CHAPTER 1

Basic definitions

The aim of this chapter is to fix the notation used in the rest of the book.

We discuss the definition of Koszul cohomology in the algebraic context (graded

modules over the symmetric algebra of a vector space) and the geometric context

(graded modules associated to a line bundle on a projective variety) and briefly dis-

cuss the relation of Koszul cohomology with minimal resolutions and its functorial

properties.

1.1. The Koszul complex

Let V be a vector space of dimension r + 1 over a field k of characteristic zero

and let

− , − : V

∨

× V → k

be the duality pairing. Given a nonzero element x ∈ V

∨,

the corresponding map

x, − : V → k extends uniquely to an anti–derivation

ιx :

∗

V →

∗

V

of the exterior algebra of degree −1. This derivation is defined inductively by

putting ιx|V = x, − : V → k and

ιx(v ∧ v1 ∧ . . . ∧ vp−1) = x, v .v1 ∧ . . . ∧ vp−1 − v ∧ ιx(v1 ∧ . . . ∧ vp−1).

The resulting maps

ιx :

p

V →

p−1

V

are called contraction (or inner product) maps; they are dual to the exterior product

maps

λx :

p−1

V

∨

∧x

−→

p

V

∨

and satisfy ιx◦ιx = 0. Hence we obtain a complex

K•(x) : (0 →

r+1

V → . . . →

p

V ιx −→ −

p−1

V ιx −→ −

p−2

V → . . . → k → 0)

called the Koszul complex.

Note that for any α ∈

k∗,

the complexes K•(x) and K•(αx) are isomorphic;

hence the Koszul complex depends only on the point [x] ∈ P(V

∨).

Lemma 1.1. Given nonzero elements ξ ∈ V , x ∈ V ∨, let λξ :

p−1

V

∧ξ

−→ −

p

V

be the map given by wedge product with ξ. We have

ιx◦λξ + λξ◦ιx = x, ξ . id .

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http://dx.doi.org/10.1090/ulect/052/01