Involutions on a Scheme and the Quotient

Introduction

An involution of a scheme is an automorphism of order two, and it determines two subschemes: the fixed subscheme, the largest closed subscheme on which the involution acts as the identity, and the quotient, the universal scheme on which the involution acts trivially. The fixed subscheme is always closed, and it is constructed from the difference of the involution with the identity; the quotient is the categorical quotient, whose existence is the finiteness theorem of invariant theory on an affine chart and a glueing in general, and whose properties — finite, surjective, with the invariants as the functions — make it the geometrical form of the descent of Real Structures on Varieties and Galois Descent. This article fixes the involution of a scheme, the fixed subscheme, the categorical quotient and its construction, and the free case in which the quotient is a torsor under the group, and it is the fifth article of the - * Theory group.

The article is the general member of the group: where Real Structures on Varieties and Galois Descent treated the involution of a variety with the semilinear twist over a Galois extension, the present article treats the involution of an arbitrary scheme over a base, with the fixed subscheme and the categorical quotient in the generality of Schemes; the quotient developed here is the one used there for a variety, and the case of a free action is the case in which the quotient is the descent to the trivial group. The invariant theory is Part I's Invariant Theory, and the descent is Part I's Descent Theory; the finite morphisms and the pushforwards are Coherent Sheaves and Schemes of this category, and the descent of the structure sheaf for the free case is The Galois Action as an Operator. The torus of The Weil Restriction and the Trace Form is the subtler quotient of a group scheme by a finite group, and only the scheme-theoretic quotient is treated here.

Throughout $S$ is a base scheme, $X$ is an $S$-scheme, and $\sigma : X\to X$ is an involution over $S$, $\sigma^2 = \mathrm{id}$, unless a field of definition is named; $G = \{\mathrm{id},\sigma\}\cong\mathbb{Z}/2$ is the group it generates. The fixed subscheme is $X^\sigma$; the quotient is $X/\sigma = X/G$.

Involutions on a Scheme

The Fixed Subscheme

Definition. A fixed subscheme of the involution $\sigma$ is a closed subscheme $Z\hookrightarrow X$ such that $\sigma|_Z = \mathrm{id}_Z$ and such that every closed subscheme $Z'$ with $\sigma|_{Z'} = \mathrm{id}_{Z'}$ factors through $Z$; equivalently, $Z$ is the equaliser of $\sigma$ and the identity, the fibre product $$ X^\sigma = X\ \times_{X\times_SX,\ (\mathrm{id},\sigma)}\ X , $$ taken along the diagonal.

Theorem (existence and the affine formula). The fixed subscheme of an involution of an $S$-scheme exists and is a closed subscheme of $X$. On an affine chart $X = \operatorname{Spec}A$ over $S$ with the involution given by the $A$-algebra involution $\theta$ of order two commuting with the structure map, $$ X^\sigma = \operatorname{Spec}\bigl(A\big/\mathrm{I}_\theta\bigr), \qquad \mathrm{I}_\theta = \bigl(\theta(a)-a\ :\ a\in A\bigr). $$

Proof. On an affine chart the equaliser of the two $\mathcal{O}_S$-algebra maps $\theta$ and the identity is the coequaliser of their difference, which is the displayed quotient; the composite $A\otimes_{\mathcal{O}_S}A\to A$, $a\otimes b\mapsto a\theta(b)$ together with the product gives the ideal generated by the differences $b - \theta(b)$, exactly as in the variety case. On a general $S$-scheme the affine construction is compatible with the restriction to an open chart, so the local closed subschemes glue to a closed subscheme of $X$.

Proposition (the fixed subscheme is the equaliser in every sense). The fixed subscheme satisfies the universal property of the equaliser among the $S$-schemes, and it is the largest closed subscheme on which the involution is the identity: $$ \operatorname{Hom}_S(T,X^\sigma) = \{\,f : T\to X \ :\ \sigma\circ f = f\,\}. $$

Proof. A morphism $f : T\to X$ with $\sigma f = f$ is a morphism to the fibre product of the diagonal and the graph, which is $X^\sigma$; this is the universal property of the equaliser, and the largest-closed-subscheme statement follows because a closed subscheme on which $\sigma$ is the identity is such a morphism and factors through the equaliser.

The Fixed Subscheme and the Local Structure

Proposition (the fixed subscheme of a linear involution of the affine line). For $X = \operatorname{Spec}k[x]$ and $\theta(x) = -x$ in a characteristic different from two, the fixed subscheme is $\operatorname{Spec}k[x]/(x)$, a single point, while the quotient below is $\operatorname{Spec}k[x^2]$, a line. The fixed subscheme is the origin of the quotient line.

Proof. The ideal $\mathrm{I}_\theta$ is generated by $\theta(x)-x = -2x$, so it is $(x)$ in characteristic different from two; the invariants are $k[x]^2 = k[x^2]$ by the elementary computation of the fixed polynomials. The closed point $x=0$ is fixed and its image in the quotient is the origin.

Remark (the fixed subscheme may be nonreduced and of codimension two). The fixed subscheme can be nonreduced, as the example $\theta(x) = -x$ on $k[x,y]$ shows: the ideal is $(x,y)$, reduced here, but the involution $x\mapsto x$, $y\mapsto y+1$ on a suitable affine chart, or the involution of a nilpotent thickening, produces fixed subschemes with nilpotents; and the fixed subscheme can have codimension greater than one, as the involution $x\mapsto-x$ of the affine plane.

Proof. The ideal is computed by the affine formula in each case; for the nilpotent thickenings the ideal $(\theta(a)-a)$ records the nilpotents, and for the affine plane the two generators $x,y$ cut out the origin of codimension two.

The Categorical Quotient

The Quotient by a Finite Group

Definition. A categorical quotient of $X$ by $G$ is an $S$-scheme $X/G$ with a morphism $\pi : X\to X/G$ over $S$ such that $\pi\circ\sigma = \pi$ and such that every $G$-invariant morphism $\psi : X\to Y$ over $S$ factors uniquely as $\psi = \phi\circ\pi$. A categorical quotient is unique up to unique isomorphism when it exists, and it is a coarse moduli of the action.

Theorem (the affine quotient). Let $X = \operatorname{Spec}A$ be affine over $S$ and let $G$ act on $X$ by $S$-automorphisms of order two, with the corresponding involution $\theta$ of $A$. Then the invariant ring $A^G = \{a\in A : \theta(a) = a\}$ is a subalgebra, the morphism $$ \pi : X = \operatorname{Spec}A\longrightarrow\operatorname{Spec}A^G $$ induced by the inclusion is a categorical quotient, and the structure sheaf of the quotient is $\mathcal{O}_{X/G} = A^G$.

Proof. For a $G$-invariant morphism $\psi : X\to Y = \operatorname{Spec}B$ the pullback $\psi^\flat : B\to A$ lands in the invariants $A^G$, since $\psi\circ\sigma = \psi$ is $\psi^\flat\circ\theta = \psi^\flat$; hence $\psi^\flat$ factors uniquely through the inclusion $A^G\hookrightarrow A$, giving $\phi : \operatorname{Spec}A^G\to Y$ with $\psi = \phi\circ\pi$. The uniqueness is the uniqueness of the factorisation through a subalgebra, and $\mathcal{O}_{X/G}$ is $A^G$ by the construction.

Theorem (the finiteness of the invariants). For a finite group $G$ acting on a finitely generated $k$-algebra $A$, the invariant ring $A^G$ is a finitely generated $k$-algebra, so the affine quotient of a variety is a variety. The bound on the degrees is independent of the group.

Proof. This is Noether's finiteness theorem for the invariants of a finite group, Part I's Invariant Theory: the invariants are generated by the $G$-averages of the monomials in a generating set up to Noether's bound. The quotient $\operatorname{Spec}A^G$ is then of finite type over $k$, hence a variety.

Existence and Properties

Theorem (the quotient of a quasi-projective variety). Let a finite group $G$ act on a quasi-projective variety $X$ over a field $k$ by $k$-automorphisms. Then the categorical quotient $X/G$ exists as a quasi-projective variety over $k$, the morphism $\pi : X\to X/G$ is finite and surjective, and $$ \mathcal{O}_{X/G} = (\pi_*\mathcal{O}_X)^G . $$ The quotient of the involution is the case $G = \mathbb{Z}/2$.

Proof. On a $G$-stable affine chart the quotient is $\operatorname{Spec}A^G$ by the affine theorem, and the finiteness of the invariants makes it a variety; a $G$-stable affine cover exists because $G$ is finite and every point has a $G$-stable affine neighbourhood, and the affine quotients glue because the quotient is unique and the invariant sheaf is defined globally. The cases of the general finite group and of the general scheme, with the categorical quotient and the existence over an algebraic space, are in the literature of Mumford's geometric invariant theory and of Keel–Mori.

Theorem (the elementary properties of the quotient). Let $\pi : X\to X/G$ be the quotient of the finite group action. Then $\pi$ is $G$-invariant by definition; the fibres of $\pi$ are the closures of the orbits, so $\pi$ separates the orbits and is surjective; the fixed subscheme $X^\sigma$ maps into $X/\sigma$ as a closed subscheme; and the quotient is separated when $X$ is separated.

Proof. The invariance is the defining property. For the fibres, two points in the same orbit have the same image because $\pi$ is invariant, and the fibres are finite because $\pi$ is finite; a finite surjective morphism separates the points into finitely many preimages, and the generic fibre is the orbit. The fixed subscheme maps to the quotient because $\pi\circ\sigma=\pi$ restricts to the identity on $X^\sigma$, and the image is closed because $\pi$ is finite. The separation of the quotient is the descent of the separatedness along the finite surjective morphism $\pi$, by Schemes.

Corollary (the invariants as the functions, and the quotient sheaf). The functions on the quotient are the invariant functions, $\Gamma(X/G,\mathcal{O}) = \Gamma(X,\mathcal{O})^G$, and more generally $\mathcal{O}_{X/G} = (\pi_*\mathcal{O}_X)^G$ as a sheaf; the quotient exists exactly when this sheaf is the structure sheaf of a scheme, which the finite generation guarantees on the affine charts.

Proof. The identification $\Gamma(X/G) = \Gamma(X)^G$ is the affine theorem applied to the global sections; the sheaf statement is the local form of the same identification, and the sheaf $(\pi_*\mathcal{O}_X)^G$ is quasi-coherent because $\pi$ is finite and the invariants are computed locally.

The Geometric Quotient and the Free Action

Free Actions and the Quotient

Definition. The action of $G$ on $X$ is free when the stabiliser of every geometric point is trivial, that is, when $\sigma$ has no fixed geometric point; equivalently, when the fixed subscheme $X^\sigma$ is empty. The quotient is a geometric quotient when the fibres of $\pi$ are exactly the orbits and $\pi$ is open.

Theorem (the quotient of a free involution). Let $\sigma$ be a free involution of an $S$-scheme $X$, with the quotient $\pi : X\to X/G$ existing. Then $\pi$ is a geometric quotient, the action is a $\mathbb{Z}/2$-torsor, $$ G\times_SX\ \xrightarrow{\ \sim\ }\ X\times_{X/G}X, $$ so the quotient is a finite surjective morphism of degree two, and the descent datum of the free action is effective.

Proof. For a free action the fibre of $\pi$ over a geometric point is the orbit, which has two elements, so the fibres are the orbits and $\pi$ is a geometric quotient. The map $G\times_SX\to X\times_{X/G}X$, $(\tau,x)\mapsto(\tau x,x)$, is injective on geometric points by freeness and finite of degree two on each side, so it is an isomorphism; this is the torsor condition. The effectiveness is the descent of Part I's Descent Theory, which applies because the torsor condition is the descent datum of The Galois Action as an Operator in the case of the trivial Galois group replaced by $\mathbb{Z}/2$.

Corollary (the double cover and the descent). A free involution on a variety $X$ over $k$ with the quotient $X/G$ realises $X$ as the base change of $X/G$ along the double cover, and the descent of Real Structures on Varieties and Galois Descent is recovered: the quotient is the descended variety, the involution is the deck transformation, and the fixed locus is empty in the free case.

Proof. The torsor condition exhibits $X\to X/G$ as a principal $G$-bundle with the group $\mathbb{Z}/2$, so the descent theorem for a finite group exhibits $X$ as a form of the trivial bundle over $X/G$ with structure group $\mathbb{Z}/2$; the descent of Real Structures on Varieties and Galois Descent is the case in which this form is the base change $X\cong(X/G)\times_kL$ with the deck transformation, and the fixed locus is empty by the freeness.

The Non-Free Case

Theorem (the branch locus of the quotient). Let $\pi : X\to X/\sigma$ be the quotient of an involution of a normal variety $X$ over a field of characteristic different from two. Then $\pi$ is finite and flat of degree two outside the fixed subscheme, the fixed subscheme $X^\sigma$ is the branch locus of $\pi$, and the quotient is nonsingular away from the image of the fixed subscheme. The quotient of a smooth variety by a non-free involution may be singular, and the singularities are exactly at the image of the fixed locus when the fixed locus is of codimension two.

Proof. At a point outside the fixed subscheme the action is free on a neighbourhood, so $\pi$ is a torsor and is finite of degree two; at a fixed point the local model is the quotient of a vector space by the linear involution, whose invariants are the ring of the fixed hyperplanes: the invariant ring has the singularity of the quotient of a vector space by a linear involution, which is smooth exactly when the fixed locus is a hyperplane. The displayed claims are the local computation; the characteristic restriction keeps the reflection and the averaging of the invariants available.

Example (the quotient of the plane and the cone). The involution $\sigma(x,y) = (-x,-y)$ of $\mathbb{A}^2_k$ has the fixed subscheme the origin and the quotient $\operatorname{Spec}k[x,y]^\sigma$ with the invariants $x^2,xy,y^2$ and the single relation $(xy)^2 = x^2y^2$, so the quotient is the quadric cone of Algebraic Geometry; the fixed loci are the codimension-two branch and the quotient is singular at the image of the origin. For $\sigma(x,y)=(x,-y)$ the fixed subscheme is the line $y=0$ of codimension one, the invariants are $k[x,y^2]$, and the quotient is smooth.

Proof. The invariants of the linear involution $x\mapsto-x$, $y\mapsto-y$ are generated by the monomials of even degree, $x^2,xy,y^2$, with the relation $(xy)^2 = x^2y^2$; the resulting ring is $k[u,v,w]/(w^2-uv)$, the affine cone. For the reflection $y\mapsto-y$ the invariants are $k[x,y^2]$, a polynomial ring, so the quotient is smooth. Both are the local computation of the theorem.

Summary

An involution of an $S$-scheme is an automorphism $\sigma$ with $\sigma^2=\mathrm{id}$. Its fixed subscheme $X^\sigma$ is the equaliser of $\sigma$ and the identity, a closed subscheme; on an affine chart $X=\operatorname{Spec}A$ with the algebra involution $\theta$ it is $\operatorname{Spec}(A/(\theta(a)-a))$, and its universal property is $\operatorname{Hom}_S(T,X^\sigma) = \{f : \sigma f = f\}$. The categorical quotient $X/\sigma$ is the universal $\sigma$-invariant morphism; on an affine chart it is $\operatorname{Spec}A^\theta$ with $A^\theta$ the invariant ring, finitely generated by Noether's theorem, and for a quasi-projective variety over a field it exists as a quasi-projective variety with $$ \mathcal{O}_{X/\sigma} = (\pi_*\mathcal{O}_X)^\sigma , \qquad \Gamma(X/\sigma,\mathcal{O}) = \Gamma(X,\mathcal{O})^\sigma , $$ the map $\pi$ finite and surjective, and the fixed subscheme mapping in as a closed subscheme. For a free involution the quotient is a geometric quotient and a $\mathbb{Z}/2$-torsor, so $\pi$ is finite of degree two and the descent of Real Structures on Varieties and Galois Descent is recovered; for a non-free involution the fixed subscheme is the branch locus of $\pi$, and the quotient of a smooth variety by a non-free involution may be singular, as the cone $\mathbb{A}^2/\!\pm1$ shows. The two objects — the fixed subscheme and the quotient — are the two halves of the involution, and they agree exactly when the involution is the identity.

Summary of Notation

Symbol	Meaning
$\sigma$, $\sigma^2=\mathrm{id}$	an involution of an $S$-scheme
$G=\{\mathrm{id},\sigma\}\cong\mathbb{Z}/2$	the group generated by the involution
$X^\sigma$	the fixed subscheme: the equaliser of $\sigma$ and the identity
$\mathrm{I}_\theta=(\theta(a)-a)$	the ideal of the fixed subscheme on an affine chart
$\operatorname{Hom}_S(T,X^\sigma)=\{f:\sigma f=f\}$	the universal property of the fixed subscheme
$X/G$, $X/\sigma$	the categorical quotient; unique when it exists
$A^G=\{a:\theta(a)=a\}$	the invariant ring; the coordinate ring of an affine quotient
$\mathcal{O}_{X/G}=(\pi_*\mathcal{O}_X)^G$	the invariant sheaf; the structure sheaf of the quotient
$\pi$ finite and surjective	the quotient map of a finite group action
free action, $X^\sigma=\varnothing$	no fixed geometric point
geometric quotient, $\mathbb{Z}/2$-torsor	the fibres are the orbits; $G\times_SX\cong X\times_{X/G}X$
branch locus	the fixed subscheme, where the quotient map is not a torsor
$\mathbb{A}^2/\!\pm1 = \operatorname{Spec}k[x^2,xy,y^2]$	the cone; a singular quotient of a smooth variety