Entropy in the category of perfect complexes with cohomology of finite length

TL;DR

This paper compares local and category-theoretical entropies for endomorphisms of finite length in Noetherian local rings, establishing inequalities, equalities in special cases, and an additive property under flat morphisms, with a formula involving Euler characteristics.

Contribution

It introduces a comparison and unification of different entropy notions in commutative algebra, including new formulas and properties for specific classes of rings.

Findings

Local entropy ≤ category-theoretical entropy.

Equality of entropies in regular rings and for Frobenius endomorphism.

Additivity of local entropy under flat morphisms.

Abstract

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic. Furthermore, given a flat morphism of Cohen-Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is "additive". Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.