Mixed Witt rings of algebras with involution

1.1 From quadratic forms to hermitian Morita theory

In the nineteenth century, quadratic forms were the object of many investigations, notably by algebraists such as Gauss, Minkowski, or Kronecker, but were mostly given an arithmetic flavor. The birth of the algebraic theory of quadratic forms over arbitrary fields (of characteristic not $2$ ) stems from the seminal 1936 paper of Ernst Witt [Reference Witt27], although a completely unnoticed 1907 paper of Dickson [Reference Dickson7] had actually already made significant advances (see Scharlau’s comment in [Reference Scharlau22]). A key insight of Witt was to shift from the study of individual quadratic forms to the study of the structure they form as a whole. He considers the set $W(K)$ of isometry classes of quadratic forms over K up to what is now called Witt equivalence, and shows that the direct sum and tensor product of quadratic spaces endow this set with a natural commutative ring structure.

Though quadratic forms over fields were still objects of study, notably through the study of various arithmetic invariants, this algebraic theory of Witt truly started expanding when it was rediscovered in the mid-60s by Pfister (see [Reference Pfister20]), and others such as Arason. It then became abundantly clear that the algebraic structure of the Witt ring was a central element of the theory, and was connected to many subjects of interest, such as Galois cohomology through the Milnor conjecture, or the theory of field orderings, in relation with the much older Artin–Schreier theory. For detailed references on the algebraic theory of quadratic forms, we can cite [Reference Lam16], [Reference Scharlau23], or [Reference Elman, Karpenko and Merkurjev9].

Around the same time, algebraic K-theory was being developed, and the connection with quadratic forms became apparent, as can be seen notably in the foundational work of Bass [Reference Bass and Roy6]. This led to working over general commutative rings, but also to consider not just the Witt ring $W(K)$ , but what we will call the Grothendieck–Witt ring $GW(K)$ (the terminology “Witt–Grothendieck ring” is equally common); one crucial reason Witt introduced the Witt equivalence is that it creates additive inverses in $W(K)$ , but K-theory taught us that it is very natural to rather add formal inverses. Shortly after that, Bak [Reference Bak, Moss and Thomas4] and Fröhlich and McEvett [Reference Fröhlich and McEvett12] independently started to generalize bilinear forms over commutative rings to $\varepsilon $ -hermitian forms over (noncommutative) rings with involution (where $\varepsilon $ is a parameter, which is simply equal to $\pm 1$ in the classical context of fields with no involution, corresponding to the theory of symmetric and antisymmetric bilinear forms); a more advanced reference is found in [Reference Knus14]. In particular, Fröhlich and McEvett define Witt and Grothendieck–Witt groups $W^\varepsilon (A,\sigma )$ and $GW^\varepsilon (A,\sigma )$ ; furthermore, they develop an analogue for hermitian modules of the classical Morita theory over rings, which they aptly call hermitian Morita theory. This theory is the central tool used in this article, and we review it in Section 3, although in a different form.

Prominent among rings with involutions are the (finite-dimensional) central simple algebras with involution over fields, which we will thereafter simply call “algebras with involution.” Their study was initiated by Albert in [Reference Albert1] with the context of Riemann surfaces in mind, but found a renewed interest when Weil [Reference Weil26] showed that they can be used to describe the simple algebraic groups over arbitrary fields. A comprehensive treatment of those algebras and their hermitian forms is given in the reference monograph [Reference Knus, Merkurjev, Rost and Tignol15] by Knus et al.

1.2 The idea of the mixed Witt ring

One important caveat in all those generalizations is that when we work over noncommutative rings, we can only speak about (Grothendieck–)Witt groups and not rings. This is simply because there is no tensor product of modules over noncommutative rings. Rather, the tensor product of two A-modules (for example, on the right) is an $(A\otimes A)$ -module. This being said, there is still some compatibility between Witt groups and tensor products: if $(A,\sigma )$ and $(B,\tau )$ are algebras with involution over some field with involution $(K,\iota )$ , the tensor product induces maps

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

(here, $\varepsilon ,\varepsilon '\in K^\times $ are parameters such that $\varepsilon \iota (\varepsilon )=1$ ). This is not surprising if one interprets the Grothendieck–Witt group as the degree $0$ part of hermitian K-theory (see [Reference Bak5] for instance), and it expresses the fact that (Grothendieck–)Witt groups are monoidal functors on some category (which will be made precise).

In the special case of algebras with involutions of the first kind (so $\iota =\operatorname {\mathrm {Id}}$ in the notation above), a remarkable phenomenon is that not only do the algebras “have order $2$ ” with respect to Brauer-equivalence, but we have an explicit canonical hermitian Morita equivalence between $(A\otimes _K A,\sigma \otimes \sigma )$ and $(K,\operatorname {\mathrm {Id}})$ , which means that by general considerations of Morita theory, we get a canonical isomorphism

$$ \begin{align*} GW(A\otimes_K A,\sigma\otimes\sigma) \stackrel{\sim}{\longrightarrow} GW(K). \end{align*} $$

If we combine this with the natural product

$$ \begin{align*} GW^\varepsilon(A,\sigma)\times GW^\varepsilon(A,\sigma) \to GW(A\otimes_K A,\sigma\otimes \sigma), \end{align*} $$

we get a canonical “mixed” product

$$ \begin{align*} GW^\varepsilon(A,\sigma)\times GW^\varepsilon(A,\sigma) \to GW(K). \end{align*} $$

In other words, we can define the product of two $\varepsilon $ -hermitian forms, but the result is not a hermitian form, but rather a quadratic form over the base field. It is in that sense that we mean the product is “mixed.” Using the basic fact that $GW(K)$ is a ring and $GW^\varepsilon (A,\sigma )$ is a $GW(K)$ -module, this allows us to construct a $\mathbb {Z}/2\mathbb {Z}$ -graded commutative ring

$$ \begin{align*} \widetilde{GW}^\varepsilon(A,\sigma) = GW(K) \oplus GW^\varepsilon(A,\sigma). \end{align*} $$

Actually, it turns out that for functoriality reasons, it is much more convenient to bundle together hermitian and anti-hermitian forms, and rather consider the mixed Grothendieck–Witt ring

$$ \begin{align*} \widetilde{GW}(A,\sigma) = GW(K) \oplus GW^{-1}(K) \oplus GW^1(A,\sigma) \oplus GW^{-1}(A,\sigma), \end{align*} $$

which is graded over $\Gamma = \mathbb {Z}/2\mathbb {Z}\times \mu _2(K)$ . Of course, the same construction holds for Witt groups, giving the mixed Witt ring

$$ \begin{align*} \widetilde{W}(A,\sigma) = W(K) \oplus W^1(A,\sigma) \oplus W^{-1}(A,\sigma). \end{align*} $$

A large part of this article is dedicated to showing that these are indeed well-defined commutative rings, and that they are functorial in the sense that any hermitian Morita equivalence between algebras with involution induces a graded ring isomorphism.

It should be mentioned that a very similar construction was made by Lewis in [Reference Lewis18], for the special case where A is a division quaternion algebra, although many key properties, such as associativity and commutativity, are stated without proof. Instead of the involution trace form, Lewis uses the norm form, which is a special feature of quaternion algebras.

We also present a less powerful construction which has the merit of working even for involutions of the second kind: in that case, $(A,\sigma )$ is not its own inverse (up to hermitian Morita equivalence); rather, its inverse is the twisted algebra $(A^\iota ,\sigma )$ , where the action of K on A is twisted by $\iota $ . Then we can define a $\mathbb {Z}$ -graded ring $\widehat {GW}(A,\sigma )$ where the component of degree n is $GW(A^{\otimes n},\sigma ^{\otimes n})$ if $n\geqslant 0$ and $GW((A^\iota )^{\otimes (-n)},\sigma ^{\otimes (-n)})$ if $n< 0$ ; we can “cancel out” the A and $A^\iota $ as many times as necessary, so that the product of $x\in GW(A^{\otimes n},\sigma ^{\otimes n})$ and $y\in GW((A^\iota )^{\otimes m},\sigma ^{\otimes m})$ is in $GW(A^{\otimes n-m},\sigma ^{\otimes n-m})$ if $n\geqslant m$ , and in $GW((A^\iota )^{\otimes m-n},\sigma ^{\otimes m-n})$ otherwise.

1.3 Content of the article

Our approach in defining and studying our mixed rings is to highlight as much as possible the formal aspects of the construction: it is a consequence of general properties of symmetric monoidal categories, applied to a certain category which encodes hermitian Morita theory. In consequence, the first part is dedicated to general considerations on monoidal categories.

Sections 2.1 and 2.2 are a basic introduction to symmetric monoidal categories, as can be found in any reference on categories, such as the classic [Reference Mac Lane19]. They are here for convenience of readers who are unfamiliar with that theory, as well as to fix some notations.

Section 2.3 contains the technical heart of the article: we explain how a special property of a symmetric monoidal category, which we call strong symmetry (Definition 2.1), allows to coherently choose inverses of objects (Theorem 2.16), and to coherently handle n-torsion of objects (Proposition 2.17). The strong symmetry property was introduced in [Reference Ulbrich24], where it was already used to get a similar result about inverses; our result is basically a reformulation of [Reference Laplaza17, Corollary 4.6 and Proposition 4.7]. This being said, the presentation we give in terms of monoidal functors $\langle \mathbb {Z}\rangle \to \mathbf {C}$ is our own, and we do believe that it sheds an interesting light on those results, by constructing a universal category $\mathbf {C}^\times $ of inverses in $\mathbf {C}$ , and defining the relevant structure as a functor $\mathbf {C}\to \mathbf {C}^\times $ which “coherently chooses inverses.” There are many variations in the literature around such “enhanced group structures” on monoidal categories (see [Reference Baez and Lauda3] for more references), and so we give yet another one. On the other hand, Proposition 2.17, which adapts the previous idea to handle n-torsion with monoidal functors $\langle \mathbb {Z}/n\mathbb {Z}\rangle \to \mathbf {C}$ , is our own, though the basic idea of the proof is very reminiscent of [Reference Joyal and Street13].

In Section 2.4, we explain how to construct graded rings (which is our end goal) from monoidal functors. In fact, we explain that an M-graded ring is essentially the same as a lax monoidal functor from the discrete category $\langle M\rangle $ to the category of abelian groups. It is a very simple idea, but we have not found previous uses in the literature. We give slightly refined versions of that statement, which will be adapted to the construction of our mixed rings, in relation with the structures described in the previous section (Corollary 2.20).

The second part is dedicated to the presentation of our version of hermitian Morita equivalence (in the framework of algebras with involution), which has a strong categorical flavor, so that we can combine it later with the result of the first part.

Section 3.1 exposes the basic theory of hermitian forms over algebras with involution, mostly to fix the terminology and notations. It does not contain anything original beyond some notation, and all the material can be found in [14, Reference Knus, Merkurjev, Rost and Tignol15]. Particular attention should be given to Example 3.5, which presents the canonical Morita equivalence alluded to earlier and which is the basis of our ring construction; it was already introduced in [Reference Fröhlich and McEvett12].

In Section 3.2, we explain how to package hermitian Morita theory in a category $\mathbf {Br}_h(K,\iota )$ . The only thing new is the presentation, as the actual mathematical results can all be found in [Reference Knus14], and actually already in [Reference Fröhlich and McEvett12]. The notion that (classical) Morita theory could be elegantly expressed in a certain $2$ -category whose morphisms are bimodules is now rather classical in the field of “higher algebra” (see, for instance, [Reference Duskin and Borceux8]) and has been generalized in all sorts of direction (for example, [Reference Vitale25]), but to the best of our knowledge, this is the first time the hermitian version is explicitly written down in practical terms. Note that indeed the natural framework would be to consider a $2$ -category, where the $2$ -morphisms are bimodule morphisms, but it is not needed for what follows, so to avoid unnecessary complications, we “truncate” the natural $2$ -category to a plain category, at the cost of having morphisms be isometry classes of hermitian bimodules. Since the non-hermitian version is sometimes called the Brauer $2$ -group of a commutative ring, we named this category the hermitian Brauer $2$ -group.

In Section 3.3, we investigate the connection between the usual automorphisms of an algebra with involution and its hermitian Morita self-equivalences. In particular, Proposition 3.9 shows that an algebraic automorphism induces the trivial Morita automorphism if and only if it is inner.

This is used in Section 3.4 to show that $\mathbf {Br}_h(K,\iota )$ is strongly symmetric (Corollary 3.13); the key point is the existence of the ubiquitous Goldman element in a central simple algebra, since it implies that the switching automorphism of $A\otimes _K A$ is actually inner. This is the crucial result upon which our construction relies, and although ultimately quite simple, it seems to have gone unnoticed until now.

We show in Section 3.5 that for involutions of the first kind, the canonical equivalence defined by the involution trace form does define a coherent $2$ -torsion structure on $\mathbf {Br}_h(K)$ (Theorem 3.14).

The third part of the article applies the general results of the first part to the category $\mathbf {Br}_h(K,\iota )$ to define our mixed rings.

Section 4.1 simply defines the underlying (Grothendieck–)Witt groups, and checks that they are functorial with respect to $\mathbf {Br}_h(K,\iota )$ . In order to make them truly functorial as graded objects, a harmless change of labeling is needed (the issue being that anti-hermitian equivalences reverse the signs of hermitian forms instead of preserving them).

In Section 4.2, we establish that those functors are actually monoidal, and thus that the whole machinery of Section 2.4 can be used. This leads to the definition of the rings $\widetilde {GW}(A,\sigma )$ and $\widetilde {W}(A,\sigma )$ (as well as the less interesting $\widehat {GW}(A,\sigma )$ and $\widehat {W}(A,\sigma )$ for unitary involutions).

Section 4.3 presents how to handle the (reduced) dimension of hermitian modules, in a way compatible with our graded ring structure.

In order to perform explicit computations in our rings, we describe in Section 4.4 how to multiply diagonal forms; it turns out the result is given by the twisted involution trace forms introduced in [Reference Knus, Merkurjev, Rost and Tignol15, §11] (Proposition 4.9). Whether or not this actually makes the product explicit is a matter of opinion, as those forms are actually quite difficult to compute in practice.

There is at least a case where we can reasonably say that the products are fully computable, namely the case of a quaternion algebra, endowed with its canonical involution. We work this out in Section 4.5 (Proposition 4.12).

Finally, the last Section 4.6, which is the most technical one of the third part, is dedicated to scalar extensions, and especially to establishing a Frobenius reciprocity formula (compare Theorem 4.16 with [Reference Scharlau23, 2.5.6]).

1.4 Perspectives

It should be noted that while in this article we work with algebras over fields, basically everything carries over to Azumaya algebras with involution over general commutative rings. The key ingredient of Corollary 3.13 will still hold in that general context thanks to the existence of the Goldman element. It can even be envisioned to work with Azumaya algebras over schemes, or even locally ringed topos, as studied in [Reference First and Williams11].

In addition to extending the algebras, we could also extend Grothendieck–Witt rings to the whole hermitian K-theory, and actually to simple algebraic K-theory if we forget about the involutions. The fact that such extensions could be easily handled with very little extra legwork is a compelling argument for our general abstract presentation of the mechanisms at play as properties of monoidal categories.

Even considering simply algebras with involutions over fields, we intend to build on the present article in future work to study, among other things: a $\lambda $ -ring structure on $\widetilde {GW}(A,\sigma )$ (the existence of which is a strong advantage over $\widetilde {W}(A,\sigma )$ ); an extension of Artin–Schreier theory for the spectrum of $\widetilde {W}(A,\sigma )$ (related to the work of Astier and Unger in [Reference Astier and Unger2]); and applications of our structure to the construction of cohomological invariants of algebras with involution and algebraic groups.

1.5 Preliminaries and conventions

We fix a base field K of characteristic not 2, and we identify symmetric bilinear forms and quadratic forms over K, through $b\mapsto q_b$ with $q_b(x)=b(x,x)$ . Diagonal quadratic forms are denoted $\langle a_1,\dots ,a_n\rangle $ , with $a_i\in K^*$ , and $\langle \!\langle a_1,\dots ,a_n\rangle \!\rangle $ is the n-fold Pfister form $\langle 1,-a_1\rangle \otimes \cdots \otimes \langle 1,-a_n\rangle $ . We always assume that bilinear forms are nondegenerate.

All rings are associative and with unit, and ring morphisms preserve the unit. The group of invertible elements of a ring A is denoted $A^\times $ . The opposite ring of A (that is, the ring with the reversed product) is denoted $A^{op}$ . Unless otherwise specified, modules are by default modules on the right, and are assumed to be nonzero. Every K-algebra and every module over such an algebra are required to have finite dimension over K. If A and B are K-algebras, a B–A-bimodule is always supposed to be over K, meaning that the right and left actions of K on V coincide. If A is a central simple algebra over K, we write $\operatorname {\mathrm {Trd}}_A:A\to K$ for the reduced trace of A.

We fix an automorphism $\iota $ of K such that $\iota ^2=\operatorname {\mathrm {Id}}$ , and we let $k=K^\iota $ be the subfield of fixed points. If $\iota =\operatorname {\mathrm {Id}}$ , then $K=k$ , but if $\iota \neq \operatorname {\mathrm {Id}}$ , then $K/k$ is a quadratic extension. Note that we do not include the split case $K=k\times k$ , to avoid having to discuss it separately. The reader is encouraged to check that everything would still work in that context.

When we say that $(A,\sigma )$ is an algebra with involution over $(K,\iota )$ , we mean that A is a central simple algebra over K, and that $\sigma $ is an involution on A with $\sigma _{|K}=\iota $ , so $\sigma $ is a k-algebra anti-automorphism of A, with $\sigma ^2=\operatorname {\mathrm {Id}}_A$ .

Recall that the involution $\sigma $ is of the first kind when $\iota = \operatorname {\mathrm {Id}}$ , and is of the second kind (or unitary) otherwise. An involution of the first kind is orthogonal if $\dim _K(\operatorname {\mathrm {Sym}}(A,\sigma )) = n(n+1)/2$ where n is the degree of A, and it is symplectic if $\dim _K(\operatorname {\mathrm {Sym}}(A,\sigma ))=n(n-1)/2$ . In particular, $(K,\operatorname {\mathrm {Id}})$ is an algebra with orthogonal involution. A quaternion algebra admits a unique symplectic involution, called its canonical involution.

We set $U(K,\iota ) = \{\varepsilon \in K^\times |\varepsilon \iota (\varepsilon )=1\}$ ; it is a subgroup of $K^\times $ , and when $\iota =\operatorname {\mathrm {Id}}$ , it is simply $\mu _2(K)$ . If $\varepsilon \in U(K,\iota )$ , we define $\operatorname {\mathrm {Sym}}^\varepsilon (A,\sigma )$ as the set of $\varepsilon $ -symmetric elements of A, which satisfy $\sigma (a)=\varepsilon a$ . We also write $\operatorname {\mathrm {Sym}}^\varepsilon (A^\times ,\sigma )$ for the set of invertible $\varepsilon $ -symmetric elements.

If $L/K$ is any field extension, and X is an object (algebra, module, involution, hermitian form, etc.) over K, then $X_L$ is the corresponding object over L, obtained by base change.

A semigroup is a set endowed with an associative binary product (so the difference to a monoid is that the semigroup might lack an identity element). The Grothendieck group $G(S)$ of S is the universal solution to the problem of finding a morphism $S\to G$ where G is a group.