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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558101.png" /> be a [[Commutative ring|commutative ring]] with unit element and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558102.png" /> a sequence of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558103.png" />. The Koszul complex defined by these data then consists of the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558105.png" /> is the canonical basis for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558106.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558107.png" />, and the differentials
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558108.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k0558109.png" /></td> </tr></table>
+
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
  
where, as usual, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581010.png" /> over a symbol means deletion. More generally one also considers the chain and cochain complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581014.png" /> consists of just two non-zero modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581016.png" /> in dimensions 0 and 1 and the only non-zero differential is multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581018.png" />. The general Koszul complex can be viewed as built up from these elementary constituents as
+
$$
 +
d _ {p} : \
 +
K _ {p} ( \underline{x} ;  R)  \rightarrow \
 +
K _ {p - 1 }  ( \underline{x} ;  R),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581019.png" /></td> </tr></table>
+
$$
 +
e _ {i _ {1}  } \wedge \dots \wedge e _ {i _ {p}  }
 +
\mapsto  \sum _ {j = 1 } ^ { p }  (- 1) ^ {j + 1 } x _ {ij} e _ {i _ {1}  } \wedge \dots \wedge \widehat{e}  _ {i _ {j}  } \wedge \dots \wedge e _ {i _ {p}  } ,
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581020.png" />, define a morphism of chain complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581021.png" /> by taking multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581022.png" /> in dimension zero and the identity in dimension 1. Taking tensor products and iterates one thus defines morphisms of chain complexes
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581023.png" /></td> </tr></table>
+
$$
 +
K _  \star  ( \underline{x} ; R)  = \
 +
K _  \star  ( x _ {1} ; R) \otimes _ {R} \dots
 +
\otimes _ {R} K _  \star  ( x _ {r} ; R) .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581024.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581025.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581026.png" />-th cohomology group of the cochain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581027.png" />. For a [[Noetherian ring|Noetherian ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581028.png" />, the local cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581029.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581030.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581031.png" /> with respect to an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581033.png" /> can then be calculated as:
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581034.png" /></td> </tr></table>
+
$$
 +
h  ^ {t} : \
 +
K _  \star  ( \underline{x}  ^ {m} ; R)  \rightarrow \
 +
K _  \star  ( \underline{x} ^ {m + t } ; R)
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581035.png" /> is a set of generators for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581036.png" />.
+
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:
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581037.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581039.png" />-regular element (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581040.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581041.png" />-module) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581042.png" /> is not a zero-divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581043.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581044.png" /> is injective. A sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581045.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581047.png" />-regular sequence of elements or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581049.png" />-sequence if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581050.png" /> is not a zero-divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581051.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581052.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581053.png" />-regular. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581054.png" /> be an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581055.png" />. Then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581056.png" />-regular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581057.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581059.png" />-regular sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581060.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581061.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581062.png" />. A maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581064.png" />-regular sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581065.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581066.png" />-regular sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581067.png" /> such that there is no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581068.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581069.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581070.png" />-regular sequence.
+
$$
 +
H _ {A}  ^ {i} ( M) = \
 +
\lim\limits _ {\begin{array}{c}
 +
\rightarrow \\
 +
t
 +
\end{array}
 +
}  H  ^ {i} ( \underline{x}  ^ {t} ;  M),
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581071.png" /> be Noetherian, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581072.png" /> a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581073.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581074.png" /> an ideal. Then the following are equivalent: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581075.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581076.png" /> and for all finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581077.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581078.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581079.png" /> (cf. [[Support of a module|Support of a module]]); ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581080.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581081.png" />; and iii) there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581082.png" />-regular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581083.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581084.png" />.
+
where  $  x _ {1} \dots x _ {r} $
 +
is a set of generators for $  \mathfrak a $.
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581086.png" />-depth of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581087.png" /> is the length of the longest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581088.png" />-regular sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581089.png" />. It is also called the grade of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581090.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581091.png" />. The [[Depth of a module|depth of a module]] is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581092.png" />-depth.
+
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.
  
The homology of the Koszul complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581093.png" /> associated with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581094.png" />-regular sequence satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581095.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055810/k05581097.png" />. 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]].
+
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====
 
====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>
 
<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>

Latest revision as of 22:15, 5 June 2020


Let $ R $ 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)
How to Cite This Entry:
Koszul complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Koszul_complex&oldid=47522