Nontriviality of the first Hochschild cohomology of some block algebras of finite groups

The first Hochschild cohomology space of a finite dimensional algebra over a field is isomorphic with the quotient space of derivations of the given algebra modulo inner derivations. It is an important tool attached to an algebra, studied in many articles over the years, which inherits a Lie algebra structure with brackets induced by the commutator derivations. One problem which is extensively studied in recent years, see [5, 6, 20] to mention just a few, is the problem of solvability of the Lie algebra structure of the first Hochschild cohomology. The investigation of the first Hochschild cohomology of block algebras of finite groups and its Lie algebra structure is done in [4, 18]. For instance in [18] the authors show that the first Hochschild cohomology space of a finite group block algebra, having a unique class of simple modules, is a simple Lie algebra if and only if the block algebra is nilpotent with elementary abelian defect group of order at least $3$.

A result of Fleischmann, Janiszczak and Lempken in [7] on centralizers in finite groups, which uses the Classification of Finite Simple Groups Theorem, implies that the first Hochschild cohomology of a group algebra, $\operatorname{HH}^{1}(kG)$ is nontrivial for any finite group $G$ of order divisible by a prime $p$, where $k$ is an algebraically closed field of characteristic $p$. The group algebra $kG$ decomposes into indecomposable factors called block algebras, which correspond to primitive idempotents of the center $Z(kG)$. Let $b$ be a block idempotent of $kG$ with $B=kGb$ its block algebra and a defect group $P$ of $b$, which is a certain $p$-subgroup in $G$ behaving similarly to a Sylow $p$-subgroup. Since $\operatorname{HH}^{1}(kG)$ decomposes into the direct product of the first Hochschild cohomology of all block algebras of $kG$, it follows that there is a block which has nonzero first Hochschild cohomology. In [15] Markus Linckelmann launches the following question:

The first cases of blocks for which Question 1.1 has positive answer are blocks of symmetric groups with abelian nontrivial defect groups, [15, Example 7.5].

Many finite group block algebras are Morita equivalent to some twisted group algebras. If $b$ has normal defect group in $G$ then $B$ is Morita equivalent with some twisted group algebra (with underlying group a semidirect product of the defect group and the so-called inertial quotient), see [13]. If $G$ is $p$-solvable, then $B$ is Morita equivalent to some twisted group algebra over a different $p$-solvable finite group. This result is obtained by Külshammer in [12]. We are now motivated to study the first Hochschild cohomology of twisted group algebras.

If $L$ is a finite group acting trivially on $k^{\times}$ (the group of multiplicative units in $k$) and $\alpha\in Z^{2}(L,k^{\times})$ is a $2$-cocycle, we denote by $k_{\alpha}L$ the twisted group algebra of $L$ with respect to $\alpha$. A twisted group algebra is invariant with respect to $2$-cocycles belonging to the same cohomology class. Since any class in $\operatorname{H}^{2}(G,k^{\times})$ can be represented by a normalized $2$-cocycle we assume in this paper that $\alpha$ is normalized ($\alpha(1,1)=1$), although we do not always need this. An element $x\in L$ is called $\alpha$-regular if

We denote by $L_{\alpha}^{\circ}$ the set of $\alpha$-regular elements in $L$. This definition and other properties can be found in [9, Chapter 3, Section 6] but, for completeness, Section 2 is devoted to the presentation of some basic facts about $2$-cocycles and $\alpha$-regular elements which are useful in this paper. For the rest of the paper we continue to assume that the prime $p$ is the characteristic of the field $k$. Recall that $L$ satisfies the commutator index property $C(p)$ if there is an element $x\in L$ such that $p$ divides $|C_{L}(x):(C_{L}(x))^{\prime}|$, see [7, Definition 1.1]. In this case we will say that $x$ is an element of $L$ which gives the Commutator index property $C(p)$ for $L$. We denote by $L_{C(p)}$ the set of all elements of $L$ which give the Commutator index property $C(p)$ for $L$. As a consequence of [7, Theorem] we obtain that $L_{C(p)}\neq\emptyset$ for any finite group $L$ such that $p$ divides the order of $L$. In the first main result of this paper we investigate when first Hochschild cohomologies of some twisted group algebras are nontrivial.

We denote by $L_{S(p)}$ the set of all $p$-elements giving the strong Non-Schur property $S(p)$ for $L$. This is the set of $p$-elements $x\in L$ such that $x\notin C_{L}(x)^{\prime}$. If $L_{S(p)}$ would be nonempty for all finite groups $L$, then Lemma 2.8 (vi) should give $L_{C(p)}\cap L_{\alpha}^{\circ}\neq\emptyset$. But, there are cases of finite groups for which $L_{S(p)}=\emptyset$, see [7, Proposition 2.2]. We do not know if any finite group $L$, with its order divisible by $p$, has a $p$-element which gives the Commutator index property $C(p)$. If this is true then any twisted group algebra of $L$ has nontrivial first Hochschild cohomology. By email discussion with Wolfgang Lempken it seems that the exceptions of [7, Proposition 2.2], the sporadic simple groups $Ru,J_{4}$ and $Th$ do not have this kind of $p$-elements for the primes $3,3,5$, respectively. So, we launch the following question:

In Proposition 2.9 we collect classes of finite groups which satisfy the assumption of Theorem 1.2. Applying Theorem 1.2 and Proposition 2.9 to a finite group $L$ with order divisible by $p$, which satisfies one of the five assumptions of Proposition 2.9 ($L$ is non-$p$-perfect; it has a normal Sylow $p$-subgroup, etc.), we obtain that Question 1.3 has a positive answer.

As application of Theorem 1.2 we obtain classes of block algebras of finite groups for which the answer of Question 1.1 is positive. An inertial block is a block which is basic Morita equivalent to its Brauer correspondent. We will discuss more details about inertial blocks and blocks of $p$-solvable groups in Section 4. Let $(P,e)$ be a fixed, maximal $b$-Brauer pair, where $e$ is a block of $kC_{G}(P)$. Recall that the set of $b$-Brauer pairs is a $G$-poset. We denote by ”$\leq$” the partial order relation on the set of $b$-Brauer pairs; see [1, 2] for the theory of $b$-Brauer pairs and their generalizations. We denote by $\mathcal{F}:=\mathcal{F}_{(P,e)}(G,b)$ the saturated fusion system associated with $b$ determined by the choice of $(P,e)$. This is the finite category with objects all subgroups of $P$ and morphisms given by conjugation between $b$-Brauer pairs, see [2, 16] for more details regarding fusion systems of blocks. If $(Q,f)$ is a Brauer $b$-subpair of $(P,e)$ (i.e. $(Q,f)\leq(P,e)$) then $Q$ is called $\mathcal{F}$-centric if $Z(Q)$ is a defect group of $kC_{G}(Q)f$, see [17, Proposition 8.5.3].

It is well known that a nilpotent block is basic Morita equivalent to its defect group algebra and has nilpotent Brauer corespondent. It follows that nilpotent blocks are inertial (see [19, 1.5] ), hence for nilpotent blocks Question 1.1 has positive answer. The proof of Corollary 1.4 is a consequence Proposition 4.10.

Recall that a finite group $L$ with a Sylow $p$-subgroup $D$ is called $p$-perfect if $L=\mathcal{O}^{p}(L)$. It is well known that this definition is equivalent to $\operatorname{H}^{1}(L,\mathbb{F}_{p})=0$ which, by some results reminded in Section 2, is the same as $\operatorname{H}^{1}(L,k)=0$. For shortness, sometimes, we will say that a group $L$ is non-$p$-perfect if $\operatorname{H}^{1}(L,k)\neq 0$, equivalently $\mathcal{O}^{p}(L)<L$. For any $p$-group $Q$ of the finite group $G$, the Scott module $Sc(G,Q)$ is the unique (up to isomorphism) indecomposable $kG$-module with vertex $Q$, with trivial source and having a submodule isomorphic to $k$ as trivial $kG$-module. In the second main result of this paper we present other blocks for which Question 1.1 has positive answer.

In Section 3 we prove Theorem 1.2 and Theorem 1.5. Section 4 is devoted to the investigation of blocks which satisfies the assumptions of Theorem 1.2 and 1.5. We end this paper with examples of blocks which satisfy the assumptions of Theorem 1.5.

We need to emphasize that although the proofs of the main results are not difficult, we were able to identify a sufficient and clearly useful criterion for the nonvanishing of the first Hochschild cohomology of twisted group algebras in terms of properties of the twisting $2$-cocycle. Also, there are only few published results on showing that the first Hochschild cohomology of a block with a nontrivial defect group is nonzero.

In this section, if otherwise is not specified, $k$ is any field and $M$ is a trivial $kL$-module, where $L$ is a finite group. It is well known that

This isomorphism can be easily described using the Universal Coefficient Theorem and, in particular we obtain

Since we work with trivial $kL$-modules we have explicit identifications of vector spaces, see [3, Theorem 3.4.1]

In the case of the extension of fields $\mathbb{F}_{p}\leq k$, with $k$ a field of characteristic $p$, by taking $\mathbb{F}_{p}$ the trivial $\mathbb{F}_{p}L$-module and $k$ the trivial $kL$-module we get

The compatibility of cohomology and flat scalar extensions (of which the following proposition is a special case) is well known. We leave the proof for the reader and just mention that, for the next statement (ii), one approach is to use statement (i) and centralizers decomposition of Hochschild cohomology of group algebras

where $X$ is a system of representatives of the conjugacy classes in $L$.

A $2$-cocycle $\alpha\in Z^{2}(L,k^{\times})$, where $L$ acts trivially on $k^{\times}$, is a map $\alpha:G\times G\rightarrow k^{\times}$ satisfying

It is well known that

The following lemma shows that $2$-cocycles behave well with respect to cyclic groups.

The above result is well known by experts. One justification is based on the fact that restriction of a $2$-cocycle to the cyclic group $<x>$ becomes a $2$-coboundary in $k$ (which is algebraically closed), and $2$-coboundaries of any abelian group are clearly symmetric in the two arguments.

One of the main goals in this paper is the searching for finite groups $L$ which have an $\alpha$-regular element $x\in L$ such that $\operatorname{H}^{1}(C_{L}(x),k)\neq 0$. For this we collect and recall some basic facts about $\alpha$-regular elements and about $L_{C(p)}$. The following lemma is based on arguments which we find in [9, Chapter 3, Lemma 6.1] and [7, Lemma 1.2].

We begin with some properties of inertial blocks and blocks of $p$-solvable groups.

Recall that the principal block of $kG$ is the unique block of $kG$ contained in the trivial $kG$-module $k$. Defect groups of principal blocks are Sylow $p$-subgroups. If $b$ is the principal block of a non-$p$-perfect finite group $G$, since $Sc(G,P)\cong k$ as trivial $kG$-module, we are under the assumptions of Theorem 1.5. There exist also non-principal blocks for which the defect groups are the Sylow $p$-subgroups, see [10, Remark 1] and [17, Example 10.2.1].

In the following remark, statement a), we mention an example of a block which is not principal but has defect group a Sylow $p$-subgroup and, for which the underlying group is non-$p$-perfect; thus, we are still under the assumptions of Theorem 1.5.