## Blocks as orders over a $$p$$-adic ring

#### Florian Eisele (City)

Let $$\mathcal{O}$$ be an extension of the $$p$$-adic integers. Let $$k$$ denote its residue field and $$K$$ its field of fractions. We consider a fixed $$p$$-block of a finite group $$G$$. An interesting problem in the modular representation theory of finite groups is
the determination of the basic algebra of such a block, which is defined to be the unique smallest (i. e. lowest-dimensional) $$k$$-algebra whose module category is equivalent to that of the block. I will talk about how to approach this problem
from an integral point of view, i. e. by looking at the corresponding block of $$\mathcal{O}G$$ as an $$O$$-order in a semisimple $$K$$-algebra. I will present some results, in particular on "tame blocks", and an approach to this problem using derived equivalences.