Involution-Invariant Ideals of the Symmetric Algebra

Introduction

The symmetric algebra of a module $M$ carries the induced involution $\sigma$ of The Symmetric Algebra with an Involution, and the ideals of $\operatorname{Sym}(M)$ that the involution preserves are the involution-invariant ideals; they are the ideals through which the involution descends to the quotient, and they are the algebraic carriers of the fixed subalgebra on the quotient side. The general theory of a $\sigma$-stable ideal and of the induced involution on the quotient belongs to Involutive Linear Algebras, which proves that a stable ideal produces an involution of the quotient and that the fixed subalgebra of the quotient is the quotient of the fixed subalgebra by the fixed part of the ideal. The purpose of this article is to read that theory in the symmetric algebra, where the grading gives a supply of stable ideals — the homogeneous ideals generated by symmetric elements — and where the fixed subalgebra of The Symmetric Algebra with an Involution controls the whole lattice.

The article defines a $\sigma$-stable ideal, proves that the induced involution of the quotient is well defined and of order two, and identifies the fixed subalgebra of the quotient as $\operatorname{Sym}(M)^\sigma/(I\cap\operatorname{Sym}(M)^\sigma)$. It shows that every ideal generated by homogeneous symmetric elements and by the pairwise products of the anti-fixed elements is stable, that the smallest stable ideal containing a given ideal is generated by the orbit of its generators under $\sigma$, and that under the descent of Commutative Algebras with an Involution the stable ideals of the symmetric algebra correspond to the ideals of the fixed subalgebra. The worked cases are the polynomial ring in two variables with the swap involution, where the stable ideals $(x^2+y^2)$, $(x-y)$ and $(x,y)$ realise the three behaviours, and the polynomial ring in one variable with $x\mapsto -x$, where the stable monomial ideals are the even and the odd ones.

The article assumes The Symmetric Algebra with an Involution for $\sigma$ and the fixed subalgebra, Involutive Linear Algebras for the stable ideals, the induced involution on the quotient and the fixed subalgebra of the quotient (cited and specialised), The Symmetric and Exterior Powers for the grading, Ideals and Quotients of Algebras for the ideals, and Commutative Algebras with an Involution for the descent in the commutative case. The symmetric powers are The Symmetric Powers of an Involutive Module, and the Jordan case is Jordan Algebras with an Involution. Throughout, $R$ is a commutative ring with identity, $M$ is an $R$-module with an $R$-linear involution $\sigma_M$, $\operatorname{Sym}(M)$ is its symmetric algebra with the induced involution $\sigma$, and $I$ is an ideal of $\operatorname{Sym}(M)$; no form, norm or distance occurs.

Stable Ideals of the Symmetric Algebra

Definition and Elementary Properties

Definition. An ideal $I$ of $\operatorname{Sym}(M)$ is $\sigma$-stable when $\sigma(I)\subseteq I$, equivalently when $\sigma(I) = I$ because $\sigma^2 = \mathrm{id}$ and $\sigma$ is bijective. An ideal that is stable is also called involution-invariant.

Proposition. The stable ideals are closed under sum, product, intersection and radical: if $I, J$ are stable then $I+J$, $IJ$, $I\cap J$ and $\sqrt I$ are stable. The image and the preimage of a stable ideal under a $\sigma$-equivariant homomorphism of symmetric algebras are stable.

Proof. $\sigma(I+J) = \sigma(I)+\sigma(J)\subseteq I+J$; $\sigma(IJ) = \sigma(I)\sigma(J)\subseteq IJ$; the intersection is immediate; $x^n\in I$ gives $\sigma(x)^n = \sigma(x^n)\in I$, so $\sigma(x)\in\sqrt I$. For a $\sigma$-equivariant $\varphi$, $\varphi(I)$ and $\varphi^{-1}(J)$ are stable by equivariance. $\square$

Corollary (the orbit). The smallest $\sigma$-stable ideal containing a given ideal $J$ is $J+\sigma(J)$; the smallest stable ideal containing an element $u$ is $(u,\sigma(u))$, and it is principal exactly when $\sigma(u)$ is a multiple of $u$.

Homogeneous Ideals and the Grading

Proposition. The symmetric algebra is graded and $\sigma$ preserves the degree, so a homogeneous ideal $I = \bigoplus_k I_k$ is stable exactly when each homogeneous piece is preserved, $\sigma(I_k)\subseteq I_k$; in particular the ideal generated by a set of homogeneous elements is stable when the set is stable under $\sigma$.

Proof. $\sigma$ preserves each $\operatorname{Sym}^k(M)$, and the homogeneous pieces of a homogeneous ideal determine it. $\square$

Theorem. An ideal generated by the symmetric elements of a set $S$ and by the pairwise products of the anti-fixed elements is stable; the ideal generated by the fixed part $M^+$ and by the products $xy$ with $x,y\in M^-$ is the stable ideal of the fixed subalgebra.

Proof. The generators are fixed by $\sigma$ — the elements of $M^+$ and the products $xy$ for $x,y\in M^-$ — and an ideal generated by fixed elements is stable because $\sigma$ maps each generator to itself. $\square$

The Induced Involution on the Quotient

Definition and the Fixed Subalgebra

Theorem. Let $I$ be a $\sigma$-stable ideal of $A = \operatorname{Sym}(M)$. Then the induced map

$$ \bar\sigma : A/I\to A/I, \qquad \bar\sigma(a+I) = \sigma(a)+I $$

is well defined, is an $R$-algebra automorphism of order two of the quotient, and its fixed subalgebra is

$$ (A/I)^{\bar\sigma} = A^\sigma/(I\cap A^\sigma), $$

where $A^\sigma = \operatorname{Sym}(M)^\sigma$ is the fixed subalgebra.

Proof. Well definedness is $\sigma(I)\subseteq I$; order two is inherited; a class $a+I$ is fixed iff $a-\sigma(a)\in I$, and then $\tfrac12(a+\sigma(a))+I$ is a fixed representative lying in $A^\sigma$, so every fixed class has a representative in $A^\sigma$; the classes that are zero in $A/I$ and represented in $A^\sigma$ are exactly $I\cap A^\sigma$. $\square$

Corollary. The fixed subalgebra of the quotient is a quotient of the fixed subalgebra, and the map $A^\sigma\to(A/I)^{\bar\sigma}$ is surjective with kernel $I\cap A^\sigma$; in particular the stable ideals of $A$ that meet $A^\sigma$ trivially produce inclusions of the fixed subalgebras.

The Lattice of the Stable Ideals

Theorem. Under the descent of Commutative Algebras with an Involution, the $\sigma$-stable ideals of $A$ correspond bijectively to the ideals of the fixed subalgebra $A^\sigma$ by

$$ I \longmapsto I\cap A^\sigma , \qquad J \longmapsto JA , $$

and the correspondence preserves the sums, the intersections and the products.

Proof. $JA$ is stable because $J\subseteq A^\sigma$; $I\cap A^\sigma$ is an ideal of $A^\sigma$; the two maps are inverse when $A$ is faithfully flat over $A^\sigma$ with the separating element of the quadratic model, by the descent of Article 13. The preservation of the lattice operations is the standard behaviour of the extension and the contraction of ideals. $\square$

Examples

Example (the swap in two variables). Let $A = R[x,y]$ with $\sigma(x) = y$, $\sigma(y) = x$. The ideal $(x^2+y^2)$ is stable, because $x^2+y^2$ is fixed; the quotient $A/(x^2+y^2)$ carries the induced involution and its fixed subalgebra is $A^\sigma/(x^2+y^2)\cap A^\sigma$, where $A^\sigma = R[x+y,xy]$. The ideal $(x-y)$ is stable because $\sigma(x-y) = y-x = -(x-y)$; the induced involution of the quotient $R[x,y]/(x-y)\cong R[x]$ is the identity. The ideal $(x)$ is not stable, because $\sigma(x) = y\notin(x)$; the smallest stable ideal containing $x$ is $(x,y)$, with quotient $R$.

Example (the negation in one variable). Let $A = R[x]$ with $\sigma(x) = -x$, so $A^\sigma = R[x^2]$. A monomial ideal $(x^k)$ is stable for every $k$, because $\sigma(x^k) = (-1)^kx^k$; the quotient $R[x]/(x^k)$ carries the induced involution $x\mapsto -x\bmod x^k$, and its fixed subalgebra is spanned by the even powers. The ideal $(x-1)$ is not stable, since $\sigma(x-1) = -x-1 = -(x+1)$; the smallest stable ideal containing $x-1$ is $(x-1,x+1)$, which is the unit ideal when $2$ is invertible in $R$.

Example (the invariant ideal of the fixed subalgebra). For $A = R[x,y]$ with the swap, $A^\sigma = R[x+y,xy]$. The ideal $J = (x+y,xy)$ of $A^\sigma$ has extension $JA = (x+y,xy)A$, the stable ideal generated by the invariants $x+y$ and $xy$; the quotient is $A/JA\cong R[x]/(x^2)$, on which $\bar\sigma$ is the identity because the two generators are fixed. This is the pattern of the descent of the stable ideals: the extension of an ideal of the fixed subalgebra is stable, and the contraction recovers it.

Summary

An ideal $I$ of the symmetric algebra is $\sigma$-stable when $\sigma(I) = I$; the stable ideals are closed under sum, product, intersection and radical, and the smallest stable ideal containing a set is generated by the orbit of the set under $\sigma$. A homogeneous ideal is stable exactly when its homogeneous pieces are preserved, and the ideal generated by the fixed part $M^+$ and by the pairwise products of the anti-fixed part $M^-$ is stable. A stable ideal produces the induced involution $\bar\sigma$ of the quotient, of order two, whose fixed subalgebra is $A^\sigma/(I\cap A^\sigma)$. Under the descent the stable ideals of $A$ correspond to the ideals of the fixed subalgebra $A^\sigma$ by extension and contraction. The swap in two variables and the negation in one variable are the worked examples, and they exhibit a stable principal ideal, a stable non-principal ideal and a non-stable ideal together with its stable hull. No form, norm or distance occurs.

Summary of Notation

Symbol	Meaning
$A = \operatorname{Sym}(M)$	Symmetric algebra with induced involution $\sigma$
$A^\sigma$	Fixed subalgebra
$I$ $\sigma$-stable	$\sigma(I) = I$
$A = \bigoplus_k\operatorname{Sym}^k(M)$	Grading preserved by $\sigma$
$(u,\sigma(u))$	Stable hull of the ideal generated by $u$
$\bar\sigma(a+I) = \sigma(a)+I$	Induced involution of the quotient
$(A/I)^{\bar\sigma} = A^\sigma/(I\cap A^\sigma)$	Fixed subalgebra of the quotient
$I\mapsto I\cap A^\sigma$, $J\mapsto JA$	Descent of the stable ideals