In mathematics, a Koszul–Tate resolution or Koszul–Tate complex of the
quotient ring ''R''/''M'' is a
projective resolution of it as an ''R''-module which also has a structure of a
dg-algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
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Definition
A differential graded alg ...
over ''R'', where ''R'' is a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
and ''M'' ⊂ ''R'' is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
. They were introduced by as a generalization of the
Koszul resolution for the quotient ''R''/(''x''
1, ...., ''x''
n) of ''R'' by a
regular sequence of elements. used the Koszul–Tate resolution to calculate
BRST cohomology
In theoretical physics, the BRST formalism, or BRST quantization (where the ''BRST'' refers to the last names of Carlo Becchi, , Raymond Stora and Igor Tyutin) denotes a relatively rigorous mathematical approach to Quantization (physics), quantizi ...
. 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 number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s ''Q''. Assume we have a
graded supercommutative ring
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have
:yx = (-1)^xy ,
where , ''x'', denotes the grade of the element and is 0 or 1 ( ...
''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''
of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s in a variable ''T'', where the differential is extended to ''T'' by
:''dT''=''x''.
(The
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
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
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have
:yx = (-1)^xy ,
where , ''x'', denotes the grade of the element and is 0 or 1 ( ...
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 (possibly 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 may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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''!.
See also
*
Lie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to prope ...
References
*
*
*
*M. Henneaux and C. Teitelboim, ''Quantization of Gauge Systems'', Princeton University Press, 1992
*
{{DEFAULTSORT:Koszul-Tate resolution
Homological algebra
Commutative algebra