Koszul-Tate Resolution - Construction

Construction

First suppose for simplicity that all rings contain the rational numbers Q. Assume we have a graded supercommutative ring X, so that

ab = (−1)deg(a)deg (b)ba,

with a differential d, with

d(ab) = d(a)b + (−1)deg(a)ad(b)),

and x ∈ X is a homogeneous cycle (dx = 0). Then we can form a new ring

Y = X

of polynomials in a variable T, where the differential is extended to T by

dT=x.

(The polynomial ring is understood in the super sense, so if T has odd degree then T2 = 0.) The result of adding the element T is to kill off the element of the homology of X represented by x, and Y is still a supercommutative ring with derivation.

A Koszul–Tate resolution of R/M can be constructed as follows. We start with the commutative ring R (graded so that all elements have degree 0). Then add new variables as above of degree 1 to kill off all elements of the ideal M in the homology. Then keep on adding more and more new variables (possible an infinite number) to kill off all homology of positive degree. We end up with a supercommutative graded ring with derivation d whose homology is just R/M.

If we are not working over a field of characteristic 0, the construction above still works, but it is usually neater to use the following variation of it. Instead of using polynomial rings X, one can use a "polynomial ring with divided powers" X〈T〉, which has a basis of elements

T(i) for i ≥ 0,

where

T(i)T(j) = ((i + j)!/i!j!)T(i+j).

Over a field of characteristic 0,

T(i) is just Ti/i!.