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)