Koszul complex
Let be a commutative ring with unit element and
a sequence of elements of
. The Koszul complex defined by these data then consists of the modules
, where
is the canonical basis for the
-module
, and the differentials
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where, as usual, a over a symbol means deletion. More generally one also considers the chain and cochain complexes
and
. If
,
consists of just two non-zero modules
and
in dimensions 0 and 1 and the only non-zero differential is multiplication by
in
. The general Koszul complex can be viewed as built up from these elementary constituents as
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For , define a morphism of chain complexes
by taking multiplication by
in dimension zero and the identity in dimension 1. Taking tensor products and iterates one thus defines morphisms of chain complexes
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where . Let
denote the
-th cohomology group of the cochain complex
. For a Noetherian ring
, the local cohomology
of an
-module
with respect to an ideal
,
can then be calculated as:
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where is a set of generators for
.
An element is called an
-regular element (where
is an
-module) if
is not a zero-divisor on
, i.e. if
is injective. A sequence of elements
is called an
-regular sequence of elements or an
-sequence if
is not a zero-divisor on
, i.e. if
is
-regular. Let
be an ideal of
. Then an
-regular sequence
is called an
-regular sequence in
if
for
. A maximal
-regular sequence in
is an
-regular sequence in
such that there is no
for which
is an
-regular sequence.
Let be Noetherian,
a finitely-generated
-module and
an ideal. Then the following are equivalent: i)
for all
and for all finitely-generated
-modules
with support in
(cf. Support of a module); ii)
for
; and iii) there exists an
-regular sequence
in
.
The -depth of a module
is the length of the longest
-regular sequence in
. It is also called the grade of
on
. The depth of a module is the
-depth.
The homology of the Koszul complex associated with an
-regular sequence satisfies
for
and
. This (and the above) makes Koszul complexes an important tool in commutative and homological algebra, for instance in dimension theory and the theory of multiplicities (and intersection theory), cf. [a1], [a2], [a3], [a4]; cf. also Depth of a module and Cohen–Macaulay ring.
References
[a1] | A. Grothendieck, "Local cohomology" , Lect. notes in math. , 41 , Springer (1967) |
[a2] | J. Herzog (ed.) E. Kunz (ed.) , Der kanonische Modul eines Cohen–Macaulay-Rings , Lect. notes in math. , 238 , Springer (1971) |
[a3] | H. Matsumura, "Commutative algebra" , Benjamin (1970) |
[a4] | D.G. Northcott, "Lessons on rings, modules, and multiplicities" , Cambridge Univ. Press (1968) |
Koszul complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Koszul_complex&oldid=15764