Koszul-Tate resolution

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 John Tate and have been used to calculate BRST cohomology. The differential of this complex is called the Koszul-Tate derivation or Koszul-Tate differential.

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" [T]

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 "T"²=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,

where:"T"^("i")"T"^("j") = (("i"+"j")!/"i"!"j"!)"T"^("i"+"j").

Over a field of characteristic 0, :"T"^("i") is just "T"^"i"/"i"!.

ee also

*Koszul complex
*Lie algebra cohomology

References

* J.L. Koszul, "Homologie et cohomologie des algèbres de Lie", "Bulletin de la Société Mathématique de France", 78, 1950, pp 65-127.
*J. Tate, "Homology of Noetherian rings and local rings", "Illinois Journal of Mathematics", 1, 1957, pp. 14-27.
*M. Henneaux and C. Teitelboim, "Quantization of Gauge Systems", Princeton University Press, 1992
*There is a jet bundle description of the Koszul-Tate complex by Verbovetsky here [http://arxiv.org/abs/math/0105207]