Injective capacity and cogeneration

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.