25 March 2015.
Abstract What is the Euler characteristic of an associative algebra? There are at least two ways to answer this. The first involves a long-range attack. For every algebra A, there is the category PI(A) of projective indecomposable A-modules, which is enriched in vector spaces. There is also a general definition of the "size" or magnitude of an enriched category. Finally, Euler characteristic deserves to be thought of as a kind of measurement of size. One could therefore define the Euler characteristic of A to be the magnitude of the category PI(A). The second answer is homological and more direct: evaluate the Euler form of A at the direct sum of the simple modules.
I will explain all this, and show that the two answers above are, in fact, equivalent. Little prior knowledge will be assumed.
Slides In this pdf file.