Koszul-Tate resolution

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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.

[edit] 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 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,

where

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

Over a field of characteristic 0,

T(i) is just Ti/i!.

[edit] See also

[edit] 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 [1]