Cancellation of finite-dimensional Noetherian modules

TL;DR

This paper proves new module cancellation theorems over Noetherian rings, weakening previous projectivity conditions and unifying existing results, with examples demonstrating the broader applicability of these new criteria.

Contribution

It establishes cancellation results for finitely generated modules over Noetherian rings under new local conditions, relaxing projectivity assumptions and unifying multiple prior theorems.

Findings

Cancellation holds under specified local conditions on modules.

Weakens projectivity assumptions in classical cancellation theorems.

Provides examples that surpass previous cancellation criteria.

Abstract

The Module Cancellation Problem solicits hypotheses that, when imposed on modules $K$, $L$, and $M$ over a ring $S$, afford the implication $K\oplus L\cong K\oplus M\Longrightarrow L\cong M$. In a well-known paper on basic element theory from 1973, Eisenbud and Evans lament the "great scarcity of strong results" in module cancellation research, expressing the wish that, "under some general hypothesis" on finitely generated modules over a commutative Noetherian ring, cancellation could be demonstrated. Singling out cancellation theorems by Bass and Dress that feature "large" projective modules, Eisenbud and Evans contend further that, although "[s]ome criteria of 'largeness' is certainly necessary in general [. . . ,] the need for projectivity is not clear." In this paper, we prove that cancellation holds if $K$, $L$, and $M$ are finitely generated modules over a commutative Noetherian ring $S$ such that $K_\mathfrak{p}^{\oplus (1+\text{dim}(S/\mathfrak{p}))}$ is a direct summand of $M_{\mathfrak{p}}$ over $S_{\mathfrak{p}}$ for every prime ideal $\mathfrak{p}$ of $S$. We also weaken projectivity conditions in the cancellation theorems of Bass and Dress and a newer theorem by De Stefani-Polstra-Yao; in fact, we obtain a statement that unifies all three of these theorems while obviating a projectivity constraint in each one. To illustrate the scope of our work, we construct a cancellation example that simultaneously eludes the three theorems just mentioned as well as many other observations from the module cancellation literature.