Brandt groupoids

Abstract

In this chapter, we study the relationship between multiplication and classes of quaternion ideals.

In this chapter, we study the relationship between multiplication and classes of quaternion ideals.

1 $\triangleright$ Composition laws and ideal multiplication

To guide our investigations, we again appeal to the quadratic case. Let $d \in \mathbb Z$ be a nonsquare discriminant. A subject of classical interest was the set of integral  primitive binary quadratic forms of discriminant d, namely

$$\begin{aligned} \mathcal Q (d) = \{ax^2+bxy+cy^2 : a,b,c \in \mathbb Z , b^2-4ac=d,\text { and }\gcd (a,b,c)=1\}. \end{aligned}$$

Of particular interest to early number theorists (Fermat, Legendre, Lagrange, and Gauss) was the set of primes represented by a quadratic form $Q \in \mathcal Q (d)$; inquiries of this nature proved to be quite deep, giving rise to the law of quadratic reciprocity and the beginnings of the theory of complex multiplication and class field theory.

An invertible, oriented change of variables on a quadratic form $Q \in \mathcal Q (d)$ does not alter the set of primes represented, so one is naturally led to study the equivalence classes of quadratic forms under the (right) action of the group ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$ given by

$$\begin{aligned} (Q \mid g)(x,y) = Q((x,y)\cdot g) \quad \text {for}\quad g \in {{\,\mathrm{SL}\,}}_2(\mathbb Z ). \end{aligned}$$

(19.1.1)

The set ${{\,\mathrm{Cl}\,}}(d)$ of ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$-classes of forms in $\mathcal Q (d)$ is finite, by reduction theory (see section 35.2): every form in $\mathcal Q (d)$ is equivalent under the action of ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$ to a unique reduced form, of which there are only finitely many. To study this finite set, Gauss defined a composition law on ${{\,\mathrm{Cl}\,}}(d)$, giving ${{\,\mathrm{Cl}\,}}(d)$ the structure of an abelian group by an explicit formula. Gauss’s composition law on binary quadratic forms can be understood using $2 \times 2 \times 2$ Rubik’s cubes, by a sublime result of Bhargava [Bha2004a].

Today, we see this composition law as a consequence of a natural bijection between ${{\,\mathrm{Cl}\,}}(d)$ and a set equipped with an obvious group structure. Let $S=S(d)$ be the quadratic ring of discriminant d. Define the narrow class group ${{\,\mathrm{Cl}\,}}^+(S)$ as

$$\begin{aligned} & \qquad\qquad\qquad\text{the group of invertible fractional ideals of} \, S \, \text{under multiplication}\\ &\text{modulo}\\ & \qquad\qquad\qquad\qquad\quad\text{the subgroup of nonzero principal fractional ideals} \\ & \qquad\qquad\qquad\qquad\qquad\quad\text{generated by a totally positive element} \end{aligned}$$

(i.e., one that is positive in every embedding into $\mathbb R$, so if $d<0$ then this is no condition). (Alternatively, ${{\,\mathrm{Cl}\,}}^+(S)$ can be thought of as the group of isomorphism classes of oriented, invertible S-modules, under a suitable notion of orientation.) Then there is a bijection between ${{\,\mathrm{Cl}\,}}(d)$ and ${{\,\mathrm{Cl}\,}}^+(S)$: explicitly, to the class of the quadratic form $Q=ax^2+bxy+cy^2 \in \mathcal Q (d)$, we associate the class of the ideal

$$\begin{aligned} \mathfrak a =a\mathbb Z + \biggl (\frac{-b+\sqrt{d}}{2}\biggr )\mathbb Z \subset S(d). \end{aligned}$$

(19.1.2)

Conversely, the quadratic form is recovered as the norm form on $K=\mathbb Q (\sqrt{d})$ restricted to $\mathfrak a$:

$$\begin{aligned} {{\,\mathrm{Nm}\,}}_{K/\mathbb Q }\biggl (ax+\frac{-b+\sqrt{d}}{2}y\biggr ) = ax^2+bxy+cy^2 \end{aligned}$$

(19.1.3)

where $c=\frac{b^2-d}{4a} \in \mathbb Z$.

Much of the same structure can be found in the quaternionic case, with several interesting twists. It was Brandt who first asked if there was a composition law for (integral, primitive) quaternary quadratic forms: it would arise naturally from some kind of multiplication of ideals in a quaternion order, with the analogous bijection furnished by the reduced norm form. Brandt started writing on composition laws for quaternary quadratic forms in 1913 [Bra13], tracing the notion of composition back to Hermite, who observed a kind of multiplication law (bilinear substitution) for quaternary forms $x_0^2+F(x_1,x_2,x_3)$ in formulas of Euler and Lagrange. He continued on this note during the 1920s [Bra24, Bra25, Bra28, Bra37], when it became clear that quaternion algebras was the right framework to place his composition laws; in 1943, he developed this theme significantly [Bra43] and defined his Brandt matrices (that will figure prominently in Chapter 41.

However, in the set of invertible lattices in B under compatible product, one cannot always multiply! However, this set has the structure of a groupoid: a nonempty set with an inverse function and a partial product that satisfies the associativity, inverse, and identity properties whenever they are defined. Groupoids now figure prominently in category theory (a groupoid is equivalently a small category in which every morphism is an isomorphism) and many other contexts; see Remark 19.3.12.

Organizing lattices by their left and right orders, which by definition are connected and hence in the same genus, we define

$$\begin{aligned} {{\,\mathrm{Brt}\,}}(\mathcal {O})=\{I : I \subset B\text { invertible }R\text {-lattice and }\mathcal {O}{}_{\textsf {\tiny {L}} }(I),\mathcal {O}{}_{\textsf {\tiny {R}} }(I) \in {{\,\mathrm{Gen}\,}}\mathcal {O}\}; \end{aligned}$$

(19.1.4)

visibly, ${{\,\mathrm{Brt}\,}}(\mathcal {O})$ depends only on the genus of $\mathcal {O}$. Organizing lattices according to the genus of orders is sensible: after all, we only apply the composition law to binary quadratic forms of the same discriminant, and in the compatible product we see precisely those classes whose left and right orders are connected. In other words, the set of invertible lattices in the quadratic field $K=\mathbb Q (\sqrt{d})$ has the structure of a groupoid if we multiply only those lattices with the same multiplicator ring.

Theorem 19.1.5

Let B be a quaternion algebra over $\mathbb Q$ and let $\mathcal {O}\subset B$ be an order. Then the set ${{\,\mathrm{Brt}\,}}(\mathcal {O})$ has the structure of a groupoid under compatible product.

We call ${{\,\mathrm{Brt}\,}}(\mathcal {O})$ the Brandt groupoid of (the genus of) $\mathcal {O}$.

We now consider classes of lattices. A lattice $I \subset B$ has the structure of a $\mathcal {O}{}_{\textsf {\tiny {L}} }(I),\mathcal {O}{}_{\textsf {\tiny {R}} }(I)$-bimodule. Two invertible lattices I, J with the same left and right orders $\mathcal {O}{}_{\textsf {\tiny {L}} }(I)=\mathcal {O}{}_{\textsf {\tiny {L}} }(J)$ and $\mathcal {O}{}_{\textsf {\tiny {R}} }(I)=\mathcal {O}{}_{\textsf {\tiny {R}} }(J)$ are isomorphic as bimodules if and only if there exists $a \in \mathbb Q ^\times$ such that $J=aI$. Accordingly, we say two lattices $I,J \subset B$ are homothetic if there exists $a \in \mathbb Q ^\times$ such that $J=aI$.

For connected orders $\mathcal {O},\mathcal {O}' \subset B$, we define

$$\begin{aligned} {{\,\mathrm{Pic}\,}}(\mathcal {O},\mathcal {O}') :=\{[ I ] : I \subset B \text { invertible and } \mathcal {O}{}_{\textsf {\tiny {L}} }(I)=\mathcal {O}\text { and } \mathcal {O}{}_{\textsf {\tiny {R}} }(I)=\mathcal {O}' \} \end{aligned}$$

(19.1.6)

to be the set of homothety classes of lattices with left order $\mathcal {O}$ and right order $\mathcal {O}'$, or equivalently the set of isomorphism classes of $\mathcal {O},\mathcal {O}'$-bimodules over R. Restricting to the subset of lattices with $\mathcal {O}=\mathcal {O}'$, and the lattices $I \subset B$ are $\mathcal {O}$-bimodules, we recover ${{\,\mathrm{Pic}\,}}(\mathcal {O},\mathcal {O})={{\,\mathrm{Pic}\,}}\mathcal {O}$ the Picard group from the previous chapter.

Now let $\mathcal {O}\subset B$ be an order and let $\mathcal {O}_i$ be representative orders for the type set ${{\,\mathrm{Typ}\,}}\mathcal {O}$. Let

$$\begin{aligned} {{\,\mathrm{BrtCl}\,}}\mathcal {O}:=\bigsqcup _{i,j} \,{{\,\mathrm{Pic}\,}}(\mathcal {O}_i,\mathcal {O}_j). \end{aligned}$$

(19.1.7)

Theorem 19.1.8

Let B be a quaternion algebra over $\mathbb Q$ and let $\mathcal {O}\subset B$ be an order. Then the set ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ has the structure of a groupoid that, up to isomorphism, is independent of the choice of the orders $\mathcal {O}_i$.

In particular, ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ depends only on the genus of $\mathcal {O}$. We call the set ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ the Brandt class groupoid of (the genus of) $\mathcal {O}$.

Returning to quadratic forms, to each R-lattice I with ${{\,\mathrm{nrd}\,}}(I)=a\mathbb Z$ and $a>0$, we associate the quadratic form

$$\begin{aligned} {{\,\mathrm{nrd}\,}}_I:I&\rightarrow \mathbb Z \\ {{\,\mathrm{nrd}\,}}_I(\mu )&={{\,\mathrm{nrd}\,}}(\mu )/a \end{aligned}$$

Alternatively, up to similarity we can just take the quadratic module ${{\,\mathrm{nrd}\,}}|_I :I \rightarrow {{\,\mathrm{nrd}\,}}(I)$ remembering that the quadratic form takes values in ${{\,\mathrm{nrd}\,}}(I)$. The discriminant of an invertible lattice $I \subset B$ is equal to the common discriminant $N^2$ of the genus of its left or right order. The quadratic forms ${{\,\mathrm{nrd}\,}}_I$ are all locally similar, respecting the canonical orientation 5.6.7 on B. Therefore, there is a map

is (well-defined and) surjective. Unfortunately, this map is not injective (a reflection of the lack of a natural quotient groupoid homomorphism): the Brandt class is a kind of rigidification of the oriented similarity class. Nevertheless, Theorem 19.1.8 can be viewed as a generalization of Gauss composition of binary quadratic forms, defining a partial composition law on (rigidified) classes of quaternary quadratic forms.

2 Example

Consider the quaternion algebra $B :=\displaystyle {\biggl (\frac{-2,-37}{\mathbb {Q}}\biggr )}$ with standard basis 1, i, j, k, and the maximal order $\mathcal {O}$ of reduced discriminant 37 defined by

$$\begin{aligned} \mathcal {O}:=\mathbb Z + i\mathbb Z +\mathbb Z \frac{1+i+j}{2}\mathbb Z + \frac{2+i+k}{4}\mathbb Z . \end{aligned}$$

(19.2.1)

The type set ${{\,\mathrm{Typ}\,}}\mathcal {O}$ of orders connected to $\mathcal {O}$ has exactly two isomorphism classes, represented by $\mathcal {O}_1=\mathcal {O}$ and

$$\begin{aligned} \mathcal {O}_2 :=\mathbb Z + 3i\mathbb Z + \frac{3-7i+j}{6}\mathbb Z + \frac{2-3i+k}{4}\mathbb Z . \end{aligned}$$

These orders are connected by the $\mathcal {O}_2,\mathcal {O}_1$-connecting ideal

$$\begin{aligned} I :=3\mathbb Z + 3i\mathbb Z + \frac{3-i+j}{2}\mathbb Z + \frac{2+3-k}{4}\mathbb Z = 3\mathcal {O}+ \frac{3-i+j}{2}\mathcal {O}. \end{aligned}$$

There are isomorphisms

$$\begin{aligned} {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \simeq {{\,\mathrm{Pic}\,}}(\mathcal {O}_2) \simeq \mathbb Z /2\mathbb Z \end{aligned}$$

with the nontrivial class in ${{\,\mathrm{Pic}\,}}(\mathcal {O}_1)$ represented by the principal two-sided ideal $J_1=j \mathcal {O}=\mathcal {O}j$ with $j \in N_{B^\times }(\mathcal {O})$, and the nontrivial class in ${{\,\mathrm{Pic}\,}}(\mathcal {O}_2)$ represented by the nonprincipal (but invertible) ideal

$$\begin{aligned} J_2 :=I J_1 I^{-1} = 37\mathcal {O}_2 + \frac{111-259i+j}{6}\mathcal {O}_2. \end{aligned}$$

In particular,

$$\begin{aligned} {{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }(\mathcal {O}_1)=\{[\mathcal {O}_1],[I],[J_2I]\}\quad \text {and}\quad {{\,\mathrm{Cls}\,}}\!{}_{\textsf {\tiny {R}} }(\mathcal {O}_2)=\{[\mathcal {O}_2],[J_2],[\overline{I}]\}, \end{aligned}$$

with $[J_1]=[\mathcal {O}_1]$.

We can visualize this groupoid as a graph as in Figure 19.2.2, with directed edges for multiplication:

Figure 19.2.2:

${{\,\mathrm{BrtCl}\,}}\mathcal {O}$, for ${{\,\mathrm{discrd}\,}}\mathcal {O}=37$

The Brandt class groupoid

$$\begin{aligned} {{\,\mathrm{BrtCl}\,}}\mathcal {O}= {{\,\mathrm{Pic}\,}}(\mathcal {O}_1) \sqcup {{\,\mathrm{Cl}\,}}(\mathcal {O}_1,\mathcal {O}_2) \sqcup {{\,\mathrm{Cl}\,}}(\mathcal {O}_2,\mathcal {O}_1) \sqcup {{\,\mathrm{Pic}\,}}(\mathcal {O}_2) \end{aligned}$$

has $2+4+4+2=12$ elements; it is generated as a groupoid by the elements $[ J_1 ],[ J_2 ],[ I ]$, with relations

$$ [ J_1 ]^2 = [ \mathcal {O}_1 ], \quad [ J_2 ]^2 = [ \mathcal {O}_2 ], \quad [ J_2 ] [ I ] = [ I ] [ J_1 ] . $$

Restricting the reduced norm to these lattices, we obtain classes of quaternary quadratic forms of discriminant $37^2$:

$$\begin{aligned} {{\,\mathrm{nrd}\,}}_{\mathcal {O}_1}&= t^2+ty+tz+2x^2+xy+2xz+5y^2+yz+10z^2 \\ {{\,\mathrm{nrd}\,}}_{\mathcal {O}_2}&= t^2+tx+tz +4x^2 - xy +4xz +5y^2 +2yz +6z^2 \\ {{\,\mathrm{nrd}\,}}_I&= 3t^2 - tx + ty + tz + 3x^2 - 3xy - xz + 4y^2 - yz + 5z^2 \\ {{\,\mathrm{nrd}\,}}_{\overline{I}} = Q(I^{-1})&= 3t^2 + tx - ty - tz + 3x^2 - 3xy - xz + 4y^2 - yz + 5z^2 \\ {{\,\mathrm{nrd}\,}}_{J_2}&= 2t^2 - tx + ty + 2x^2 - 2xy + xz + 3y^2 + 2yz + 10z^2 \end{aligned}$$

The quadratic forms ${{\,\mathrm{nrd}\,}}_I$ and ${{\,\mathrm{nrd}\,}}_{\overline{I}}$ are isometric but not by an oriented isometry.

3 Groupoid structure

We begin with some generalities on groupoids.

Definition 19.3.1

A partial function $f:X \rightarrow Y$ is a function defined on a subset of the domain X.

Definition 19.3.2

A groupoid G is a set with a unary operation ${}^{-1}:G \rightarrow G$ and a partial function $*:G \times G \rightarrow G$ such that $*$ and ${}^{-1}$ satisfy the associativity, inverse, and identity properties (as in a group) whenever they are defined:

• (a) [Associativity] For all $a,b,c \in G$ such that $a*b$ is defined and $(a*b)*c$ is defined, both $b*c$ and $a*(b*c)$ are defined and

  $$\begin{aligned} (a*b)*c=a*(b*c). \end{aligned}$$

• (b) [Inverses] For all $a \in G$, there exists $a^{-1} \in G$ such that $a*a^{-1}$ and $a^{-1}*a$ are defined (but not necessarily equal).

• (c) [Identity] For all $a,b \in G$ such that $a*b$ is defined, we have

  $$\begin{aligned} (a*b)*b^{-1}=a \quad \text {and} \quad a^{-1}*(a*b)=b. \end{aligned}$$

  (19.3.3)

A homomorphism $\phi :G \rightarrow G'$ of groupoids is a map satisfying

$$\begin{aligned} \phi (a*b)=\phi (a)*\phi (b) \end{aligned}$$

for all $a,b \in G$.

19.3.4

Let G be a groupoid. Then the products in the identity law (19.3.3) are defined by the associative and inverse laws, and it follows that $e=a*a^{-1}$, the left identity of a, and $f=a^{-1}*a$ the corresponding right identity of a, satisfy $e*a=a=a*f$ for all $a \in G$. (We may have that $e \ne f$, i.e., the left and right identities for $a \in G$ disagree.) The right identity of $a \in G$ is the left identity of $a^{-1} \in G$, so we call the set

$$\begin{aligned} \{e=a*a^{-1} : a \in G\} \end{aligned}$$

the set of identity elements in G.

19.3.5

Equivalently, a groupoid is a small category (the class of objects in the category is a set) such that every morphism is an isomorphism: given a groupoid, we associate the category whose objects are the elements of the set $S:=\{e=a*a^{-1} : a \in G\}$ of identity elements in G and the morphisms between $e,f \in S$ are the elements $a \in G$ such that $e*a$ and $a*f$ are defined (see Proposition 19.3.9 below). Conversely, to a category in which every morphism is an isomorphism, we associate the groupoid whose underlying set is the union of all morphisms under inverse and composition.

Example 19.3.6

The set of homotopy classes of paths in a topological space X forms a groupoid under composition: the paths $\gamma _1,\gamma _2:[0,1] \rightarrow X$ can be composed to a path $\gamma _2\circ \gamma _1:[0,1] \rightarrow X$ if and only if $\gamma _2(0)=\gamma _1(1)$.

Example 19.3.7

A disjoint union of groups is a groupoid, with the product defined if and only if the elements belong to the same group; the set of identities is canonically in bijection with the index set of the disjoint union.

19.3.8

Let G be a groupoid and let $e,f \in G$ be identity elements. We say that e is connected  to f if there exists $a \in G$ such that a has left identity e and right identity f. The relation of being connected defines an equivalence relation on the set of identity elements in G, and the resulting equivalence classes are called connected components of G. We say G is connected if all identity elements $e,f \in G$ are connected; connected components of a groupoid are connected.

Viewing the groupoid G as a small category as in 19.3.5, we say two objects are connected if there exists a morphism between them, and the category is connected if every two objects are connected.

If $e \in G$ is an identity element in a groupoid G, then the set of elements $a \in G$ with left and right identity equal to e has the structure of a group; for the associated category, this is the automorphism group of the object. More generally, the following structural result holds.

Proposition 19.3.9

Let G be a connected groupoid, and let e, f be identity elements in G. Let

$$\begin{aligned} G(e,f):=\{a \in G : e*a\,\mathrm {and}\,a*f\,\mathrm {are\,\, defined}\}. \end{aligned}$$

(19.3.10)

Then the following statements hold.

• (a) The set G(e, e) is a group under $*$.

• (b) There is a (noncanonical) isomorphism $G(e,e) \simeq G(f,f)$.

• (c) The set G(e, f) is a principal homogeneous space for $G(e,e) \simeq G(f,f)$.

Proof. The set G(e, e) is nonempty, has the identity element $e \in G$, and if $a \in G(e,e)$ then $a*a^{-1}=a^{-1}*a = e$. If e, f are identity elements, since G is connected there exists $a \in G(e,f)$, so $a^{-1} \in G(f,e)$ and the map $G(e,e) \rightarrow G(f,f)$ by $x \mapsto a * x * a^{-1}$ is an isomophism of groups. Similarly, the set G(e, f) has a right, simply transitive action of G(e, e) under right multiplication by $*$. $\square$

19.3.11

The moral of Proposition 19.3.9 is that the only two interesting invariants of a connected groupoid are the number of identity elements (objects in the category) and the group of elements with a common left and right identity (the automorphism group of every one of the objects). A connected groupoid is determined up to isomorphism of groupoids by these two properties.

Remark 19.3.12. After seeing its relevance in the context of composition of quaternary forms, Brandt set out general axioms for his notion of a groupoid [Bra27, Bra40]. (Brandt’s original definition of groupoid is now called a connected groupoid.) This notion has blossomed into an important structure in mathematics that sees quite general use, especially in homotopy theory and category theory. It is believed that the groupoid axioms influenced the work of Eilenberg–Mac Lane [EM45] in the first definition of a category: see e.g., Brown [Bro87] for a survey, Bruck [Bruc71] for context in the theory of binary structures, as well as the article by Weinstein [Wein96].

Groupoids exhibit many facets of mathematics, arising naturally in functional analysis ($C^*$-algebras) and group representations, as Figure 19.3.13 indicates (appearing in Williams [Will2001, p. 21], and attributed to Arlan Ramsay).

Figure 19.3.13:

Groupoids, as they relate to other mathematical objects

(In this diagram, for example, a set X is a groupoid with only the multiplications $x*x=x$ for $x \in X$. The corner between sets and groups can be explained by a set with one element which can be made into a group in a unique way.)

4 Brandt groupoid

Let R be a Dedekind domain with field of fractions F and let B be a quaternion algebra over F.

Proposition 19.4.1

The set of invertible R-lattices in B is a groupoid under inverse and compatible product; the R-orders in B are the identity elements in this groupoid.

Proof. For the associative law, suppose I, J, K are invertible R-lattices with IJ and (IJ)K compatible products. Then $\mathcal {O}{}_{\textsf {\tiny {R}} }(I)=\mathcal {O}{}_{\textsf {\tiny {L}} }(J)=\mathcal {O}{}_{\textsf {\tiny {L}} }(JK)$ and $\mathcal {O}{}_{\textsf {\tiny {R}} }(IJ)=\mathcal {O}{}_{\textsf {\tiny {R}} }(J)=\mathcal {O}{}_{\textsf {\tiny {L}} }(K)$ by Lemma 16.5.11, so the products JK and I(JK) are compatible. Multiplication is associative in B, and it follows that $I(JK)=(IJ)K$. Inverses exist exactly because we restrict to the invertible lattices.

The law of identity holds as follows: if I, J are invertible R-lattices such that IJ is a compatible product, then $(IJ)J^{-1}$ is a compatible product since $\mathcal {O}{}_{\textsf {\tiny {R}} }(IJ)=\mathcal {O}{}_{\textsf {\tiny {R}} }(J)=\mathcal {O}{}_{\textsf {\tiny {L}} }(J^{-1})$, and by associativity

$$\begin{aligned} (IJ)J^{-1} = I(JJ^{-1})=I \mathcal {O}{}_{\textsf {\tiny {L}} }(J)=I\mathcal {O}{}_{\textsf {\tiny {R}} }(I)=I, \end{aligned}$$

with a similar argument on the left. If I is an invertible R-lattice, then $II^{-1}=\mathcal {O}{}_{\textsf {\tiny {L}} }(I)$ is an R-order in B, and every R-order $\mathcal {O}$ arises by taking $I=\mathcal {O}$ itself, so the R-orders are the identity elements in the groupoid. $\square$

Lemma 19.4.2

The connected components of the groupoid of invertible R-lattices in B are identified by the genus of the (left or) right order, and the group defined on such a component corresponding to an order $\mathcal {O}$ is ${{\,\mathrm{Idl}\,}}(\mathcal {O})$, the group of invertible two-sided $\mathcal {O}$-ideals.

Proof. By Proposition 19.4.1, the identity elements correspond to orders, and two orders are connected if and only if there is a (invertible, equivalently locally principal) connecting ideal if and only if they are in the same genus, as in section 17.4. The second statement follows immediately. $\square$

As a consequence of Lemma 19.4.2, the subset of R-lattices whose (left or) right order belong to a specified genus of orders is a connected subgroupoid.

Definition 19.4.3

Let $\mathcal {O}\subseteq B$ be an R-order. The Brandt groupoid of (the genus of) $\mathcal {O}$ is

$$\begin{aligned} {{\,\mathrm{Brt}\,}}(\mathcal {O})=\{I : I \subset B\text { invertible }R\text {-lattice and }\mathcal {O}{}_{\textsf {\tiny {L}} }(I),\mathcal {O}{}_{\textsf {\tiny {R}} }(I) \in {{\,\mathrm{Gen}\,}}\mathcal {O}\}. \end{aligned}$$

In the next section, we consider a variant that considers classes of lattices, giving rise to a finite groupoid.

5 Brandt class groupoid

We now organize lattices up to isomorphism as bimodules for their left and right orders.

Lemma 19.5.1

Let $I,J \subset B$ be lattices with $\mathcal {O}{}_{\textsf {\tiny {L}} }(I)=\mathcal {O}{}_{\textsf {\tiny {L}} }(J)=\mathcal {O}$ and $\mathcal {O}{}_{\textsf {\tiny {R}} }(I)=\mathcal {O}{}_{\textsf {\tiny {R}} }(J)=\mathcal {O}'$. Then I is isomorphic to J as $\mathcal {O},\mathcal {O}'$-bimodules if and only if there exists $a \in F^\times$ such that $J=aI$.

Proof. We have $F=Z(B)$. If $J=aI$ with $a \in F^\times$, then multiplication by a gives an R-module isomorphism $I \rightarrow J$ that commutes with the left and right actions and so defines a $\mathcal {O},\mathcal {O}'$-bimodule isomorphism.

Conversely, suppose that is a $\mathcal {O},\mathcal {O}'$-bimodule isomorphism. Then $\phi (\mu \alpha \nu )=\mu \phi (\alpha )\nu$ for all $\alpha \in I$ and $\mu ,\nu \in \mathcal {O}$. Extending scalars to B, we obtain a B-bimodule isomorphism $\phi :IF=B \rightarrow JF=B$. Let $\phi (1)=\beta$. Then for all $\alpha \in B$, we have $\phi (\alpha )=\phi (1)\alpha =\beta \alpha$; but by the same token, $\phi (\alpha )=\alpha \beta$ for all $\alpha \in B$, so $\beta \in Z(B)=F$. $\square$

Definition 19.5.2

Let $I,J \subseteq B$ be R-lattices. We say that I is homothetic to J if there exists $a \in F^\times$ such that $J=aI$.

Homothety defines an equivalence relation, and we let [I] denote the homothety class of an R-lattice I. The left and right order of a homothety class is well-defined.

19.5.3

The set of homothety classes of invertible R-lattices $I \subseteq B$ has the structure of a groupoid under compatible product, since the compatible product [IJ] is well-defined: if $I'=aI$ and $J'=bJ$ with $a,b \in F^\times$, then $[I'J']=[abIJ]=[IJ]$ since a, b are central.

The map which takes an invertible lattice to its homothety class yields a surjective homomorphism of groupoids. Taking connected components we obtain a connected groupoid associated to a (genus of an) R-order $\mathcal {O}$. Recalling 19.3.11, we note that the group at an R-order $\mathcal {O}$ is ${{\,\mathrm{Pic}\,}}_R(\mathcal {O})$, but there are still infinitely many orders (objects in the category).

In order to whittle down to a finite groupoid, we fix representatives of the type set, and make the following definitions.

19.5.4

For R-orders $\mathcal {O},\mathcal {O}' \subseteq B$, let

$$\begin{aligned} {{\,\mathrm{Pic}\,}}_R(\mathcal {O},\mathcal {O}')=\{[ I ] : I \subset B \text { invertible and }\mathcal {O}{}_{\textsf {\tiny {L}} }(I)=\mathcal {O}, \mathcal {O}{}_{\textsf {\tiny {R}} }(I)=\mathcal {O}' \} \end{aligned}$$

be the set of homothety classes of R-lattices in B with left order $\mathcal {O}$ and right order $\mathcal {O}'$; equivalently, by Lemma 19.5.1, ${{\,\mathrm{Pic}\,}}(\mathcal {O},\mathcal {O}')$ is the set of isomorphism classes of invertible $\mathcal {O},\mathcal {O}'$-bimodules over R. In particular, ${{\,\mathrm{Pic}\,}}_R(\mathcal {O})={{\,\mathrm{Pic}\,}}_R(\mathcal {O},\mathcal {O})$.

We have ${{\,\mathrm{Pic}\,}}_R(\mathcal {O},\mathcal {O}') \ne \emptyset$ if and only if $\mathcal {O}$ is connected to $\mathcal {O}'$.

Let $\mathcal {O}\subset B$ be an order and let $\mathcal {O}_i$ be representative orders for the type set ${{\,\mathrm{Typ}\,}}\mathcal {O}$. We define

$$\begin{aligned} {{\,\mathrm{BrtCl}\,}}\mathcal {O}:=\bigsqcup _{i,j} {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i,\mathcal {O}_j). \end{aligned}$$

Theorem 19.5.5

Let R be a Dedekind domain with field of fractions F, and let B be a quaternion algebra over F. Let $\mathcal {O}\subset B$ be an order. Then the set ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ has the structure of a finite groupoid that, up to isomorphism, is independent of the choice of the orders $\mathcal {O}_i$.

In particular, by Theorem 19.5.5 ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ depends only on the genus of $\mathcal {O}$ up to groupoid isomorphism. We call the set ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ the Brandt class groupoid  of (the genus of) $\mathcal {O}$.

Proof. The groupoid structure is compatible multiplication, with

$$\begin{aligned} {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i,\mathcal {O}_j) {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_j,\mathcal {O}_k) \subseteq {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i,\mathcal {O}_k) \end{aligned}$$

for all i, j, k; in other words, ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ is a connected subgroupoid of the groupoid of homothety classes of R-lattices 19.5.4.

The groupoid is finite, by 19.3.11: the type set ${{\,\mathrm{Typ}\,}}\mathcal {O}$ is finite by Main Theorem 17.7.1 and ${{\,\mathrm{Pic}\,}}_R(\mathcal {O})$ is finite by Proposition 18.4.10. Explicitly, if $[I_{ij}] \in {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i,\mathcal {O}_j)$ then the map

$$\begin{aligned} {{\,\mathrm{Pic}\,}}_R(\mathcal {O}) \simeq {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i)&\rightarrow {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i,\mathcal {O}_j) \\ [I]&\mapsto [II_{ij}] \end{aligned}$$

is a bijection of sets, just as in the proof of Proposition 19.3.9. Therefore

$$\begin{aligned} \#{{\,\mathrm{BrtCl}\,}}\mathcal {O}=\#{{\,\mathrm{Pic}\,}}_R(\mathcal {O})\#{{\,\mathrm{Typ}\,}}\mathcal {O}. \end{aligned}$$

(19.5.6)

Finally, this subgroupoid is independent of the choices of the orders $\mathcal {O}_i$ as follows: all other choices correspond to $\mathcal {O}_i'=\alpha _i \mathcal {O}_i \alpha _i^{-1}$ with $\alpha _i \in B^\times$, and the induced maps

$$\begin{aligned} {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i,\mathcal {O}_j)&\rightarrow {{\,\mathrm{Pic}\,}}(\mathcal {O}_i',\mathcal {O}_j') \\ [I]&\mapsto [\alpha _i I \alpha _j^{-1}]=[I'] \end{aligned}$$

together give an isomorphism of groupoids, since

$$\begin{aligned}{}[I'J']=[\alpha _i I \alpha _j^{-1} \alpha _j J \alpha _k^{-1}] = [\alpha _i IJ \alpha _k^{-1} \end{aligned}$$

for all $[I] \in {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_i,\mathcal {O}_j)$ and $[J] \in {{\,\mathrm{Pic}\,}}_R(\mathcal {O}_j,\mathcal {O}_k)$. $\square$

Remark 19.5.7. Unfortunately, there is not in general a natural equivalence relation on ${{\,\mathrm{Brt}\,}}(\mathcal {O})$ giving rise to a quotient groupoid homomorphism ${{\,\mathrm{Brt}\,}}(\mathcal {O}) \rightarrow {{\,\mathrm{BrtCl}\,}}\mathcal {O}$. Rather, we find that ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ is naturally a subgroupoid of ${{\,\mathrm{Brt}\,}}(\mathcal {O})$.

Turning to the invariants 19.3.11, we see that the Brant class groupoid ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$ encodes two things: the group ${{\,\mathrm{Pic}\,}}_R(\mathcal {O})$ and the type set ${{\,\mathrm{Typ}\,}}\mathcal {O}$.

Remark 19.5.8. The modern theory of Brandt composition was investigated by Kaplansky [Kap69] and generalized to Azumaya quaternion algebras over commutative rings by Kneser–Knus–Ojanguren–Parimala–Sridharan [KKOPS86] for a generalization of the composition law to Azumaya algebras over rings.

6 Quadratic forms

We now connect the Brandt class groupoid to quadratic forms. For simplicity, we suppose ${{\,\mathrm{char}\,}}F \ne 2$ throughout this section.

19.6.1

We begin by recalling Proposition 4.5.17: for the quaternary quadratic form ${{\,\mathrm{nrd}\,}}:B \rightarrow F$, every oriented similarity of ${{\,\mathrm{nrd}\,}}$ is of the form

$$\begin{aligned} B&\mapsto B \\ x&\mapsto \alpha x \beta ^{-1} \end{aligned}$$

with $\alpha ,\beta \in B$ (in particular, respecting the canonical orientation 5.6.7 of B); the similitude factor of such a map is $u = {{\,\mathrm{nrd}\,}}(\alpha )/{{\,\mathrm{nrd}\,}}(\beta )$.

Let $I \subset B$ be a projective R-lattice.

19.6.2

Generalizing Exercise 10.2, the reduced norm restricts to give a quadratic form on I. We are given that I is projective of rank 4 as an R-module. Therefore the map

$$\begin{aligned} {{\,\mathrm{nrd}\,}}_I:I \rightarrow L={{\,\mathrm{nrd}\,}}(I) \end{aligned}$$

is a quaternary quadratic module over R.

If $J \subset B$ is another projective R-lattice, and f is an oriented similarity from ${{\,\mathrm{nrd}\,}}_I$ to ${{\,\mathrm{nrd}\,}}_J$, then extending scalars by F we obtain a oriented self-similarity of ${{\,\mathrm{nrd}\,}}:B \rightarrow B$; by 19.6.1, we conclude that $J=\alpha I \beta ^{-1}$ for some $\alpha ,\beta \in B^\times$:

(19.6.3)

19.6.4

Suppose that ${{\,\mathrm{nrd}\,}}(I)=L=aR$ is principal. Then there is a similarity

(19.6.5)

In other words, if every value of the quadratic form is divisible by a, then we up to similarity it is equivalent to consider the quadratic form $a^{-1}{{\,\mathrm{nrd}\,}}$, taking values in R.

Lemma 19.6.6

Suppose I is invertible. Then the quadratic form ${{\,\mathrm{nrd}\,}}_I:I \rightarrow L$ is locally oriented similar to ${{\,\mathrm{nrd}\,}}_{\mathcal {O}}:\mathcal {O}\rightarrow R$, where $\mathcal {O}=\mathcal {O}{}_{\textsf {\tiny {R}} }(I)$.

Proof. By 19.6.3, if $I=\alpha \mathcal {O}$ is principal, then ${{\,\mathrm{nrd}\,}}_I$ is similar to ${{\,\mathrm{nrd}\,}}_\mathcal {O}$. If I is invertible, then I is locally principal, so for all primes $\mathfrak p$ of R the quadratic form ${{\,\mathrm{nrd}\,}}:I_\mathfrak p \rightarrow L_\mathfrak p$ is similar to ${{\,\mathrm{nrd}\,}}:\mathcal {O}_\mathfrak p \rightarrow R_\mathfrak p$ where $\mathcal {O}_\mathfrak p$ is the left (or right) order of $I_\mathfrak p$. $\square$

19.6.7

From Lemma 15.3.6, it follows from Lemma 19.6.6 that

$$\begin{aligned} {{\,\mathrm{disc}\,}}({{\,\mathrm{nrd}\,}}_I)={{\,\mathrm{disc}\,}}(\mathcal {O}) \end{aligned}$$

and in particular this discriminant is a square.

The quadratic forms ${{\,\mathrm{nrd}\,}}_I$ are all locally similar, respecting the canonical orientation 5.6.7 on B. Therefore, there is a map

is (well-defined and) surjective.

Remark 19.6.8. The Brandt groupoid is connected as a groupoid. This can also be viewed in the language of quadratic forms: a connected class of orders is equivalently a genus of integral ternary quadratic forms, and this is akin to a resolvent for the quaternary norm forms. We refer to Chapter 23 for further discussion.

Exercises

• 1. Verify the computational details in the example of section 19.2.

• 2. Let $B=({-1,-11} \mid \mathbb{Q })$ with ${{\,\mathrm{disc}\,}}B = 11$ and $\mathcal {O}=\mathbb Z \langle i,(1+j)/2 \rangle$ a maximal order. Compute ${{\,\mathrm{BrtCl}\,}}\mathcal {O}$, in a manner analogous to the example of section 19.2.

• 3. Let G be a groupoid.

  • (a) Show that if $a,b,c \in G$ and both $a*b$ and $a*c$ are defined, then $b*b^{-1}=c*c^{-1}$ (and both are defined).

  • (b) Show that for all $a \in G$ we have $(a^{-1})^{-1}$.

• 4. Let G be a group acting on a nonempty set X. Let

  $$\begin{aligned} A(G,X) = \{(g,x) : g \in G, x \in X\}. \end{aligned}$$

  Show that A(G, X) has a natural groupoid structure with $(g,x) * (h,y) = (gh,y)$ defined if and only if $x=hy$. What are the identity elements?

• 5. Show that in a homomorphism $\phi :G \rightarrow G'$ of groupoids, the set of identity elements of G maps to the set of identity elements of $G'$.

• 6. Let $\mathcal {C}$ be a small category. Show that there is a unique maximal subcategory that is a groupoid. [Hint: Discard all nonisomorphisms.]

• 7. Let X be a set and let $\sim$ be an equivalence relation on X, thought of as a subset $S \subseteq X \times X$. Equip S with the partial binary operation $*$ defined by $(x,y)*(y,z)=(x,z)$ for $(x,y),(y,z) \in S$ (and $(x,y)*(w,z)$ is not defined if $y \ne w$). Show that S is a groupoid. [This shows that “equivalence relations are groupoids”, cf. (19.3.13).]

• 8. Let F be a field and let ${{\,\mathrm{GL}\,}}(F)=\bigcup _{n=1}^{\infty } {{\,\mathrm{GL}\,}}_n(F)$. Show that ${{\,\mathrm{GL}\,}}(F)$ has a natural structure of groupoid, sometimes called the general linear groupoid over F.

• 9. Show that the reduced norm is a homomorphism from the groupoid of invertible R-lattices in B to the group(oid) of fractional R-ideals in F.

• 10. Let X be a nonempty topological space, and let $x,y \in X$. Recall that a path from x to y is a continuous map $\upsilon _0:[0,1] \rightarrow X$ with $\upsilon (0)=x$ and $\upsilon (1)=y$. We say that paths $\upsilon _0,\upsilon _1:[0,1] \rightarrow X$ from x to y are homotopic if there exists a continuous map $H :[0,1] \times [0,1] \rightarrow X$ such that $H(0,s)=x$ and $H(1,s)=y$ for all $s \in [0,1]$ and $H(t,0)=\upsilon _0$ and $H(t,1)=\upsilon _1(t)$ for all $t \in [0,1]$. [So each H(t, s) for fixed $t \in [0,1]$ is a path from x to y, and this set of paths varies continuously.]

  • (a) Check that being homotopic defines an equivalence relation on the set of continuous paths from x to y.

  • (b) Check that paths can be composed (going at twice speed) and that composition of paths is well-defined on homotopy classes.

  • (c) Show that composition of homotopy classes of continuous paths is associative.

  Let $\Pi (X)$ be the category whose objects are the points of X and with morphisms to be the set of homotopy classes of continuous paths from x to y under composition.

  • (d) Show that $\Pi (X)$ is a category.

  • (e) Show that $\Pi (X)$ is a groupoid, called the fundamental groupoid of X.

  • (f) Finally, for all $x \in X$, show the set of all morphisms from x to x in $\Pi (X)$ is a group (the more familiar fundamental group $\pi _1(X,x)$ with base point x).

• 11. Continuing the previous exercise, show that if X is path-connected, then $\Pi (X)$ is equivalent as a category to a groupoid with one object. [Hint: choose a point $x \in X$, look at the group(oid) $\pi _1(X,x)$.]