Koszul complex
Let
be a commutative ring with unit element and \underline{x} = ( x _ {1} \dots x _ {r} )
a sequence of elements of R .
The Koszul complex defined by these data then consists of the modules K _ {p} ( \underline{x} ; R) = \wedge ^ {p} ( R ^ {r} ) = \oplus _ {i _ {1} < \dots < i _ {p} } R ( e _ {i _ {1} } \wedge \dots \wedge e _ {i _ {p} } ) ,
where \{ e _ {1} \dots e _ {r} \}
is the canonical basis for the R -
module R ^ {r} ,
and the differentials
d _ {p} : \ K _ {p} ( \underline{x} ; R) \rightarrow \ K _ {p - 1 } ( \underline{x} ; R),
e _ {i _ {1} } \wedge \dots \wedge e _ {i _ {p} } \mapsto \sum _ {j = 1 } ^ { p } (- 1) ^ {j + 1 } x _ {ij} e _ {i _ {1} } \wedge \dots \wedge \widehat{e} _ {i _ {j} } \wedge \dots \wedge e _ {i _ {p} } ,
where, as usual, a \widehat{ {}} over a symbol means deletion. More generally one also considers the chain and cochain complexes K _ \star ( \underline{x} ; M) = K _ {p} ( \underline{x} ; R) \otimes _ {R} M and K ^ \star ( \underline{x} ; M) = \mathop{\rm Hom} _ {R} ( K _ \star ( x; R); M) . If r = 1 , K _ \star ( x _ {1} ; R) consists of just two non-zero modules R and R in dimensions 0 and 1 and the only non-zero differential is multiplication by x _ {1} in R \rightarrow R . The general Koszul complex can be viewed as built up from these elementary constituents as
K _ \star ( \underline{x} ; R) = \ K _ \star ( x _ {1} ; R) \otimes _ {R} \dots \otimes _ {R} K _ \star ( x _ {r} ; R) .
For x _ {1} \in R , define a morphism of chain complexes h _ {x _ {1} } : K _ \star ( x _ {1} ; R) \rightarrow K _ \star ( x _ {1} ^ {2} ; R) by taking multiplication by x _ {1} in dimension zero and the identity in dimension 1. Taking tensor products and iterates one thus defines morphisms of chain complexes
h ^ {t} : \ K _ \star ( \underline{x} ^ {m} ; R) \rightarrow \ K _ \star ( \underline{x} ^ {m + t } ; R)
where \underline{x} ^ {m} = ( x _ {1} ^ {m} \dots x _ {r} ^ {m} ) . Let H ^ {i} ( \underline{x} ; M) denote the i - th cohomology group of the cochain complex K ^ \star ( \underline{x} ; M) . For a Noetherian ring R , the local cohomology H _ {A} ^ {i} ( M) of an R - module M with respect to an ideal \mathfrak a , A = V ( \mathfrak a ) = \{ {\mathfrak p } : {\mathfrak p \textrm{ is a prime ideal and } \mathfrak p \supset \mathfrak a } \} can then be calculated as:
H _ {A} ^ {i} ( M) = \ \lim\limits _ {\begin{array}{c} \rightarrow \\ t \end{array} } H ^ {i} ( \underline{x} ^ {t} ; M),
where x _ {1} \dots x _ {r} is a set of generators for \mathfrak a .
An element x \in R is called an M - regular element (where M is an R - module) if x is not a zero-divisor on M , i.e. if M \rightarrow ^ {x} M is injective. A sequence of elements x _ {1} \dots x _ {r} is called an M - regular sequence of elements or an M - sequence if x _ {i} is not a zero-divisor on M ( x _ {1} M + \dots + x _ {i - 1 } M) , i.e. if x _ {i} is M/( x _ {1} M + \dots + x _ {i - 1 } M) - regular. Let I be an ideal of R . Then an M - regular sequence x _ {1} \dots x _ {r} is called an M - regular sequence in I if x _ {i} \in I for i = 1 \dots r . A maximal M - regular sequence in I is an M - regular sequence in I such that there is no y \in I for which x _ {1} \dots x _ {r} , y is an M - regular sequence.
Let A be Noetherian, M a finitely-generated A - module and I an ideal. Then the following are equivalent: i) \mathop{\rm Ext} _ {A} ^ {i} ( N, M) = 0 for all i = 0 \dots r and for all finitely-generated A - modules N with support in I ( cf. Support of a module); ii) \mathop{\rm Ext} _ {A} ^ {i} ( A/I, M) = 0 for i = 0 \dots r ; and iii) there exists an M - regular sequence x _ {1} \dots x _ {r} in I .
The I - depth of a module M is the length of the longest M - regular sequence in I . It is also called the grade of I on M . The depth of a module is the A - depth.
The homology of the Koszul complex K _ \star ( \underline{x} ; M) associated with an M - regular sequence satisfies H _ {i} ( K _ \star ( x; M)) = 0 for i > 0 and H _ {0} ( K _ \star ( x; M)) = M / \sum _ {i = 1 } ^ {r} x _ {i} M . 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=47522