Injective capacity and cogeneration

TL;DR

This paper introduces the concepts of injective capacity and cogeneration for modules over a commutative ring, establishing their global and local relationships and extending the theory to graded modules.

Contribution

It defines injective capacity and cogeneration for modules, proving their global-local equivalences and extending results to graded modules.

Findings

Global injective capacity equals the infimum of local capacities

Global cogeneration number equals the supremum of local invariants

Enhanced versions of the main theorems are provided

Abstract

Let $M$ and $N$ be modules over a commutative ring $R$ with $N$ Noetherian. We define the injective capacity of $M$ with respect to $N$ over $R$ to be the supremum of the values $t$ for which $N^{\oplus t}$ embeds into $M$. In a dual fashion, we deem the number of cogenerators of $N$ with respect to $M$ over $R$ to be the infimum of the numbers $t$ for which $N$ embeds into $M^{\oplus t}$. We demonstrate that the global injective capacity is the infimum of its local analogues and that the global number of cogenerators is the supremum of the corresponding local invariants. We also prove enhanced versions of these statements and consider the graded case.