Noncommutative geometry and cayley smooth orders

Noncommutative Geometry
and Cayley-smooth Orders

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Noncommutative Geometry
and Cayley-smooth Orders

Lieven Le Bruyn
Universiteit Antwerpen
Belgium

Boca Raton London New York

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Taylor & Francis Group, an informa business

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Library of Congress Cataloging‑in‑Publication Data
Le Bruyn, Lieven, 1958‑
Noncommutative geometry and Cayley‑smooth orders / Lieven Le Bruyn.
p. cm. ‑‑ (Pure and applied mathematics ; 290)

Includes bibliographical references and index.
ISBN 978‑1‑4200‑6422‑3 (alk. paper)
1. Noncommutative differential geometry. 2. Cayley‑Hamilton theorem. I.
Title. II. Series.
QC20.7.D52L4 2007
512’.55‑‑dc22

2007019964

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This book is dedicated to the women in my life
Simonne Stevens (1926-2004), Ann, Gitte & Bente

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Contents

Preface

xiii

Introduction
xv
I.1 Noncommutative algebra . . . . . . . . . . . . . . . . . . . . xvii
I.2 Noncommutative geometry . . . . . . . . . . . . . . . . . . . xxxv
I.3 Noncommutative desingularizations . . . . . . . . . . . . . .
l
About the Author

lxiii

1 Cayley-Hamilton Algebras
1.1 Conjugacy classes of matrices .
1.2 Simultaneous conjugacy classes
1.3 Matrix invariants and necklaces
1.4 The trace algebra . . . . . . .
1.5 The symmetric group . . . . .
1.6 Necklace relations . . . . . . .
1.7 Trace relations . . . . . . . . .
1.8 Cayley-Hamilton algebras . . .
2 Reconstructing Algebras
2.1 Representation schemes . .
2.2 Some algebraic geometry .
2.3 The Hilbert criterium . . .
2.4 Semisimple modules . . . .
2.5 Some invariant theory . . .
2.6 Geometric reconstruction .
2.7 The Gerstenhaber-Hesselink

2.8 The real moment map . . .

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3 Etale Technology
3.1 Etale topology . . . . . . . . . . .
3.2 Central simple algebras . . . . . .
3.3 Spectral sequences . . . . . . . . .
3.4 Tsen and Tate fields . . . . . . . .
3.5 Coniveau spectral sequence . . . .
3.6 The Artin-Mumford exact sequence
3.7 Normal spaces . . . . . . . . . . .

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ix
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x

Contents
3.8

Knop-Luna slices . . . . . . . . . . . . . . . . . . . . . . . . .

155

4 Quiver Technology
4.1 Smoothness . . . . . . . . .
4.2 Local structure . . . . . . .
4.3 Quiver orders . . . . . . . .
4.4 Simple roots . . . . . . . .
4.5 Indecomposable roots . . .
4.6 Canonical decomposition .
4.7 General subrepresentations
4.8 Semistable representations

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5 Semisimple Representations
5.1 Representation types . . .
5.2 Cayley-smooth locus . . . .
5.3 Reduction steps . . . . . .
5.4 Curves and surfaces . . . .
5.5 Complex moment map . .
5.6 Preprojective algebras . . .
5.7 Central smooth locus . . .
5.8 Central singularities . . . .

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6 Nilpotent Representations
6.1 Cornering matrices . . . . . . .
6.2 Optimal corners . . . . . . . . .
6.3 Hesselink stratification . . . . .
6.4 Cornering quiver representations
6.5 Simultaneous conjugacy classes .
6.6 Representation fibers . . . . . .
6.7 Brauer-Severi varieties . . . . . .
6.8 Brauer-Severi fibers . . . . . . .

7 Noncommutative Manifolds
7.1 Formal structure . . . . .
7.2 Semi-invariants . . . . . .
7.3 Universal localization . .
7.4 Compact manifolds . . .
7.5 Differential forms . . . .
7.6 deRham cohomology . .
7.7 Symplectic structure . . .
7.8 Necklace Lie algebras . .

© 2008 by Taylor & Francis Group, LLC

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Contents
8 Moduli Spaces
8.1 Moment maps . . . . . . . . . .
8.2 Dynamical systems . . . . . . .
8.3 Deformed preprojective algebras
8.4 Hilbert schemes . . . . . . . . .
8.5 Hyper Kă
ahler structure . . . . .
8.6 Calogero particles . . . . . . . .

8.7 Coadjoint orbits . . . . . . . . .
8.8 Adelic Grassmannian . . . . . .

xi

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References

© 2008 by Taylor & Francis Group, LLC

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452
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464
469
482
487
493
497
505

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Preface

This book explains the theory of Cayley-smooth orders in central simple algebras over functionfields of varieties. In particular, we will describe the
´etale local structure of such orders as well as their central singularities and
finite dimensional representations. There are two major motivations to study
Cayley-smooth orders.
A first application is the construction of partial desingularizations of (commutative) singularities from noncommutative algebras. This approach is summarized in the introduction and can be read independently, modulo technical
details and proofs, which are deferred to the main body of the book. A second motivation stems from noncommutative algebraic geometry as developed
by Joachim Cuntz, Daniel Quillen, Maxim Kontsevich, Michael Kapranov
and others. One studies formally smooth algebras or quasi-free algebras (in
this book we will call them Quillen-smooth algebras), which are huge, nonNoetherian algebras, the free associative algebras being the archetypical examples. One attempts to study these algebras via their finite dimensional
representations which, in turn, are controlled by associated Cayley-smooth
algebras. In the final two chapters, we will give an introduction to this fast

developing theory.
This book is based on a series of courses given since 1999 in the ”advanced
master program on noncommutative geometry” organized by the NOncommutative Geometry (NOG) project, sponsored by the European Science Foundation (ESF). As the participating students came from different countries there
was a need to include background information on a variety of topics including
invariant theory, algebraic geometry, central simple algebras and the representation theory of quivers. In this book, these prerequisites are covered in
chapters 1 to 4.
Chapters 1 and 2 contain the invariant theoretic description of orders and
their centers, due to Michael Artin and Claudio Procesi. Chapter 3 contains
an introduction to ´etale topology and its use in noncommutative algebra,
in particular to the study of Azumaya algebras and to the local description
of algebras via Luna slices. Chapter 4 collects the necessary material on
representations of quivers, including the description of their indecomposable
roots, due to Victor Kac, the determination of dimension vectors of simple
representations, and results on general quiver representations, due to Aidan
Schofield. The results in these chapters are due to many people and the
presentation is influenced by a variety of sources.
Chapters 5 and 6 contain the main results on Cayley-smooth orders. In

xiii
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xiv

Noncommutative Geometry and Cayley-smooth orders

chapter 5, we describe the ´etale local structure of a Cayley-smooth order in
a semisimple representation and classify the associated central singularity up

to smooth equivalence. This is done by associating to a semisimple representation a combinatorial gadget, a marked quiver setting, which encodes the
tangent-space information to the noncommutative manifold in the cluster of
points determined by the simple factors of the representation. In chapter 6 we
will describe the nullcone of these marked quiver representations and relate
them to the study of all isomorphism classes of n-dimensional representations
of a Cayley-smooth order.

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Introduction

Ever since the dawn of noncommutative algebraic geometry in the midseventies, see for example the work of P. Cohn [21], J. Golan [38], C. Procesi [86], F. Van Oystaeyen and A. Verschoren [103],[105], it has been ring
theorists’ hope that this theory might one day be relevant to commutative
geometry, in particular to the study of singularities and their resolutions.
Over the last decade, noncommutative algebras have been used to construct
canonical (partial) resolutions of quotient singularities. That is, take a finite
group G acting on Cd freely away from the origin then its orbit-space Cd /G
✲ Cd /G have been constructed
is an isolated singularity. Resolutions Y ✲
using the skew group algebra
C[x1 , . . . , xd ]#G
which is an order with center C[Cd /G] = C[x1 , . . . , xd ]G or deformations of it.
In dimension d = 2 (the case of Kleinian singularities) this gives us minimal
resolutions via the connection with the preprojective algebra, see for example
[27]. In dimension d = 3, the skew group algebra appears via the superpotential and commuting matrices setting (in the physics literature) or via the
McKay quiver, see for example [23]. If G is Abelian one obtains from this
study crepant resolutions but for general G one obtains at best partial resolutions with conifold singularities remaining. In dimension d > 3 the situation

is unclear at this moment.
Usually, skew group algebras and their deformations are studied via homological methods as they are Serre-smooth orders, see for example [102]. In
this book, we will follow a different approach.
We want to find a noncommutative explanation for the omnipresence of
conifold singularities in partial resolutions of three-dimensional quotient singularities. One may argue that they have to appear because they are somehow
the nicest singularities. But then, what is the corresponding list of ”nice” singularities in dimension four? or five, six...?
The results contained in this book suggest that the nicest partial resolutions
of C4 /G should only contain singularities that are either polynomials over the
conifold or one of the following three types
C[[a, b, c, d, e, f ]]
(ae − bd, af − cd, bf − ce)

C[[a, b, c, d, e]]
(abc − de)

C[[a, b, c, d, e, f, g, h]]
I

where I is the ideal of all 2 ì 2 minors of the matrix
ab cd
ef gh

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xvi

Noncommutative Geometry and Cayley-smooth Orders

FIGURE I.1:

Local structure of Cayley-smooth orders

In dimension d = 5 there is another list of ten new specific singularities that
will appear; in dimension d = 6 another 63 new ones appear and so on.
How do we arrive at these specific lists? The hope is that any quotient
singularity X = Cd /G has associated to it a ”nice” order A with center
R = C[X] such that there is a stability structure θ such that the scheme of
all θ-semistable representations of A is a smooth variety (all these terms will
be explained in the main body of the book). If this is the case, the associated
moduli space will be a partial resolution
✲ X = Cd /G
moduliθα A ✲
and has a sheaf of Cayley-smooth orders A over it, allowing us to control its
singularities in a combinatorial way as depicted in figure I.1.
If A is a Cayley-smooth order over R = C[X] then its noncommutative
variety max A of maximal twosided ideals is birational to X away from the
ramification locus. If P is a point of the ramification locus ram A then there is
a finite cluster of infinitesimally nearby noncommutative points lying over it.
The local structure of the noncommutative variety max A near this cluster can
be summarized by a (marked) quiver setting (Q, α), which in turn allows us to
compute the ´etale local structure of A and R in P . The central singularities
that appear in this way have been classified in [14] (see also section 5.8) up to
smooth equivalence giving us the small lists of singularities mentioned before.

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Introduction

xvii

In this introduction we explain this noncommutative approach to the desingularization project of commutative singularities. Proofs and more details will
be given in the following chapters.

I.1

Noncommutative algebra

Let me begin by trying to motivate why one might be interested in noncommutative algebra if you want to understand quotient singularities and
their resolutions. Suppose we have a finite group G acting on d-dimensional
affine space Cd such that this action is free away from the origin. Then the
orbit-space, the so called quotient singularity Cd /G, is an isolated singularity
Cd

❄
❄
res
✛
C /G ✛
d

Y

and we want to construct ”minimal” or ”canonical” resolutions (so called

crepant resolutions) of this singularity. In his Bourbaki talk [89] Miles Reid
asserts that McKay correspondence follows from a much more general principle
Miles Reid’s Principle: Let M be an algebraic manifold, G a group of
✲ X a resolution of singularities of X = M/G.
automorphisms of M , and Y ✲
Then the answer to any well-posed question about the geometry of Y is the
G-equivariant geometry of M .
Applied to the case of quotient singularities, the content of his slogan is that
the G-equivariant geometry of Cd already knows about the crepant resolution
✲ Cd /G. Let us change this principle slightly: assume we have an affine
Y ✲
variety M on which a reductive group (we will take P GLn ) acts with algebraic
quotient variety M/P GLn Cd /G
Cd

M

✲
✲ M/P GLn

❄
❄
res
d
✛
C /G ✛

Y

then, in favorable situations, we can argue that the P GLn -equivariant geometry of M knows about good resolutions Y . One of the key lessons to be learned

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xviii

Noncommutative Geometry and Cayley-smooth Orders

from this book is that P GLn -equivariant geometry of M is roughly equivalent
to the study of a certain noncommutative algebra over Cd /G. In fact, an
order in a central simple algebra of dimension n2 over the function field of the
quotient singularity. Hence, if we know of good orders over Cd /G, we might
get our hands on ”good” resolutions Y by noncommutative methods.
We will work in the following, quite general, setting:
• X will be a normal affine variety, possibly having singularities.
• R will be the coordinate ring C[X] of X.
• K will be the function field C(X) of X.
If you are only interested in quotient singularities, you should replace X by
Cd /G, R by the invariant ring C[x1 , . . . , xd ]G and K by the invariant field
C(x1 , . . . , xd )G in all statements below.
Our goal will be to construct lots of R-orders A in a central simple Kalgebra Σ.
✲ Σ⊂
✲ Mn (K)
A⊂
✻
✻
✻
∪

∪

∪

✲ K⊂
✲ K
R
A central simple algebra is a noncommutative K-algebra Σ with center Z(Σ) =
K such that over the algebraic closure K of K we obtain full n × n matrices
⊂

Σ ⊗K K

Mn (K)

(more details will be given in section 3.2). There are plenty such central
simple K-algebras:
Example I.1
For any nonzero functions f, g ∈ K ∗ , the cyclic algebra
Σ = (f, g)n

defined by

(f, g)n =

K x, y
(xn − f, y n − g, yx − qxy)

with q is a primitive n-th root of unity, is a central simple K-algebra of

dimension n2 . Often, (f, g)n will even be a division algebra, that is a noncommutative algebra such that every nonzero element has an inverse.
For example, this is always the case when E = K[x] is a (commutative) field
extension of dimension n and if g has order n in the quotient K ∗ /NE/K (E ∗ )
where NE/K is the norm map of E/K.
Fix a central simple K-algebra Σ, then an R-order A in Σ is a subalgebras
A ⊂ Σ with center Z(A) = R such that A is finitely generated as an R-module

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Introduction

xix

and contains a K-basis of Σ, that is
A ⊗R K

Σ

The classic reference for orders is Irving Reiner’s book [90] but it is somewhat
outdated and focuses mainly on the one-dimensional case. With this book we
hope to remedy this situation somewhat.
Example I.2
In the case of quotient singularities X = Cd /G a natural choice of R-order
might be the skew group ring : C[x1 , . . . , xd ]#G, which consists of all formal
sums g∈G rg #g with multiplication defined by
(r#g)(r #g ) = rφg (r )#gg
where φg is the action of g on C[x1 , . . . , xd ]. The center of the skew group

algebra is easily verified to be the ring of G-invariants
R = C[Cd /G] = C[x1 , . . . , xd ]G
Further, one can show that C[x1 , . . . , xd ]#G is an R-order in Mn (K) with
n the order of G. Later we will give another description of the skew group
algebra in terms of the McKay-quiver setting and the variety of commuting
matrices.
However, there are plenty of other R-orders in Mn (K), which may or may
not be relevant in the study of the quotient singularity Cd /G.
Example I.3
If f, g ∈ R−{0}, then the free R-submodule of rank n2 of the cyclic K-algebra
Σ = (f, g)n of example I.1
n−1

Rxi y j

A=
i,j=0

is an R-order. But there is really no need to go for this ”canonical” example.
Someone more twisted may take I and J any two nonzero ideals of R, and
consider
n−1

I i J j xi y j

AIJ =
i,j=0

which is also an R-order in Σ, far from being a projective R-module unless I
and J are invertible R-ideals.

For example, in Mn (K) we can take the ”obvious” R-order Mn (R) but one
might also take the subring
R I
J R

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xx

Noncommutative Geometry and Cayley-smooth Orders

which is an R-order if I and J are nonzero ideals of R.
From a geometric viewpoint, our goal is to construct lots of affine P GLn varieties M such that the algebraic quotient M/P GLn is isomorphic to X
and, moreover, such that there is a Zariski open subset U ⊂ X
M ✛

⊃

π −1 (U )

π

X

principal P GLn -fibration

❄

❄
M/P GLn ✛

⊃

❄
❄
U

for which the quotient map is a principal P GLn -fibration, that is, all fibers
π −1 (u)
P GLn for u ∈ U . For the connection between such varieties M
and orders A in central simple algebras think of M as the affine variety of
n-dimensional representations repn A and of U as the Zariski open subset of
all simple n-dimensional representations.
Naturally, one can only expect the R-order A (or the corresponding P GLn variety M ) to be useful in the study of resolutions of X if A is smooth in
some appropriate noncommutative sense. There are many characterizations
of commutative smooth domains R:
• R is regular, that is, has finite global dimension
• R is smooth, that is, X is a smooth variety
and generalizing either of them to the noncommutative world leads to quite
different concepts. We will call an R-order A a central simple K-algebra Σ:
• Serre-smooth if A has finite global dimension together with some extra
features such as Auslander regularity or Cohen-Macaulay property, see
for example [80].
• Cayley-smooth if the corresponding P GLn -affine variety M is a
smooth variety as we will clarify later.
For applications of Serre-smooth orders to desingularizations we refer to the
paper [102]. We will concentrate on the properties of Cayley-smooth orders
instead. Still, it is worth pointing out the strengths and weaknesses of both

definitions.
Serre-smooth orders are excellent if you want to control homological properties, for example, if you want to study the derived categories of their modules.
At this moment there is no local characterization of Serre-smooth orders if
dimX ≥ 3. Cayley-smooth orders are excellent if you want to have smooth
moduli spaces of semistable representations. As we will see later, in each dimension there are only a finite number of local types of Cayley-smooth orders

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Introduction

xxi

and these will be classified in this book. The downside of this is that Cayleysmooth orders are less versatile than Serre-smooth orders. In general though,
both theories are quite different.
Example I.4
The skew group algebra C[x1 , . . . , xd ]#G is always a Serre-smooth order but
it is virtually never a Cayley-smooth order.
Example I.5
Let X be the variety of matrix-invariants, that is
X = Mn (C) ⊕ Mn (C)/P GLn
where P GLn acts on pairs of n×n matrices by simultaneous conjugation. The
trace ring of two generic n × n matrices A is the subalgebra of Mn (C[Mn (C) ⊕
Mn (C)]) generated over C[X] by the two generic matrices


x11 . . .
 ..

X= .


x1n
.. 
. 



y11 . . .
 ..
and Y =  .

xn1 . . . xnn


y1n
.. 
. 

yn1 . . . ynn

Then, A is an R-order in a division algebra of dimension n2 over K, called
the generic division algebra. Moreover, A is a Cayley-smooth order but is
Serre-smooth only when n = 2, see [78].
Descent theory allows construction of elaborate examples out of trivial ones
by bringing in topology and enables one to classify objects that are only locally
(but not necessarily globally) trivial. For applications to orders there are two
topologies to consider : the well-known Zariski topology and the perhaps
lesser-known ´etale topology. Let us try to give a formal definition of Zariski

and ´etale covers aimed at ring theorists. Much more detail on ´etale topology
will be given in section 3.1.
A Zariski cover of X is a finite product of localizations at elements of R
k

Sz =

R fi

such that

(f1 , . . . , fk ) = R

i=1

and is therefore a faithfully flat extension of R. Geometrically, the ring✲ Sz defines a cover of X = spec R by k disjoint sheets
morphism R
spec Sz = i spec Rfi , each corresponding to a Zariski open subset of X, the
complement of V(fi ), and the condition is that these closed subsets V(fi ) do
not have a point in common. That is, we have the picture of figure I.2.

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xxii

Noncommutative Geometry and Cayley-smooth Orders

FIGURE I.2:

A Zariski cover of X = spec R.

Zariski covers form a Grothendieck topology, that is, two Zariski covers
k
l
Sz1 = i=1 Rfi and Sz2 = j=1 Rgj have a common refinement
k

l

Sz = Sz1 ⊗R Sz2 =

R fi g j
i=1 j=1

For a given Zariski cover Sz =
product

k
i=1

Rfi a corresponding ´etale cover is a
 ∂g(i)1
∂x(i)1

k

Se =

Rfi [x(i)1 , . . . , x(i)ki ]
(g(i)1 , . . . , g(i)ki )
i=1

with





...

..
.
∂g(i)ki
∂x(i)1

∂g(i)1
∂x(i)ki

..
.

...

∂g(i)ki
∂x(i)ki







a unit in the i-th component of Se . In fact, for applications to orders it is
usually enough to consider special etale extensions
k

Se =

Rfi [x]
(xki − ai )
i=1

where

ai is a unit in Rfi

Geometrically, an ´etale cover determines for every Zariski sheet spec Rfi a
locally isomorphic (for the analytic topology) multicovering and the number
of sheets may vary with i (depending on the degrees of the polynomials g(i)j ∈
Rfi [x(i)1 , . . . , x(i)ki ]. That is, the mental picture corresponding to an ´etale
cover is given in figure I.3.

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Introduction

FIGURE I.3:

xxiii

An ´etale cover of X = spec R.

Again, ´etale covers form a Zariski topology as the common refinement Se1 ⊗R
Se2 of two ´etale covers is again ´etale because its components are of the form
Rfi gj [x(i)1 , . . . , x(i)ki , y(j)1 , . . . , y(j)lj ]
(g(i)1 , . . . , g(i)ki , h(j)1 , . . . , h(j)lj )
and the Jacobian-matrix condition for each of these components is again satisfied. Because of the local isomorphism property many ring theoretical local
properties (such as smoothness, normality, etc.) are preserved under ´etale
covers.
For a fixed R-order B in some central simple K-algebra Σ, then a Zariski
twisted form A of B is an R-algebra such that
A ⊗R Sz

B ⊗R Sz

for some Zariski cover Sz of R. If P ∈ X is a point with corresponding
maximal ideal m, then P ∈ spec Rfi for some of the components of Sz and
as Afi Bfi we have for the local rings at P
Am

Bm

that is, the Zariski local information of any Zariski-twisted form of B is that
of B itself.

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xxiv

Noncommutative Geometry and Cayley-smooth Orders

Likewise, an ´etale twisted form A of B is an R-algebra such that
A ⊗R Se

B ⊗R Se

for some ´etale cover Se of R. This time the Zariski local information of A
and B may be different at a point P ∈ X but we do have that the m-adic
completions of A and B
ˆm
Aˆm B
ˆ m -algebras. Thus, the Zariski local structure of A deterare isomorphic as R
mines the localization Am , the ´etale local structure determines the completion
Aˆm .
Descent theory allows us to classify Zariski- or ´etale twisted forms of an
R-order B by means of the corresponding cohomology groups of the automorphism schemes. For more details on this please read [57], [82] or section 3.1. If
one applies descent to the most trivial of all R-orders, the full matrix algebra
Mn (R), one arrives at Azumaya algebras. A Zariski twisted form of Mn (R) is
an R-algebra A such that
k

A ⊗R Sz

Mn (Rfi )

Mn (Sz ) =
i=1

Conversely, you can construct such twisted forms by gluing together the matrix
rings Mn (Rfi ). The easiest way to do this is to glue Mn (Rfi ) with Mn (Rfj )
over Rfi fj via the natural embeddings
Rfi

⊂

✲ Rfi fj ✛

⊃

R fj

Not surprisingly, we obtain in this way Mn (R) back. However there are more
clever ways to perform the gluing by bringing in the noncommutativity of
matrix-rings. We can glue
Mn (Rfi )

⊂

✲ Mn (Rfi fj )

−1
gij .gij

✲ Mn (Rfi fj ) ✛

⊃

Mn (Rfj )

over their intersection via conjugation with an invertible matrix gij in
GLn (Rfi fj ). If the elements gij for 1 ≤ i, j ≤ k satisfy the cocycle condition (meaning that the different possible gluings are compatible over their
common localization Rfi fj fl ), we obtain a sheaf of noncommutative algebras
A over X = spec R such that its global sections are not necessarily Mn (R).
PROPOSITION I.1
Any Zariski twisted form of Mn (R) is isomorphic to EndR (P ) where P is
a projective R-module of rank n. Two such twisted forms are isomorphic as
R-algebras
EndR (P ) EndR (Q) iff P Q ⊗ I

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Introduction

xxv

for some invertible R-ideal I.
PROOF

We have an exact sequence of group schemes

1

✲ GLn

✲ Gm

✲ PGLn

✲ 1

(here, Gm is the sheaf of units) and taking Zariski cohomology groups over X
we have a sequence
1

1
✲ HZar
(X, Gm )

1
✲ HZar
(X, GLn )

1
✲ HZar
(X, PGLn )

where the first term is isomorphic to the Picard group P ic(R) and the second
term classifies projective R-modules of rank n upto isomorphism. The final
term classifies the Zariski twisted forms of Mn (R) as the automorphism group
of Mn (R) is P GLn .

Example I.6
Let I and J be two invertible ideals of R, then
EndR (I ⊕ J)
and if IJ −1 = (r) then I⊕J
1 0
0 r−1

R I −1 J
⊂ M2 (K)
IJ −1 R

(Rr⊕R)⊗J and indeed we have an isomorphism
R I −1 J
IJ −1 R

10
RR
=
0r
RR

The situation becomes a lot more interesting when we replace the Zariski
topology by the ´etale topology.
DEFINITION I.1 An n-Azumaya algebra over R is an ´etale twisted form
A of Mn (R). If A is also a Zariski twisted form we call A a trivial Azumaya
algebra.
LEMMA I.1
If A is an n-Azumaya algebra over R, then:
1. The center Z(A) = R and A is a projective R-module of rank n2 .
2. All simple A-representations have dimension n and for every maximal

ideal m of R we have
A/mA Mn (C)

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xxvi

Noncommutative Geometry and Cayley-smooth Orders

PROOF For (2) take M ∩ R = m where M is the kernel of a simple
ˆ m ) it follows that
✲ Mk (C), then as Aˆm Mn (R
representation A ✲
A/mA

Mn (C)

and hence that k = n and M = Am.
It is clear from the definition that when A is an n-Azumaya algebra and A
is an m-Azumaya algebra over R, A ⊗R A is an mn-Azumaya and also that
A ⊗R Aop

EndR (A)

where Aop is the opposite algebra (that is, equipped with the reverse multiplication rule). These facts allow us to define the Brauer group BrR to be the
set of equivalence classes [A] of Azumaya algebras over R where
[A] = [A ]

iff

A ⊗R A

EndR (P )

for some projective R-module P and where multiplication is induced from the
rule
[A].[A ] = [A ⊗R A ]
One can extend the definition of the Brauer group from affine varieties to
arbitrary schemes and A. Grothendieck has shown that the Brauer group of
a projective smooth variety is a birational invariant, see [40]. Moreover, he
conjectured a cohomological description of the Brauer group BrR, which was
subsequently proved by O. Gabber in [34].
THEOREM I.1
The Brauer group is an ´etale cohomology group
BrR

2
Het
(X, Gm )torsion

where Gm is the unit sheaf and where the subscript denotes that we take only
2
torsion elements. If R is regular, then Het
(X, Gm ) is torsion so we can forget
the subscript.
This result should be viewed as the ring theory analogon of the crossed product theorem for central simple algebras over fields. Observe that in Gabber’s
result there is no sign of singularities in the description of the Brauer group.

In fact, with respect to the desingularization project, Azumaya algebras are
only as good as their centers.
PROPOSITION I.2
If A is an n-Azumaya algebra over R, then
1. A is Serre-smooth iff R is commutative regular.

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Introduction

xxvii

2. A is Cayley-smooth iff R is commutative regular.
PROOF (1) follows from faithfully flat descent and (2) from lemma I.1,
which asserts that the P GLn -affine variety corresponding to A is a principal
P GLn -fibration in the ´etale topology, which shows that both n-Azumaya algebras and principal P GLn -fibrations are classified by the ´etale cohomology
1
group Het
(X, PGLn ). More details are given in chapter 3.
In the correspondence between R-orders and P GLn -varieties, Azumaya algebras correspond to principal P GLn -fibrations over X and with respect to
desingularizations, Azumaya algebras are of little use. So let us bring in ramification in order to construct orders that may be more useful.
Example I.7
Consider the R-order in M2 (K)
A=

RR
I R

where I is some ideal of R and let P ∈ X be a point with corresponding
maximal ideal m. For I not contained in m we have Am M2 (Rm ) whence A
is an Azumaya algebra in P . For I ⊂ m we have
Rm Rm
= M2 (Rm )
Im Rm

Am
whence A is not Azumaya in P .

DEFINITION I.2 The ramification locus of an R-order A is the Zariski
closed subset of X consisting of those points P such that for the corresponding
maximal ideal m
A/mA Mn (C)
That is, ram A is the locus of X where A is not an Azumaya algebra. Its
complement azu A is called the Azumaya locus of A, which is always a Zariski
open subset of X.
DEFINITION I.3
algebra iff

An R-order A is said to be a reflexive n-Azumaya

1. ram A has codimension at least two in X, and
2. A is a reflexive R-module

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