 First Tor of a cyclic quotient

ID: first-tor-of-a-cyclic-quotient

A / I

with

P_{0} = A

P_{1}

surjecting onto

I

. The next differential presents the relations in

I

. Right exactness of the tensor product of modules identifies

coker (P_{2} \otimes_{A} M \to P_{1} \otimes_{A} M)

with

I \otimes_{A} M

. Taking degree-one homology then gives the displayed kernel formula. Hence vanishing for every ideal is equivalent to the flatness criterion using ideals.

 New to topics? Read the docs here!