The Divisor Operator

Introduction

A rational function on a variety has zeros and poles along the subvarieties of codimension one, and the map that records them is additive where the function is multiplicative: the product of two functions has, along each divisor, the sum of the orders of its factors. That map is the divisor operator, and it is the fifth operator of the category. It carries the multiplicative group $k(X)^\times$ of the function field onto the group $\operatorname{Div}(X)$ of divisors generated by the codimension-one subvarieties, and the quotient by its image is the class group $\operatorname{Cl}(X)$, which is the Picard group $\operatorname{Pic}(X)$ of invertible sheaves, the same group on which the pullback of the previous article acts. This article fixes the divisors of a variety, the divisor map, the class group and the Picard group with the exact sequence that relates them, the invertible sheaf $\mathcal{O}_X(D)$ of a divisor, and the pullback of a divisor.

The article is the fifth of the - Operator Theory group. It depends on Schemes and Coherent Sheaves for the structure sheaf, the invertible sheaves and their cohomology, and on the previous article, The Pullback Operator of a Morphism, for the pullback on $\operatorname{Pic}$; the valuations it uses are the discrete valuations of Part I's Valuation Theory and Henselian Rings and Localization and the Fraction Field. The arithmetic curve theory — the divisors on a curve, the Riemann–Roch theorem, the group law on an elliptic curve — is Part I's, in Algebraic Curves, The Riemann–Roch Theorem for Curves and Elliptic Curves, and is referred to rather than developed.

Throughout $X$ is an integral separated variety over a field $k$, regular in codimension one — that is, the local ring at every codimension-one point is a discrete valuation ring — which holds for a normal variety and in particular for a regular one. The field of rational functions is $k(X)$, written $\mathcal{K}$, and $X^{(1)}$ is the set of prime divisors of $X$, the irreducible closed subvarieties of codimension one.

Divisors on a Variety

Definition. A prime divisor of $X$ is an irreducible closed subvariety $Y\subseteq X$ of codimension one, $Y\in X^{(1)}$. The group of divisors of $X$ is the free abelian group $$ \operatorname{Div}(X) = \bigoplus_{Y\in X^{(1)}}\mathbb{Z}\cdot Y , $$ whose elements are the finite formal sums $D = \sum_{Y}n_Y\,Y$ with $n_Y\in\mathbb{Z}$. A divisor is effective, written $D\geq0$, if every $n_Y\geq0$; the effective divisors are the additive monoid generated by $X^{(1)}$.

Definition. Let $Y\in X^{(1)}$. The local ring $\mathcal{O}_{X,Y}$ is a discrete valuation ring with fraction field $k(X)$, and its valuation is the order of vanishing along $Y$, $$ v_Y : k(X)^\times\longrightarrow\mathbb{Z}, $$ normalised by $v_Y(\mathcal{O}_{X,Y}\setminus\{0\}) = \mathbb{N}$. For $f\in k(X)^\times$ the value $v_Y(f)$ is positive at a zero of $f$ along $Y$ and negative at a pole, and $v_Y$ is the valuation of Part I's Valuation Theory and Henselian Rings.

Proposition (the stalks of a codimension-one point). For $Y\in X^{(1)}$ the local ring $\mathcal{O}_{X,Y}$ is a discrete valuation ring, its maximal ideal is generated by a uniformiser $t_Y$, and every $f\in k(X)^\times$ is uniquely $f = u\,t_Y^{v_Y(f)}$ with $u$ a unit of $\mathcal{O}_{X,Y}$. The residue field of $Y$ is $\kappa(Y) = \mathcal{O}_{X,Y}/t_Y\mathcal{O}_{X,Y}$.

Proof. The codimension-one local ring of a variety is one-dimensional and Noetherian and, by the hypothesis that $X$ is regular in codimension one, regular; a regular local ring of dimension one is a discrete valuation ring, by Part I's Valuation Theory and Henselian Rings. The unique factorisation of an element of the fraction field of a discrete valuation ring into a power of a uniformiser and a unit is the standard structure of the valuation ring.

Example (the affine line and the affine plane). On $X = \mathbb{A}^1_k$ the prime divisors are the closed points, $\operatorname{Div}(\mathbb{A}^1) = \bigoplus_{a\in k}\mathbb{Z}\cdot(a)$, and for $f\in k(t)^\times$ the coefficient at $a$ is the order of vanishing $\operatorname{ord}_a(f)$ of $f$ at $a$. On $X = \mathbb{A}^2_k$ the prime divisors are the irreducible plane curves, and $\operatorname{Div}(\mathbb{A}^2)$ is free on them; the divisor of the coordinate $x$ is the line $V(x)$ with multiplicity one, and the divisor of $y/x$ is $V(y)-V(x)$.

The Divisor Map

Definition. The divisor map is the group homomorphism $$ \operatorname{div} : k(X)^\times\longrightarrow\operatorname{Div}(X), \qquad \operatorname{div}(f) = \sum_{Y\in X^{(1)}}v_Y(f)\,Y , $$ the sum being finite because a rational function has only finitely many zeros and poles. The value $\operatorname{div}(f)$ is a principal divisor, also written $(f)$.

Theorem (the divisor map is a homomorphism). For all $f,g\in k(X)^\times$, $$ \operatorname{div}(fg) = \operatorname{div}(f) + \operatorname{div}(g), \qquad \operatorname{div}(f^{-1}) = -\operatorname{div}(f), \qquad \operatorname{div}(1) = 0 . $$ Consequently the divisor map is a homomorphism of abelian groups whose source is written multiplicatively and whose target is written additively: the operator converts the product of functions into the sum of divisors.

Proof. Each $v_Y$ is a valuation and therefore satisfies $v_Y(fg) = v_Y(f)+v_Y(g)$; summing over the finitely many $Y$ for which either term is nonzero gives the first identity, and the other two are its special cases.

Theorem (the divisor of a global function). Let $f\in\Gamma(X,\mathcal{O}_X)$ be a nonzero global function on a complete variety $X$. Then $\operatorname{div}(f)\geq0$, and $\operatorname{div}(f)=0$ if and only if $f$ is a unit.

Proof. A regular function on a variety has no pole at any point, so $v_Y(f)\geq0$ for every $Y\in X^{(1)}$; this is the statement that $\mathcal{O}_{X,Y}$ contains $f$. If in addition $\operatorname{div}(f) = 0$ then $f$ and $f^{-1}$ are both regular on a complete variety, hence both constant, and $f$ is a nonzero scalar, a unit.

Remark (the operator is not linear over the functions). The divisor map is defined on the multiplicative group of the function field and is not additive there: $\operatorname{div}(f+g)$ has no expression in terms of $\operatorname{div}(f)$ and $\operatorname{div}(g)$. It is therefore not an operator of the layer of Operators on a Variety, which is linear over the structure sheaf, but a homomorphism from the multiplicative structure of the function field to the additive structure of the divisors, and this change of operation is what makes it an operator of a different kind.

Example (the projective line). On $X = \mathbb{P}^1_k$ the prime divisors are the closed points of the affine line together with the point at infinity, and for $f\in k(t)^\times$ a rational function of the form $f = c\prod_i(t-a_i)^{m_i}$ one has $$ \operatorname{div}(f) = \sum_im_i\,(a_i) - \Bigl(\sum_im_i\Bigr)\,(\infty), $$ with the coefficient at $\infty$ negative when $f$ has a pole there; the degree of $\operatorname{div}(f)$, the sum of its coefficients, is $0$. On $\mathbb{P}^1$ every divisor of degree $0$ is principal, so $\operatorname{Cl}(\mathbb{P}^1)\cong\mathbb{Z}$.

The Class Group and the Picard Group

Definition. The class group of $X$ is the quotient $$ \operatorname{Cl}(X) = \operatorname{Div}(X)\big/\operatorname{div}\bigl(k(X)^\times\bigr), $$ the divisors modulo the principal ones. Two divisors with the same class are linearly equivalent. The class of $D$ is written $[D]$, and the class group is also the divisor class group.

Definition. An $\mathcal{O}_X$-module $\mathcal{L}$ is invertible if it is locally free of rank one; the Picard group $\operatorname{Pic}(X)$ is the group of isomorphism classes of invertible sheaves under the tensor product, with inverse $\mathcal{L}^{-1} = \mathcal{H}om(\mathcal{L},\mathcal{O}_X)$ and unit $\mathcal{O}_X$.

Theorem (the invertible sheaf of a divisor). For a divisor $D$ let $$ \mathcal{O}_X(D)(U) = \{\,f\in k(X)^\times : \operatorname{div}(f)|_U + D|_U\geq0\,\}\cup\{0\} . $$ Then $\mathcal{O}_X(D)$ is an invertible sheaf, the assignment $D\mapsto\mathcal{O}_X(D)$ is additive up to canonical isomorphism, $$ \mathcal{O}_X(D_1+D_2)\cong\mathcal{O}_X(D_1)\otimes\mathcal{O}_X(D_2), \qquad \mathcal{O}_X(0)\cong\mathcal{O}_X, \qquad \mathcal{O}_X(-D)\cong\mathcal{O}_X(D)^{-1}, $$ and it depends on $D$ only through its class: $\mathcal{O}_X(\operatorname{div}(f))\cong\mathcal{O}_X$ for every $f$.

Proof. On an affine open $U$ on which $D$ is the divisor of a rational section, $\mathcal{O}_X(D)|_U$ is generated by one rational function, hence free of rank one; the compatibility of the local trivialisations is the additivity of the valuations. The multiplicativity is the identity $\operatorname{div}(f_1f_2) = \operatorname{div}(f_1)+\operatorname{div}(f_2)$ applied to the defining condition, and the last statement is the case $D = \operatorname{div}(f)$, where the defining condition becomes $f\in\Gamma(U,\mathcal{O}_U)$ after multiplication by $1/f$, giving $\mathcal{O}_X$ itself.

Theorem (the class group computes the Picard group). The map $D\mapsto[\mathcal{O}_X(D)]$ induces a homomorphism $$ \operatorname{cl} : \operatorname{Cl}(X)\longrightarrow\operatorname{Pic}(X), $$ and it is the connecting map of the exact sequence of sheaves of the next section. When $\Gamma(X,\mathcal{O}_X^\times) = k^\times$ — in particular when $X$ is complete — this homomorphism is an isomorphism $$ \operatorname{Cl}(X)\ \cong\ \operatorname{Pic}(X). $$

Proof. The last theorem shows that the assignment factors through the class group and is multiplicative. For the isomorphism, the sheaf sequence below gives the exact sequence $k(X)^\times\to\Gamma(X,\mathcal{K}^\times/\mathcal{O}_X^\times)\to\operatorname{Pic}(X)\to H^1(X,\mathcal{K}^\times)$; the group $\Gamma(X,\mathcal{K}^\times/\mathcal{O}_X^\times)$ is the group of Cartier divisors, which for a variety regular in codimension one equals $\operatorname{Div}(X)$, and $H^1(X,\mathcal{K}^\times) = 0$ because the constant sheaf $k(X)^\times$ on the irreducible space $X$ is flabby and hence acyclic. The image of $k(X)^\times$ is the principal divisors, so the cokernel is $\operatorname{Cl}(X)$, and this cokernel injects into $\operatorname{Pic}(X)$; injectivity of the last map follows from the vanishing of $H^1(X,\mathcal{K}^\times)$.

The Sheaf Sequence of the Divisor Operator

Definition. Let $\mathcal{K}$ be the constant sheaf of the function field and $\mathcal{K}^\times$ its sheaf of units, so that $\mathcal{K}^\times(U) = k(X)^\times$ for every nonempty open $U$. The sheaf of divisors is the quotient sheaf $$ \mathcal{D}iv = \mathcal{K}^\times/\mathcal{O}_X^\times . $$

Theorem (the exact sequence). There is a short exact sequence of sheaves of abelian groups $$ 0\longrightarrow\mathcal{O}_X^\times\longrightarrow\mathcal{K}^\times\xrightarrow{\ \operatorname{div}\ }\mathcal{D}iv\longrightarrow0 , $$ in which the middle arrow is the divisor map on sections, and its long exact sequence of cohomology begins $$ 0\longrightarrow\Gamma\bigl(X,\mathcal{O}_X^\times\bigr)\longrightarrow k(X)^\times\xrightarrow{\ \operatorname{div}\ }\operatorname{Div}(X)\longrightarrow\operatorname{Pic}(X)\longrightarrow0 , $$ the last term being $H^1(X,\mathcal{O}_X^\times)=\operatorname{Pic}(X)$ and the preceding one $H^0(X,\mathcal{D}iv)=\operatorname{Div}(X)$ for a variety regular in codimension one.

Proof. The sequence of sheaves is exact at $\mathcal{O}_X^\times$ by definition and at the quotient because the map is the quotient map; exactness at $\mathcal{K}^\times$ is the statement that a function with $\operatorname{div}(f) = 0$ is a unit, which is the theorem on the divisor of a global function applied on each open set. The long exact sequence of Sheaf Cohomology then gives the displayed beginning, with $H^1(X,\mathcal{K}^\times) = 0$ by the flabbiness of a constant sheaf on an irreducible space, and $H^0(X,\mathcal{D}iv) = \operatorname{Div}(X)$ because a Cartier divisor on a variety regular in codimension one is a Weil divisor.

Corollary (the structure of the Picard group). The Picard group fits into the exact sequence $$ 0\longrightarrow\frac{\Gamma(X,\mathcal{O}_X^\times)}{k^\times}\longrightarrow\operatorname{Cl}(X)\longrightarrow\operatorname{Pic}(X)\longrightarrow0 $$ when the constants are separated by the choice of a normalisation, so that for a complete variety $$ \operatorname{Pic}(X)\cong\operatorname{Div}(X)\big/\operatorname{div}(k(X)^\times) = \operatorname{Cl}(X) . $$

Proof. Divide the long exact sequence by the constants $k^\times$ in the first two terms; the identification is the previous theorem.

The Degree and the Pullback of a Divisor

Definition. Let $X$ be a complete variety of dimension one — a projective curve — and let $D = \sum_Pn_PP\in\operatorname{Div}(X)$. The degree of $D$ is $$ \deg(D) = \sum_Pn_P\,[\kappa(P):k] , $$ the sum being finite. A divisor of degree zero is a principal divisor when it has the form $\operatorname{div}(f)$ for $f\in k(X)^\times$. The curve theory — the Riemann–Roch theorem, the Jacobian and the group law — is Part I's Algebraic Curves and The Riemann–Roch Theorem for Curves, and is referenced here only for the degree.

Proposition (the degree is a homomorphism vanishing on the principal divisors). The degree is a homomorphism $\deg : \operatorname{Div}(X)\to\mathbb{Z}$, and $\deg(\operatorname{div}(f)) = 0$ for every $f\in k(X)^\times$, so that the degree descends to a homomorphism $\deg : \operatorname{Pic}(X)\to\mathbb{Z}$.

Proof. The degree is additive by definition. That the divisor of a rational function has degree zero is the statement that a rational function on a projective curve has as many zeros as poles, counted with multiplicity, which is Part I's The Riemann–Roch Theorem for Curves; it is the reason the degree is well defined on classes.

Theorem (the pullback of a divisor). Let $f : X\to Y$ be a dominant morphism of varieties regular in codimension one. Then the pullback of functions $k(Y)^\times\to k(X)^\times$ composed with the divisor map defines a homomorphism $$ f^* : \operatorname{Cl}(Y)\longrightarrow\operatorname{Cl}(X), \qquad f^*[D] = \bigl[\operatorname{div}(f^\sharp g)\bigr] $$ for a local equation $g\in k(Y)^\times$ of $D$, and it agrees with the pullback on the Picard group of The Pullback Operator of a Morphism under the identification $\operatorname{Cl}\cong\operatorname{Pic}$, whenever the two identifications hold. In the formula the valuation $v_{Y'}(f^\sharp g)$ at a prime divisor $Y'$ mapping dominantly to $Y$ is the ramification multiplicity of $f$ at $Y'$ when $g$ is a uniformiser at $Y$.

Proof. The map is the composite of the pullback of rational functions with the divisor map, and it is well defined on classes because $\operatorname{div}(f^\sharp\mathrm{id}_{k(Y)}) = 0$; the agreement with the sheaf-theoretic pullback is the description of $f^*\mathcal{O}_Y(D)$ by the local equations $f^\sharp g$ together with the ramification multiplicities of Valuation Theory and Henselian Rings.

Example (the degree on the projective line and its pullback). On $\mathbb{P}^1$ the class group is $\mathbb{Z}$ under the degree, $\operatorname{Cl}(\mathbb{P}^1)\cong\mathbb{Z}$, and the class of a point is the generator. For the map $f : \mathbb{P}^1\to\mathbb{P}^1$ of degree $d$, the pullback of a point has degree $d$; in the language of the previous article this is the pullback of the invertible sheaf $\mathcal{O}_{\mathbb{P}^1}(1)$, of degree one, whose degree multiplies by $d$.

Summary

For a variety $X$ regular in codimension one, the prime divisors are the codimension-one subvarieties and they generate the free abelian group $\operatorname{Div}(X)$ of divisors; the order of vanishing $v_Y$ along a prime divisor is the valuation of the discrete valuation ring at its generic point, and the divisor map $\operatorname{div} : k(X)^\times\to\operatorname{Div}(X)$, $\operatorname{div}(f) = \sum_Yv_Y(f)Y$, is a homomorphism from the multiplicative group of the function field to the additive group of divisors, whose values are the principal divisors. The class group $\operatorname{Cl}(X) = \operatorname{Div}(X)/\operatorname{div}(k(X)^\times)$ is the quotient by the principal divisors; the invertible sheaves $\mathcal{O}_X(D)$ of the divisors are the line bundles, the assignment $D\mapsto\mathcal{O}_X(D)$ is additive and factors through the class group, and the induced map is the connecting map of the exact sequence $0\to\mathcal{O}_X^\times\to\mathcal{K}^\times\to\mathcal{D}iv\to0$, which identifies $\operatorname{Cl}(X)$ with $\operatorname{Pic}(X)$ when the only globally invertible functions are the constants. The degree of a divisor on a complete curve is a homomorphism vanishing on principal divisors and hence descends to the Picard group, and the pullback of the previous article is the pullback on divisor classes, with the ramification multiplicities as the correction.

Summary of Notation

Symbol	Meaning
$X^{(1)}$	the set of prime divisors, the codimension-one subvarieties of $X$
$v_Y$, $t_Y$	order of vanishing along $Y$; uniformiser of $\mathcal{O}_{X,Y}$
$\operatorname{Div}(X)=\bigoplus_{Y\in X^{(1)}}\mathbb{Z}Y$	group of divisors; free on the prime divisors
$\operatorname{div}(f)=\sum_Yv_Y(f)Y$	the divisor map; the principal divisor of $f$
$\operatorname{Cl}(X)=\operatorname{Div}(X)/\operatorname{div}(k(X)^\times)$	class group (divisor class group)
$\operatorname{cl}:\operatorname{Cl}(X)\to\operatorname{Pic}(X)$	class of a divisor to the class of its invertible sheaf
$\operatorname{Pic}(X)$	Picard group of invertible sheaves under tensor product
$\mathcal{O}_X(D)$	invertible sheaf of a divisor; $\mathcal{O}_X(D_1+D_2)\cong\mathcal{O}_X(D_1)\otimes\mathcal{O}_X(D_2)$
$\mathcal{K}^\times$, $\mathcal{D}iv=\mathcal{K}^\times/\mathcal{O}_X^\times$	constant sheaf of the function field; sheaf of divisors
$0\to\mathcal{O}_X^\times\to\mathcal{K}^\times\to\mathcal{D}iv\to0$	the exact sequence of the divisor operator
$\operatorname{Cl}(X)\cong\operatorname{Pic}(X)$	for $\Gamma(X,\mathcal{O}_X^\times)=k^\times$
$\deg(D)=\sum_Pn_P[\kappa(P):k]$	degree of a divisor on a complete curve
$f^*[D]$	pullback of a divisor class along a dominant morphism
$D\geq0$	an effective divisor