Semicontinuous maps on module varieties

TL;DR

This paper investigates semicontinuous maps on module varieties over finite-dimensional algebras, establishing their properties and implications for invariants like g-vectors and E-invariants, with applications to inequalities of generic values.

Contribution

It proves that truncated Euler maps are semicontinuous and shows that g-vectors and E-invariants are upper semicontinuous, advancing understanding of module invariants.

Findings

Truncated Euler maps are upper or lower semicontinuous.

g-vectors and E-invariants are upper semicontinuous.

Discusses inequalities of generic values of upper semicontinuous maps.

Abstract

We study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that $g$-vectors and $E$-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps.