# Commutative algebra

Yoshifumi Tsuchimoto

LEMMA 04.1   Let be a commutative ring. Then for the polynomial ring of -variables over , the module of -differentials of over is equal to a free module generated by . Namely, we have

LEMMA 04.2   Let be a ring. Let be -algebras. Then for any -algebra homomorphism we have

LEMMA 04.3   Let be a commutative ring. Let be a commutative -algebra. Then for any ideal of , we have the following exact sequence:

where the first arrow maps to .

2011-05-26