Hochschild Cohomology of the Integral Group Ring of the Dihedral Group. I: Even Case

A free bimodule resolution is constructed for the integral group ring of the dihedral group of order 4m. This resolution is applied for a description, in terms of generators and defining relations, of the Hochschild cohomology algebra of this group ring. Introduction Let K be a commutative ring with unity, let R be an associative K-algebra that is a finitely generated projective K-module, let Λ = R = R ⊗K R be its enveloping algebra, and let HH(R) = ExtΛ(R,R) be the nth Hochschild cohomology group of the algebra R (with coefficients in the R-bimodule R). On the Abelian group HH∗(R) = ⊕ n≥0 HH(R) = ⊕ n≥0 ExtΛ(R,R), we introduce a structure of an associative algebra with the -product (see [1, §5], [2, Chapter XI], and [3]); this algebra is called the Hochschild cohomology algebra. The algebra HH∗(R) is a graded commutative algebra, see [3]; moreover, it is well known (see, e.g., [4, p. 120]) that the -product on HH∗(R) coincides with the Yoneda product. In recent years, interest has grown in the investigation of the multiplicative structure of the Hochschild cohomology algebra, and appreciable success has been made in the solution of this problem for finite-dimensional algebras. In [5], a description was obtained of the Hochschild cohomology algebra for the symmetric group S3 over the field F3 and for the alternating group A4 and the dihedral 2-groups over the field F2. In [6], the algebra HH∗(R) was described in the case where R is a self-injective Nakayama algebra. In the author’s papers [7, 8], a description of the Hochschild cohomology algebra was given for algebras of dihedral type in the family D(3K) over an algebraically closed ground field of characteristic two and for a family of local algebras of quaternion type, respectively. We recall that algebras of dihedral, semidihedral, and quaternion types appeared in the work of K. Erdmann on classification of group blocks of tame representation type (see [9]). Some partial results were obtained in the description of the algebra HH∗(R) for the so-called Möbius algebra (see [10, 11]) and for group blocks of tame representation type that have one or three simple modules (see [12]). On the other hand, the Hochschild cohomology algebra was calculated recently in [13] for the integral group ring of the generalized quaternion group (see also [14]). It should be noted that a ring isomorphism HH∗(K[G]) H(G)⊗KK[G], where K is a commutative 2000 Mathematics Subject Classification. Primary 13D03.