Koszul-Tate resolution
Encyclopedia
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.
s Q. 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 we can form a new ring
of polynomial
s 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
where
Over a field of characteristic 0,
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
and M is an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
). They were introduced by John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
and have been used to calculate BRST cohomology. The differential
Differential (mathematics)
In mathematics, the term differential has several meanings.-Basic notions:* In calculus, the differential represents a change in the linearization of a function....
of this complex is called the Koszul-Tate derivation or Koszul-Tate differential.
Construction
First suppose for simplicity that all rings contain the rational numberRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s 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 polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s in a variable T, where the differential is extended to T by
- dT=x.
(The polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
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
Supercommutative algebra
In mathematics, a supercommutative algebra is a superalgebra such that for any two homogeneous elements x, y we haveyx = ^In mathematics, a supercommutative algebra is a superalgebra In mathematics, a supercommutative algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two...
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
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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!.