & if j even \\ 1 (mod 2) & if j odd \end{matrix}

Consequently, Q_{1} ∪ Q_{2} = Q_{1}Q_{2}(∑_{j odd} f_{2j}X^{2j−1}). □

We are now able to describe the cohomology rings HH^{∗}(A_{f}).

Theorem 5.2. Let k be a commutative ring with characteristic p > 0. Assume f ∈ k[X] is monic and f′ = 0, i.e., f = ∑_{j=0}^{m} f_{jp}X^{jp} ∈ k[X]. Then the Hochschild cohomology ring of A_{f} = k[X]/(f) has the following structure: If p ≠ 2 then HH^{∗}(A_{f}) ≅ A_{f}[y, z]/(y^{2}) where deg y = 1 and deg z = 2. If p = 2 then HH^{∗}(A_{f}) ≅ k[x, y, z]/(f(x), y^{2}−(∑_{j odd} f_{2j}x^{2j−2})z) where deg x = 0, deg y = 1 and deg z = 2.

Proof. Take y ∈ HH^{1}(A_{f}) to be the coset of the constant polynomial 1 ∈ k[X]. In both cases the cup product ∪ : HH^{1}(A_{f}) × HH^{0}(A_{f}) → HH^{1}(A_{f}) is multiplication in A_{f} and therefore HH^{1}(A_{f}) = {y ∪ Q | Q ∈ HH^{0}(A_{f})} is generated by y and HH^{0}(A_{f}).

First we consider the case p ≠ 2 or p = 2 and ∑_{j odd} f_{2j}X^{2j−2} = 0. Then y ∪ y = 0 by Lemma 5.1. Therefore one needs another generator z in degree two. Take z ∈ HH^{2}(A_{f}) as the coset of 1 ∈ k[X]. For all i ≥ 0 the i-fold cup product z^{i} = z ∪ … ∪ z corresponds to 1. Thus HH^{2i}(A_{f}) = {z^{i} ∪ Q | Q ∈ HH^{0}(A_{f})} and HH^{2i+1}(A_{f}) = {z^{i} ∪ P | P ∈ HH^{1}(A_{f})} are generated by y and z over HH^{0}(A_{f}). Obviously the only relation is y^{2} = 0.

Assume now p = 2 and ∑_{j odd} f_{2j}X^{2j−2} ≠ 0. Then y ∪ y ∈ HH^{2}(A_{f}) is the coset of the polynomial ∑_{j odd} f_{2j}X^{2j−2} ∈ k[X]. We shall need a generator z in degree two corresponding to the constant polynomial 1 ∈ k[X]. As in the previous case, HH^{∗}(A_{f}) is generated by y and z over HH^{0}(A_{f}). Obviously, the only defining relation is y^{2} − (∑_{j odd} f_{2j}x^{2j−2})z = 0. □

6. The general case (in characteristic ≠ 2)

Let f = g_{1}^{a_{1}} ⋯ g_{r}^{a_{r}} be the decomposition of f in irreducible factors in k[X]. By the Chinese Remainder Theorem A_{f} ≅ A_{g_{1}^{a_{1}}} × … × A_{g_{r}^{ar}}. It is well known that Hochschild cohomology behaves well with respect to direct products, i.e.

HH^{∗}(A_{f}) ≅ HH^{∗}(A_{g_{1}^{a_{1}}}) × … × HH^{∗}(A_{g_{r}^{ar}}).

So the general case reduces to the study of HH^{∗}(A_{f}) where f = g^{a} for an irreducible g ∈ k[X]. This will be done in this section. But for technical reasons we have to exclude the case of characteristic 2.

So assume char k ≠ 2. Let g ∈ k[X] be irreducible and monic and f = g^{a} for some a ∈ ℕ. If a = 1 then HH^{∗}(A_{f}) ≅ A_{f} concentrated in degree zero. So we may assume a ≥ 2. One has f′ = ag^{a−1}g′; if f′ = 0 then we already know the structure of HH^{∗}(A_{f}) by Theorem 5.2. Thus we may assume f′ ≠ 0. In particular, a ≢ 0 (mod p), g′ ≠ 0 and g is separable. Then HH^{2i+1}(A_{f}) = Ann(f′) = (g)/(g^{a}) ≤ A_{f} for all i ≥ 0 and as before HH^{2i}(A_{f}) = A_{f}/(f′) for i > 0.

After these introductory remarks we begin with building up a presentation for HH^{∗}(A_{f}). Let x ∈ HH^{0}(A_{f}) be the coset of the polynomial X ∈ k[X]. Obviously, x generates HH^{0}(A_{f}) as a k-algebra; the only relation is f(x) = 0. Take t ∈ HH^{1}(A_{f}) corresponding to g + (g^{a}) ∈ Ann(f′). Then HH^{1}(A_{f}) = (g)/(g^{a}) is generated by t and x; an obvious relation is tg^{a−1}(x) = 0. To deal with powers of t we need the following auxiliary result.

Lemma 6.1. Let k be a commutative ring with characteristic ≠ 2 in which 2 is invertible. Assume g ∈ k[X] is irreducible, f = g^{a} = ∑_{i=0}^{n} f_{i}X^{i} for some a ∈ ℕ and set S = ∑_{j=2}^{n} f_{j} ∑_{k=1}^{j−1} kX^{j−2} ∈ k[X]. Then g^{a−2} divides S.

Proof. As the characteristic of k is not 2 one gets

S = ∑_{j=2}^{n} f_{j} ∑_{k=1}^{j−1} kX^{j−2} = ∑_{j=2}^{n} f_{j} \frac{j(j − 1)}{2}X^{j−2} = \frac{1}{2}f′′.

Every root of g is a root of f with multiplicity a and thus a root of S = \frac{1}{2}f′′ with multiplicity a − 2. □

The expression S appears in the formula for the cup product of elements of odd degree (cf. Lemma 4.1). The element t^{2} = t ∪ t ∈ HH^{2}(A_{f}) is represented by −g^{2}S. By the preceding lemma f = g^{a} divides −g^{2}S and therefore t^{2} = 0 in HH^{2}(A_{f}) = A_{f}/(f′). So one needs at least one additional generator to generate the cohomology ring. Take z ∈ HH^{2}(A_{f}) to be the element represented by the constant polynomial 1 ∈ k[X]. Then HH^{2}(A_{f}) is generated by z and x because the cup product in even degrees is induced by multiplication. One gets the relation zf′(x) = 0. The set {x, t, z} generates HH^{∗}(A_{f}) as a graded k-algebra. In fact, the i-fold cup product z^{i} ∈ HH^{2i}(A_{f}) is represented by 1 ∈ k[X]; therefore HH^{2i}(A_{f}) is generated by z^{i} and x and HH^{2i+1}(A_{f}) is generated by z^{i} ∪ t and x. It is not hard to see that all the relations occuring in higher degrees are consequences of the relations given above. Thus we have proved the following result.

Theorem 6.2. Let k be a commutative ring with characteristic ≠ 2 in which 2 is invertible. Let g ∈ k[X] be irreducible, separable and monic and let f = g^{a} for some a ≥ 2. If f′ ≠ 0 then the Hochschild cohomology ring of A_{f} = k[X]/(f) has the following structure

HH^{∗}(A_{f}) ≅ k[x, t, z]/(f(x), tg^{a−1}(x), t^{2}, zf′(x))

where deg x = 0, deg t = 1 and deg z = 2. □

Remark 6.3. 1. If one restricts the above result to the even subring, i.e. set t = 0, then HH^{∗}(A_{f}) ≅ k[x, z]/(f(x), zf′(x)) as in Theorem 3.2.

2. Note that the case a = 1 which is not covered by Theorem 6.2 was already dealt with in Section 2. If a = 1 then HH^{∗}(A_{f}) ≅ k[X]/(f) concentrated in degree zero.

It remains open to deal with the case of characteristic 2. The problem seems to be that one does not have a result like Lemma 6.1 at hand. This makes it difficult to handle the powers of the generator t. Although it may be possible to compute specific examples we were not able to find a general presentation for HH^{∗}(A_{f}).

7. Truncated polynomial algebras

In this final section we apply our results to an important class of examples. For n ≥ 2 let A_{n} = k[X]/(X^{n}) be the truncated polynomial algebra. Note that this class contains modular group algebras of cyclic p-groups, whose Hochschild cohomology rings were determined in [4].

The structure of the even Hochschild cohomology rings of A_{n} are given by Theorem 3.2:

HH^{ev}(A_{n}) ≅ k[x, z]/(x^{n}, nx^{n−1}z)

where deg x = 0 and deg z = 2. The structure of the odd part depends on the characteristic of the ground ring k. Case 1: p := char k | n. The structure of HH^{∗}(A_{n}) is given by Theorem 5.2:

HH^{∗}(A_{n}) ≅ \{ \begin{matrix}k[x, y, z]/(x^{n}, y^{2}) & if p ≠ 2 or p = 2 and t ≡ 0 (mod 4) \\ k[x, y, z]/(x^{n}, y^{2} − x^{n−2}z) & if p = 2 and t ≡ 2 (mod 4) \end{matrix}

where deg x = 0, deg y = 1 and deg z = 2. Case 2: p := char k ∤ n. In this case we have the following result:

HH^{∗}(A_{n}) ≅ k[x, t, z]/(x^{n}, nx^{n−1}z, tx^{n−1}, t^{2}).

For p ≠ 2 this follows from Theorem 6.2. But the presentation is also true in characteristic 2; in fact, for the special polynomial X^{n} the formula in 4.1 for multiplication of odd degree elements becomes in characteristic 2

Q_{1} ∪ Q_{2} = \{ \begin{matrix}0 & if n ≡ 0, 1 (mod 4) \\ Q_{1}Q_{2}X^{n−2} & if n ≡ 2, 3 (mod 4) \end{matrix}

It follows that the square of the generator t chosen in Section 6 is 0. The other relations are proved verbatim as for p ≠ 2. We summarize the results on the structure of the Hochschild cohomology rings of truncated polynomial algebras.

Theorem 7.1. Let k be a commutative ring with char k = p ≥ 0. For n ≥ 2 consider the truncated polynomial algebra A_{n} = k[X]/(X^{n}). Then the Hochschild cohomology ring of A_{n} has the following structure

HH^{∗}(A_{n}) ≅ \{ \begin{matrix}k[x, y, z]/(x^{n}, y^{2}) & if p | n, and \\ & if p ≠ 2 or p = 2 and t ≡ 0 (mod 4) \\ k[x, y, z]/(x^{n}, y^{2} − x^{n−2}z) & if p | n, and p = 2 and t ≡ 2 (mod 4) \\ k[x, y, z]/(x^{n}, nx^{n−1}z, yx^{n−1}, y^{2}) & if p ∤ n \end{matrix}

where deg x = 0, deg y = 1 and deg z = 2.

References

[1] Benson, D. J.: Representations and Cohomology II: Cohomology of Groups and Modules. Cambridge Studies in Advanced Mathematics. Cambridge University Press 1991.

[2] Cibils, C.; Solotar, A.: Hochschild cohomology algebra and Hopf bimodules of an abelian group. Arch. Math. 68 (1997), 17–21.

[3] Cibils, C.; Solotar, A.: The Buenos Aires Cyclic Homology Group. Cyclic homology of algebras with one generator. K-Theory 5 (1991), 51–69.

[4] Holm, Th.: The Hochschild cohomology ring of a modular group algebra: The commutative case. Comm. Algebra 24(6) (1996), 1957–1969.

Received October 20, 1998; revised version April 30, 1999