In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Koszul complex was first introduced to define a
cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
for
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, by
Jean-Louis Koszul
Jean-Louis Koszul (; January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki.
Biography
Koszul was educated at the in ...
(see
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 p ...
). It turned out to be a useful general construction in
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an
M-regular sequence, and hence it can be used to prove basic facts about the
depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ...
. Moreover, in certain circumstances, the complex is the complex of
syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.
Definition
Let ''R'' be a commutative ring and ''E'' a free module of finite rank ''r'' over ''R''. We write
for the ''i''-th
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of ''E''. Then, given an
''R''-linear map ,
the Koszul complex associated to ''s'' is the
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
of ''R''-modules:
:
,
where the differential
is given by: for any
in ''E'',
:
.
The superscript
means the term is omitted. To show that
, use the
self-duality of a Koszul complex.
Note that
and
. Note also that
; this isomorphism is not canonical (for example, a choice of a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
in differential geometry provides an example of such an isomorphism.)
If
(i.e., an ordered basis is chosen), then, giving an ''R''-linear map
amounts to giving a finite sequence
of elements in ''R'' (namely, a row vector) and then one sets
If ''M'' is a finitely generated ''R''-module, then one sets:
:
,
which is again a chain complex with the induced differential
.
The ''i''-th homology of the Koszul complex
:
is called the ''i''-th Koszul homology. For example, if
and