Open AccessThe equality of Hochschild cohomology group and module cohomology group for semigroup algebras
Author(s) -
Ebrahim Nasrabadi
Publication year - 2022
Publication title -
boletim da sociedade paranaense de matemática
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.347
H-Index - 15
eISSN - 2175-1188
pISSN - 0037-8712
DOI - 10.5269/bspm.44931
Subject(s) - mathematics , semigroup , cohomology , group (periodic table) , idempotence , commutative property , group cohomology , inverse , pure mathematics , combinatorics , discrete mathematics , algebra over a field , physics , geometry , quantum mechanics
Let $S$ be a commutative inverse semigroup with idempotent set $E$. In this paper, we show that for every $n\in \mathbb{N}_0$, $n$-th Hochschild cohomology group of semigroup algebra $\ell^1(S)$ with coefficients in $\ell^\infty(S)$ and its $n$-th $\ell^1(E)$-module cohomology group, are equal. Indeed, we prove that \[ \HH^{n}(\ell^1(S),\ell^\infty(S))=\HH^{n}_{\ell^1(E)}(\ell^1(S),\ell^\infty(S)),\] for all $n\geq 0$.