Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A × DA in terms of the combinatorics of the quiver and relations is determined.
XU YungeDepartment of Mathematics & Computer Sciences, Hubei University, Wuhan 430062, China
Let C be a finite dimensional directed algebra over an algebraically closed field k and A=A(C) the dual extension of C.The characteristic modules of A are constructed explicitly for a class of directed algebras,which generalizes the results of Xi.Furthermore,it is shown that the characteristic modules of dual extensions of a certain class of directed algebras admit the left Gr(?)bner basis theory in the sense of E.L.Green.
For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.
