# Tag Archives: right-exact

## Commutative Algebra 52

Direct Limits of Rings Let be a directed system of rings. Regard them as a directed system of abelian groups (i.e. ℤ-modules) and take the direct limit A. Proposition 1. The abelian group A has a natural structure of a … Continue reading

## Commutative Algebra 51

Limits Are Left-Exact By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in is a right-exact functor, but there is a rather huge issue here: the functors are between and … Continue reading

## Commutative Algebra 50

Adjoint Functors Adjoint functors are a general construct often used for describing universal properties (among other things). Take two categories and . Definition. Covariant functors and are said to be adjoint if we have isomorphisms which are natural in A and … 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