Koszul Complex - Introduction

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\to R\xrightarrow{\ x\ }R\to0.

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)) = AnnR(x).

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

Now, if x1, x2, ..., xn are elements of R, the Koszul complex of R with respect to x1, x2, ..., xn, usually denoted K•(x1, x2, ..., xn), 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 jth degree in the complex (0 ≤ j ≤ n). The matrices involved in the maps can be written down precisely. Letting denote a free-basis generator in Kp, d: Kp ↦ Kp − 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 \to R \xrightarrow{\ d_2\ } R^2 \xrightarrow{\ d_1\ } R\to 0,

with the matrices and given by

d_1 = \begin{bmatrix} x & y\\ \end{bmatrix} and
d_2 = \begin{bmatrix} -y\\ x\\ \end{bmatrix}.

Note that di 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 x1, x2, ..., xn form a regular sequence, the higher homology modules of the Koszul complex are all zero.

Read more about this topic:  Koszul Complex

Famous quotes containing the word introduction:

    “Such is oftenest the young man’s introduction to the forest, and the most original part of himself. He goes thither at first as a hunter and fisher, until at last, if he has the seeds of a better life in him, he distinguishes his proper objects, as a poet or naturalist it may be, and leaves the gun and fish-pole behind. The mass of men are still and always young in this respect.”
    —Henry David Thoreau (1817–1862)

    “The role of the stepmother is the most difficult of all, because you can’t ever just be. You’re constantly being tested—by the children, the neighbors, your husband, the relatives, old friends who knew the children’s parents in their first marriage, and by yourself.”
    —Anonymous Stepparent. Making It as a Stepparent, by Claire Berman, introduction (1980, repr. 1986)

    “For the introduction of a new kind of music must be shunned as imperiling the whole state; since styles of music are never disturbed without affecting the most important political institutions.”
    —Plato (c. 427–347 B.C.)