Vanishing of Hochschild cohomologies and directed graphs with polynomial weights
In what follows, \(\Lambda\) denotes a finite dimensional algebra over an algebraically closed field \(K\). This paper addresses a question raised by \textit{H. Tachikawa} [``Quasi-Frobenius Rings and Generalizations'' (Lect. Notes Math. 351, 1973; Zbl 0271.16004)], namely: Is \(\Lambda\) self- injective if the Hochschild cohomologies \(H^ i(\Lambda \otimes \Lambda)\) are zero for all \(i \geq 1\)? The author [Tsukuba J. Math. 14, No. 2, 263-273 (1990; Zbl 0735.13008)] has previously provided an affirmative answer in the special case when \(\Lambda\) is a local commutative algebra over \(K\) of characteristic 0 which is a homomorphic image of \(K[x,y]\) and satisfies \((\text{rad }\Lambda)^ 4 = 0\). Here he concentrates on removing this latter nilpotency condition on \(\text{rad }\Lambda\). He defines \(\Lambda\) to be generalized biserial if \(\Lambda\) is a commutative local algebra over \(K\), where \(\text{char }K \neq 2\), such that for all \(j \geq 1\) the lengths of the \(\Lambda\)-modules \((\text{rad }\Lambda)^ j / (\text{rad }\Lambda)^{j + 1}\) are at most two. His main result is that if \(\Lambda\) is generalized biserial then \(\Lambda\) is self-injective if and only if \(H^ 1(\Lambda \otimes \Lambda) = 0\). To prove this he first shows that if \(\Lambda\) is generalized biserial but not serial in which the nilpotency index of the radical is \(n\) then \(\Lambda\) is isomorphic to one of the following four algebras: (i) \(K[x,y]/((x,y)^ n,x^ 2 - y^ p)\); (ii) \(K[x,y]/((x,y)^ n,x^ 2 - y^ p,xy^ s)\); (iii) \(K[x,y]/((x,y)^ n\), \(x^ 2 - y^ p,y^{n- 1})\); (iv) \(K[x,y]/((x,y)^ n,x^ 2 - y^ p,xy^ q - cy^ r)\), where \(p \geq 2\), \(s\geq 1\), \(n > r > q \geq 1\), and \(c = \sum c_ i y^ i \in K[y]\) with \(c_ 0 \neq 0\). He then employs a direct weighted graph associated with \(\Lambda\) to prove his main result for each of the cases (i)-(iv). Finally he gives an example of a generalized biserial algebra which is not gradable by the non-negative integers.
scientific article; zbMATH DE number 168589