# Commutative algebra

Yoshifumi Tsuchimoto

Let be an abelian category. For any object of , the extension group is defined to be the derived functor of the hom'' functor

We note that the functor is a bifunctor''. We may thus consider the right derived functor of and that of . Fortunately, both coincide: The extension group may be calculated by using either an injective resolution of the second variable or a projective resoltuion of the first variable .

EXAMPLE 09.1   Let us compute the extension groups .
1. We may compute them by using an injective resolution

of .
2. We may compute them by using a free resolution

of .

EXERCISE 09.1   Compute an extension group for modules of your choice. (Please choose a non-trivial example).

DEFINITION 09.2   Let be an associative unital (but not necessarily commutative) ring. Let be a right -module. Let be a left -module. For any ( -)module , an map

is called an -balanced biadditive map if
1.      .
2.      .
3.      .

PROPOSITION 09.3   Let be an associative unital (but not necessarily commutative) ring. Then for any right -module and for any left -module , there exists a ( -)module together with a -balanced map

which is universal amoung -balanced maps.

DEFINITION 09.4   We employ the assumption of the proposition above. By a standard argument on universal objects, we see that such object is unique up to a unique isomorphism. We call it the tensor product of and and denote it by

LEMMA 09.5   Let be an associative unital ring. Then:
1. .
2. For any , is a right exact functor.
3. For any right ideal of and for any -module , we have

In particular, if the ring is commutative, then for any ideals of , we have

DEFINITION 09.6   For any left -module , the left derived functor of is called the Tor functor and denoted by .

By definition, may be computed by using projective resolutions of .

EXERCISE 09.2   Compute for .