2.1 Monoidal categories

A monoidal category $(\mathbf {C},\otimes ,I)$ (more precisely $(\mathbf {C},\otimes ,I,\alpha ,\lambda ,\rho )$ ) consists of the data of:

• • a category C;

• • a bifunctor $\otimes : \mathbf {C}\times \mathbf {C}\to \mathbf {C}$ (i.e., the expression $x\otimes y$ is functorial in each variable);

• • a distinguished unit object $I\in \mathbf {C}$ ;

• • for each triple $x,y,z\in \mathbf {C}$ , a natural isomorphism $\alpha _{x,y,z}: x\otimes (y\otimes z) \stackrel {\sim }{\longrightarrow } (x\otimes y)\otimes z$ called associator;

• • for each object $x\in \mathbf {C}$ , a natural isomorphism $\lambda _x: I\otimes x\stackrel {\sim }{\longrightarrow } x$ called left unitor; and

• • for each object $x\in \mathbf {C}$ , a natural isomorphism $\rho _x: x\otimes I\stackrel {\sim }{\longrightarrow } x$ called right unitor.

The fact that the product is bifunctorial means that for any morphisms $f:x\to y$ and $g:x'\to y'$ , we get a well-defined $f\otimes g: x\otimes x'\to y\otimes y'$ .

Those various isomorphisms are required to satisfy certain coherence laws relating one another: the triangle diagram

and the pentagon diagram

The associators and unitors are very much part of the data, although in most natural examples they are the “obvious” isomorphisms, and are rarely actually spelled out in practice. Furthermore, MacLane’s coherence theorem [Reference Mac Lane19, VII.2] guarantees that under the above axioms, there is no ambiguity arising if we omit all parentheses and simplify all products with I, as any two identifications between two different expressions using associators and unitors will actually be equal. Therefore, we will use a more relaxed style of notation, and leave all associators/unitors completely in the background. In particular, we allow ourself to write $x^{\otimes n}$ for any object x and any $n\in \mathbb {N}$ (with the convention $x^{\otimes 0} = I$ ). We also often simply say that “ $\mathbf {C}$ is a monoidal category” when the notations are clear.

If $(\mathbf {C},\otimes , I)$ and $(\mathbf {D},\otimes , J)$ are monoidal categories, a (lax) monoidal functor from $\mathbf {C}$ to $\mathbf {D}$ consists of the data of:

• • a functor $F: \mathbf {C}\to \mathbf {D}$ ;

• • a morphism $J\to F(I)$ ; and

• • for each $x,y\in \mathbf {C}$ , a morphism $F(x)\otimes F(y)\to F(x\otimes y)$ .

Those morphisms are once again required to satisfy some coherence laws, which we will not spell out here, in the same spirit as the ones above. When those morphisms are isomorphisms, the monoidal functor is called strong (if they are equalities, it is called strict). Again, the structural morphisms in the definition are often “the obvious ones” and are often left unnamed in practice.

A natural transformation $\varphi : F\to G$ between two monoidal functors $\mathbf {C}\to \mathbf {D}$ is called monoidal if it satisfies the commutative diagrams

and

Note that while being a monoidal category and being a monoidal functor are structures put on top of categories/functors, being a monoidal natural transformation is a property.

Monoidal functors and natural transformations can be composed in the obvious way (the behavior of the structural morphisms in those compositions is straightforward). In particular, we get the expected notions of monoidally isomorphic monoidal functors, and of monoidally equivalent categories (in fact, monoidal categories form a $2$ -category, so all the usual notions can apply without change). The composition of two strong (resp. strict) monoidal functors is again strong (resp. strict).

Given two (small) monoidal categories $\mathbf {C}$ and $\mathbf {D}$ , we define $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ (resp. $\operatorname {\mathrm {LaxHom}}_{\otimes }(\mathbf {C},\mathbf {D})$ ) as the category of strong (resp. lax) monoidal functors between the two, with monoidal natural transformations as morphisms. Note that any $F\in \operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ is canonically isomorphic to some $F'$ such that the structural isomorphism $J\to F'(I)$ is actually the identity, and the structural isomorphisms $F'(I)\otimes F'(x)\to F(I\otimes x)\stackrel {\sim }{\rightarrow } F(x)$ and $F'(x)\otimes F'(I)\to F(x\otimes I)\stackrel {\sim }{\rightarrow } F(x)$ are then identified with the unitors in $\mathbf {D}$ . Therefore, we will usually implicitly restrict to those functors, which brings no noticeable change to the theory (they form a full subcategory of $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ such that the inclusion functor is an equivalence, with a canonical quasi-inverse).

2.2 Symmetric monoidal categories

If $(\mathbf {C},\otimes ,I)$ is a monoidal category, a symmetric structure on $\mathbf {C}$ is the data of an isomorphism natural in x and y

(2.1) $$ \begin{align} s_{x,y}: x\otimes y \stackrel{\sim}{\longrightarrow} y\otimes x, \end{align} $$

for all $x,y\in \mathbf {C}$ , which we call the switching morphism (also called “symmetry isomorphism”), satisfying some coherence axioms so that it is compatible with associators and unitors, and very importantly the involution axiom stating that

(2.2) $$ \begin{align} s_{y,x}\circ s_{x,y} = \operatorname{\mathrm{Id}}_{x\otimes y}, \end{align} $$

in other words, the composition

$$ \begin{align*} x\otimes y\xrightarrow{s_{x,y}} y\otimes x \xrightarrow{s_{y,x}} x\otimes y \end{align*} $$

is the identity of $x\otimes y$ (without this last axiom, the category is only called braided).

In a symmetric monoidal category, for any objects $x_1,\dots ,x_n\in \mathbf {C}$ and any permutation $g\in \mathfrak {S}_n$ , there is a well-defined induced isomorphism

(2.3) $$ \begin{align} g_*: x_1\otimes\dots \otimes x_n \stackrel{\sim}{\longrightarrow} x_{g^{-1}(1)}\otimes\dots\otimes x_{g^{-1}(n)}, \end{align} $$

which can be described by applying a switching morphism for each transposition, and in particular there is a canonical group morphism

(2.4) $$ \begin{align} \mathfrak{S}_n\to \operatorname{\mathrm{Aut}}_{\mathbf{C}}(x^{\otimes n}) \end{align} $$

for each object $x\in \mathbf {C}$ (in a braided category, we get a morphism from the braid group instead, hence the name).

A monoidal functor F between two symmetric monoidal categories $\mathbf {C}$ and $\mathbf {D}$ is called symmetric (this time, it is a property and not a structure) if the following diagram commutes for all $x,y\in \mathbf {C}$ :

(2.5)

A “symmetric monoidal natural transformation” is nothing but a monoidal natural transformation between two symmetric monoidal functors (there is no extra condition involving the symmetric structure). Therefore, we can drop the adjective “symmetric” in that case.

The composition of symmetric monoidal functors is symmetric monoidal (there is again a $2$ -category of symmetric monoidal categories). We adapt the earlier notation, and write $\operatorname {\mathrm {Hom}}_{\otimes }^s(\mathbf {C},\mathbf {D})$ (resp. $\operatorname {\mathrm {LaxHom}}_{\otimes }^s(\mathbf {C},\mathbf {D})$ ) for the full subcategory of $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ (resp. $\operatorname {\mathrm {LaxHom}}_{\otimes }(\mathbf {C},\mathbf {D})$ ) consisting of the symmetric monoidal functors (provided that $\mathbf {C}$ and $\mathbf {D}$ have a symmetric structure, of course).

The set of functions from any set X to some monoid M has an obvious pointwise monoid structure, but if X is a monoid itself, then the subset of monoid morphisms is in general not a submonoid, unless M is commutative. Likewise, for any (small) category $\mathbf {C}$ and any (small) monoidal category $\mathbf {D}$ , the category of functors $\mathrm {Fun}(\mathbf {C},\mathbf {D})$ has a natural “pointwise” monoidal structure, with unit the constant functor to the unit object of $\mathbf {D}$ . If $\mathbf {C}$ is itself monoidal, then there is a natural forgetful functor $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})\to \mathrm {Fun}(\mathbf {C},\mathbf {D})$ , but $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ is not monoidal; however, if $\mathbf {D}$ is symmetric monoidal, then $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ becomes a (symmetric) monoidal category, $\mathrm {Fun}(\mathbf {C},\mathbf {D})$ is naturally symmetric, and $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})\to \mathrm {Fun}(\mathbf {C},\mathbf {D})$ is symmetric monoidal. To be precise, given two monoidal functors F and G in $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ , the functor $F\otimes G: x\mapsto F(x)\otimes G(x)$ is always well defined in $\mathrm {Fun}(\mathbf {C},\mathbf {D})$ (and this does not use that F and G are monoidal), but when $\mathbf {D}$ is symmetric, we give it a monoidal structure by

$$ \begin{align*} (F(x)\otimes G(x))\otimes (F(y)\otimes G(y)) &\stackrel{\sim}{\longrightarrow} (F(x)\otimes F(y))\otimes (G(x)\otimes G(y))\\ &\qquad \to F(x\otimes y)\otimes G(x\otimes y), \end{align*} $$

where we use the symmetric structure in the first arrow. Therefore, we may speak about the monoidal functor $F\otimes G\in \operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ . Note that $\operatorname {\mathrm {Hom}}_{\otimes }^s(\mathbf {C},\mathbf {D})$ is also then a (symmetric) monoidal subcategory of $\operatorname {\mathrm {Hom}}_{\otimes }(\mathbf {C},\mathbf {D})$ .

2.3 Strong symmetry, inverses and torsion

For any monoid M, the set of monoid morphisms $N\to M$ has a special interpretation when $N=\mathbb {N}$ , $\mathbb {Z}$ or $\mathbb {Z}/n\mathbb {Z}$ : it gives, respectively, M, the set $M^\times $ of invertible elements, and the set $M[n]\subset M^\times $ of invertible elements of order dividing n. When M is commutative, those identifications are compatible with the natural monoid structure on the set of morphisms $N\to M$ .

We extend those considerations to monoidal categories; an additional layer of difficulty comes from the fact that one may consider either monoidal or symmetric monoidal functors.

2.3.1 Strong symmetry

If $\mathbf {C}$ is symmetric and we want to characterize similarly $\operatorname {\mathrm {Hom}}_{\otimes }^s(\langle \mathbb {N}\rangle , \mathbf {C})$ , we need a definition.

Definition 2.1 Let $(\mathbf {C},\otimes ,I)$ be a symmetric monoidal category. An object $x\in \mathbf {C}$ is called strongly symmetric if the switching map $s_{x,x}$ is the identity of $x\otimes x$ .

We write $\mathbf {C}^{ss}$ for the full subcategory of $\mathbf {C}$ consisting of the strongly symmetric elements; we say that $\mathbf {C}$ is strongly symmetric if $\mathbf {C}^{ss}=\mathbf {C}$ .

The definition is equivalent to requiring that the canonical group morphism $\mathfrak {S}_n\to \operatorname {\mathrm {Aut}}_{\mathbf {C}}(x^{\otimes n})$ is trivial for $n=2$ (and thus for all n).

Example 2.12 If M is a commutative monoid, then $\langle M\rangle $ is strongly symmetric.

Lemma 2.13 Let $(\mathbf {C},\otimes ,I)$ be a symmetric monoidal category. If $x\in \mathbf {C}$ is isomorphic to a strongly symmetric element, it is strongly symmetric.

Proof Let $f: x\stackrel {\sim }{\longrightarrow } y$ with y strongly symmetric. Since the switching morphism is natural in each variable, the following diagram commutes:

Since $s_{y,y}$ is the identity, so is $s_{x,x}$ .▪

Lemma 2.14 Let M be a commutative monoid, and $\mathbf {C}$ a symmetric monoidal category. If $F\in \operatorname {\mathrm {Hom}}_{\otimes }^s(\langle M\rangle , \mathbf {C})$ , then for any $x\in M$ , the object $F(x)\in \mathbf {C}$ is strongly symmetric.

Proof Since F is symmetric, we get the commutative diagram (2.5) (with $y=x$ ). The vertical arrows are equal isomorphisms because F is strongly symmetric, and the bottom horizontal arrow is the identity because x is strongly symmetric; therefore, the top horizontal arrow is also the identity, which exactly means that $F(x)$ is strongly symmetric.▪

When $M=\mathbb {N}$ , there is a form of converse.

Proposition 2.15 Under the equivalence $\operatorname {\mathrm {Hom}}_{\otimes }(\langle \mathbb {N}\rangle , \mathbf {C})\simeq \mathbf {C}$ , the essential image of the subcategory $\operatorname {\mathrm {Hom}}_{\otimes }^s(\langle \mathbb {N}\rangle , \mathbf {C})$ is exactly $\mathbf {C}^{ss}$ .

Proof If $x\in \mathbf {C}^{ss}$ , then the canonical $\Phi _x: n\mapsto x^{\otimes n}$ is clearly symmetric, so x is in the essential image. Conversely, if x is in the essential image, it is isomorphic to some $F(1)$ , which is strongly symmetric by Lemma 2.14, so x is strongly symmetric by Lemma 2.13.▪

In particular, we see that $\mathbf {C}^{ss}$ is a monoidal subcategory of $\mathbf {C}$ (which can be shown directly).

2.3.2 Inverses

Definition 2.2 Let $\mathbf {C}$ be a monoidal category. We define its category of inverses $\mathbf {C}^\times $ as $\operatorname {\mathrm {Hom}}_{\otimes }(\langle \mathbb {Z}\rangle , \mathbf {C})$ . If $\mathbf {C}$ is symmetric, its category of symmetric inverses $\mathbf {C}^{\times ,s}$ is $\operatorname {\mathrm {Hom}}_{\otimes }^s(\langle \mathbb {Z}\rangle , \mathbf {C})$ .

There is an obvious functor $\mathbf {C}^\times \to \mathbf {C}$ sending F to $F(1)$ . Clearly, if $F\in \mathbf {C}^\times $ , $x=F(1)$ and $\overline {x}=F(-1)$ must be “weak inverses,” in the sense that $x\otimes \overline {x}\simeq \overline {x}\otimes x\simeq I$ . If one unfolds the definition of F, one realizes it is exactly the data of an “adjoint equivalence” $(x,\overline {x},\varphi ,\psi )$ where $\varphi : x\otimes \overline {x}\stackrel {\sim }{\rightarrow } I$ and $\psi : \overline {x}\otimes x\stackrel {\sim }{\rightarrow } I$ fit into some commutative diagrams:

It turns out that it is enough to satisfy one of them [Reference Baez and Lauda3]. These kinds of data are well known in the literature, and can be considered as a sort of “coherent inverse” of x. In particular, it follows immediately that the choice of a functor $\mathbf {C}\to \mathbf {C}^\times $ such that the composition $\mathbf {C}\to \mathbf {C}^\times \to \mathbf {C}$ is isomorphic to the identity (which of course can only exist if all objects in $\mathbf {C}$ are weakly invertible) is equivalent to the choice of a “group structure” (or “gs-category” structure) on $\mathbf {C}$ in the sense of [Reference Laplaza17] (see also [Reference Ulbrich24]). It corresponds to a coherent (functorial) choice of inverses for all objects. In that spirit, $\mathbf {C}^\times $ is a category which embodies all possible such choices, in a canonical way.

Similarly, if $\mathbf {C}$ is symmetric, a symmetric strong monoidal functor $\mathbf {C}\to \mathbf {C}^{\times ,s}$ such that the composition $\mathbf {C}\to \mathbf {C}^{\times ,s}\to \mathbf {C}$ is monoidally isomorphic to the identity is the same as an “abelian gs-category” structure in the sense of [Reference Laplaza17]. Laplaza shows the following in [Reference Laplaza17, Corollary 4.6 and Proposition 4.7].

Theorem 2.16 (Laplaza)

Let $\mathbf {C}$ be a monoidal category with every object weakly invertible. Then if we choose for each $x\in \mathbf {C}$ a weak inverse $\overline {x}$ and an isomorphism $x\otimes \overline {x}\stackrel {\sim }{\rightarrow } I$ , there is a unique way to extend that to a gs-category structure $\mathbf {C}\to \mathbf {C}^\times $ .

Furthermore, it defines an abelian gs-structure $\mathbf {C}\to \mathbf {C}^{\times ,s}$ if and only if $\mathbf {C}$ is strongly symmetric.

2.3.3 Torsion

Definition 2.3 Let $\mathbf {C}$ be a symmetric monoidal category. Then, for any $n\geqslant 2$ , we define the symmetric monoidal category $\mathbf {C}[n]$ as $\operatorname {\mathrm {Hom}}_{\otimes }^s(\langle \mathbb {Z}/n\mathbb {Z}\rangle , \mathbf {C})$ . We call it the n-torsion category of $\mathbf {C}$ .

Of course, we could have looked at the intermediary step where we only consider $\operatorname {\mathrm {Hom}}_{\otimes }(\langle \mathbb {Z}/n\mathbb {Z}\rangle , \mathbf {C})$ for any monoidal $\mathbf {C}$ , but we will not need it, and already in the case of ordinary groups, the notion of n-torsion is much better behaved in a commutative context.

To any $F\in \mathbf {C}[n]$ is associated a choice of isomorphism $\varphi _x: x^{\otimes n}\stackrel {\sim }{\rightarrow } I$ satisfying some coherence conditions, where $x=F(1)$ . In particular, x must have weak n-torsion, meaning that $x^{\otimes n}\simeq I$ .

As before, there is a natural symmetric strong monoidal functor $\mathbf {C}[n]\to \mathbf {C}$ , and we are interested in the choice of a symmetric strong monoidal functor $\mathbf {C}\to \mathbf {C}[n]$ such that $\mathbf {C}\to \mathbf {C}[n]\to \mathbf {C}$ is monoidally isomorphic to the identity. We call such a choice a “coherent n-torsion” structure on $\mathbf {C}$ . Clearly, it can only exist if $\mathbf {C}$ is strongly symmetric and all elements have weak n-torsion. Unfortunately, this is no longer sufficient, but we do have the following proposition.

Proposition 2.17 Let $\mathbf {C}$ be a symmetric monoidal category. If $x\in \mathbf {C}$ is strongly symmetric, any choice of isomorphism $\varphi _x: x^{\otimes n}\stackrel {\sim }{\rightarrow } I$ extends to a unique $F\in \mathbf {C}[n]$ .

Furthermore, if $\mathbf {C}$ is strongly symmetric, a coherent n-torsion structure is equivalent to the choice of such a $\varphi _x$ for all $x\in \mathbf {C}$ , such that for all $x,y\in \mathbf {C}$ , we have a commutative diagram

(2.6)

and for each morphism $f:x\to y$ in $\mathbf {C}$ :

(2.7)

Proof It is easy to see that the extension of $\varphi _x$ to F is necessarily unique (up to isomorphism of course), and that it exists exactly when we have the commutative diagram

where the morphism in the top row is given by the identification of $x^{\otimes n}\otimes x$ and $x\otimes x^{\otimes n}$ with $x^{\otimes n+1}$ . However, up to the action of the permutation group $\mathfrak {S}_{n+1}$ on the top row, that diagram always commutes, and since x is strongly symmetric, that action is trivial.

It is clear that the two diagrams in the statement are necessary to get a monoidal functor $\mathbf {C}\to \mathbf {C}[n]$ . Conversely, if they hold, and if we send each $x\in \mathbf {C}$ to the unique $F_x\in \mathbf {C}[n]$ extending $\varphi _x$ , to get a functor $\mathbf {C}\to \mathbf {C}[n]$ , we need that each $f:x\to y$ induces a monoidal transformation $F_x\to F_y$ , and since the composition $\mathbf {C}\to \mathbf {C}[n]\to \mathbf {C}$ should be isomorphic to the identity, the component $F_x(1)\to F_y(1)$ has to correspond to f, so it is entirely determined, and is only well defined if the diagram (2.7) holds. It is straightforward that the diagram (2.6) will guarantee that the functor $\mathbf {C}\to \mathbf {C}[n]$ is monoidal.▪

Note that if $\mathbf {C}$ is a groupoid (so all morphisms are invertible), then it is enough to fix one $\varphi _x$ to determine the whole structure (if it exists).

2.4 Lax monoidal functors and graded rings

Up until now, we have only considered strong monoidal functors, but lax monoidal functors are also key to our central construction. Recall the definition of graded algebras.

In the literature, the most common cases are $M=\mathbb {N}$ and $M=\mathbb {Z}/2\mathbb {Z}$ . Note that if A is a graded algebra, $A_0$ is always a ring, and if A is commutative, then A is a graded $A_0$ -algebra. There is an obvious notion of graded module/algebra morphisms, which are just morphisms preserving the decomposition, and therefore we get categories $R\text {-}\mathbf {Mod}_M$ , $R\text {-}\mathbf {Alg}_M$ , and $R\text {-}\mathbf {CommAlg}_M$ . These categories are actually themselves naturally monoidal, using the tensor product

$$ \begin{align*} (A\otimes B)_x = \bigoplus_{y+z=x} A_y\otimes_R B_z, \end{align*} $$

and they have a natural symmetric structure which consists in applying the switching morphism to each $A_y\otimes _R B_z$ .

Then we can make the following fundamental observation: let M and N be commutative monoids; to any lax monoidal functor $F: \langle M\rangle \to R\text {-}\mathbf {Mod}_N$ , we can associate the R-module $A_F = \bigoplus _{x\in M} F(x)$ , which is an $(M\times N)$ -graded module (since each $F(x)$ is itself N-graded). Then we can define an R-bilinear product $A_F\otimes _R A_F\to A_F$ using the maps $F(x)\otimes F(y)\to F(x+y)$ corresponding to the monoidal structure on F.

Proposition 2.19 Let M be a monoid. Then, for any lax monoidal functor $F: \langle M\rangle \to R\text {-}\mathbf {Mod}_N$ , the associated graded object $A_F$ is an $(M\times N)$ -graded R-algebra, and the association $F\mapsto A_F$ defines an equivalence of categories

$$ \begin{align*} \operatorname{\mathrm{LaxHom}}_\otimes(\langle M\rangle,R\text{-}\mathbf{Mod}_N) \stackrel{\sim}{\longrightarrow} R\text{-}\mathbf{Alg}_{M\times N}. \end{align*} $$

This also induces an equivalence

$$ \begin{align*} \operatorname{\mathrm{LaxHom}}_\otimes^s(\langle M\rangle,R\text{-}\mathbf{Mod}_N) \stackrel{\sim}{\longrightarrow} R\text{-}\mathbf{CommAlg}_{M\times N}. \end{align*} $$

This gives us the following construction: if $\mathbf {C}$ is any monoidal category, and $K: \mathbf {C}\to R\text {-}\mathbf {Mod}_N$ is a lax monoidal functor, then the composition with K defines a functor

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\otimes}(\langle M\rangle, \mathbf{C}) \xrightarrow{K\circ - } \operatorname{\mathrm{LaxHom}}_\otimes(\langle M\rangle, R\text{-}\mathbf{Mod}_N) \simeq R\text{-}\mathbf{Alg}_{M\times N}, \end{align*} $$

and when K is symmetric, we get

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\otimes}^s(\langle M\rangle, \mathbf{C}) \xrightarrow{K\circ - } \operatorname{\mathrm{LaxHom}}_\otimes^s(\langle M\rangle, R\text{-}\mathbf{Mod}_N) \simeq R\text{-}\mathbf{CommAlg}_{M\times N}. \end{align*} $$

Actually, we get something a little better: the objects we get are graded $K(I)$ -algebras, since any commutative graded algebra A is in fact an $A_0$ -algebra. Let us emphasize what we really want to use in the end.

Corollary 2.20 Let $\mathbf {C}$ be a symmetric monoidal category, let N be a commutative monoid, and let $K: \mathbf {C}\to \mathbb {Z}\text {-}\mathbf {Mod}_N$ be a symmetric lax monoidal functor. Then, if $\mathbf {C}$ has an abelian gs-structure, composition with K induces a functor

$$ \begin{align*} \mathbf{C}\to K(I)\text{-}\mathbf{CommAlg}_{\mathbb{Z}\times N}, \end{align*} $$

and if $\mathbf {C}$ has a coherent n-torsion structure, it induces a functor

$$ \begin{align*} \mathbf{C}\to K(I)\text{-}\mathbf{CommAlg}_{\mathbb{Z}/n\mathbb{Z} \times N}. \end{align*} $$

Proof The discussion just above shows that from K we get functors

$$ \begin{align*} \mathbf{C}^{\times,s}\longrightarrow K(I)\text{-}\mathbf{CommAlg}_{\mathbb{Z}\times N},\, \mathbf{C}[n]\longrightarrow K(I)\text{-}\mathbf{CommAlg}_{\mathbb{Z}/n\mathbb{Z} \times N}, \end{align*} $$

simply using $M=\mathbb {Z}$ and $M=\mathbb {Z}/n\mathbb {Z}$ . Now, we just compose with the structural functors $\mathbf {C}\to \mathbf {C}^{\times , s}$ and $\mathbf {C}\to \mathbf {C}[n]$ , respectively.▪

3.1 Hermitian modules and involutions

We start, for the reader’s convenience as well as for establishing notations, by reviewing basic facts about hermitian modules (see [Reference Knus, Merkurjev, Rost and Tignol15] for a reference, or [Reference Knus14] for an account over general rings with involution).

3.1.1 Morita equivalence

Let A and B be central simple algebras over K. We say that a B–A-bimodule V is a Morita equivalence if the following equivalent conditions hold (see [Reference Knus, Merkurjev, Rost and Tignol15, 1.10]):

• • the left action of B gives an isomorphism $B\simeq \operatorname {\mathrm {End}}_A(V)$ and

• • the right action of A gives an isomorphism $A\simeq \operatorname {\mathrm {End}}_B(V)$ .

We use the notation $B\stackrel {V}{\rightsquigarrow } A$ to state that V is such a bimodule. Then A and B are Brauer-equivalent iff there exists a Morita equivalence, which is then unique up to isomorphism.

We can always consider A as a tautological A–A-bimodule, and it defines a Morita equivalence. We will often write $|A|$ when we consider A as a vector space or a module, so we can write $A\stackrel {|A|}{\rightsquigarrow } A$ .

Example 3.1 It is a defining property of Azumaya algebras that the natural “sandwich” map

(3.1) $$ \begin{align} \begin{array}{rcl} A\otimes_K A^{op} & \longrightarrow & \operatorname{\mathrm{End}}_K(|A|) \\ a\otimes b & \longmapsto & (x\mapsto axb) \end{array} \end{align} $$

is a K-algebra isomorphism, so we get $(A\otimes _K A^{op})\stackrel {|A|}{\rightsquigarrow } K$ .

3.1.2 Hermitian forms

Recall that if $(A,\sigma )$ is an algebra with involution over $(K,\iota )$ and $\varepsilon \in U(K,\iota )$ , an $\varepsilon $ -hermitian module $(V,h)$ over $(A,\sigma )$ is a (right) A-module V equipped with an $\varepsilon $ -hermitian form h, meaning that

$$ \begin{align*} h: V\times V\to A \end{align*} $$

is bi-additive and satisfies for all $x,y\in V$ and all $a,b\in A$ :

$$ \begin{align*} h(xa,yb) &= \sigma(a)h(x,y)b, \\ h(y,x) &= \varepsilon \sigma(h(x,y)). \end{align*} $$

We always assume that $\varepsilon $ -hermitian forms are regular, meaning that the induced map $V\to \operatorname {\mathrm {Hom}}_A(V,A)$ given by $x\mapsto h(x,-)$ is bijective. An isometry between two $\varepsilon $ -hermitian modules is a module isomorphism which preserves the hermitian forms. We call $\varepsilon $ the sign of h, and use the notation $\varepsilon _h$ to refer to it when it is not already introduced.

Example 3.2 An $\varepsilon $ -hermitian module over $(K,\operatorname {\mathrm {Id}})$ is simply an $\varepsilon $ -symmetric bilinear module over K (and in that case $\varepsilon =\pm 1$ ).

Let $(A,\sigma )$ be an algebra with involution over K, and let $a\in \operatorname {\mathrm {Sym}}^\varepsilon (A^\times ,\sigma )$ (for instance, $a\in K^\times $ and $\varepsilon =1$ ). Then we define an $\varepsilon $ -hermitian form over $|A|$ , by:

$$ \begin{align*} \begin{array}{rrcl} \langle a\rangle_\sigma : & |A|\times |A| & \longrightarrow & A \\ & (x,y) & \longmapsto & \sigma(x)ay. \end{array} \end{align*} $$

We call such a form elementary diagonal. We will write $\langle a_1,\dots ,a_n\rangle _\sigma $ for an orthogonal sum $\langle a_1\rangle _\sigma \perp \dots \perp \langle a_n\rangle _\sigma $ (where all $a_i$ are $\varepsilon $ -symmetric), and call such a form diagonal.

Remark 3.3 Not every form is diagonalizable. In fact, for $(V,h)$ to be diagonalizable, it is clearly necessary that V is a free A-module (which is in this case equivalent to $\dim _K(V)$ being a multiple of $\dim _K(A)$ ). It turns out that this condition is actually sufficient, except in the special case where $(A,\sigma )=(K,\operatorname {\mathrm {Id}})$ and $\varepsilon =-1$ , which is the only case where $h(x,x)=0$ for all $x\in V$ .

3.1.3 Hermitian Morita equivalences

If $(V,h)$ is an $\varepsilon $ -hermitian module over $(A,\sigma )$ , then the adjoint involution $\sigma _h$ on $\operatorname {\mathrm {End}}_A(V)$ is defined (see [Reference Knus, Merkurjev, Rost and Tignol15, 4.1]) by the fact that for all $x,y\in V$ and all $f\in \operatorname {\mathrm {End}}_A(V)$ ,

$$ \begin{align*} h(f(x),y) = h(x,\sigma_h(f)(y)). \end{align*} $$

If $(A,\sigma )$ and $(B,\tau )$ are algebras with involution over $(K,\iota )$ , we say that $(V,h)$ is an $\varepsilon $ -hermitian Morita equivalence between $(B,\tau )$ and $(A,\sigma )$ , which we write

$$ \begin{align*} (B,\tau) \stackrel{(V,h)}{\rightsquigarrow} (A,\sigma), \end{align*} $$

if $B\stackrel {V}{\rightsquigarrow } A$ , h is an $\varepsilon $ -hermitian form over $(A,\sigma )$ on V, and $\tau $ corresponds to $\sigma _h$ through the natural isomorphism $B\simeq \operatorname {\mathrm {End}}_A(V)$ . In particular, any $\varepsilon $ -hermitian module $(V,h)$ over $(A,\sigma )$ defines an equivalence $(\operatorname {\mathrm {End}}_A(V),\sigma _h) \stackrel {(V,h)}{\rightsquigarrow } (A,\sigma )$ . An isomorphism of $\varepsilon $ -hermitian Morita equivalences is a bimodule isomorphism which is also an isometry.

If $B\stackrel {V}{\rightsquigarrow } A$ , then there always exists an $\varepsilon $ -hermitian form h on V such that $(B,\tau ) \stackrel {(V,h)}{\rightsquigarrow } (A,\sigma )$ (see [Reference Knus, Merkurjev, Rost and Tignol15, 4.2]). When the involutions are unitary (so $\iota \neq \operatorname {\mathrm {Id}}$ ), we can take any $\varepsilon \in U(K,\iota )$ ; when the involutions are of the first kind (so $\iota =\operatorname {\mathrm {Id}}$ ), then we must take $\varepsilon =1$ if $\sigma $ and $\tau $ have the same type (orthogonal or symplectic), and $\varepsilon =-1$ otherwise. Moreover, in any case, if h and $h'$ are two possible choices, then there is $\lambda \in K^\times $ such that $h'=\langle \lambda \rangle h$ .

Example 3.4 If $a\in \operatorname {\mathrm {Sym}}^\varepsilon (A^\times , \sigma )$ , then $(A,\sigma _a)\stackrel {(|A|,\langle a\rangle _\sigma )}{\rightsquigarrow } (A,\sigma )$ , where $\sigma _a(x) = a^{-1}\sigma (x)a$ . In particular, if $a\in K^\times $ , then $\sigma _a=\sigma $ .

Example 3.5 Using the involution $\sigma $ on A, we can twist the usual sandwich map (3.1) to

(3.2) $$ \begin{align} \begin{array}{rcl} A\otimes_K A^{\iota} & \longrightarrow & \operatorname{\mathrm{End}}_K(|A|) \\ a\otimes b & \longmapsto & (x\mapsto ax\sigma(b)), \end{array} \end{align} $$

where $A^{\iota }$ is the same ring as A, but with the twisted K-algebra structure $K\xrightarrow {\iota } K\to A$ .

We call this action of $A\otimes _K A^{\iota }$ on $|A|$ the “twisted sandwich action” (relative to $\sigma $ ). We will write $|A|_\sigma $ instead of $|A|$ when we see it as a left $A\otimes _K A^{\iota }$ -module with this action. It is shown in [Reference Knus, Merkurjev, Rost and Tignol15, 11.1] that the so-called involution trace form

(3.3) $$ \begin{align} \begin{array}{rrcl} T_\sigma : & |A|\times |A| & \longrightarrow & K \\ & (x,y) & \longmapsto & \operatorname{\mathrm{Trd}}_A(\sigma(x)y) \end{array} \end{align} $$

is a hermitian form over $(K,\iota )$ on the K-vector space $|A|$ , such that

(3.4) $$ \begin{align} (A\otimes_K A^{\iota},\sigma\otimes \sigma) \stackrel{(|A|_\sigma,T_\sigma)}{\rightsquigarrow} (K,\iota). \end{align} $$

3.1.4 Tensor products

If $(V_i,h_i)$ is an $\varepsilon _i$ -hermitian module over an algebra with involution $(A_i,\sigma _i)$ , for $i=1,2$ , then $(V_1\otimes _K V_2,h_1\otimes h_2)$ is an $\varepsilon _1\varepsilon _2$ -hermitian module over ( $A_1\otimes _K A_2$ , $\sigma _1\otimes \sigma _2$ ). In particular, $\langle a\rangle _{\sigma _1}\otimes \langle b\rangle _{\sigma _2}\simeq \langle a\otimes b\rangle _{\sigma _1\otimes \sigma _2}$ .

If, for $i=1,2$ , we have $(B_i,\tau _i) \stackrel {(V_i,h_i)}{\rightsquigarrow } (A_i,\sigma _i)$ , then

$$ \begin{align*} (B_1\otimes_K B_2,\tau_1\otimes\tau_2) \stackrel{(V_1\otimes_K V_2,h_1\otimes h_2)}{\rightsquigarrow} (A_1\otimes_K A_2,\sigma_1\otimes\sigma_2). \end{align*} $$

3.1.5 Conjugate form

If $(B,\tau ) \stackrel {(V,h)}{\rightsquigarrow } (A,\sigma )$ , then we define $(\overline {V},\overline {h})$ , where $\overline {V}$ is the A–B-bimodule defined as V as a K-vector space, with the action

$$ \begin{align*} a\cdot v\cdot b = \tau(b)\cdot v\cdot \sigma(a), \end{align*} $$

and $\overline {h}: \overline {V}\times \overline {V}\to B$ is characterized by

$$ \begin{align*} \overline{h}(x,y)z = xh(y,z), \end{align*} $$

for all $x,y,z\in V$ .

Then it is easy to see that $(A,\sigma ) \stackrel {(\overline {V},\overline {h})}{\rightsquigarrow } (B,\tau )$ .

3.2 The category $\mathbf {Br}_h(K,\iota )$

We now define a category $\mathbf {Br}_h(K,\iota )$ , which we call the hermitian Brauer 2-group of $(K,\iota )$ , such that:

• • the objects are the algebras with involution $(A,\sigma )$ over $(K,\iota )$ and

• • the morphisms from $(B,\tau )$ to $(A,\sigma )$ are the isomorphism classes of $\varepsilon $ -hermitian Morita equivalences between $(B,\tau ) \stackrel {(V,h)}{\rightsquigarrow } (A,\sigma )$ .

We will usually identify an $\varepsilon $ -hermitian bimodule and its isomorphism class when no confusion is possible. If $f:(B,\tau )\to (A,\sigma )$ is a morphism in $\mathbf {Br}_h(K,\iota )$ , we will sometimes write $(V_f,h_f)$ for the corresponding $\varepsilon $ -hermitian bimodule. We usually write $\mathbf {Br}_h(K)$ for $\mathbf {Br}_h(K,\operatorname {\mathrm {Id}})$ .

To properly define $\mathbf {Br}_h(K,\iota )$ as a category, we need to specify how to compose morphisms. Given two morphisms $(C,\theta )\stackrel {(U,h)}{\rightsquigarrow } (B,\tau )$ and $(B,\tau ) \stackrel {(V,h')}{\rightsquigarrow } (A,\sigma )$ , we define a C–A-bimodule $V\circ U = U\otimes _B V$ , and a map $h'\circ h: (V\circ U)\times (V\circ U) \to A$ by

(3.5) $$ \begin{align} (h'\circ h)(u\otimes v, u'\otimes v') = h'(v, h(u,u')v'). \end{align} $$

Proposition 3.6 In the situation just above, if h and $h'$ are, respectively, $\varepsilon $ and $\varepsilon '$ -hermitian, then $h'\circ h$ is an $\varepsilon \varepsilon '$ -hermitian form over $(A,\sigma )$ , which defines an equivalence

$$ \begin{align*} (C,\theta) \stackrel{(V\circ U,h'\circ h)}{\rightsquigarrow} (A,\sigma) \end{align*} $$

whose isomorphism class depends only on the isomorphism classes of $(U,h)$ and $(V,h')$ .

Example 3.7 Let $f: (B,\tau )\stackrel {(V,h)}{\rightsquigarrow } (A,\sigma )$ be a morphism in $\mathbf {Br}_h(K,\iota )$ , and let $b\in \operatorname {\mathrm {Sym}}^\varepsilon (B^\times , \tau )$ and $a\in \operatorname {\mathrm {Sym}}^{\varepsilon '}(A^\times , \sigma )$ . Then the underlying space of $\langle a\rangle _\sigma \circ f\circ \langle b\rangle _\tau $ is $|B|\otimes _B V\otimes _A |A|$ , which is canonically identified with V, and the $\varepsilon \varepsilon _f\varepsilon '$ -hermitian form is:

$$ \begin{align*} \begin{array}{rcl} V\times V & \longrightarrow & A \\ (x,y) & \longmapsto & h(xa,by). \end{array} \end{align*} $$

In particular, $f\circ \langle 1\rangle _\tau = \langle 1\rangle _\sigma \circ f = f$ .

We can then state the following proposition.

3.3 Automorphisms and isometries

We defined a category $\mathbf {Br}_h(K,\iota )$ in which the (iso)morphisms between two algebras with involution correspond to hermitian Morita equivalences. However, obviously, there is a more elementary notion of isomorphism between algebras with involution. To be precise, we may define a category $\mathbf {AlgInv}(K,\iota )$ , with the same objects as $\mathbf {Br}_h(K,\iota )$ , but where the morphisms are K-algebra isomorphisms which are compatible with the involutions.

Then, to any algebra isomorphism $\varphi : (B,\tau )\to (A,\sigma )$ , we can associate $(B,\tau ) \stackrel {(|A|_\varphi ,\langle 1\rangle _\sigma )}{\rightsquigarrow } (A,\sigma )$ where $|A|_\varphi =|A|$ as a right A-module, and the left action of B is given by $b\cdot a = \varphi (b)a$ for any $b\in B$ and $a\in |A|$ . It is easily seen that this is indeed a hermitian Morita equivalence, and that this defines a functor

$$ \begin{align*} \Theta: \mathbf{AlgInv}(K,\iota)\to \mathbf{Br}_h(K,\iota), \end{align*} $$

which is the identity on objects.

Obviously, this functor is far from being full, since there are isomorphisms in $\mathbf {Br}_h(K,\iota )$ between $(B,\tau )$ and $(A,\sigma )$ whenever A and B are Brauer-equivalent. On the other hand, we can look at the situation where $(A,\sigma )=(B,\tau )$ . We call

$$ \begin{align*} \operatorname{\mathrm{Aut}}_{\mathcal{M}}(A,\sigma) := \operatorname{\mathrm{Aut}}_{\mathbf{Br}_h(K,\iota)}(A,\sigma) \end{align*} $$

the group of Morita automorphisms of $(A,\sigma )$ , and

$$ \begin{align*} \operatorname{\mathrm{Aut}}_K(A,\sigma) := \operatorname{\mathrm{Aut}}_{\mathbf{AlgInv}(K,\iota)}(A,\sigma) \end{align*} $$

the group of algebraic automorphisms of $(A,\sigma )$ . By extension, we say that a morphism in $\mathbf {Br}_h(K,\iota )$ is algebraic if it is in the image of $\Theta $ . Then we have a group morphism

$$ \begin{align*} \Theta: \operatorname{\mathrm{Aut}}_K(A,\sigma) \longrightarrow \operatorname{\mathrm{Aut}}_{\mathcal{M}}(A,\sigma), \end{align*} $$

and we want to understand the groups involved, as well as the kernel and cokernel.

Recall from [Reference Knus, Merkurjev, Rost and Tignol15, 12.14] that the subgroup $\operatorname {\mathrm {Sim}}(A,\sigma )\subset A^\times $ of similitudes of $(A,\sigma )$ is defined as

$$ \begin{align*} \operatorname{\mathrm{Sim}}(A,\sigma) = \{ a\in A\,|\, a\sigma(a) \in k^\times\}. \end{align*} $$

When $a\in \operatorname {\mathrm {Sim}}(A,\sigma )$ , we can define its multiplier $\mu (a)=a\sigma (a)=\sigma (a)a\in k^\times $ , and $\mu $ is a group morphism $\operatorname {\mathrm {Sim}}(A,\sigma )\to k^\times $ . Then $\operatorname {\mathrm {Ker}}(\mu )$ is the subgroup $\operatorname {\mathrm {Iso}}(A,\sigma )$ of isometries of $(A,\sigma )$ , and $\operatorname {\mathrm {Im}}(\mu )=G(A,\sigma )$ is the group of multipliers of $(A,\sigma )$ . The following proposition gives a complete picture of the situation.

Proposition 3.9 Let $(A,\sigma )$ be an algebra with involution over $(K,\iota )$ . We write $N: K^\times \to k^\times $ for the group morphism $x\mapsto x\iota (x)$ (when $\iota \neq \operatorname {\mathrm {Id}}$ , this is the Galois norm of $K/k$ ), and $\operatorname {\mathrm {int}}(a)\in \operatorname {\mathrm {Aut}}_K(A)$ for the inner automorphism of A induced by some $a\in A^\times $ .

Then we have a commutative diagram of groups with exact rows and columns:

In particular, there is a canonical isomorphism $\operatorname {\mathrm {Aut}}_{\mathcal {M}}(A,\sigma )\simeq K^\times /N(K^\times )$ , and when $\iota =\operatorname {\mathrm {Id}}$ , this gives $\operatorname {\mathrm {Aut}}_{\mathcal {M}}(A,\sigma )\simeq K^\times /K^{\times 2}$ .

Proof To prove that the diagram commutes, the only nontrivial thing to show is that for any $a\in \operatorname {\mathrm {Sim}}(A,\sigma )$ , $\Theta (\operatorname {\mathrm {int}}(a)) \simeq (|A|, \langle \mu (a)\rangle _\sigma )$ . By definition, $\Theta (\operatorname {\mathrm {int}}(a))$ corresponds to the hermitian A–A-bimodule $(|A|_a,\langle 1\rangle _\sigma )$ , where $|A|_a=|A|$ as a right A-module, and the action on the left is given by $x\cdot y = axa^{-1}y$ for all $x\in A$ , $y\in |A|_a$ . Then we easily see that

$$ \begin{align*} \begin{array}{rrcl} f : & |A|_a & \longrightarrow & |A| \\ & y & \longmapsto & a^{-1}y \end{array} \end{align*} $$

is a bimodule isomorphism. Since

$$ \begin{align*} \langle \mu(a)\rangle_\sigma(f(x),f(y)) = \mu(a)\sigma(f(x))f(y) = \mu(a)\sigma(x)\sigma(a)^{-1}a^{-1}y=\sigma(x)y, \end{align*} $$

the map f induces an isometry from $\langle 1\rangle _\sigma $ to $\langle \mu (a)\rangle _\sigma $ , which shows that the diagram commutes.

The exactness of each row is clear. The exactness of the second column is a consequence of [Reference Knus, Merkurjev, Rost and Tignol15, 12.15]. For the third column, the surjectivity comes from the fact that any automorphism of $(A,\sigma )$ in $\mathbf {Br}_h(K,\iota )$ has the form $(|A|,h)$ , and since $h=\langle 1\rangle _\sigma $ is a possible choice, any other choice must have the form $h=\langle \lambda \rangle _\sigma $ . To prove exactness at $K^\times $ , we must also show that $(|A|,\langle \lambda \rangle _\sigma )$ and $(|A|,\langle 1\rangle _\sigma )$ are isomorphic iff $\lambda \in N(K^\times )$ . However, any bimodule isomorphism $f:|A|\to |A|$ has the form $f(x)=ax$ for some $a\in K^\times $ , so if it induces an isometry from $\langle \lambda \rangle _\sigma $ to $\langle 1\rangle _\sigma $ , we must have $\lambda =a\iota (a)$ .

For the first and fourth columns, the only thing to show is the surjectivity, but it is easily obtained by a diagram chase now that we have shown exactness everywhere else.▪

3.4 Monoidal structure and the Goldman element

The tensor product of algebras with involution endows $\mathbf {AlgInv}(K,\iota )$ with a natural symmetric monoidal structure, with unit object $(K,\iota )$ , such that the obvious forgetful functor $\mathbf {AlgInv}(K,\iota )\to K\text {-}\mathbf {Alg}$ (which is faithful but not full) is symmetric strong monoidal.

Likewise, the tensor product of algebras with involution (and of hermitian modules, for the morphisms) defines a monoidal structure on $\mathbf {Br}_h(K,\iota )$ , with unit object $(K,\iota )$ , such that $\Theta : \mathbf {AlgInv}(K,\iota )\to \mathbf {Br}_h(K,\iota )$ is a strict monoidal functor. We can also transport the symmetric structure.

Proof If we want $\Theta $ to be symmetric, we need to define the switching map in $\mathbf {Br}_h(K,\iota )$ as

$$ \begin{align*} (A,\sigma)\otimes_K (B,\tau) \xrightarrow{\Theta(s)} (B,\tau)\otimes_K (A,\sigma), \end{align*} $$

where $s: (A,\sigma )\otimes _K (B,\tau )\to (B,\tau )\otimes _K (A,\sigma )$ is the switching map in $\mathbf {AlgInv}(K,\iota )$ .

Since the switching maps, the associators, and the unit maps are all images of morphisms in $\mathbf {AlgInv}(K,\iota )$ by $\Theta $ , all the coherence axioms can be verified in $\mathbf {AlgInv}(K,\iota )$ .▪

In particular, this means that for any algebra with involution $(A,\sigma )$ over $(K,\iota )$ and any $n\in \mathbb {N}$ , there are natural group morphisms

A remarkable feature of Azumaya algebras is the existence of the so-called Goldman element (see [Reference Knus, Merkurjev, Rost and Tignol15, 3.A]).

It is shown in [Reference Knus, Merkurjev, Rost and Tignol15, 10.1] that the natural morphism $\mathfrak {S}_n\to \operatorname {\mathrm {Aut}}_K(A^{\otimes n})$ admits a lift

where $\mathfrak {S}_n\to (A^{\otimes n})^\times $ is uniquely characterized by

$$ \begin{align*} (i, i+1) \mapsto 1\otimes\cdots \otimes g_A \otimes 1\otimes \dots \otimes 1 \end{align*} $$

(with $g_A$ occupying the i and $i+1$ slots).

Proposition 3.12 Let $(A,\sigma )$ be an algebra with involution over $(K,\iota )$ . Then the natural morphism $\mathfrak {S}_n\to (A^{\otimes n})^\times $ gives a commutative diagram of groups:

The crucial consequence is the following.

3.5 Inverses and coherent $2$ -torsion

For any small category C, let us write $\pi _0(C)$ for its set of isomorphism classes. When C is monoidal, $\pi _0(C)$ inherits a natural monoid structure, which is commutative when $\mathbf {C}$ is symmetric. As we have already stated, there is a morphism in $\mathbf {Br}_h(K,\iota )$ between $(B,\tau )$ and $(A,\sigma )$ if and only if A and B are Brauer-equivalent, which means that we get a canonical embedding $\pi _0(\mathbf {Br}_h(K,\iota ))\to \operatorname {\mathrm {Br}}(K)$ . In particular, all objects in $\mathbf {Br}_h(K,\iota )$ are weakly invertible. When $\iota \neq \operatorname {\mathrm {Id}}$ , $\pi _0(\mathbf {Br}_h(K,\iota ))$ is identified with the kernel of the norm map $\operatorname {\mathrm {Br}}(K)\to \operatorname {\mathrm {Br}}(k)$ ; when $\iota =\operatorname {\mathrm {Id}}$ , $\pi _0(\mathbf {Br}_h(K))$ is identified with the $2$ -torsion subgroup $\operatorname {\mathrm {Br}}(K)[2]$ (in particular, every object has weak $2$ -torsion).

In fact, Example 3.5 gives a more precise statement about invertibility and torsion, as it gives an explicit weak inverse of each $(A,\sigma )$ , namely $(A^{\iota },\sigma )$ , which is indeed equal to $(A,\sigma )$ when $\iota =\operatorname {\mathrm {Id}}$ , and an explicit equivalence from their product to the unit object $(K,\iota )$ , given by the involution trace form $(|A|_\sigma ,T_\sigma )$ .

According to Theorem 2.16, since $\mathbf {Br}_h(K,\iota )$ is strongly symmetric by Corollary 3.13, the data of all those equivalences define a canonical abelian gs-structure $\mathbf {Br}_h(K,\iota ) \to \mathbf {Br}_h(K,\iota )^{\times ,s}$ . This is always the one we implicitly refer to when we mention such a structure.

When $\iota =\operatorname {\mathrm {Id}}$ , we can hope to give $\mathbf {Br}_h(K,\iota )$ a coherent $2$ -torsion structure since all objects have weak $2$ -torsion, and indeed:

Proof According to Proposition 2.17, we have to verify the two diagrams (2.6) and (2.7). The first one is straightforward, using the fact that the reduced trace $\operatorname {\mathrm {Trd}}_{A\otimes _K B}$ of $A\otimes _K B$ is nothing but $\operatorname {\mathrm {Trd}}_A\otimes \operatorname {\mathrm {Trd}}_B$ .

The second one requires more work; it states that for any $f:(B,\tau )\stackrel {(V,h)}{\rightsquigarrow } (A,\sigma )$ in $\mathbf {Br}_h(K)$ , the diagram

commutes. We define the following $(B\otimes _K B)$ –K-bimodule morphism:

$$ \begin{align*} \begin{array}{rrcl} \psi : & (V\otimes_K V)\otimes_{A\otimes_K A}A & \longrightarrow & B \\ & {(v\otimes w)\otimes a} & \longmapsto &{\varphi_h(va\otimes w),} \end{array} \end{align*} $$

where $\varphi _h: V\otimes _K V\to B$ (see [Reference Knus, Merkurjev, Rost and Tignol15, 5.1]) is given, identifying $B=\operatorname {\mathrm {End}}_A(V)$ , by

$$ \begin{align*} \varphi_h(v\otimes w)(x) = vh(w,x). \end{align*} $$

Then $\psi $ is well defined since for $x,y\in A$ :

$$ \begin{align*} \psi((v\otimes w)\otimes (xa\sigma(y)) &= \varphi_h(vxa\sigma(y)\otimes w) \\ &= \varphi_h(vxa\otimes wy) \\ &= \psi ((vx\otimes wy)\otimes a), \end{align*} $$

and it is a bimodule morphism since for $x,y\in B$ :

$$ \begin{align*} \psi((xv\otimes yw)\otimes a) &= \varphi_h(xva\otimes yw) \\ &= x\varphi_h(va\otimes v)\tau(y). \end{align*} $$

To show that $\psi $ is an isometry, we must establish equality between on the one hand

$$ \begin{align*} & \operatorname{\mathrm{Trd}}_B\left(\tau \left( \psi((v\otimes w)\otimes a)\right)\cdot \psi(((v'\otimes w')\otimes b))\right) \\ &= \operatorname{\mathrm{Trd}}_B\left( \tau(\varphi_h(va\otimes w))\cdot \varphi_h(v'b\otimes w')\right) \end{align*} $$

and on the other hand

$$ \begin{align*} & \operatorname{\mathrm{Trd}}_A \left(\sigma(a)(h\otimes h)(v\otimes w,v'\otimes w')\cdot b\right) \\ &= \varepsilon \operatorname{\mathrm{Trd}}_A(\sigma(a)h(v,v')bh(w',w)) \end{align*} $$

with $\varepsilon =\varepsilon _h$ . Now, applying successively the formulas in theorem [Reference Knus, Merkurjev, Rost and Tignol15, 5.1], we get:

$$ \begin{align*} & \operatorname{\mathrm{Trd}}_B\left( \tau(\varphi_h(va\otimes w)) \cdot \varphi_h(v'b\otimes w')\right) \\ &= \varepsilon \operatorname{\mathrm{Trd}}_B\left(\varphi_h(w\otimes va) \cdot \varphi_h(v'b\otimes w')\right) \\ &= \varepsilon \operatorname{\mathrm{Trd}}_B\left(\varphi_h( wh(va,v'b) \otimes w') \right) \\ &= \varepsilon \operatorname{\mathrm{Trd}}_A\left(h(w',wh(va,v'b))\right) \\ &= \varepsilon \operatorname{\mathrm{Trd}}_A\left(h(w',w)\sigma(a)h(v,v')b\right)\kern-1pt.\\[-34pt] \end{align*} $$

▪

4.1 The Witt groups

We start by defining the underlying group structures.

We often make the slight abuse of notations $SW_\varepsilon (K)$ for $SW_\varepsilon (K,\operatorname {\mathrm {Id}})$ ; note that $SW_\varepsilon (K)=SW^\varepsilon (K)$ .

We can observe that there is a natural functoriality for those semigroups: if $f: (B,\tau ) \to (A,\sigma )$ is a morphism in $\mathbf {Br}_h(K,\iota )$ with sign $\varepsilon '\in U(K,\iota )$ , then it induces a function

(4.1) $$ \begin{align} f_*: SW^{\varepsilon}(B,\tau)\longrightarrow SW^{\varepsilon\varepsilon'}(A,\sigma) \end{align} $$

given by $g\mapsto f\circ g$ , where we see any $(V,h)\in SW^{\varepsilon }(B,\tau )$ as a morphism $g: (\operatorname {\mathrm {End}}_K(V),\tau _h)\stackrel {(V,h)}{\rightsquigarrow } (B,\tau )$ .

To simplify things a bit, and with no real consequence to the theory, we are going to consider the subcategory $\mathbf {Br}_h(K,\iota )'$ of $\mathbf {Br}_h(K,\iota )$ , which is equal to $\mathbf {Br}_h(K,\iota )$ when $\iota =\operatorname {\mathrm {Id}}$ , but only includes the morphisms which are $1$ -hermitian when $\iota \neq \operatorname {\mathrm {Id}}$ . The restriction is mostly harmless, since any two objects which are isomorphic in $\mathbf {Br}_h(K,\iota )$ are still isomorphic in $\mathbf {Br}_h(K,\iota )'$ .

Proof First, we need to establish that $f_*$ does define a function from $SW_\varepsilon (B,\tau )$ to $SW_\varepsilon (A,\sigma )$ . When $\iota \neq \operatorname {\mathrm {Id}}$ , this comes from the restriction to $\mathbf {Br}_h(K,\iota )'$ , since in (4.1) we have $\varepsilon '=1$ . When $\iota =\operatorname {\mathrm {Id}}$ , there is no a priori restriction on $\varepsilon '$ , but it is actually determined by the types of $\sigma $ and $\tau $ . In fact, we get

$$ \begin{align*} f_*: SW_{t(\tau)\varepsilon}(B,\tau)\longrightarrow SW_{t(\sigma)\varepsilon\varepsilon'}(A,\sigma), \end{align*} $$

and since $t(\tau ) = \varepsilon 't(\sigma )$ , the signs do correspond on both sides.

The functoriality is a direct consequence of the associativity of the composition in $\mathbf {Br}_h(K,\iota )$ . The fact that it is a semigroup morphism amounts to the formula $h\circ (h_1\perp h_2) \simeq (h\circ h_1)\perp (h\circ h_2)$ which can be checked directly on the definition of the composition of hermitian forms.

Since all morphisms in $\mathbf {Br}_h(K,\iota )'$ are invertible, all $f_*$ are invertible too by functoriality.▪

The reason we introduce $SW_\varepsilon $ (instead of the more obvious $SW^\varepsilon $ ) is precisely in order to get the functoriality in the previous proposition.

Of course, we also have groups $GW^\varepsilon $ and $W^\varepsilon $ , but we are less interested in them because they lack the functoriality we are looking for. We also have in an obvious way $GW_\pm $ and $W_\pm $ .

Note that since by Witt’s theorem $SW_\varepsilon (A,\sigma )$ satisfies the cancellation property, it embeds in $GW_\varepsilon (A,\sigma )$ .

Clearly, we get functors $GW_\pm $ and $W_\pm $ from $\mathbf {Br}_h(K,\iota )$ to $\mu _2(K)$ -graded abelian groups.

4.2 The mixed Witt ring

It is easy to see that the natural maps

$$ \begin{align*} SW^\varepsilon(A,\sigma) \times SW^{\varepsilon'}(B,\tau) \to SW^{\varepsilon\varepsilon'}(A\otimes_K B, \sigma\otimes \tau) \end{align*} $$

given by the tensor products of hermitian modules define maps

$$ \begin{align*} SW_\varepsilon(A,\sigma) \times SW_{\varepsilon'}(B,\tau) \to SW_{\varepsilon\varepsilon'}(A\otimes_K B, \sigma\otimes \tau). \end{align*} $$

This is because $t(\sigma \otimes \tau )=t(\sigma )\cdot t(\tau )$ . Since these maps are additive in each variable, they induce group morphisms

(4.2) $$ \begin{align} GW_\varepsilon(A,\sigma) \otimes GW_{\varepsilon'}(B,\tau) \to GW_{\varepsilon\varepsilon'}(A\otimes_K B, \sigma\otimes \tau) \end{align} $$

by the universal property of the Grothendieck groups.

When $\iota \neq \operatorname {\mathrm {Id}}$ , we are not really interested in $GW_\pm $ and $W_\pm $ but rather simply in the neutral components $GW_1$ and $W_1$ (since $GW_{-1}\approx GW_1$ in that case, albeit noncanonically), which give symmetric monoidal functors to the category of abelian groups.

We can therefore apply Corollary 2.20. Using, respectively, $K=GW_1$ and $K=W_1$ and the canonical abelian gs-structure of $\mathbf {Br}_h(K,\iota )'$ we described in 3.5, we get functors:

(4.3) $$ \begin{align} \widehat{GW}: \mathbf{Br}_h(K,\iota)' \to GW(K,\iota)\text{-}\mathbf{CommAlg}_{\mathbb{Z}}, \end{align} $$

(4.4) $$ \begin{align} \widehat{W}: \mathbf{Br}_h(K,\iota)' \to W(K,\iota)\text{-}\mathbf{CommAlg}_{\mathbb{Z}}. \end{align} $$

In concrete terms:

$$ \begin{align*} \widehat{GW}(A,\sigma) = \bigoplus_{n\in \mathbb{N}^*} GW_1((A^\iota)^{\otimes n},\sigma^{\otimes n}) \oplus GW_1(K,\iota) \oplus \bigoplus_{n\in \mathbb{N}^*} GW_1(A^{\otimes n},\sigma^{\otimes n}) \end{align*} $$

and likewise for $\widehat {W}(A,\sigma )$ . An element in $GW_1((A^\iota )^{\otimes n},\sigma ^{\otimes n})$ has degree $-n$ , and when an element of positive degree is multiplied with an element of negative degree, the difference is canceled using the hermitian Morita equivalence $(A\otimes A^\iota ,\sigma \otimes \sigma ) \stackrel {}{\rightsquigarrow } (K,\iota )$ as many times as necessary.

The point of the machinery developed in Section 2.4 is that it guarantees that this is well defined, and gives a commutative $\mathbb {Z}$ -graded ring, functorial in $(A,\sigma )$ (none of which is trivial). Although this ring has interesting applications for the theory of hermitian forms over algebras with unitary involution, we will put it aside in the remainder of the article, and focus on the case of involutions of the first kind, for which a more powerful construction is available.

If we now specialize to $\iota =\operatorname {\mathrm {Id}}$ , we can use $K=GW_\pm $ and $K=W_\pm $ in Corollary 2.20, as well as the canonical coherent $2$ -torsion structure of $\mathbf {Br}_h(K)$ , to get functors

(4.5) $$ \begin{align} \widetilde{GW}: \mathbf{Br}_h(K) \to GW(K)\text{-}\mathbf{CommAlg}_{\Gamma}, \end{align} $$

(4.6) $$ \begin{align} \widetilde{W}: \mathbf{Br}_h(K) \to W(K)\text{-}\mathbf{CommAlg}_{\Gamma}, \end{align} $$

where $\Gamma = \mathbb {Z}/2\mathbb {Z} \times \mu _2(K)$ .

In concrete terms, for any algebra with involution (of the first kind), we have the mixed Grothendieck–Witt ring

(4.7) $$ \begin{align} \widetilde{GW}(A,\sigma) = GW_1(K) \oplus GW_{-1}(K) \oplus GW_1(A,\sigma) \oplus GW_{-1}(A,\sigma) \end{align} $$

and the mixed Witt ring

(4.8) $$ \begin{align} \widetilde{W}(A,\sigma) = W_1(K) \oplus W_1(A,\sigma) \oplus W_{-1}(A,\sigma) \end{align} $$

(note that $W_{-1}(K)=0$ because all antisymmetric bilinear forms are hyperbolic). The word “mixed” is here to indicate that the ring structure mixes bilinear and hermitian forms. Of course, given its construction, $\widetilde {W}(A,\sigma )$ is the quotient of $\widetilde {GW}(A,\sigma )$ by the ideal of hyperbolic forms.

A lot of the ring structure of $\widetilde {GW}(A,\sigma )$ comes from the fact that $GW_\pm (K)$ is a commutative ring and that $GW_\pm (A,\sigma )$ is a module over this ring, which is not something new. The novel part is the product

$$ \begin{align*} GW_\varepsilon(A,\sigma)\otimes GW_{\varepsilon'}(A,\sigma)\to GW_{\varepsilon\varepsilon'}(K), \end{align*} $$

which simply comes from the natural product $GW_\varepsilon (A,\sigma )\otimes GW_{\varepsilon '}(A,\sigma ) \to GW_{\varepsilon \varepsilon '}(A^{\otimes 2},\sigma ^{\otimes 2})$ composed with the isomorphism $GW_{\varepsilon \varepsilon '}(A^{\otimes 2},\sigma ^{\otimes 2})\simeq GW_{\varepsilon \varepsilon '}(K)$ coming from the canonical hermitian Morita equivalence between $(A^{\otimes 2},\sigma ^{\otimes 2})$ and $(K,\operatorname {\mathrm {Id}})$ , given by $(|A|_\sigma ,T_\sigma )$ . Once again, the machinery of Section 2.4 guarantees that everything is well defined and functorial. (Of course, similar statements hold for $\widetilde {W}(A,\sigma )$ .)

We will call $GW_\pm (K)$ the even part of $\widetilde {GW}(A,\sigma )$ , and $GW_\pm (A,\sigma )$ its odd part. This corresponds to considering the $\mathbb {Z}/2\mathbb {Z}$ -grading canonically induced by the $\Gamma $ -grading on the ring. Then the functoriality of $\widetilde {GW}$ behaves very differently on the even and odd parts: for any $f: (B,\tau )\to (A,\sigma )$ in $\mathbf {Br}_h(K)$ , the induced $f_*: \widetilde {GW}(B,\tau ) \to \widetilde {GW}(A,\sigma )$ consists of the identity on the even part, and of the morphism described in (4.1) on the odd part.

For instance, any automorphism of $(A,\sigma )$ in $\mathbf {Br}_h(K)$ induces a graded ring automorphism on $\widetilde {GW}(A,\sigma )$ ; such an automorphism will be called standard. Recall from Proposition 3.9 that all Morita automorphisms of $(A,\sigma )$ are of the form $(|A|,\langle \lambda \rangle _\sigma )$ with $\lambda \in K^\times $ , so the associated standard automorphism of $\widetilde {GW}(A,\sigma )$ is the identity on $GW_\pm (K)$ , and is the multiplication of hermitian forms by the scalar $\lambda $ on $GW_\pm (A,\sigma )$ .

Since all morphisms in $\mathbf {Br}_h(K)$ are invertible, this means that all the induced morphisms between mixed Witt rings are actually isomorphisms. In particular, given some algebra with involution $(A,\sigma )$ , for any other $(B,\tau )$ such that A and B are Brauer-equivalent, there are isomorphisms $\widetilde {GW}(A,\sigma ) \approx \widetilde {GW}(B,\tau )$ , well defined up to a standard automorphism of $\widetilde {GW}(B,\tau )$ . This means that we can translate any computation in $\widetilde {GW}(A,\sigma )$ to a computation in $\widetilde {GW}(B,\tau )$ ; the even component of the result does not depend on any choice of equivalence, whereas the odd part is well defined up to multiplication by a scalar (which corresponds to a choice of equivalence). For instance, it is often convenient to choose for B a division algebra, so that we can reduce all computations to the case of diagonal forms.

Of course, everything we said in the last paragraphs also holds for $\widetilde {W}(A,\sigma )$ (for the same reasons, or by seeing it as a quotient of $\widetilde {GW}(A,\sigma )$ ).

4.3 The reduced dimension map

In the classical theory of quadratic forms, the dimension maps $GW(K)\to \mathbb {Z}$ and $W(K)\to \mathbb {Z}/2\mathbb {Z}$ play an important role (especially through their kernel). We can generalize that to mixed Witt rings: to any $\varepsilon $ -hermitian module $(V,h)$ over $(A,\sigma )$ , we can associate its reduced dimension $\operatorname {\mathrm {rdim}}(V)$ (or $\operatorname {\mathrm {rdim}}(h)$ ), defined as the degree of the algebra $\operatorname {\mathrm {End}}_A(V)$ . It can also be characterized as $\dim _K(V)/\deg (A)$ (see [Reference Knus, Merkurjev, Rost and Tignol15]).

This defines a semigroup morphism $\operatorname {\mathrm {rdim}}: SW_\varepsilon (A,\sigma )\to \mathbb {N}$ , which extends to a group morphism $\operatorname {\mathrm {rdim}}: GW_\varepsilon (A,\sigma )\to \mathbb {Z}$ . If we use these morphisms componentwise, we get a $\Gamma $ -graded group morphism

(4.9) $$ \begin{align} \widetilde{\operatorname{\mathrm{rdim}}}: \widetilde{GW}(A,\sigma)\to \mathbb{Z}[\Gamma]. \end{align} $$

Since hyperbolic forms always have an even reduced dimension, this also induces a group morphism

(4.10) $$ \begin{align} \widetilde{\operatorname{\mathrm{rdim}}}_2: \widetilde{W}(A,\sigma)\to \mathbb{Z}/2\mathbb{Z}[\Gamma]. \end{align} $$

Proposition 4.8 The morphisms ( 4.9 ) and ( 4.10 ) are actually $\Gamma $ -graded ring morphisms, and for any $f: (B,\tau )\to (A,\sigma )$ in $\mathbf {Br}_h(K)$ , we get a commutative diagram

4.4 Products of diagonal forms

Now that we have established the formal properties of our mixed rings, we would like to be able to perform explicit computations. Obviously, the only products which present any difficulty are the products of two elements in $GW_\pm (A,\sigma )$ . Actually, since the elements in $GW_{-1}(K)$ are all hyperbolic and characterized by their (reduced) dimension, and we know from Proposition 4.8 how reduced dimensions multiply, the only nontrivial products to compute are those of two elements in the same $GW_\varepsilon (A,\sigma )$ .

Furthermore, it is enough to know how to multiply elementary diagonal forms $\langle a\rangle _\sigma $ , since we know that if needed we can perform these computations in $\widetilde {GW}(D,\theta )$ where D is the division algebra Brauer-equivalent to our algebra A (and $\theta $ is any involution). The result will even be independent of the choice of equivalence between $(A,\sigma )$ and $(D,\theta )$ as it lies in the even component.

In [Reference Knus, Merkurjev, Rost and Tignol15, §11], given $a\in \operatorname {\mathrm {Sym}}^\varepsilon (A^\times ,\sigma )$ , a symmetric bilinear form $T_{\sigma ,a}: A\times A\to K$ is introduced, called a twisted involution trace form. We generalize that slightly by taking $a,b\in \operatorname {\mathrm {Sym}}^\varepsilon (A^\times ,\sigma )$ and defining $T_{\sigma ,a,b}: A\times A\to K$ as

(4.11) $$ \begin{align} T_{\sigma,a,b}(x,y) = \operatorname{\mathrm{Trd}}_A(\sigma(x)ay\sigma(b)). \end{align} $$

We recover the initial $T_{\sigma ,a}$ as $T_{\sigma ,a,1}$ , and also the usual involution trace form $T_\sigma $ as $T_{\sigma ,1,1}$ . Then we get the following proposition.

Proposition 4.9 Let $(A,\sigma )$ be an algebra with involution over K, and let $a,b\in \operatorname {\mathrm {Sym}}^\varepsilon (A^\times ,\sigma )$ . Then in $\widetilde {GW}(A,\sigma )$ we have

$$ \begin{align*} \langle a\rangle_\sigma \cdot \langle b\rangle_\sigma = T_{\sigma,a,b} \in GW(K). \end{align*} $$

Proof By definition, $\langle a\rangle _\sigma \cdot \langle b\rangle _\sigma $ is the bilinear form given by the composition $T_\sigma \circ \langle a\otimes b\rangle _{\sigma \otimes \sigma }$ . According to Example 3.7, this is

$$ \begin{align*} (x,y)\mapsto T_\sigma(x, (a\otimes b)\cdot y), \end{align*} $$

where the action of $A\otimes _K A$ on A is the twisted sandwich action (3.2). Since $T_\sigma (x,y)=\operatorname {\mathrm {Trd}}_A(\sigma (x)y)$ and $(a\otimes b)\cdot y = ay\sigma (b)$ , we may conclude.▪

Corollary 4.11 Let $(A,\sigma )$ be an algebra with involution over K, V be a right A-module, and $h,h'$ be two $\varepsilon $ -hermitian forms on V. If we set $B=\operatorname {\mathrm {End}}_A(V)$ and $\tau =\sigma _h$ , then there is a unique $u\in B^\times $ such that for all $x,y\in V$ , $h'(x,y)=h(ux,y)$ . Furthermore, $\tau (u)=u$ , and

$$ \begin{align*} h'\cdot h = T_{\tau,u} \end{align*} $$

as a product in $\widetilde {GW}(A,\sigma )$ . In particular, $h^2=T_\tau $ .

This means that we can reinterpret twisted involution forms in the sense of [Reference Knus, Merkurjev, Rost and Tignol15, §11] as being exactly the products of $\varepsilon $ -hermitian forms defined on the same module. The previous computations show that understanding the product in $\widetilde {GW}(A,\sigma )$ and $\widetilde {W}(A,\sigma )$ amounts to understanding those twisted involution trace forms (usually for involutions different from $\sigma $ ).

4.5 Quaternion algebras

We can in particular perform these computations for quaternion algebras, imitating the proof of [Reference Knus, Merkurjev, Rost and Tignol15, 11.6]. Recall that if Q is a quaternion algebra, then its reduced norm map is a quadratic form on Q, denoted $n_Q\in GW(K)$ , and it is the unique 2-Pfister form whose Clifford invariant $e_2(n_Q)\in H^2(K,\mu _2)$ is the Brauer class of Q.

For any pure quaternions $z_1,z_2\in Q^\times$ , the Brauer class $[Q]$ and the symbol $(z_1^2,z_2^2)\in H^2(K,\mu _2)$ have a common slot (for instance, $z_1^2$ ), so $[Q]+(z_1^2,z_2^2)$ is a symbol. We write $\varphi _{z_1,z_2}$ for the unique 2-Pfister form whose Clifford invariant is this symbol. In particular, if $z_1$ and $z_2$ anticommute, $\varphi _{z_1,z_2}$ is hyperbolic.

Proposition 4.12 Let $(Q,\gamma )$ be a quaternion algebra over K endowed with its canonical symplectic involution. Then, for any $a,b\in K^*$ , we have in $\widetilde {GW}(Q,\gamma )$ :

$$ \begin{align*} \langle a\rangle_\gamma\cdot \langle b\rangle_\gamma = \langle 2ab\rangle n_Q \in GW(K). \end{align*} $$

Furthermore, for any pure quaternions $z_1,z_2\in Q^\times $ , $\langle z_1\rangle _\gamma \cdot \langle z_2\rangle _\gamma $ is similar to $\varphi _{z_1,z_2}$ . When $z_1$ and $z_2$ anticommute, this means $\langle z_1\rangle _\gamma \cdot \langle z_2\rangle _\gamma $ is hyperbolic. When they do not anticommute, we get:

$$ \begin{align*} \langle z_1\rangle_\gamma\cdot \langle z_2\rangle_\gamma = \langle -\operatorname{\mathrm{Trd}}_Q(z_1z_2)\rangle\varphi_{z_1,z_2} \in GW(K). \end{align*} $$

Proof From Proposition 4.9, we see that

$$ \begin{align*} \langle a\rangle_\gamma\cdot \langle b\rangle_\gamma = \langle ab\rangle T_\gamma, \end{align*} $$

and it is easy to see that $T_\gamma = \langle 2\rangle n_Q$ (for instance, by taking a standard quaternionic basis of Q). For the second formula, we can check that if $z_1$ and $z_2$ anticommute, then $1$ and $z_1$ span a Lagrangian in $(Q,T_{\gamma ,z_1,z_2})$ , and otherwise we can check that if $z\in Q^\times $ anticommutes with $z_1$ , then the basis $(1,z_1,zz_2,z_1zz_2)$ is orthogonal for $T_{\gamma ,z_1,z_2}$ , giving the diagonalization

$$ \begin{align*} \langle -\operatorname{\mathrm{Trd}}_Q(z_1z_2)\rangle\langle 1,-z_1^2,-z_2^2z^2,z_1^2z_2^2z^2\rangle, \end{align*} $$

which shows that $T_{\gamma ,z_1,z_2}\simeq \langle -\operatorname {\mathrm {Trd}}_Q(z_1z_2)\rangle \langle \!\langle z_1,z_2z\rangle \!\rangle $ , and it is easy to see that by definition $\varphi _{z_1,z_2}=\langle \!\langle z_1,z_2z\rangle \!\rangle $ .▪

Using this result, we can also compute products in $(Q,\sigma )$ for orthogonal involutions $\sigma $ . Indeed, we know that $\sigma =\gamma _u$ for some pure quaternion $u\in Q^\times $ ; then we have a Morita equivalence $f: (Q,\sigma )\stackrel {\langle u\rangle _\gamma }{\rightsquigarrow } (Q,\gamma )$ . From Example 3.7, we deduce that $f_*(\langle z\rangle _\sigma ) = \langle uz\rangle _\gamma $ . So we can simply compute $\langle z\rangle _\sigma \cdot \langle z'\rangle _\sigma = \langle uz\rangle _\gamma \cdot \langle uz'\rangle _\gamma $ using Proposition 4.12 (it does not give especially enlightening formulas).

4.6 Scalar extension and reciprocity

Scalar extension is a standard tool in the theory of algebras with involution, in particular when extending the scalars to a splitting field to reduce to the classical theory of bilinear forms over fields.

When studying the usual Witt rings of fields, it is also standard to consider the ring morphisms $\rho : GW(K)\to GW(L)$ induced by a field extension $L/K$ . One of the basic facts (see [Reference Scharlau23, 2.5.6]) is that we can also go in the other direction provided the extension is finite: if $s:L\to K$ is a nonzero K-linear map, then one can define a group morphism

$$ \begin{align*} \begin{array}{rrcl} s_* : & GW(L) & \longrightarrow & GW(K) \\ & (V,b) & \longmapsto & (V,s\circ b), \end{array} \end{align*} $$

usually called a Scharlau transfer map, and the Reciprocity Theorem states that this is a $GW(K)$ -module morphism. In concrete terms, this means that if $x\in GW(K)$ and $y\in GW(L)$ , then

$$ \begin{align*} s_*(\rho(x)\cdot y) = x\cdot s_*(y). \end{align*} $$

Of course, this is reminiscent of other such reciprocity phenomena, such as Frobenius reciprocity for induction/restriction of group representations, or the projection formula for cup-products in cohomology.

Let us explain how such reciprocity formulas arise inside the framework we established in Section 2.4 to construct graded rings. Consider commutative monoids M and N, and $F,G\in \operatorname {\mathrm {LaxHom}}_{\otimes }^s(\langle M\rangle , \mathbb {Z}\text {-}\mathbf {Mod}_N)$ . Then, if $\varphi : F\to G$ is a monoidal transformation, it induces an $(M\times N)$ -graded ring morphism $\varphi _*: A_F\to A_G$ , where $A_F$ and $A_G$ are the graded rings corresponding to F and G. In particular, $A_G$ is a (graded) module over $A_F$ . Now, if $\psi : G\to F$ is a natural transformation (not monoidal) in the other direction, it induces a graded group morphism $\psi _*: A_G\to A_F$ . When is this an $A_F$ -module morphism? It is easy to check that the relevant condition is that the following diagram commutes for all $x,y\in M$ :

(4.12)

Now, let us apply this to our situation. Let $L/K$ be any field extension. We have an obvious scalar extension functor $\mathbf {Br}_h(K)\to \mathbf {Br}_h(L)$ , which fits in commutative diagrams of functors

and

using the canonical coherent $2$ -torsion structures.

For any algebra with involution $(A,\sigma )$ , scalar extension of hermitian modules yields group morphisms

(4.13) $$ \begin{align} \rho: GW_\varepsilon(A,\sigma)\longrightarrow GW_\varepsilon(A_L,\sigma_L) \end{align} $$

which fit into a $2$ -cell

In fact, as scalar extension is compatible with tensor products, $\rho $ is even a monoidal natural transformation.

Now, let us assume that $L/K$ is finite, and let $s:L\to K$ be a nonzero K-linear form. Then, for any K-algebra A, s extends naturally to a K-linear map $s_A: A_L\to A$ by $\operatorname {\mathrm {Id}}_A\otimes s$ , and this defines an involution trace in the sense of [Reference Knus, Merkurjev, Rost and Tignol15, 4.3]. This shows (see also [Reference Knus14, I.7.2,I.7.3.2]) that we can define group morphisms, which are the analogues of the Scharlau transfer,

(4.14) $$ \begin{align} s_*: GW_\varepsilon(A_L,\sigma_L)\longrightarrow GW_\varepsilon(A,\sigma) \end{align} $$

by sending $(V,h)$ over $(A_L,\sigma _L)$ to $(V,s_A\circ h)$ .

Proposition 4.14 The morphisms in ( 4.14 ) form a natural transformation which fits into a $2$ -cell

Proof If $f: (B,\tau )\stackrel {(V,h)}{\rightsquigarrow } (A,\sigma )$ is a morphism in $\mathbf {Br}_h(K)$ , we need to show that we get a commutative diagram

Let $(U,g)\in GW_\varepsilon (B_L,\tau _L)$ . Then we need to give an isometry from $(U\otimes _B V,\alpha )$ to $(U\otimes _{B_L} V_L,\beta )$ with

$$ \begin{align*} \alpha(u\otimes v, u'\otimes v') = h(v, s_B(g(u,u'))\cdot v') \end{align*} $$

and

$$ \begin{align*} \beta(u\otimes (v\otimes \lambda), u'\otimes (v'\otimes \mu)) = s_A(h_L(v\otimes \lambda, g(u,u')\cdot (v'\otimes \mu))). \end{align*} $$

We consider $\varphi : U\otimes _B V\to U\otimes _{B_L} V_L$ defined by $u\otimes v\mapsto u\otimes (v\otimes 1)$ ; to show that it is an isometry from $\alpha $ to $\beta $ , we need to show the equality:

$$ \begin{align*} s_A(h_L(v\otimes 1, g(u,u')\cdot (v'\otimes 1))) = h(v, s_B(g(u,u'))\cdot v'). \end{align*} $$

We can write $g(u,u')=\sum b_i\otimes \lambda _i$ , and since both expressions are additive in $g(u,u')$ , we may actually assume that $g(u,u')=b\otimes \lambda $ . But then, the left-hand side is

$$ \begin{align*} s_A(h_L(v\otimes 1,bv'\otimes \lambda)) = s_A(h(v,bv')\otimes \lambda) = h(v,bv')s(\lambda),\end{align*} $$

and the right-hand side is

$$ \begin{align*} h(v, s_B(b\otimes \lambda)\cdot v') = h(v,s(\lambda)bv'), \end{align*} $$

which concludes.▪

Thus, we have at our disposal two natural transformations $\rho $ and $s_*$ going in opposite directions between the same functors, and one of them, namely $\rho $ is monoidal. The following proof is essentially an adaptation of [Reference Scharlau23, 2.5.6].

Proof We need to show that if $(V,h)\in GW_\varepsilon (A,\sigma )$ and $(U,g)\in GW_{\varepsilon '}(B_L,\tau _L)$ , then

$$ \begin{align*} s_{A\otimes_K B}\circ (h_L\otimes g) = h\otimes (s_B\circ g) \in GW_{\varepsilon\varepsilon'}(A\otimes_K B,\sigma\otimes \tau). \end{align*} $$

It is easy to see that the map

$$ \begin{align*} \begin{array}{rcl} (V\otimes_K L)\otimes_L U & \longrightarrow & V \otimes_K U \\ {(x\otimes \lambda)\otimes y} & \longmapsto &{x\otimes (\lambda y)} \end{array} \end{align*} $$

defines an isometry.▪

We finally establish the desired reciprocity result.

Theorem 4.16 (Frobenius reciprocity)

Let $L/K$ be a finite field extension, and let $s:L\to K$ be a nonzero K-linear form. Then, for any algebra with involution $(A,\sigma )$ over $(K,\operatorname {\mathrm {Id}})$ , the scalar extension map $\rho : \widetilde {GW}(A,\sigma )\to \widetilde {GW}(A_L,\sigma _L)$ is a $\Gamma $ -graded ring morphism, and the transfer $s_*: \widetilde {GW}(A_L,\sigma _L)\to \widetilde {GW}(A,\sigma )$ is a morphism of graded $\widetilde {GW}(A,\sigma )$ -modules.

In practice, this means that for all $x\in \widetilde {GW}(A,\sigma )$ and $y\in \widetilde {GW}(A_L,\sigma _L)$ , we have:

$$ \begin{align*} s_*(\rho(x)\cdot y) = x\cdot s_*(y). \end{align*} $$

Proof The canonical coherent $2$ -torsion structures associate to each $(A,\sigma )$ symmetric monoidal functors

If we use the notation in the statement of Proposition 4.15, then by composition, we get symmetric monoidal functors $F',G': \langle \mathbb {Z}/2\mathbb {Z}\rangle \to \mathbb {Z}\text {-}\mathbf {Mod}_{\mu _2(L)}$ , and $\rho $ and $s_*$ give natural transformation $\varphi : F'\to G'$ and $\psi : G'\to F'$ which satisfy the relation in (4.12). By construction, the graded rings associated to $F'$ and $G'$ are precisely $\widetilde {GW}(A,\sigma )$ and $\widetilde {GW}(A_L,\sigma _L)$ , and the natural transformations $\varphi $ and $\psi $ correspond to the maps given by $\rho $ and $s_*$ .

Then, as discussed above, the relation (4.12) exactly gives the fact that $\rho $ is a ring morphism and $s_*$ is a $\widetilde {GW}(A,\sigma )$ -module morphism.▪

Since both $\rho $ and $s_*$ send hyperbolic forms to hyperbolic forms, they also induce maps $\rho : \widetilde {W}(A,\sigma )\to \widetilde {W}(A_L,\sigma _L)$ and $s_*: \widetilde {W}(A_L,\sigma _L)\to \widetilde {W}(A,\sigma )$ , which also satisfy the reciprocity formula, since it holds in $\widetilde {GW}$ .