Blocks in flat families of finite-dimensional algebras

TL;DR

This paper investigates how blocks in flat families of finite-dimensional algebras vary across the base scheme, introducing a graph-based stratification and analyzing the semicontinuity of the number of blocks.

Contribution

It constructs a finite directed graph encoding block structures in families and relates fiber blocks to atomic structures on Weil divisor components, advancing understanding of block variation.

Findings

Block structures are encoded by a finite directed graph.

Number of blocks varies lower semicontinuously across the base.

Block structure of any fiber is determined by atomic structures on divisor components.

Abstract

We study the behavior of blocks in flat families of finite-dimensional algebras. In a general setting we construct a finite directed graph encoding a stratification of the base scheme according to the block structures of the fibers. This graph can be explicitly obtained when the central characters of simple modules of the generic fiber are known. We show that the block structure of an arbitrary fiber is completely determined by "atomic" block structures living on the components of a Weil divisor. As a byproduct, we deduce that the number of blocks of fibers defines a lower semicontinuous function on the base scheme. We furthermore discuss how to obtain information about the simple modules in the blocks by generalizing and establishing several properties of decomposition matrices by Geck and Rouquier.