View source for Koszul complex

<!--
k0558101.png
$#A+1 = 91 n = 0
$#C+1 = 91 : ~/encyclopedia/old_files/data/K055/K.0505810 Koszul complex
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

Let  $  R $
be a [[Commutative ring|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|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|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|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|intersection theory]]), cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]]; cf. also [[Depth of a module|Depth of a module]] and [[Cohen–Macaulay ring|Cohen–Macaulay ring]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Grothendieck,   "Local cohomology" , ''Lect. notes in math.'' , '''41''' , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Herzog (ed.)  E. Kunz (ed.) , ''Der kanonische Modul eines Cohen–Macaulay-Rings'' , ''Lect. notes in math.'' , '''238''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Matsumura,   "Commutative algebra" , Benjamin  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.G. Northcott,   "Lessons on rings, modules, and multiplicities" , Cambridge Univ. Press  (1968)</TD></TR></table>