In mathematics, a Koszul–Tate resolution or Koszul–Tate complex is a projective resolution of R/M that is an R-algebra (where R is a commutative ring and M is an ideal). They were introduced by Tate (1957) as a generalization of the Koszul complex. Friedemann Brandt, Glenn Barnich, and Marc Henneaux (2000) used the Koszul–Tate resolution to calculate BRST cohomology. The differential[disambiguation needed] of this complex is called the Koszul–Tate derivation or Koszul–Tate differential.
- 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[T]
of polynomials in a variable T, where the differential is extended to T by
(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[T], one can use a "polynomial ring with divided powers" X〈T〉, which has a basis of elements
- T(i) for i ≥ 0,
- T(i)T(j) = ((i + j)!/i!j!)T(i+j).
Over a field of characteristic 0,
- T(i) is just Ti/i!.
- Brandt, Friedemann; Barnich, Glenn; Henneaux, Marc (2000), "Local BRST cohomology in gauge theories", Physics Reports. A Review Section of Physics Letters 338 (5): 439–569, doi:10.1016/S0370-1573(00)00049-1, ISSN 0370-1573, MR 1792979
- Koszul, Jean-Louis (1950), "Homologie et cohomologie des algèbres de Lie", Bulletin de la Société Mathématique de France 78: 65–127, ISSN 0037-9484, MR 0036511
- Tate, John (1957), "Homology of Noetherian rings and local rings", Illinois Journal of Mathematics 1: 14–27, ISSN 0019-2082, MR 0086072
- M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, 1992
- Verbovetsky, Alexander (2002), "Remarks on two approaches to the horizontal cohomology: compatibility complex and the Koszul–Tate resolution", Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications 72 (1): 123–131, doi:10.1023/A:1015276007463, ISSN 0167-8019, MR 1907621