Hochschild cohomology for a class of some self-injective special biserial algebras of rank four

In this paper, we construct an explicit minimal projective bimodule resolution of a self-injective special biserial algebra AT (T≥0) whose Grothendieck group is of rank 4. As the main result, we determine the dimension of the Hochschild cohomology group HHi(AT) of AT for i≥0, completely. Moreover we give a presentation of the Hochschild cohomology ring modulo nilpotence HH⁎(AT)/NAT of AT by generators and relations in the case where T=0.