Koszul complex

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra.

Introduction

In commutative algebra, if "x" is an element of the ring "R", multiplication by "x" is "R"-linear and so represents an "R"-module homomorphism "x":"R" →"R" from "R" to itself. It is useful to throw in zeroes on each end and make this a (free) "R"-complex:

:$0\; o\; Rxrightarrow\{\; x\; \}R\; o0.$

Call this chain complex "K"_•("x").

Counting the right-hand copy of "R" as the zeroth degree and the left-hand copy as the first degree, this chain complex neatly captures the most important facts about multiplication by "x" because its zeroth homology is exactly the homomorphic image of "R" modulo the multiples of "x", H₀("K"_•("x")) = "R"/"xR", and its first homology is exactly the annihilator of "x", H₁("K"_•("x")) = Ann_"R"("x").

This chain complex "K"_•("x") is called the Koszul complex of "R" with respect to "x".

Now, if "x"₁, "x"₂, ..., "x"_"n" are elements of "R", the Koszul complex of "R" with respect to "x"₁, "x"₂, ..., "x"_"n", usually denoted "K"_•("x"₁, "x"₂, ..., "x"_"n"), is the tensor product in the category of "R"-complexes of the Koszul complexes defined above individually for each "i".

The Koszul complex is a free chain complex. There are exactly ("n" choose "j") copies of the ring "R" in the "j"th degree in the complex (0 ≤ "j" ≤ "n"). The matrices involved in the maps can be written down precisely. Letting $e\_\{i\_1...i\_p\}$ denote a free-basis generator in"K"_"p", "d": "K"_"p" $mapsto$ "K"_{"p" − 1} is defined by:

:$d(e\_\{i\_1...i\_p\})\; :=\; sum\; \_\{j=1\}^\{p\}(-1)^\{j-1\}x\_\{i\_j\}e\_\{i\_1...widehat\{i\_j\}...i\_p\}.$

For the case of two elements "x" and "y", the Koszul complex can then be written down quite succinctly as:$0\; o\; R\; xrightarrow\{\; d\_2\; \}\; R^2\; xrightarrow\{\; d\_1\; \}\; R\; o\; 0,$with the matrices $d\_1$ and $d\_2$ given by

: and:Note that "d_i" is applied on the left. The cycles in degree 1 are then exactly the linear relations on the elements "x" and "y", while the boundaries are the trivial relations. The first Koszul homology H₁("K"_•("x", "y")) therefore measures exactly the relations mod the trivial relations. With more elements the higher-dimensional Koszul homologies measure the higher-level versions of this.

In the case that the elements "x"₁, "x"₂, ..., "x"_"n" form a regular sequence, the higher homology modules of the Koszul complex are all zero, so "K"_•("x"₁, "x"₂, ..., "x"_"n") forms a free resolution of the "R"-module "R"/("x"₁, "x"₂, ..., "x"_"n")"R".

Example

If "k" is a field and "X"₁, "X"₂, ..., "X"_"d" are indeterminates and "R" is the polynomial ring "k" ["X"₁, "X"₂, ..., "X"_"d"] , the Koszul complex "K"_•("X"_"i") on the "X"_"i"'s forms a concrete free "R"-resolution of "k".

Theorem

If ("R", "m") is a local ring and "M" is a finitely-generated "R"-module with "x"₁, "x"₂, ..., "x"_"n" in "m", then the following are equivalent:

# The ("x"_"i") form a regular sequence on "M",
# H₁("K"_•("x"_"i")) = 0,
# H_"j"("K"_•("x"_"i")) = 0 for all "j" ≥ 1.

Applications

The Koszul complex is essential in defining the joint spectrum of a tuple of bounded linear operators in a Banach space.

References

* David Eisenbud, "Commutative Algebra. With a view toward algebraic geometry", Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. ISBN 0-387-94268-8