Krull Dimension for Differential Graded Algebras

TL;DR

This paper explores the concept of Krull dimension in the context of differential graded algebras, comparing different notions of systems of parameters and establishing their equivalence in specific cases.

Contribution

It introduces a naive system of parameters for complexes, compares it with Christensen's, and proves their equivalence for DG R-algebras, linking it to prime ideal chains.

Findings

Naive and Christensen's systems of parameters differ generally.

Both notions agree for DG R-algebras.

Krull dimension via parameters matches chain-based definitions.

Abstract

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring, and compare it to the system of parameters defined by Christensen. We show that these notions differ in general, but that they agree when the complex in question is a DG R-algebra. In this case we also show that the Krull dimension defined in terms of the lengths of such systems of parameters agrees with Krull dimensions defined in terms of certain chains of prime ideals.