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
1.1. The Koszul complex
Let V be a vector space of dimension r + 1 over a field k of characteristic zero
− , − : 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
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
are called contraction (or inner product) maps; they are dual to the exterior product
and satisfy ιx◦ιx = 0. Hence we obtain a complex
K•(x) : (0 →
V → . . . →
V ιx −→ −
V ιx −→ −
V → . . . → k → 0)
called the Koszul complex.
Note that for any α ∈
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 :
be the map given by wedge product with ξ. We have
ιx◦λξ + λξ◦ιx = x, ξ . id .