# Tag Archives: induced modules

## Commutative Algebra 56

Throughout this article, A denotes a noetherian ring and is a fixed ideal. All A-modules are finitely generated. Consequences of Artin-Rees Lemma Suppose we have an exact sequence of finitely generated A-modules Let M be given the 𝔞-adic filtration; the induced filtration on … Continue reading

## Commutative Algebra 33

Snake Lemma Let us introduce a useful tool for computing kernels and cokernels in a complicated diagram of modules. Although it is only marginally useful for now, it will become a major tool in homological algebra. Snake Lemma. Suppose we … Continue reading

## Commutative Algebra 29

Distributivity Finally, tensor product is distributive over arbitrary direct sums. Proposition 1. Given any family of modules , we have: Proof Take the map which takes . Note that this is well-defined: since only finitely many are non-zero, only finitely … Continue reading