12. Differentials

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In this section, we sketch the theory of differentials. We allow k to be an arbitrary

field.

Let A be a k-algebra, and let M be an A-module. Recall (from §4) that a kderivation

is a k-linear map D : A → M such that

D(fg) = f ◦ Dg + g ◦ Df (Leibniz’s rule).

A pair (Ω1

A/k, d) comprising an A-module Ω1

A/k and a k-derivation d : A → Ω1

A/k is

called the module of differential one-forms for A over kal if it is universal:

A

d ✲ Ω1

❅

❅

D ❅❘

M

❄

∃!k-linear

Example 12.1. Let A = k[X1, ...,Xn]; then Ω1

A/k is the free A-module with basis

the symbols dX1, ..., dXn, and df =

_

∂f/∂Xi · dXi.

Example 12.2. Let A = k[X1, ...,Xn]/a; then Ω1

A/k is the free A-module with

basis the symbols dX1, ..., dXn modulo the relations: df = 0 for all f ∈ a.

Proposition 12.3. Let V be a variety. For each n ≥ 0, there is a unique sheaf of

OV -modules Ωn

V/k on V such that Ωn

V/k(U) = ΛnΩ1

A/k whenever U = SpecmA is an

open affine of V .

Proof. Omitted.

The sheaf Ωn

V/k is called the sheaf of differential n-forms on V .

Example 12.4. Let E be the affine curve

Y 2 = X3 + aX + b,

and assume X3 + aX + b has no repeated roots (so that E is nonsingular). Write x

and y for regular functions on E defined by X and Y. On the open set D(y) where

y _= 0, let ω1 = dx/y, and on the open set D(3x2 + a), let ω2 = 2dy/(3x2 + a). Since

y2 = x3 + ax + b,

2ydy = (3x2 + a)dx.

and so ω1 and ω2 agree on . Since E = D(y) ∩ D(3x2 + a), we see that there

is a differential ω on E whose restrictions to D(y) and D(3x2 + a) are ω1 and ω2

respectively. It is an easy exercise in working with projective coordinates to show

that ω extends to a differential one-form on the whole projective curve

Y 2Z = X3 + aXZ2 + bZ3.

In fact, Ω1

C/k(C) is a one-dimensional vector space over k, with ω as basis. More

generally, if C is a complete nonsingular absolutely irreducible curve of genus g, then

Ω1

C/kC) is a vector space of dimension g over k. Note that ω = dx/y = dx/(x3+ax+

b)

1

2 , which can’t be integrated in terms of elementary functions. Its integral is called

an elliptic integral (integrals of this form arise when one tries to find the arc length

150 Algebraic Geometry: 12. Differentials

of an ellipse). The study of elliptic integrals was one of the starting points for the

study of algebraic curves.

Proposition 12.5. If V is nonsingular, then Ω1

V/k is a locally free sheaf of rank

dim(V ) (that is, every point P of V has a neighbourhood U such that Ω1

V/k

|U ≈

(OV |U)dim(V )).

Proof. Omitted.

Let C be a complete nonsingular absolutely irreducible curve, and let ω be a nonzero

element of Ω1

k(C)/k. We define the divisor (ω) of ω as follows:let P ∈ C; if t is a

uniformizing parameter at P, then dt is a basis for Ω1

k(C)/k as a k(C)-vector space,

and so we can write ω = fdt, f ∈ k(V )×; define _ ordP (ω) = ordP (f), and (ω) =

ordP (ω)P. Because k(C) has transcendence degree 1 over k, Ω1

k(C)/k is a k(C)-

vector space of dimension one, and so the divisor (ω) is independent of the choice of

ω up to linear equivalence. By an abuse of language, one calls (ω) for any nonzero

element of Ω1

k(C)/k a canonical class K on C. For a divisor D on C, let B(D) =

dimk(L(D)).

Theorem 12.6 (Riemann-Roch). Let C be a complete nonsingular absolutely irreducible

curve over k.

(a) The degree of a canonical divisor is 2g − 2.

(b) For any divisor D on C,

B(D) − B(K − D) = 1+g − deg(D).

More generally, if V is a smooth complete variety of dimension d, it is possible to

associate with the sheaf of differential d-forms on V a canonical linear equivalence

class of divisors K. This divisor class determines a rational map to projective space,

called the canonical map.

References

Shafarevich, 1994, III.5.

Mumford 1966, III.4.

Algebraic Geometry: 13. Algebraic Varieties over the Complex Numbers 151

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