The homological quadratic form of a biextension algebra

Let A be a finite dimensional algebra over an algebraically closed field k. It is conjectured that A has to be of tame representation type provided A is strongly simply connected and its Tits quadratic form is weakly non-nega tive. In the paper a partial result in this direction is proved. Instead of the T its form the Euler form chi(A) is considered. Let R-1,..., R-t and R-1'...; R-s' be two sequences of modules over an algebra B. We consider R = [GRAPHICS] R-i as a B - k(t)-bimodule and R' = [GRAPHICS] R-j' as a B - k(s)-bimodule. The biextension [R']B[R] of B by the two seque nces is by definition the matrix algebra [GRAPHICS] equipped with the obvious addition and multiplication, where we denote by D =Hom(k)(-,k) the usual duality. For any set of pairs of indices L subset of {1,...,t} x {1,...,s}, consider the subspace V = [GRAPHICS] DRj x(B) R-i of the space DR' x(B) R and the associated ideal J(V) in [R']B [R], The algebra A = [R']B[R]/J(V) is called a truncated biextension of B. The main result of the paper says: If B is a strongly simply connected alge bra with at least 6 vertices and R-1,..., R-t; R-1',..., R-s' are two seque nces of indecomposable B-modules such that chi(A) is non-negative with cora nk chi(A) = 1 + s + t and corank chi(B) = 1, then A = [R']B[R] is of tame r epresentation type.