LOCAL LIMITATIONS OF THE Ext FUNCTOR DO NOT EXIST

Abstract

In this note it is shown that for $k$ a field, and for the four-dimensional algebra $\Lambda=k\langle x,y\rangle /\langle x^2,y^2,xy+qyx\rangle$ when $q^n\neq 1,0$ for all $n$, there exist a two-dimensional module $M$ and a family of two-dimensional modules $M_i$, $i=1,2,\ldots$, such that $\dim_k\Ext^i_\Lambda(M,M_j)=1$ for $i$ equal to 0, $j$ and $j+1$, and $\dim_k\Ext^i_\Lambda(M,M_j)=0$ otherwise. This is probably the most straightforward example giving a negative answer to a question raised by Maurice Auslander.