be an exact sequence of -modules. Then we have
We call the Hilbert polynomial of .
holds. If an ideal of satisfies
then there exists an element such that
holds. If furthermore is contained in (the Jacobson radical of ), then we have .
holds. In a matrix notation, this may be rewritten as
with , . Using the unit matrix one may also write :
Now let be the adjugate matrix of . In other words, it is a matrix which satisfies
Then we have
On the other hand, since modulo , we have for some . This clearly satisfies