The Euler characteristic of an algebra
Tom Leinster (University of Edinburgh)
Wednesday 25th March, 2015 16:00-17:00 Maths 516
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. This is joint work with Joe Chuang and Alastair King. Little prior knowledge will be assumed.