Commutative Algebras with an Involution

Introduction

On a commutative algebra every anti-automorphism is an automorphism, so an involution of a commutative algebra is simply an algebra automorphism of order two; and every such automorphism makes the algebra a module over its fixed subalgebra $A^\sigma$ with a descent datum. The two subjects of this article are the fixed subalgebra and the descent to the self-adjoint part: the fixed subalgebra is the ring of invariants, the skew part is a module over it whose products fall back into it, and a module over the involutive algebra with a compatible semilinear involution descends to a module over the fixed subalgebra. This is the commutative case of the picture of Involutive Linear Algebras, specialised and read with the descent, and it is the case that the symmetric algebra of The Symmetric Algebra with an Involution and the polynomial ring exhibit.

The article shows that the fixed subalgebra of an involution of a commutative algebra is a subalgebra, that the algebra decomposes as $A = A^\sigma\oplus A^-$ when $2$ is invertible, and that $A^-$ is an $A^\sigma$-module with $A^-\cdot A^-\subseteq A^\sigma$, so that $A$ is generated over $A^\sigma$ by $A^-$ as a square-zero-compatible extension. When $A^-$ is a free $A^\sigma$-module of rank one, $A = A^\sigma\oplus A^\sigma t$ with $\sigma(t) = -t$ and $t^2 \in A^\sigma$, and $A$ is a quadratic algebra over $A^\sigma$; this is the model for the descent. The article then states the descent: the category of $A$-modules with a $\sigma$-semilinear involution $\tau$ is equivalent to the category of $A^\sigma$-modules, by the functors $M\mapsto H(M) = \{m : \tau(m) = m\}$ and $N\mapsto N\otimes_{A^\sigma}A$, the equivalence holding under the flatness that the quadratic model provides. The worked examples are the polynomial ring with $\sigma(x) = -x$, whose fixed subalgebra is the ring of even polynomials, and the quadratic field extension, which is the classical Galois descent of a vector space with a semilinear involution.

The article assumes Involutive Linear Algebras for the involution of an associative algebra and the symmetric and skew elements, Algebras and Commutative Algebras for the objects, Modules over a Ring and Extension of Scalars for the module theory, Tensor Products of Modules for the tensor product, and Galois Theory for the quadratic extension. The Jordan case is Jordan Algebras with an Involution, the symmetric algebra is The Symmetric Algebra with an Involution, and the real forms are Real Forms and the Descent of an Algebra, in this group. Throughout, $R$ is a commutative ring with identity, $A$ is a commutative unital $R$-algebra, $\sigma$ is an $R$-algebra involution (an automorphism of order two) of $A$, $A^\sigma = A^+$ is the fixed subalgebra, and $A^-$ is the skew part; no form, norm or distance occurs.

Involution of a Commutative Algebra

The Two Notions Coincide

Proposition. Let $A$ be a commutative unital $R$-algebra. A map $\sigma : A\to A$ is an anti-automorphism of order two if and only if it is an automorphism of order two. Hence the involutions of a commutative algebra are exactly its involutive automorphisms, and the distinction of Involutive Linear Algebras between the involution and the involutive automorphism is void.

Proof. In a commutative algebra $ba = ab$, so multiplicativity and anti-multiplicativity are the same condition. $\square$

Definition. An involutive commutative algebra is a commutative unital $R$-algebra $A$ with an $R$-algebra automorphism $\sigma$ of order two; its fixed subalgebra is $A^\sigma = \{a : \sigma(a) = a\}$, its skew part is $A^- = \{a : \sigma(a) = -a\}$, and an element of $A^\sigma$ is symmetric.

The Decomposition

Theorem. If $2$ is invertible in $R$, then

$$ A = A^\sigma \oplus A^- , \qquad \sigma(a^\sigma + a^-) = a^\sigma - a^- , $$

and both summands are $A^\sigma$-submodules of $A$.

Proof. The projections $a\mapsto\tfrac12(a+\sigma(a))$ and $a\mapsto\tfrac12(a-\sigma(a))$ are $A^\sigma$-linear and split the identity; their images are $A^\sigma$ and $A^-$. $\square$

Proposition. The fixed subalgebra is a subalgebra of $A$, and it is generated as an $R$-algebra by the images $\tfrac12(a+\sigma(a))$ of the generators of $A$; the skew part is an $A^\sigma$-module and it is closed under the commutator trivially, while

$$ A^-\cdot A^- \subseteq A^\sigma , \qquad A^\sigma\cdot A^- \subseteq A^- . $$

Proof. Products of fixed elements are fixed and the skew part is carried by the sign rule: $(-1)(-1) = +1$, $(+1)(-1) = -1$; these are the inclusions. $\square$

Corollary (the square of a skew element). For $t \in A^-$ one has $t^2 \in A^\sigma$; the skew part obeys the parity rule $A^-\cdot A^-\subseteq A^\sigma$, and $A$ is generated over $A^\sigma$ by $A^-$ together with $A^\sigma$ itself.

The Quadratic Model

Theorem. Suppose $A = A^\sigma\oplus A^-$ and $A^- = A^\sigma t$ is a free $A^\sigma$-module of rank one with $\sigma(t) = -t$. Then $t^2 \in A^\sigma$, every element of $A$ is uniquely $a^\sigma + b^\sigma t$ with $a^\sigma, b^\sigma \in A^\sigma$, and

$$ (a^\sigma + b^\sigma t)(c^\sigma + d^\sigma t) = (a^\sigma c^\sigma + b^\sigma d^\sigma t^2) + (a^\sigma d^\sigma + b^\sigma c^\sigma)t . $$

Thus $A$ is a quadratic algebra over $A^\sigma$ in the sense of the extension of scalars, and $\sigma$ is the nontrivial automorphism of the quadratic pair; when $A$ is a field, $A/A^\sigma$ is a quadratic extension.

Proof. $t^2$ is fixed by the sign rule $A^-\cdot A^-\subseteq A^\sigma$; the multiplication formula is bilinearity and $t^2 \in A^\sigma$; uniqueness of the coordinates is the directness of the sum. $\square$

The Descent to the Self-Adjoint Part

Semilinear Modules and the Self-Adjoint Part

Definition. Let $(A,\sigma)$ be an involutive commutative algebra. A $\sigma$-semilinear involution of an $A$-module $M$ is an additive map $\tau : M \to M$ with

$$ \tau(am) = \sigma(a)\,\tau(m), \qquad \tau^2 = \mathrm{id} . $$

The self-adjoint part of $(M,\tau)$ is $H(M) = \{m \in M : \tau(m) = m\}$.

Proposition. $H(M)$ is an $A^\sigma$-submodule of $M$, and the map $h\mapsto h$ identifies it with a subset of $M$ on which $A^\sigma$ acts; if $2$ is invertible then $M = H(M)\oplus S(M)$ with $S(M) = \{m : \tau(m) = -m\}$, as $A^\sigma$-modules, and $\tau$ is the multiplication by $-1$ on the skew part.

Proof. If $a \in A^\sigma$ and $h \in H(M)$ then $\tau(ah) = \sigma(a)\tau(h) = ah$, so $H(M)$ is an $A^\sigma$-submodule; the projections $\tfrac12(m\pm\tau(m))$ give the decomposition as in the algebra case. $\square$

The Descent

Theorem (descent). Let $(A,\sigma)$ be an involutive commutative algebra with $A = A^\sigma\oplus A^-$ and $A^-$ generated by a single element $t$ with $\sigma(t) = -t$ and $t^2 \in A^\sigma$ invertible or, more generally, with $A$ faithfully flat over $A^\sigma$. Then the functors

$$ \Phi : M \longmapsto H(M) = \{m : \tau(m) = m\}, \qquad \Psi : N \longmapsto N\otimes_{A^\sigma}A \ \text{ with } \ \tau(n\otimes a) = n\otimes\sigma(a) , $$

are inverse equivalences between the category of $A$-modules with $\sigma$-semilinear involution and the category of $A^\sigma$-modules.

Proof. $\Phi\Psi(N) = H(N\otimes_{A^\sigma}A)$ is $N$ when $A$ is faithfully flat over $A^\sigma$ with the separating element $t$: a self-adjoint element $n\otimes1 + n'\otimes t$ of $N\otimes A$ is fixed precisely when $n'\otimes t = 0$ and equals a pure tensor $n\otimes1$, using $t^2\in A^\sigma$ and flatness. Conversely $\Psi\Phi(M) = H(M)\otimes_{A^\sigma}A\to M$ is the multiplication map, which is an isomorphism because $M = H(M)\oplus S(M)$ and $S(M) = t\,H(M)$ for the quadratic model. The two composites are the identities on objects, and they are natural in the morphisms, which are the $A$-linear maps commuting with $\tau$ and the $A^\sigma$-linear maps respectively. $\square$

Corollary (the classical case). If $A = K$ is a field, a quadratic extension of $F = A^\sigma$ with nontrivial automorphism $\sigma$, then the descent is the classical Galois descent: a $K$-linear space with a $\sigma$-semilinear involution is the scalar extension of its self-adjoint $F$-subspace, and $\dim_F H(M) = \dim_K M$. This is the descent to the self-adjoint part.

Corollary (the fixed subalgebra of a tensor product). For an $A^\sigma$-algebra $B$, the $A$-algebra $B\otimes_{A^\sigma}A$ carries the involution $\sigma_B = \mathrm{id}\otimes\sigma$, and its fixed subalgebra is $B\otimes_{A^\sigma}A^\sigma = B$; the descent recovers $B$.

Examples

Example (the polynomial ring). Let $A = R[x]$ with $\sigma(x) = -x$. Then $A^\sigma$ is the subring $R[x^2]$ of the even polynomials, $A^- = xR[x^2]$, and $A = R[x^2]\oplus xR[x^2]$ with $t = x$ and $t^2 = x^2 \in R[x^2]$. The descent says that a module over $R[x]$ with a semilinear involution is the scalar extension of a module over $R[x^2]$; the involution is the substitution $x\mapsto -x$ and the self-adjoint part is the set of the even power series in the completion. This is the commutative model of Involutions of a Polynomial Ring and the Symmetric Part.

Example (the quadratic field). Let $A = F(\sqrt d)$ with $\sigma(\sqrt d) = -\sqrt d$ and $d \in F$ not a square when $F$ is a field of characteristic not two. Then $A^\sigma = F$, $A^- = F\sqrt d$, $t = \sqrt d$, $t^2 = d$, and the descent is the classical identification of the $F$-spaces with the $A$-spaces with a semilinear involution. The self-adjoint part is the real form of the complex-like structure.

Summary

An involution of a commutative algebra $A$ is the same thing as an involutive automorphism, and its fixed subalgebra $A^\sigma$ is a subalgebra; the algebra splits as $A = A^\sigma\oplus A^-$ with $A^-$ an $A^\sigma$-module satisfying $A^-\cdot A^-\subseteq A^\sigma$, and when $A^- = A^\sigma t$ is free of rank one the algebra is the quadratic algebra $A^\sigma\oplus A^\sigma t$ with $t^2 \in A^\sigma$. A module $M$ over the involutive algebra with a $\sigma$-semilinear involution $\tau$ has a self-adjoint part $H(M)$, an $A^\sigma$-module, and the descent identifies the category of such modules with the category of $A^\sigma$-modules by $M\mapsto H(M)$ and $N\mapsto N\otimes_{A^\sigma}A$; in the quadratic-extension case this is Galois descent, and the dimension is preserved. The polynomial ring with $x\mapsto -x$ and the quadratic field extension are the worked models. No form, norm or order occurs.

Summary of Notation

Symbol	Meaning
$A$	Commutative unital $R$-algebra
$\sigma$	Involutive automorphism of $A$
$A^\sigma = A^+$	Fixed subalgebra
$A^-$	Skew part, $\sigma(a) = -a$
$A = A^\sigma\oplus A^-$	Decomposition when $2$ invertible
$A^-\cdot A^-\subseteq A^\sigma$	Skew elements square into the fixed subalgebra
$A = A^\sigma\oplus A^\sigma t$, $t^2\in A^\sigma$	Quadratic model
$\tau(am) = \sigma(a)\tau(m)$, $\tau^2=\mathrm{id}$	Semilinear involution of a module
$H(M)$	Self-adjoint part, an $A^\sigma$-module
$M\mapsto H(M)$, $N\mapsto N\otimes_{A^\sigma}A$	The descent equivalence