They were introduced by Tate (1957) as a generalization of the Koszul resolution for the quotient R/(x1, ...., xn) of R by a regular sequence of elements.
Friedemann Brandt, Glenn Barnich, and Marc Henneaux (2000) used the Koszul–Tate resolution to calculate BRST cohomology.
Assume we have a graded supercommutative ring X, so that with a differential d, with and x ∈ X is a homogeneous cycle (dx = 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 (possibly an infinite number) to kill off all homology of positive degree.