Koszul complex

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)