Derived complete intersections and polynomial growth of Betti numbers over dg-algebras

TL;DR

This paper extends Gulliksen's theorem to derived settings, characterizing dg-algebras with polynomial Betti number growth and establishing foundational results on minimal models and acyclic closures.

Contribution

It proves a derived version of Gulliksen's theorem, providing a structure theorem for dg-algebras with polynomial Betti growth and extending key properties of minimal models.

Findings

Proved a structure theorem for dg-algebras with polynomial Betti growth

Established existence and uniqueness of minimal models and acyclic closures in broader settings

Extended Halperin's theorem to dg-algebras, recovering Gulliksen's theorem

Abstract

A theorem of Gulliksen states that a local ring is a complete intersection if and only if the Betti numbers of its finitely generated modules grow polynomially. We prove a derived version of Gulliksen's Theorem. More precisely, we prove a structure theorem for dg-algebras whose modules exhibit polynomial Betti growth. As a key ingredient in the proof, we establish the existence and uniqueness of minimal models and acyclic closures of morphisms of dg-algebras in a broader setting than was previously known. We also extend to dg-algebras a theorem of Halperin on the vanishing of deviations of local rings, recovering Gulliksen's Theorem as an immediate consequence.