arXiv:math/0312221v1 [math.RA] 11 Dec 2003 · 2018. 11. 9. · lecture 1. non-commutative algebra 8 with qis a primitive n-th root of unity, is a central simpleK-algebra of dimensionn2

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3 three talks onnoncommutative geometry@n

lieven le bruyn

ram AX

max A

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x2kk

Âm ≃

R̂m R̂my1 + R̂mx3x2 R̂mx3 + R̂my1y2R̂mx1 + R̂my1y3 R̂m R̂my2 + R̂mx1x3R̂my3 + R̂mx2x1 R̂mx2 + R̂my3y1 R̂m

R̂m = C[[x1y1, x2y2, x3y3, x1x2x3, y1y2y3]]

2003

university of antwerp

http://arxiv.org/abs/math/0312221v1

CONTENTS

1. non-commutative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1 Why non-commutative algebra? . . . . . . . . . . . . . . . . . . . . . .. 6

1.2 What non-commutative algebras? . . . . . . . . . . . . . . . . . . . .. . 7

1.3 Constructing orders by descent . . . . . . . . . . . . . . . . . . . . .. . . 10

1.4 Azumaya algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Reflexive Azumaya algebras . . . . . . . . . . . . . . . . . . . . . . . . .15

1.6 Cayley-Hamilton algebras . . . . . . . . . . . . . . . . . . . . . . . . .. 17

1.7 Smooth orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2. non-commutative geometry . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1 Why non-commutative geometry? . . . . . . . . . . . . . . . . . . . . .. 23

2.2 What non-commutative geometry? . . . . . . . . . . . . . . . . . . . .. . 25

2.3 Marked quiver and Morita settings . . . . . . . . . . . . . . . . . . .. . . 27

2.4 Local classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

2.5 A two-person game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Central singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 33

2.7 Isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 36

3. non-commutative desingularizations . . . . . . . . . . . . . . . . . . . 38

3.1 Quotient singularities . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 38

3.2 Stability structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40

3.3 Partial resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43

3.4 Going fromalg@n to alg@α . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 The affine opensXD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6 TheC[XD]- ordersAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7 Non-commutative desingularizations . . . . . . . . . . . . . . .. . . . . . 49

INTRODUCTION

Ever since the dawn of non-commutative algebraic geometry in the mid seventies, see forexample the work of P. Cohn [11], J. Golan [17], C. Procesi [38], F. Van Oystaeyen andA. Verschoren [47],[48], it has been ringtheorists’ wet dream that this theory may one daybe relevant to commutative geometry, in particular to the study of singularities and theirresolutions.

Over the last decade, non-commutativealgebras have been used to construct canonical (par-tial) resolutions of quotient singularities. That is, takea finite groupG acting onCd freelyaway from the origin then the orbit-spaceCd/G is an isolated singularity. ResolutionsY ✲✲ Cd/G have been constructed using the skew group algebra

C[x1, . . . , xd]#G

which is an order with centerC[Cd/G] = C[x1, . . . , xd]G or deformations of it. In di-mensiond = 2 (the case of Kleinian singulariies) this gives us minimal resolutions viathe connection with the preprojective algebra, see for example [14]. In dimensiond = 3,the skew group algebra appears via the superpotential and commuting matrices setting (inthe physics literature) or via the McKay quiver, see for example [13]. If G is Abelian oneobtains from this study crepant resolutions but for generalG one obtains at best partial res-olutions with conifold singularities remaining. In dimension d > 3 the situation is unclearat this moment. Usually, skew group algebras and their deformations are studied via ho-mological methods as they are Regular orders, see for example [46]. Here, we will followa different approach.

My motivation was to find a non-commutative explanation for the omnipresence of conifoldsingularities in partial resolutions of three-dimensional quotient singularities. Of courseyou 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... ?? If my conjectural explanation has any merit the nicest partial resolutions ofC4/Gshould contain only singularities which are either polynomials over the conifold or one ofthe 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

whereI is the ideal of all2× 2 minors of the matrix[a b c de f g h

]

In dimensiond = 5 the conjecture is that another list of ten new specific singularities willappear, in dimensiond = 6 another63 new ones appear and so on.

How do we come to these outlandish conjectures and specific lists? The hope is that anyquotient singularityX = Cd/G has associated to it a ’nice’ orderA with centerR = C[X ]such that there is a stability structureθ with the scheme of allθ-semistable representations

Contents 4

of A being a smooth variety (all these terms will be explained in the main text). If this isthe case, the associated moduli space will be a partial resolution

moduliθα A ✲✲ X = Cd/G

and has a sheaf of Smooth ordersA over it, allowing us to control its singularities in acombinatorial way as depicted in the frontispiece.

If A is a Smooth order overR = C[X ] then its non-commutative varietymaxA of maximaltwosided ideals is birational toX away from the ramification locus. IfP is a point ofthe ramification locusram A then there is a finite cluster of infinitesimally close non-commutative points lying over it. The local structure of thenon-commutative varietymaxAnear this cluster can be summarized by a (marked) quiver setting (Q,α) which in turnallows us to compute the étale local structure ofA andR in P . The central singularitieswhich appear in this way have been classified in [6] up to smooth equivalence giving us thesmall lists of conjectured singularities.

In these talks I have tried to include background information which may or may not beuseful to you. I suggest to browse through the notes by reading the ’jotter-notes’ (grey-shaded). If the remark seems obvious to you, carry on. If it puzzles you this may be agood point to enter the main text. More information can be found in the never-endingbookproject [28].

ACKNOWLEDGEMENT

These notes are (hopefully) a streamlined version of three talks given at the workshop”Schémas de Hilbert, algèbre noncommutative et correspondance de McKay” held atCIRM in Luminy (France), october 27-31, 2003.

I like to thank the organizers, Jacques Alev, Bernhard Keller and Thierry Levasseur forthe invitation (and the possibility to have a nice vacation with part of my family) and theparticipants for their patience.

lecture 1

NON-COMMUTATIVE ALGEBRA

The organizers of this conference on ”Hilbert schemes, non-commutative algebra and theMcKay correspondence” are perfectly aware of my ignorance on Hilbert schemes andMcKay correspondence. I therefore have to assume that I was hired in to tell you some-thing about non-commutative algebra and that is precisely what I intend to do in these threetalks.

1.1 Why non-commutative algebra?

Let me begin by trying to motivate why you might get interested in non-commutative alge-bra if you want to understand quotient singularities and their resolutions.

So let us take a setting which will be popular this week : we have a finite groupG actingon d-dimensional affine spaceCd and this action is free away from the origin. Then theorbit-space, the so calledquotient singularityCd/G, is an isolated singularity

Cd

Cd/G

❄❄✛✛res Y

and we want to construct ’minimal’ or ’canonical’ resolutions of this singularity. The buzz-word seems to be ’crepant’ in these circles. In his Bourbaki talk [40] Miles Reid assertsthat McKay correspondence follows from a much more general principle

Miles Reid’s Principle : LetM be an algebraic manifold,G a group of automorphismsof M , andY ✲✲ X a resolution of singularities ofX = M/G. Then the answer to anywell posed question about the geometry ofY is theG-equivariant geometry ofM .

Applied to the case of quotient singularities, the content of his slogan is that theG-equivariant geometry ofCd alreadyknowsabout the crepant resolutionY ✲✲ Cd/G.

Men having principles are an easy target for abuse. So let us change this principle slightly :assume we have an affine varietyM on which a reductive group (and for definiteness take

lecture 1. non-commutative algebra 7

PGLn) acts with algebraic quotient varietyM//PGLn ≃ Cd/G

Cd

M ✲✲ M//PGLn ≃Cd/G

❄❄✛✛res Y

then, in favorable situations, we can argue that thePGLn-equivariant geometry ofMknows about good resolutionsY . This brings us to our first entry in our

jotter :

One of the key lessons to be learned from this talk is thatPGLn-equivariant geometry ofM is roughlyequivalent to the study of a certain non-commutative algebra overCd/G.In fact, anorder in a central simple algebra of dimensionn2 over the function field of thequotient singularity.

Hence, if we know ofgoodorders overCd/G, we might get our hands at ’good’ resolu-tionsY by non-commutative methods.

1.2 What non-commutative algebras?

For the duration of these talks, we will work in the following, quite general, setting :

• X will be anormalaffine variety, possibly having singularities.

• R will be the coordinate ringC[X ] of X .

• K will be the function fieldC(X) of X .

If you are only interested in quotient singularities, you should replaceX byCd/G,R by theinvariant ringC[x1, . . . , xd]G andK by the invariant fieldC(x1, . . . , xd)G in all statementsbelow.

If you are an algebraist, you have my sympathy and our goal will be to construct lots ofR-ordersA in a central simpleK-algebraΣ.

A ⊂ ✲ Σ ⊂ ✲ Mn(K)

R∪

✻

⊂ ✲ K∪

✻

⊂ ✲ K∪

✻

If you do not know what acentral simple algebrais, take any non-commutativeK-algebraΣ with centerZ(Σ) = K such that over the algebraic closureK ofK we obtain fulln×nmatrices

Σ⊗K K ≃Mn(K)

There are plenty such central simpleK-algebras :

Example 1.1 For any non-zero functionsf, g ∈ K∗, thecyclic algebra

Σ = (f, g)n defined by (f, g)n =K〈x, y〉

(xn − f, yn − g, yx− qxy)


with q is a primitiven-th root of unity, is a central simpleK-algebra of dimensionn2.Often,(f, g)n will even be adivision algebra, that is a non-commutative algebra such thatevery non-zero element has an inverse.

For example, this is always the case whenE = K[x] is a (commutative) field extension ofdimensionn and if g has ordern in the quotientK∗/NE/K(E∗) whereNE/K is thenormmapofE/K. See for example [37, Chp. 15] for more details, but if your German is á pointI strongly suggest you to read Ina Kersten’s book [21] instead.

Now, fix such a central simpleK-algebraΣ. An R-orderA in Σ is a subalgebrasA ⊂ Σwith centerZ(A) = R such thatA is finitely generated as anR-module and contains aK-basis ofΣ, that is

A⊗R K ≃ Σ

The classic reference for orders is Irving Reiner’s book [41] but it is hopelessly outdatedand focusses too much on the one-dimensional case. Here is a gap in the market for some-one to fill...

Example 1.2 In the case of quotient singularitiesX = Cd/G a natural choice ofR-order might be theskew 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′

whereφg is the action ofg onC[x1, . . . , xd]. The center of the skew group algebra is easilyverified to be the ring ofG-invariants

R = C[Cd/G] = C[x1, . . . , xd]G

Further, one can show thatC[x1, . . . , xd]#G is anR-order inMn(K) with n the order ofG. If we ever get to the third lecture, we will give another description of the skew groupalgebra in terms of the McKay-quiver setting and the varietyof commuting matrices.

However, there are plenty of otherR-orders inMn(K) which may or may not be relevantin the study of the quotient singularityCd/G.

Example 1.3 If f, g ∈ R − {0}, then the freeR-submodule of rankn2 of the cyclicK-algebraΣ = (f, g)n of example 1.1

A =

n−1∑

i,j=0

Rxiyj

is anR-order. But there is really no need to go for this ’canonical’example. Someonemore twisted may takeI andJ any two non-zero ideals ofR, and consider

AIJ =n−1∑

i,j=0

IiJjxiyj

which is anR-order too inΣ and which is far from being a projectiveR-module unlessIandJ are invertibleR-ideals.

For example, inMn(K) we can take the ’obvious’R-orderMn(R) but one might also takethe subring

[R IJ R

]

which is anR-order if I andJ are non-zero ideals ofR.


If you are a geometer (and frankly we are all wannabe geometers these days), our goal isto construct lots of affinePGLn-varietiesM such that the algebraic quotientM//PGLnis isomorphic toX and, moreover, such that there is a Zariski open subsetU ⊂ X

M ✛ ⊃ π−1(U)

X ≃M//PGLn

π

❄❄✛ ⊃ U

principalPGLn-fibration

❄❄

for which the quotient map is a principalPGLn-fibration, that is, all fibersπ−1(u) ≃PGLn for u ∈ U .

The connection between such varietiesM and ordersA in central simple algebras may notbe clear at first sight. To give you at least an idea that there is a link, think ofM as theaffine variety ofn-dimensional representationsrepn A and ofU as the Zariski open subsetof all simplen-dimensional representations.

Naturally, one can only expect theR-orderA (or the correspondingPGLn-varietyM )to be useful in the study of resolutions ofX if A is smoothin some appropriate non-commutative sense.

Now, there are many characterizations ofcommutativeregular domainsR :

• 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 non-commutative world leads to quite different con-cepts.

We will call anR-orderA is a central simpleK-algebraΣ :

• Regular if A has finite global dimension together with some extra features such asAuslander regularity or Cohen-Macaulay property, see for example [33].

• Smoothif the correspondingPGLn-affine varietyM is a smooth variety as we willclarify later in this talk.

For applications of Regular orders to desingularizations we refer to the talks by MichelVan den Bergh at this conference or to his paper [46] on this topic. I will concentrate onthe properties of Smooth orders instead. Still, it is worth pointing out the strengths andweaknesses of both definitions right now

jotter :

Regular orders are excellent if you want to control homological properties, for exampleif you want to study the derived categories of their modules.At this moment there is nolocal characterization of Regular orders ifdimX ≥ 2.

Smooth orders are excellent if you want to have smooth modulispaces of semi-stablerepresentations. As we will see later, in each dimension there are only a finite number oflocal types of Smooth orders and these are classified. The downside of this is that Smoothorders are less versatile as Regular orders.

In applications to canonical desingularizations, one often needs the good properties ofboth so there is a case for investigating SmoothRegular orders better than has been donein the past.


In general though, both theories are quite different.

Example 1.4 The skew group algebraC[x1, . . . , xd]#G is always a Regular order but wewill see in the next lecture, it is virtually never a Smooth order.

Example 1.5 LetX be the variety of matrix-invariants, that is

X =Mn(C)⊕Mn(C)//PGLn

wherePGLn acts on pairs ofn× n matrices by simultaneous conjugation. Thetrace ringof two genericn× n matricesA is the subalgebra ofMn(C[Mn(C)⊕Mn(C)]) generatedoverC[X ] by the twogeneric matrices

X =

x11 . . . x1n...

...xn1 . . . xnn

and Y =

y11 . . . y1n...

...yn1 . . . ynn

Then,A is anR-order in a division algebra of dimensionn2 overK, called thegenericdivision algebra. Moreover,A is a Smooth order but is Regular only whenn = 2, see [30].

1.3 Constructing orders by descent

jotter :

French mathematicians have developed in the sixties an elegant theory, calleddescenttheory, which allows one to construct elaborate examples out of trivial ones by bringing intopology. This theory allows to classify objects which are only locally (but not necessarilyglobally) trivial.

For applications to orders there are two topologies to consider : the well-known Zariskitopology and the perhaps lesser-known étale topology. Letus try to give a formal definitionof Zariski and étalecoversaimed at ringtheorists.

A Zariski coverof X is a finite product of localizations at elements ofR

Sz =

k∏

i=1

Rfi such that (f1, . . . , fk) = R

and is therefore a faithfully flat extension ofR. Geometrically, the ringmorphismR ✲ Sz defines a cover ofX = spec R by k disjoint sheetsspec Sz = ⊔ispec Rfi ,each corresponding to a Zariski open subset ofX , the complement ofV(fi) and the condi-tion is that these closed subsetsV(fi) do not have a point in common. That is, we have thepicture of figure 1.1 :

Zariski covers form aGrothendieck topology, that is, two Zariski coversS1z =∏ki=1Rfi

andS2z =∏lj=1 Rgj have a common refinement

Sz = S1z ⊗R S

2z =

k∏

i=1

l∏

j=1

Rfigj

For a given Zariski coverSz =∏ki=1 Rfi a correspondinǵetale coveris a product

Se =k∏

i=1

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

(g(i)1, . . . , g(i)ki)with

∂g(i)1∂x(i)1

. . . ∂g(i)1∂x(i)ki...

...∂g(i)ki∂x(i)1

. . .∂g(i)ki∂x(i)ki


spec R

spec Rfk

spec Rf2

spec Rf1

...

Fig. 1.1: A Zariski cover ofX = spec R

a unit in thei-th component ofSe. In fact, for applications to orders it is usually enough toconsiderspecial etale extensions

Se =

k∏

i=1

Rfi [x]

(xki − ai)where ai is a unit inRfi

Geometrically, an étale cover determines for every Zariski sheetspec Rfi a locally iso-morphic(for the analytic topology) multi-covering and the number of sheets may vary withi (depending on the degrees of the polynomialsg(i)j ∈ Rfi [x(i)1, . . . , x(i)ki ]. That is, themental picture corresponding to an étale cover is given in figure 1.2 below.

Again, étale covers form a Zariski topology as the common refinementS1e ⊗R S2e of two

étale covers is again étale because its components are of the form

Rfigj [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. Becauseof the local isomorphism property many ringtheoretical local properties (such as smooth-ness, normality etc.) are preserved under étale covers.

Now, fix anR-orderB in some central simpleK-algebraΣ, then aZariski twisted formAof B is anR-algebra such that

A⊗R Sz ≃ B ⊗R Sz

for some Zariski coverSz of R.

If P ∈ X is a point with corresponding maximal idealm, thenP ∈ spec Rfi for some ofthe components ofSz and asAfi ≃ Bfi we have for the local rings atP

Am ≃ Bm

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

Likewise, anétale twisted formA of B is anR-algebra such that

A⊗R Se ≃ B ⊗R Se


spec R

specRfk [x(k)1,...,x(k)kk ]

(g(k)1,...,g(k)kk )

specRf2 [x(2)1,...,x(2)k2 ]

(g(2)1,...,g(2)k2 )

specRf1 [x(1)1,...,x(1)k1 ]

(g(1)1,...,g(1)k1 )

...

Fig. 1.2: An étale cover ofX = spec R

for some étale coverSe of R.

This time the Zariski local information ofA andB may be different at a pointP ∈ X butwe do have that them-adic completions ofA andB

Âm ≃ B̂m

are isomorphic aŝRm-algebras.

jotter :

The Zariski local structure ofA determines the localizationAm, the étale local structuredetermines the completion̂Am.

Descent theory allows to classify Zariski- or étale twisted forms of anR-orderB by meansof the corresponding cohomology groups of the automorphismschemes. For more detailson this please read the book [23] by M. Knus and M. Ojanguren ifyou are a ringtheoristand that of S. Milne [35] if you are more of a geometer.

If one applies descent to the most trivial of allR-orders, the full matrix algebraMn(R),one arrives at

1.4 Azumaya algebras

A Zariski twisted form ofMn(R) is anR-algebraA such that

A⊗R Sz ≃Mn(Sz) =

k∏

i=1

Mn(Rfi )


Conversely, you can construct such twisted forms bygluing togetherthe matrix ringsMn(Rfi). The easiest way to do this is to glueMn(Rfi) with Mn(Rfj ) overRfifj viathe natural embeddings

Rfi⊂ ✲ Rfifj ✛ ⊃ Rfj

Not surprisingly, we obtain in this wayMn(R) back.

But there are more clever ways to perform the gluing by bringing in the non-commutativityof matrix-rings. We can glue

Mn(Rfi)⊂ ✲ Mn(Rfifj )

gij .g−1ij

≃✲ Mn(Rfifj ) ✛ ⊃ Mn(Rfj )

over their intersection via conjugation with an invertiblematrix gij ∈ GLn(Rfifj ). If theelementsgij for 1 ≤ i, j ≤ k satisfy thecocycle condition(meaning that the differentpossible gluings are compatible over their common localizationRfifjfl), we obtain a sheafof non-commutative algebrasA overX = spec R such that its global sections are notnecessarilyMn(R).

Proposition 1.6 Any Zariski twisted form ofMn(R) is isomorphic to

EndR(P )

whereP is a projectiveR-module of rankn. Two such twisted forms are isomorphic asR-algebras

EndR(P ) ≃ EndR(Q) iff P ≃ Q⊗ I

for some invertibleR-idealI.

Proof. [sketch] We have an exact sequence of groupschemes

1 ✲ Gm ✲ GLn ✲ PGLn ✲ 1

(here,Gm is the sheaf of units) and taking Zariski cohomology groups overX we have asequence

1 ✲ H1Zar(X,Gm) ✲ H1Zar(X, GLn)

✲ H1Zar(X, PGLn)

where the first term is isomorphic to the Picard groupPic(R) and the second term classifiesprojectiveR-modules of rankn upto isomorphism. The final term classifies the Zariskitwisted forms ofMn(R) as the automorphism group ofMn(R) is PGLn. �

Example 1.7 Let I andJ be two invertible ideals ofR, then

EndR(I ⊕ J) ≃

[R I−1J

IJ−1 R

]

⊂M2(K)

and ifIJ−1 = (r) thenI ⊕ J ≃ (Rr ⊕R)⊗ J and indeed we have an isomorphism[1 00 r−1

] [R I−1J

IJ−1 R

] [1 00 r

]

=

[R RR R

]

Things get a lot more interesting in the étale topology.

Definition 1.8 An n-Azumaya algebraoverR is an étale twisted formA of Mn(R). If Ais also a Zariski twisted form we callA a trivial Azumaya algebra.


From the definition and faithfully flat descent, the following facts follow :

Lemma 1.9 If A is ann-Azumaya algebra overR, then :

1. The centerZ(A) = R andA is a projectiveR-module of rankn2.

2. All simpleA-representations have dimensionn and for every maximal idealm ofRwe have

A/mA ≃Mn(C)

Proof. For (2) takeM ∩ R = m whereM is the kernel of a simple representationA ✲✲ Mk(C), then asÂm ≃Mn(R̂m) it follows that

A/mA ≃Mn(C)

and hence thatk = n andM = Am. �

It is clear from the definition that whenA is ann-Azumaya algebra andA′ is anm-Azumaya algebra overR,A⊗R A′ is anmn-Azumaya and also that

A⊗R Aop ≃ EndR(A)

whereAop is theoppositealgebra (that is, equipped with the reverse multiplicationrule).

These facts allow us to define theBrauer groupBrR to be the set of equivalence classes[A] of Azumaya algebras overR where

[A] = [A′] iff A⊗R A′ ≃ EndR(P )

for some projectiveR-moduleP 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 schemesand A. Grothendieck has shown that the Brauer group of a projective smooth variety is abirational invariant, see [19]. Moreover, he conjectured acohomological description of theBrauer groupBrR which was subsequently proved by O. Gabber in [16].

Theorem 1.10 The Brauer group is ańetale cohomology group

BrR ≃ H2et(X,Gm)torsion

whereGm is the unit sheaf and where the subscript denotes that we takeonly torsionelements. IfR is regular, thenH2et(X,Gm) is torsion so we can forget the subscript.

This result should be viewed as the ringtheory analogon of thecrossed product theoremforcentral simple algebras over fields, see for example [37].

Observe that in Gabber’s result there is no sign of singularities in the description of theBrauer group. In fact, with respect to the desingularization problem, Azumaya algebras areonly as good as their centers.

Proposition 1.11 If A is ann-Azumaya algebra overR, then

1. A is Regular iffR is commutative regular.


2. A is Smooth iffR is commutative regular.

Proof. (1) follows from faithfully flat descent and(2) from lemma 1.9 which asserts thatthe PGLn-affine variety corresponding toA is a principalPGLn-fibration in the étaletopology, which shows that bothn-Azumaya algebras and principalPGLn-fibrations areclassified by the étale cohomology groupH1et(X, PGLn). �

jotter :

In the correspondence betweenR-orders andPGLn-varieties, Azumaya algebras corre-spond toprincipal PGLn-fibrations overX . With respect to desingularizations, Azu-maya algebras are therefore only as good as their centers.

1.5 Reflexive Azumaya algebras

So let us bring inramificationin order to construct orders which may be more useful in ourdesingularization project.

Example 1.12 Consider theR-order inM2(K)

A =

[R RI R

]

whereI is some ideal ofR. If P ∈ X is a point with corresponding maximal idealm wehave that :

For I not contained inm we haveAm ≃M2(Rm) whenceA is an Azumaya algebra inP .

For I ⊂ m we have

Am ≃

[Rm RmIm Rm

]

6=M2(Rm)

whenceA is not Azumaya inP .

Definition 1.13 Theramification locusof anR-orderA is the Zariski closed subset ofXconsisting of those pointsP such that for the corresponding maximal idealm

A/mA 6≃Mn(C)

That is,ram A is the locus ofX whereA is not an Azumaya algebra. Its complementazu A is called theAzumaya locusof A which is always a Zariski open subset ofX .

Definition 1.14 An R-orderA is said to be areflexiven-Azumaya algebraiff

1. ram A has codimension at least two inX , and

2. A is a reflexiveR-module

that is,A ≃ HomR(HomR(A,R), R) = A∗∗.

The origin of the terminology is that whenA is a reflexiven-Azumaya algebra we havethatAp is n-Azumaya for every height one prime idealp of R and thatA = ∩pAp wherethe intersection is taken over all height one primes.


For example, in example 1.12 ifI is a divisorial ideal ofR, thenA is not reflexive AzumayaasAp is not Azumaya forp a height one prime containingI and ifI has at least height two,thenA is often not a reflexive Azumaya algebra becauseA is not reflexive as anR-module.For example take

A =

[C[x, y] C[x, y](x, y) C[x, y]

]

then the reflexive closure ofA isA∗∗ =M2(C[x, y]).

Sometimes though, we get reflexivity ofA for free, for example whenA is a Cohen-MacaulayR-module. An other important fact to remember is that forA a reflexive Azu-maya,A is Azumaya if and only ifA is projective as anR-module. If you want to knowmore about reflexive Azumaya algebras you may want to read [36] or my Ph.D. thesis [24].

Example 1.15 Let A = C[x1, . . . , xd]#G thenA is a reflexive Azumaya algebra when-everG acts freely away from the origin andd ≥ 2. Moreover,A is never an Azumayaalgebra as its ramification locus is the isolated singularity.

In analogy with the Brauer group one can define thereflexive Brauer groupβ(R) whoseelements are the equivalence classes[A] for A a reflexive Azumaya algebra overR withequivalence relation

[A] = [A′] iff A⊗R A′ ≃ EndR(M)

whereM is a reflexiveR-module and with multiplication induced by the rule

[A].[A′] = [(A⊗R A′)∗∗]

In [26] it was shown that the reflexive Brauer group does have acohomological description

Proposition 1.16 The reflexive Brauer group is ańetale cohomology group

β(R) ≃ H2et(Xsm,Gm)

whereXsm is the smooth locus ofX .

This time we see that the singularities ofX do appear in the description so perhaps reflexiveAzumaya algebras are a class of orders more suitable for our project. This is even moreevident if we impose non-commutative smoothness conditions onA.

Proposition 1.17 LetA be a reflexive Azumaya algebra overR, then :

1. ifA is Regular, thenram A = Xsing , and

2. ifA is Smooth, thenXsing is contained inram A.

Proof. (1) was proved in [27] the essential point being that ifA is Regular thenA is aCohen-MacaulayR-module whence it must be projective over a smooth point ofX butthen it is not just an reflexive Azumaya but actually an Azumaya algebra in that point. Thesecond statement can be further refined as we will see in the next lecture. �

Many classes of well-studied algebras are reflexive Azumayaalgebras,

• Trace ringsTm,n of m genericn× n matrices (unless(m,n) = (2, 2)), see [25].


• Quantum enveloping algebrasUq(g) of semi-simple Lie algebras at roots of unity,see for example [8].

• Quantum function algebrasOq(G) for semi-simple Lie groups at roots of unity, seefor example [9].

• Symplectic reflection algebrasAt,c, see [10].

jotter :

Many interesting classes of Regular orders are reflexive Azumaya algebras. As a conse-quence their ramification locus coincides with the singularity locus of the center.

1.6 Cayley-Hamilton algebras

It is about time to clarify the connection withPGLn-equivariant geometry. We will intro-duce a class of non-commutative algebras, the so calledCayley-Hamilton algebraswhichare the leveln generalization of the category of commutative algebras andwhich containall R-orders.

A trace maptr is aC-linear functionA ✲ A satisfying for alla, b ∈ A

tr(tr(a)b) = tr(a)tr(b) tr(ab) = tr(ba) and tr(a)b = btr(a)

so in particular, the imagetr(A) is contained in the center ofA.

If M ∈Mn(R) whereR is a commutativeC-algebra, then its characteristic polynomial

χM = det(t1n −M) = tn + a1t

n−1 + a2tn−2 + . . .+ an

has coefficientsai which are polynomials with rational coefficients in traces of powers ofM

ai = fi(tr(M), tr(M2), . . . , tr(Mn−1)

Hence, if we have an algebraA with a trace maptr we can define aformal characteristicpolynomialof degreen for everya ∈ A by taking

χa = tn + f1(tr(a), . . . , tr(a

n−1)tn−1 + . . .+ fn(tr(a), . . . , tr(an−1)

which allows us to define the categoryalg@n of Cayley-Hamilton algebras of degreen.

Definition 1.18 An objectA in alg@n is a Cayley-Hamilton algebra of degreen, that is, aC-algebra with trace maptr : A ✲ A satisfying

tr(1) = n and ∀a ∈ A : χa(a) = 0

MorphismsA ✲ B in alg@n are trace preservingC-algebra morphisms, that is,

A ✲ B

A

trA

❄✲ B

trB

❄

is a commutative diagram.


Example 1.19 Azumaya algebras, reflexive Azumaya algebras and more generally everyR-orderA in a central simpleK-algebra of dimensionn2 is a Cayley-Hamilton algebra ofdegreen. For, consider the inclusions

A ⊂ ✲ Σ ⊂ ✲ Mn(K)

R

tr

❄

................⊂ ✲ K

tr

❄

................⊂ ✲ K

tr

❄

Here,tr : Mn(K) ✲ K is the usual trace map. By Galois descent this induces a tracemap, the so calledreduced trace, tr : Σ ✲ K. Finally, becauseR is integrally closedin K andA is a finitely generatedR-module it follows thattr(a) ∈ R for every elementa ∈ A.

If A is a finitely generated object inalg@n, we can define an affinePGLn-scheme,

trepn A, classifying all trace preservingn-dimensional representationsAφ✲ Mn(C)

of A. The action ofPGLn ontrepn A is induced by conjugation in the target space, thatis g.φ is the trace preserving algebra map

Aφ✲ Mn(C)

g−1g✲ Mn(C)

Orbits under this action correspond precisely to isomorphism classes of representations.The schemetrepn A is a closed subscheme ofrepn A the more familiarPGLn-affinescheme of alln-dimensional representations ofA. In general, both schemes may be differ-ent.

Example 1.20 LetA be the quantum plane at−1, that is

A =C〈x, y〉

(xy + yx)

thenA is an order with centerR = C[x2, y2] in the quaternion algebra(x, y)2 = K1 ⊕Ku⊕Kv ⊕Kuv overK = C(x, y) whereu2 = x.v2 = y anduv = −vu. Observe thattr(x) = tr(y) = 0 as the embeddingA ⊂ ✲ (x, y)2 ⊂ ✲ M2(C[u, y]) is given by

x 7→

[u 00 −u

]

and y 7→

[0 1y 0

]

Therefore, a trace preserving algebra mapA ✲ M2(C) is fully determined by theimages ofx andy which are trace zero2× 2 matrices

φ(x) =

[a bc −a

]

and φ(y) =

[d ef −d

]

satisfying bf + ce = 0

That is,trep2 A is the hypersurfaceV(bf + ce) ⊂ A6 which has a unique isolated singu-

larity at the origin. However,rep2 A contains more points, for example

φ(x) =

[a 00 b

]

and φ(y) =

[0 00 0

]

is a point inrep2 A− trep2 A wheneverb 6= −a.

A functorial description oftrepn A is given by the following universal property proved byC. Procesi [39]


Theorem 1.21 LetA be aC-algebra with trace maptrA, then there is a trace preservingalgebra morphism

jA : A ✲ Mn(C[trepn A])

satisfying the following universal property. IfC is a commutativeC-algebra and there is

a trace preserving algebra mapAψ✲ Mn(C) (with the usual trace onMn(C)), then

there is a unique algebra morphismC[trepn A]φ✲ C such that the diagram

Aψ✲ Mn(C)

��

Mn(φ)

✒

Mn(C[trepn A])

jA

❄

is commutative. Moreover,A is an object inalg@n if and only ifjA is a monomorphism.

The PGLn-action on trepn A induces an action ofPGLn by automorphisms onC[trepn A]. On the other hand,PGLn acts by conjugation onMn(C) so we have acombined action onMn(C[trepn A]) = Mn(C) ⊗ C[trepn A] and it follows from theuniversal property that the image ofjA is contained in the ring ofPGLn-invariants

AjA✲ Mn(C[trepn A])

PGLn

which is an inclusion ifA is a Cayley-Hamilton algebra. In fact, C. Procesi proved in [39]the following important result which allows to reconstructorders and their centers fromPGLn-equivariant geometry.

Theorem 1.22 The functor

trepn : alg@n✲ PGL(n)-affine

has aleft inverseA− : PGL(n)-affine ✲ alg@n

defined byAY =Mn(C[Y ])PGLn . In particular, we have for anyA in alg@n

A =Mn(C[trepn A])PGLn and tr(A) = C[trepn A]

PGLn

That is the central subalgebratr(A) is the coordinate ring of the algebraic quotient variety

trepn A//PGLn = tissn A

classifying isomorphism classes of trace preserving semi-simplen-dimensional represen-tations ofA.

However, these functors donotgive an equivalence betweenalg@n andPGLn-equivariantaffine geometry. There are plenty morePGLn-varieties than Cayley-Hamilton algebras.

Example 1.23 Conjugacy classes of nilpotent matrices inMn(C) correspond bijective topartitionsλ = (λ1 ≥ λ2 ≥ . . .) of n (theλi determine the sizes of the Jordan blocks). Itfollows from the Gerstenhaber-Hesselink theorem that the closures of such orbits

Oλ = ∪µ≤λOµ

where≤ is the dominance order relation. EachOλ is an affinePGLn-variety and thecorresponding algebra is

AOλ = C[x]/(xλ1)

whence many orbit closures (all of which are affinePGLn-varieties) correspond to thesame algebra.


jotter :

The categoryalg@n of Cayley-Hamilton algebras is tononcommutative geometry@nwhat commalg, the category of all commutative algebras is to commutativealgebraicgeometry.

In fact, alg@1 ≃ commalg by taking as trace maps the identity on every commutativealgebra. Further we have a natural commutative diagram of functors

alg@ntrepn ✲✛A−

PGL(n)-aff

commalg

tr

❄

spec

✲ aff

quot

❄

where the bottom map is the equivalence between affine algebras and affine schemes andthe top map is the correspondence between Cayley-Hamilton algebras and affinePGLn-schemes, which isnotan equivalence of categories.

1.7 Smooth orders

To finish this talk let us motivate and define the notion of aSmooth orderproperly. Amongthe many characterizations of commutative regular algebras is the following due to A.Grothendieck.

Theorem 1.24 A commutativeC-algebraA is regular if and only if it satisfies the followinglifting property : if (B, I) is a test-object such thatB is a commutative algebra andI is anilpotent ideal ofB, then for any algebra mapφ, there exists a lifted algebra morphism̃φ

A ....................∃φ̃

✲ B

❅❅❅❅

φ

❘B/I

π

❄❄

making the diagram commutative.

As the categorycommalg of all commutativeC-algebras is justalg@1 it makes sense todefine Smooth Cayley-Hamilton algebras by the same lifting property. This was done firstby W. Schelter [42] in the category of all algebras satisfying all polynomial identities ofn× n matrices and later by C. Procesi [39] inalg@n.

Definition 1.25 A Smooth Cayley-Hamilton algebraA is an object inalg@n satisfying thefollowing lifting property. If (B, I) is a test-object inalg@n, that is,B is an object inalg@n, I is a nilpotent ideal inB such thatB/I is an object inalg@n and such that the

natural mapBπ✲✲ B/I is trace preserving, then every trace preserving algebra map φ


has a liftφ̃

A ....................∃φ̃

✲ B

❅❅❅❅

φ

❘B/I

π

❄❄

making the diagram commutative. IfA is in addition an order, we say thatA is aSmoothorder.

Next talk we will give a large class of Smooth orders but againit should be stressed thatthere is no connection between this notion of non-commutative smoothness and the morehomological notion of Regular orders (except in dimension one when all notions coincide).

Still, in the context ofPGLn-equivariant affine geometry this notion of non-commutativesmoothness is quite natural as illustrated by the followingresult due to C. Procesi [39].

Theorem 1.26 An objectA in alg@n is Smooth if and only if the corresponding affinePGLn-schemetrepn A is smooth (and hence, in particular, reduced).

Proof. (One implication) AssumeA is Smooth, then to prove thattrepn A is smoothwe have to prove thatC[trepn A] satisfies Grothendieck’s lifting property. So let(B, I)be a test-object incommalg and take an algebra morphismφ : C[trepn A] ✲ B/I.Consider the following diagram

A..............

(1)

❘Mn(C[trepn A])

jA

❄

∩

.......(2)✲ Mn(B)

❅❅❅❅

Mn(φ)

❘Mn(B/I)

❄❄

the morphism(1) follows from Smoothness ofA applied to the morphismMn(φ) ◦ jA.From the universal property of the mapjA it follows that there is a morphism(2) which isof the formMn(ψ) for some algebra morphismψ : C[trepn A] ✲ B. Thisψ is therequired lift. �

Example 1.27 Trace ringsTm,n are the free algebras generated bym elements inalg@nand as such trivially satisfy the lifting property so are Smooth orders. Alternatively, because

trepn Tm,n ≃Mn(C)⊕ . . .⊕Mn(C) = Cmn2

is a smoothPGLn-variety,Tm,n is Smooth by the previous result.

Any commutative algebraC can be viewed as an element ofalg@n via the diagonal embed-dingC ⊂ ✲ Mn(C). However, ifC is a regular commutative algebra it isnot true thatCis Smooth inalg@n. For example, takeC = C[x1, . . . , xd] and consider the4-dimensionalnon-commutative local algebra

B =C〈x, y〉

(x2, y2, xy + yx)= C⊕ Cx⊕ Cy ⊕ Cxy


with the obvious trace map so thatB ∈ alg@2. B has a nilpotent idealI = B(xy − yx)such that the quotientB/I is a3-dimensional commutative algebra. Consider the algebramap

C[x1, . . . , xd]φ✲ B

Idefined by x1 7→ x x2 7→ y and xi 7→ 0 for i ≥ 3

This map has no lift as for any potential lifted morphism̃φ we have

[φ̃(x), φ̃(y)] 6= 0

whenceC[x1, . . . , xd] is not Smooth inalg@2.

Example 1.28 Consider again the quantum plane at−1

A =C〈x, y〉

(xy + yx)

then we have seen thattrep2 A = V(bf + ce) ⊂ A6 has a unique isolated singularity at

the origin. Hence,A is not a Smooth order.

jotter :

Under the correspondence betweenalg@n andPGL(n)-aff, Smooth Cayley-Hamiltonalgebras correspond to smoothPGLn-varieties.

lecture 2

NON-COMMUTATIVE GEOMETRY

Last time we introducedalg@n as a leveln generalization ofcommalg, the variety of allcommutative algebras. Today we will associate to anyA ∈ alg@n a non-commutativevariety max A and argue that this gives a non-commutative manifold ifA is a Smoothorder. In particular we will show that for fixedn and central dimensiond there are a finitenumber of étale types of such orders. This fact is the non-commutative analogon of thefact that every manifold is locally diffeomorphic to affine space or, in ringtheory terms,that them-adic completion of a regular algebraC of dimensiond has just one étale type :Ĉm ≃ C[[x1, . . . , xd]].

2.1 Why non-commutative geometry?

jotter :

There is one new feature that non-commutative geometry has to offer compared to com-mutative geometry : distinct points can lie infinitesimallyclose to each other. As desin-gularization is the process of separating bad tangents, this fact should be useful somehowin our project.

Recall that ifX is an affine commutative variety with coordinate ringR, then to each pointP ∈ X corresponds a maximal idealmP ⊳ R and a one-dimensional simple representation

SP =R

mP

A basic tool in the study of Hilbert schemes is that finite closed subschemes ofX can bedecomposed according to their support. In algebraic terms this means that there are noextensions between different points, that ifP 6= Q then

Ext1R(SP , SQ) = 0 whereas Ext1R(SP , SP ) = TP X

In more plastic lingo : all infinitesimal information ofX nearP is contained in the self-extensions ofSP and distinct points do not contribute. This is no longer the case fornon-commutative algebras.

Example 2.1 Take the path algebraA of thequiver oo , that is

A ≃

[C C

0 C

]

ThenA has two maximal ideals and two corresponding one-dimensional simple represen-tations

S1 =

[C

0

]

=

[C C

0 C

]

/

[0 C0 C

]

and S2 =

[0C

]

=

[C C

0 C

]

/

[C C

0 0

]

lecture 2. non-commutative geometry 24

Then, there is a non-split exact sequence with middle term the second column ofA

0 ✲ S1 =[C

0

]

✲ M =[C

C

]

✲ S2 =[0C

]

✲ 0

WhenceExt1A(S2, S1) 6= 0 whereasExt1A(S1, S2) = 0. It is no accident that these two

facts are encoded into the quiver.

Definition 2.2 ForA an algebra inalg@n, define itsmaximal ideal spectrummax A to bethe set of all maximal twosided idealsM of A equipped with thenon-commutative Zariskitopology, that is, a typical open set ofmax A is of the form

X(I) = {M ∈ max A | I 6⊂M}

Recall that for everyM ∈ max A the quotient

A

M≃Mk(C) for somek ≤ n

that is,M determines a uniquek-dimensional simple representationSM of A.

As every maximal idealM of A intersects the centerR in a maximal idealmP = M ∩ Rwe get, in the case of anR-orderA a continuous map

max Ac✲ X defined by M 7→ P whereM ∩R = mP

Ringtheorists have studied the fibersc−1(P ) of this map in the seventies and eighties inconnection with localization theory. The oldest description is theBergman-Smalltheorem,see for example [2]

Theorem 2.3 (Bergman-Small)If c−1(P ) = {M1, . . . ,Mk} then there are natural num-bersei ∈ N+ such that

n =

k∑

i=1

eidi wheredi = dimC SMi

In particular, c−1(P ) is finite for allP .

Here is a modern proof of this result based on the results of the previous lecture. BecauseX is the algebraic quotienttrepn A//GLn, points ofX correspond toclosedGLn-orbitsin repn A. By a result of M. Artin [1] closed orbits are precisely the isomorphism classesof semi-simplen-dimensional representations, and therefore we denote thequotient variety

X = trepn A//GLn = tissn A

So, a pointP determines a semi-simplen-dimensionalA-representation

MP = S⊕e11 ⊕ . . .⊕ S

⊕ekk

with theSi the distinct simple components, say of dimensiondi = dimC Si and occurringin MP with multiplicity ei ≥ 1. This givesn =

∑eidi and clearly the annihilator ofSi is

a maximal idealMi of A lying overmP .

Another interpretation ofc−1(P ) follows from the work of A. V. Jategaonkar and B. Müller.Define alink diagramon the points ofmax A by the rule

M M ′ ⇔ Ext1A(SM , SM ′) 6= 0

In fancier language,M M ′ if and only ifM andM ′ lie infinitesimally close together inmaxA. In fact, the definition of the link diagram in [20, Chp. 5] or [18, Chp. 11] is slightlydifferent but amounts to the same thing.


Theorem 2.4 (Jategaonkar-Müller) The connected components of the link diagram onmax A are all finite and are in one-to-one correspondence withP ∈ X . That is, if

{M1, . . . ,Mk} = c−1(P ) ⊂ max A

then this set is a connected component of the link diagram.

jotter :

In maxA there is a Zariski open set ofAzumaya points, that is thoseM ∈ maxA such thatA/M ≃ Mn(C). It follows that each of these maximal ideals is a singleton connectedcomponent of the link diagram. So on this open set there is a one-to-one correspondencebetween points ofX and maximal ideals ofA so we can say thatmax A andX arebirational. However, over the ramification locus there may be several maximal ideals ofA lying over the same central maximal ideal and these points should be thought of aslying infinitesimally close to each other.

ram AX

max A

One might hope that the cluster of infinitesimally points ofmax A lying over a centralsingularityP ∈ X can be used to separate tangent information inP rather than having toresort to the blowing-up process to achieve this.

2.2 What non-commutative geometry?

As anR-orderA in a central simpleK-algebraΣ of dimensionn2 is a finiteR-module, wecan associate toA the sheafOA of non-commutativeOX -algebras using central localiza-tion. That is, the section over a basic affine open pieceX(f) ⊂ X are

Γ(X(f),OA) = Af = A⊗R Rf

which is readily checked to be a sheaf with global sectionsΓ(X,OA) = A. As we willinvestigate Smooth orders via their (central) étale structure, that is information about̂AmP ,we will only need the structure sheafOA overX .

In the ’70-ties F. Van Oystaeyen [47] and A. Verschoren [48] introduced genuine non-commutative structure sheaves associated to anR-orderA. It is not my intention to promotenostalgia here but perhaps these non-commutative structure sheavesOncA onmaxA deserverenewed investigation.

Definition 2.5 OncA is defined by taking as the sections over the typical open setX(I) (forI a twosided ideal ofA) in max A

Γ(X(I),OncA ) = {δ ∈ Σ | ∃l ∈ N : Ilδ ⊂ A }

By [47] this defines a sheaf of non-commutative algebras overmax A with global sectionsΓ(max A,OncA ) = A. The stalk of this sheaf at a pointM ∈ max A is the symmetriclocalization

OncA,M = QA−M (A) = {δ ∈ Σ | Iδ ⊂ A for some idealI 6⊂ P }


Often, these stalks have no pleasant properties but in some examples, these non-commutative stalks are nicer than those of the central structure sheaf.

Example 2.6 LetX = A1, that is,R = C[x] and consider the order

A =

[R Rm R

]

wherem = (x)⊳R. A is an hereditary order so is both a Regular order and a Smooth order.The ramification locus ofA is P0 = V(x) so over anyP0 6= P ∈ A1 there is a uniquemaximal ideal ofA lying overmP and the corresponding quotient isM2(C). However,overm there are two maximal ideals ofA

M1 =

[m Rm R

]

and M2 =

[R Rm m

]

BothM1 andM2 determine a one-dimensional simple representation ofA, so the Bergman-Small number aree1 = e2 = 1 andd1 = d2 = 1. That is, we have the following picture

m

A1

max AM1

M2

There is one non-singleton connected component in the link diagram ofA namely

!!


2.3 Marked quiver and Morita settings

Consider the continuous map for the Zariski topology

max Ac✲ X

and let for a central pointP ∈ X the fiber be{M1, . . . ,Mk} where theMi are maximalideals ofA with corresponding simpledi-dimensional representationSi. In the previoussection we have introduced theBergman-Small data, that is

α = (e1, . . . , ek) and β = (d1, . . . , dk) ∈ Nk+ satisfying α.β =k∑

i=1

eidi = n

(recall thatei is the multiplicity of Si in the semi-simplen-dimensional representationcorresponding toP . Moreover, we have theJategaonkar-M̈uller datawhich is a directedconnected graph on the vertices{v1, . . . , vk} (corresponding to theMi) with an arrow

vi vj iff Ext1A(Si, Sj) 6= 0

We now want to associate combinatorial objects to this localdata.

To start, introduce a quiver setting(Q,α) whereQ is aquiver(that is, a directed graph) onthe vertices{v1, . . . , vk} with the number of arrows fromvi to vj equal to the dimensionof Ext1A(Si, Sj),

# ( vi ✲ vj ) = dimC Ext1A(Si, Sj)

and whereα = (e1, . . . , ek) is thedimension vectorof the multiplicitiesei.

Recall that the representation spacerepα Q of a quiver-setting is⊕aMei×ej (C) where thesum is taken over all arrowsa : vj ✲ vi of Q. On this space there is a natural actionby the group

GL(α) = GLe1 × . . .×GLek

by base-change in the vertex-spacesVi = Cei (actually this is an action ofPGL(α) whichis the quotient ofGL(α) by the central subgroupC∗(1e1 , . . . , 1ek)).

The ringtheoretic relevance of the quiver-setting(Q,α) is that

repα Q ≃ Ext1A(MP ,MP ) asGL(α)-modules

whereMP is the semi-simplen-dimensionalA-module corresponding toP

MP = S⊕e11 ⊕ . . .⊕ S

⊕ekk

and becauseGL(α) is the automorphism group ofMP there is an induced action onExt1A(MP ,MP ).

BecauseMP is n-dimensional, an elementψ ∈ Ext1A(MP ,MP ) defines an algebra mor-phism

Aρ✲ Mn(C[ǫ])

whereC[ǫ] = C[x]/(x2) is the ring ofdual numbers. As we are working in the categoryalg@n we need the stronger assumption thatρ is trace preserving. For this reason we haveto consider theGL(α)-subspace

tExt1A(MP ,MP ) ⊂ Ext1A(MP ,MP )

of trace preserving extensions. As traces only use blocks on the diagonal (correspondingto loops inQ) and as any subspaceMei(C) of repα Q decomposes as aGL(α)-module insimple representations

Mei(C) =M0ei(C)⊕ C


whereM0ei(C) is the subspace of trace zero matrices, we see that

repα Q∗ ≃ tExt1A(MP ,MP ) asGL(α)-modules

whereQ∗ is a marked quiverthat has the same number of arrows between distinct ver-tices asQ has, but may have fewer loops and some of these loops may acquire amarkingmeaning that their corresponding component inrepα Q

∗ isM0ei(C) instead ofMei(C).

jotter :

Let the local structure of the non-commutative varietymax A near the fiberc−1(P ) of apointP ∈ X be determined by the Bergman-Small data

α = (e1, . . . , ek) and β = (d1, . . . , dk)

and by the Jategoankar-Müller data which is encoded in the marked quiverQ∗ on k-vertices. Then, we associate toP the combinatorial data

type(P ) = (Q∗, α, β)

We call (Q∗, α) the marked quiver settingassociated toA in P ∈ X . The dimensionvectorβ = (d1, . . . , dk) will be called theMorita settingassociated toA in P .

Example 2.7 If A is an Azumaya algebra overR. then for every maximal idealm corre-sponding to a pointP ∈ X we have that

A/mA =Mn(C)

so there is a unique maximal idealM = mA lying overm whence the Jategaonkar-Müllerdata areα = (1) andβ = (n). If SP = R/m is the simple representation ofR we have

Ext1A(MP ,MP ) ≃ Ext1R(SP , SP ) = TP X

and as all the extensions come from the center, the corresponding algebra representationsA ✲ Mn(C[ǫ]) are automatically trace preserving. That is, the marked quiver-settingassociated toA in P is

1((

hh

where the number of loops is equal to the dimension of the tangent spaceTP X in P atXand the Morita-setting associated toA in P is (n).

Example 2.8 Consider the order of example 2.6 which is generated as aC-algebra by theelements

a =

[1 00 0

]

b =

[0 10 0

]

c =

[0 0x 0

]

d =

[0 00 1

]

and the2-dimensional semi-simple representationMP0 determined bym is given by thealgebra morphismA ✲ M2(C) sendinga andd to themselves andb andc to the zeromatrix. A calculation shows that

Ext1A(MP0 ,MP)) = repα Q for (Q,α) = 1u **

1v

jj


and as the correspondence with algebra maps toM2(C[ǫ]) is given by

a 7→

[1 00 0

]

b 7→

[0 ǫv0 0

]

c 7→

[0 0ǫu 0

]

d 7→

[0 00 1

]

each of these maps is trace preserving so the marked quiver setting is(Q,α) and the Morita-setting is(1, 1).

2.4 Local classification

jotter :

Because the combinatorial datatype(P ) = (Q∗, α, β) encodes the infinitesimal in-formation of the cluster of maximal ideals ofA lying over the central pointP ∈ X ,(repα Q

∗, β) should be viewed as analogous to the usual tangent spaceTP X .

If P ∈ X is a singular point, then the tangent space is too large so we have to imposeadditional relations to describe the varietyX in a neighborhood ofP , but ifP is a smoothpoint we can recover the local structure ofX from TP X .

Here we might expect a similar phenomenon : in general the data (repα Q∗, β) will be

too big to describêAmP unlessA is a Smooth order inP in which case we can recoverÂmP .

We begin by defining some algebras which can be described combinatorially from(Q∗, α, β).

For every arrowa : vi ✲ vj define ageneric rectangular matrixof sizeej × ei

Xa =

x11(a) . . . . . . x1ei(a)...

...xej1(a) . . . . . . xejei(a)

(and ifa is a marked loop takexeiei(a) = −x11(a)−x22(a)− . . .−xei−1ei−1(a)) then thecoordinate ringC[repα Q

∗] is the polynomial ring in the entries of allXa. For an orientedpathp in the marked quiverQ∗ with starting vertexvi and terminating vertexvj

vi ........p✲ vj = vi

a1✲ vi1a2✲ . . .

al−1✲ vilal✲ vj

we can form the squareej × ei matrix

Xp = XalXal−1 . . .Xa2Xa1

which has all its entries polynomials inC[repα Q∗]. In particular, if the path is an oriented

cyclec in Q∗ starting and ending invi thenXc is a squareei × ei matrix and we can takeits tracetr(Xc) ∈ C[repα Q

∗] which is a polynomial invariant under the action ofGL(α)onrepα Q

∗.

In fact, it was proved in [31] that thesetraces along oriented cyclesgenerate the invariantring

RαQ∗ = C[repα Q∗]GL(α) ⊂ C[repα Q

∗]

Next we bring in the Morita-settingβ = (d1, . . . , dk) and define a block-matrix ring

Aα,βQ∗ =

Md1×d1(P11) . . . Md1×dk(P1k)...

...Mdk×d1(Pk1) . . . Mdk×dk(Pkk)

⊂Mn(C[repα Q

∗])


wherePij is theRαQ∗ -submodule ofMej×ei(C[repα Q∗]) generated by allXp wherep is

an oriented path inQ∗ starting invi and ending invk.

Observe that for triples(Q∗, α, β1) and(Q∗, α, β2) we have that

Aα,β1Q∗ is Morita-equivalent to Aα,β2Q∗

whence the name Morita-setting forβ.

Before we can state the next result we need theEuler-formof the underlying quiverQof Q∗ (that is, forgetting the markings of some loops) which is thebilinear formχQ on

Zk determined by the matrix having as its(i, j)-entry δij − #{a : via✲ vj}. The

statements below can be deduced from those of [31]

Theorem 2.9 For a triple (Q∗, α, β) withα.β = n we have

1. Aα,βQ∗ is anRαQ-order in alg@n if and only ifα is the dimension vector of a simple

representation ofQ∗, that is, for all vertex-dimensionsδi we have

χQ(α, δi) ≤ 0 and χQ(δi, α) ≤ 0

unlessQ∗ is an oriented cycle of typẽAk−1 thenα must be(1, . . . , 1).

2. If this condition is satisfied, the dimension of the centerRαQ∗ is equal to

dim RαQ∗ = 1− χQ(α, α) −#{marked loops inQ∗}

These combinatorial algebras determine the étale local structure of Smooth orders as wasproved in [29]. The principal technical ingredient in the proof is theLuna slice theorem,see for example [45] or [34].

Theorem 2.10 LetA be a Smooth order overR in alg@n and letP ∈ X with correspond-ing maximal idealm. If the marked quiver setting and the Morita-setting associated toA inP is given by the triple(Q∗, α, β), then there is a Zariski open subsetX(fi) containingPand anétale extensionS of bothRfi and the algebraR

αQ∗ such that we have the following

diagramAfi ⊗Rfi S ≃ A

α,βQ∗ ⊗RαQ∗ S

��

�✒ ■❅❅❅

❅Afi S

✻

Aα,βQ∗

��

�

etale

✒ ■❅❅❅

❅etale

Rfi

✻

RαQ∗

✻

In particular, we have

R̂m ≃ R̂αQ∗ and Âm ≃ Â

α,βQ∗

where the completions at the right hand sides are with respect to the maximal (graded)ideal ofRαQ∗ corresponding to the zero representation.


Example 2.11 From example 2.7 we recall that the triple(Q∗, α, β) associated to an Azu-maya algebra in a pointP ∈ X is given by

1((

hh and β = (n)

where the number of arrows is equal todimC TPX . In caseP is a smooth point ofX thisnumber is equal tod = dimX . Observe thatGL(α) = C∗ acts trivially onrepα Q

∗ = Cd

in this case. Therefore we have that

RαQ∗ ≃ C[x1, . . . , xd] and Aα,βQ∗ =Mn(C[x1, . . . , xd])

BecauseA is a Smooth order in such points we get that

ÂmP ≃Mn(C[[x1, . . . , xd]])

consistent with our étale local knowledge of Azumaya algebras.

jotter :

Becauseα.β = n, the number of vertices ofQ∗ is bounded byn and as

d = 1− χQ(α, α) −#{marked loops}

the number of arrows and (marked) loops is also bounded. Thismeans that for a particulardimensiond of the central varietyX there are only a finite number of étale local types ofSmooth orders inalg@n.

This fact might be seen as a non-commutative version of the fact that there is just oneétale type of a smooth variety in dimensiond namelyC[[x1, . . . , xd]]. At this moment asimilar result for Regular orders seems to be far out of reach.

2.5 A two-person game

Starting with a marked quiver setting(Q∗, α) we will play a two-person game. Left will beallowed to make one of the reduction steps to be defined below if the condition on Leavingarrows is satisfied, Red on the other hand if the condition on aRRiving arrows is satisfied.Although we will not use combinatorial game theory in any way, it is a very pleasant topicand the interested reader is referred to [12] or [3].

The reduction steps below were discovered by R. Bocklandt inhis Ph.D. thesis [4] (see also[5]) in which he classifies quiver settings having a regular ring of invariants. These stepswere slightly extended in [6] in order to classify central singularities of Smooth orders. Allreductions are made locally around a vertex in the marked quiver. There are three types ofallowed moves

Vertex removal

Assume we have a marked quiver setting(Q∗, α) and a vertexv such that the local structureof (Q∗, α) nearv is indicated by the picture on the left below, that is, insidethe verticeswe have written the components of the dimension vector and the subscripts of an arrowindicate how many such arrows there are inQ∗ between the indicated vertices. Definethe new marked quiver setting(Q∗R, αR) obtained by the operationR

vV which removes

the vertexv and composes all arrows throughv, the dimensions of the other vertices are


unchanged :

u1 · · · uk

αv

b1

aaBBBBBBBBB bk

==|||||||||

i1

a1

>>}}}}}}}}}· · · il

al

``AAAAAAAAA

RvV✲

u1 · · · uk

i1

c11

OO

c1k

==zzzzzzzzzzzzzzzzzzzz· · · il

clk

OO

cl1

aaDDDDDDDDDDDDDDDDDDDDD

.

wherecij = aibj (observe that some of the incoming and outgoing vertices maybe thesame so that one obtains loops in the corresponding vertex).Left (resp. Right) is allowedto make this reduction step provided the following condition is met

(Left) χQ(α, ǫv) ≥ 0 ⇔ αv ≥

l∑

j=1

ajij

(Right) χQ(ǫv, α) ≥ 0 ⇔ αv ≥

k∑

j=1

bjuj

(observe that if we started off from a marked quiver setting(Q∗, α) coming from an order,then these inequalities must actually be equalities).

loop removal

If v is a vertex with vertex-dimensionαv = 1 and havingk ≥ 1 loops. Let(Q∗R, αR) bethe marked quiver setting obtained by the loop removal operationRvl

1

k

�

Rvl✲

1

k−1

�

.

removing one loop inv and keeping the same dimension vector. Both Left and Right areallowed to make this reduction step.

Loop removal

If the local situation inv is such that there is exactly one (marked) loop inv, the dimensionvector inv is k ≥ 2 and there is exactly one arrow Leavingv and this to a vertex withdimension vector1, then Left is allowed to make the reductionRvL indicated below

k

~~~~

~~~~

~

•

1 u1

OO

· · · um

ggPPPPPPPPPPPPPPPPP

RvL✲

k

k

{� ~~~~

~~~~

~

~~~~

~~~~

~

1 u1

OO

· · · um

ggPPPPPPPPPPPPPPPPP

.

k

~~~~

~~~~

~

1 u1

OO

· · · um

ggPPPPPPPPPPPPPPPPP

RvL✲

k

k

{� ~~~~

~~~~

~

~~~~

~~~~

~

1 u1

OO

· · · um

ggPPPPPPPPPPPPPPPPP

.


Similarly, if there is one (marked) loop inv andαv = k ≥ 2 and there is only one arrowaRRiving atv coming from a vertex of dimension vector1, then Right is allowed to makethe reductionRvL

k

�� ''PPPP

PPPP

PPPP

PPPP

P

•

1

??~~~~~~~~~u1 · · · um

RvL✲

k

�� ''PPPP

PPPP

PPPP

PPPP

P

1

k

;C~~~~~~~~~

~~~~~~~~~

u1 · · · um

k

�� ''PPPP

PPPP

PPPP

PPPP

P

1

??~~~~~~~~~u1 · · · um

RvL✲

k

�� ''PPPP

PPPP

PPPP

PPPP

P

1

k

;C~~~~~~~~~

~~~~~~~~~

u1 · · · um

In accordance with combinatorial game theory we call a marked quiver setting(Q∗, α) azero settingif neither Left nor Right has a legal reduction step. The relevance of this gameon marked quiver settings is that if

(Q∗1, α1) (Q∗2, α2)

is a sequence of legal moves (both Left and Right are allowed to pass), then

Rα1Q∗1≃ Rα2Q∗2

[y1, . . . , yz]

wherez is the sum of all loops removed inRvl reductions plus the sum ofαv for eachreduction stepRvL involving a genuine loop and the sum ofαv − 1 for each reduction stepRvL involving a marked loop. That is, marked quiver settings which below to the samegame tree have smooth equivalent invariant rings.

In general games, a position can reduce to several zero-positions depending on the chosenmoves. For this reason the next result, proved in [6] is somewhat surprising

Theorem 2.12 Let (Q∗, α) be a marked quiver setting, then there is a unique zero-setting(Q∗0, α0) for which there exists a reduction procedure

(Q∗, α) (Q∗0, α0)

We will denote this unique zero-setting byZ(Q∗, α).

jotter :

Therefore it is sufficient to classify the zero-positions ifwe want to characterize all centralsingularities of a Smooth order in a given central dimensiond.

2.6 Central singularities

Let A be a SmoothR-order inalg@n andP a point in the central varietyX with corre-sponding maximal idealm ⊳ R. We now want to classify the types of singularities ofX inP , that is to classifyR̂m.


To start, can we decide whenP is a smooth point ofX ? In the case thatA is an Azumayaalgebra inP , we know already thatA can only be a Smooth ifR is regular inP . Moreoverwe have seen forA a Regular reflexive Azumaya algebra that the non-Azumaya points inX are precisely the singularities ofX .

For Smooth orders the situation is more delicate but as mentioned before we have a com-plete solution in terms of the two-person game by a slight adaptation of Bocklandt’s mainresult [5].

Theorem 2.13 If A is a SmoothR-order and(Q∗, α, β) is the combinatorial data associ-ated toA in P ∈ X . Then,P is a smooth point ofX if and only if the unique associatedzero-setting

Z(Q∗, α) ∈ { 1 k

k

•

2(( vv

2((

•vv

2•((

•vv

}

The Azumaya points are such thatZ(Q∗, α) = 1 hence the singular locus ofX is

contained in the ramification locusram A but may be strictly smaller.

To classify the central singularities of Smooth orders we may reduce to zero-settings(Q∗, α) = Z(Q∗, α). For such a setting we have for all verticesvi the inequalities

χQ(α, δi) < 0 and χQ(δi, α) < 0

and the dimension of the central variety can be computed fromthe Euler-formχQ. Thisgives us an estimate ofd = dim X which is very efficient to classify the singularities inlow dimensions.

Theorem 2.14 Let (Q∗, α) = Z(Q∗, α) be a zero-setting onk ≥ 2 vertices. Then,

dimX ≥ 1 +

a≥1∑

a

a+

a>1∑

a• 55

(2a− 1) +

a>1∑

a55

(2a) +

a>1∑

a• 55 •ii

(a2 + a− 2)+

a>1∑

a• 55 ii

(a2 + a− 1) +

a>1∑

a55 ii

(a2 + a) + . . .+

a>1∑

a•k 55 lii

((k + l− 1)a2 + a− k) + . . .

In this sum the contribution of a vertexv with αv = a is determined by the number of(marked) loops inv. By the reduction steps (marked) loops only occur at vertices whereαv > 1.

Let us illustrate this result by classifying the central singularities in low dimensions

Example 2.15 (dimension2) Whendim X = 2 no zero-position on at least two verticessatisfies the inequality of theorem 2.14, so the only zero-position possible to be obtainedfrom a marked quiver-setting(Q∗, α) in dimension two is

Z(Q∗, α) = 1

and therefore the central two-dimensional varietyX of a Smooth order is smooth.


Example 2.16 (dimension3) If (Q∗, α) is a zero-setting for dimension≤ 3 thenQ∗ canhave at most two vertices. If there is just one vertex it must have dimension1 (reducing

again to 1 whence smooth) or must be

Z(Q∗, α) = 2• 66 •hh

which is again a smooth setting. If there are two vertices both must have dimension1 andboth must have at least two incoming and two outgoing arrows (for otherwise we couldperform an additional vertex-removal reduction). As thereare no loops possible in thesevertices for zero-settings, it follows from the formulad = 1 − χQ(α, α) that the onlypossibility is

Z(Q∗, α) = 1a ))

b

##1cii

d

cc

The ring of polynomial invariantsRαQ∗ is generated by traces along oriented cycles inQ∗

so in this case it is generated by the invariants

x = ac, y = ad, u = bc and v = bd

and there is one relation between these generators, so

RαQ∗ ≃C[x, y, u, v]

(xy − uv)

Therefore, the only étale type of central singularity in dimension three is theconifold sin-gularity.

Example 2.17 (dimension4) If (Q∗, α) is a zero-setting for dimension4 thenQ∗ can haveat most three vertices. If there is just one, its dimension must be1 (smooth setting) or2 inwhich case the only new type is

Z(Q∗, α) = 266 •hh

which is again a smooth setting.

If there are two vertices, both must have dimension1 and have at least two incoming andoutgoing arrows as in the previous example. The only new typethat occurs is

Z(Q∗, α) = 1++&&1kkiiff

for which one calculates as before the ring of invariants to be

RαQ∗ =C[a, b, c, d, e, f ]

(ae− bd, af − cd, bf − ce)

If there are three vertices all must have dimension1 and each vertex must have at least twoincoming and two outgoing vertices. There are just two such possibilities in dimension4

Z(Q∗, α) ∈ { 1++

��

1kk

xx1

88XX1

'/1

t|1

T\}


The corresponding rings of polynomial invariants are

RαQ∗ =C[x1, x2, x3, x4, x5]

(x4x5 − x1x2x3)resp. RαQ∗ =

C[x1, x2, x3, x4, y1, y2, y3, y4]

R2

whereR2 is the ideal generated by all2× 2 minors of the matrix[x1 x2 x3 x4y1 y2 y3 y4

]

In [6] it was proved that there are exactly ten types of Smoothorder central singularitiesin dimensiond = 5 and53 in dimensiond = 6. The strategy to prove such a result is asfollows.

First one makes a full list of all zero-settings(Q∗, α) = Z(Q∗, α) such thatd = 1 −χQ(α, α) −# marked loops, using theorem 2.14.

Next, one has to weed out zero-settings having isomorphic rings of polynomial invariants(or rather, having the samem-adic completion wherem ⊳ RαQ∗ is the unique graded max-imal ideal generated by all generators). There are two invariants to separate two rings ofinvariants.

One is the sequence of numbers

dimCmn

mn+1

which can sometimes be computed easily (for example if all dimension vector componentsare equal to1).

The other invariant is what we call thefingerprintof the singularity. In most cases, therewill be other types of singularities (necessarily also of Smooth order type) in the vari-ety corresponding toRαQ∗ and the methods of [29] allow us to determine their associatedmarked quiver settings as well as the dimensions of these strata.

In most cases these two methods allow to separate the different types of singularities. Inthe few remaining cases it is then easy to write down an explicit isomorphism. We refer to(the published version of) [6] for the full classification ofthese singularities in dimension5 and6.

jotter :

In low dimensions there is a full classification of all central singularitiesR̂m of a Smoothorder inalg@n. However, at this moment no such classification exists forÂm. That is,under the game rules it is not clear what structural results of the ordersAαQ∗ are preserved.

2.7 Isolated singularities

In the classification of central singularities of Smooth orders, isolated singularities standout as the fingerprinting method to separate them clearly fails. Fortunately, we do have by[7] a complete classification of these (in all dimensions).

Theorem 2.18 LetA be a Smooth order overR and let(Q∗, α, β) be the combinatorialdata associated to aA in a pointP ∈ X . Then,P is an isolated singularity if and only if


Z(Q∗, α) = T (k1, . . . , kl) where

T (k1, . . . , kl) = 1 1

1

1

11

kl +3

k1 ;C

k2

KS

k3

[c????k4

ks

""

with d = dimX =∑

i ki − l + 1.

Moreover, two such singularities, corresponding toT (k1, . . . , kl) andT (k′1, . . . , k′l′), are

isomorphic if and only ifl = l′ and k′i = kσ(i)

for some permutationσ ∈ Sl.

The results we outlined in this talk are good as well as bad news.

jotter :

On the positive side we have very precise information on the types of singularities whichcan occur in the central variety of a Smooth order (certainlyin low dimensions) in sharpcontrast to the case of Regular orders.

However, because of the scarcity of such types most interesting quotient singularitiesCd/G will nothave a Smooth order over their coordinate ringR = C[Cd/G].

So, after all this hard work we seem to have come to a dead end with respect to the desin-gularization problem as there are no Smooth orders with centerC[Cd/G]. Fortunately, wehave one remaining trick available : to bring in astability structure.

lecture 3

NON-COMMUTATIVEDESINGULARIZATIONS

In the first talk I claimed that in order to find good desingularizations of quotient singular-itiesCd/G we had to find Smooth orders inalg@n with centerR = C[Cd/G]. Last timewe have seen that Smooth orders can be described and classified locally in a combinatorialway but also that there can be no Smooth order with centerC[Cd/G]. So maybe you beginto feel that I don’t know what I’m talking about.

Fine, but give me one last chance to show that the overall strategy may still have some valuein the desingularization project of quotient singularities. What we will see today is thatthere are ordersA overR which may not be Smooth but are Smooth on a sufficiently largeZariski open subset ofrepα A. Here ’sufficiently large’ means determined by astabilitystructure. Whenever this is the case we can apply the results of last time to construct nice(partial) desingularizations ofCd/G and if you are in for non-commutative geometry, evena genuine non-commutative desingularization.

3.1 Quotient singularities

Last time we associated to a combinatorial triple(Q∗, α, β) a Smooth orderAα,βQ∗ withcenter the ring of polynomial quiver-invariantsRαQ∗ . As we were able to classify the quiver-

invariants it followed that there is no triple such that the center ofAα,βQ∗ is the coordinateringR = C[Cd/G] of the quotient singularity. However, thereare nice orders of the form

A =Aα,βQ∗

I

for some idealI of relations which do have centerR are have been used in studying quotientsingularities.

Example 3.1 (Kleinian singularities) For a Kleinian singularity, that is, a quotient singu-larity C2/G with G ⊂ SL2(C) there is an extended Dynkin diagramD associated.

LetQ be thedouble quiverofD, that is to each arrow x // inD we adjoin an arrowx∗oo in Q in the opposite direction and letα be the unique minimal dimension

vector such thatχD(α, α) = 0. Further, consider themoment element

m =∑

x∈D

[x, x∗]

in the orderAαQ then

A =AαQ(m)

lecture 3. non-commutative desingularizations 39

is an order with centerR = C[C2/G] which is isomorphic to the skew-group algebraC[x, y]#G. Moreover,A is Morita equivalent to thepreprojective algebrawhich is thequotient of the path algebra ofQ by the ideal generated by the moment element

Π0 = CQ/(∑

[x, x∗])

For more details we refer to the lecture notes by W. Crawley-Boevey [14].

Example 3.2 Consider a quotient singularityX = Cd/G with G ⊂ SLd(C) andQ be theMcKay quiverof G acting onV = Cd.

That is, the vertices{v1, . . . , vk} of Q are in one-to-one correspondence with the irre-ducible representations{R1, . . . , Rk} of G such thatR1 = Ctriv is the trivial representa-tion. Decompose the tensorproduct in irreducibles

V ⊗C Rj = R⊕j11 ⊕ . . .⊕R

⊕jkk

then the number of arrows inQ from vi to vj

# (vi ✲ vj) = ji

is the multiplicity ofRi in V ⊗ Rj . Letα = (e1, . . . , ek) be the dimension vector whereei = dimC Ri.

The relevance of this quiver-setting is that

repα Q = HomG(R,R⊗ V )

whereR is theregular representation, see for example [13]. ConsiderY ⊂ repα Q theaffine subvariety of allα-dimensional representations ofQ for which the correspondingG-equivariant mapB ∈ HomG(R, V ⊗R) satisfies

B ∧B = 0 ∈ HomG(R,∧2V ⊗R)

Y is called thevariety of commuting matricesand its defining relations can be expressedas linear equations between paths inQ evaluated inrepα Q, say(l1, . . . , lz). Then,

A =AαQ

(l1, . . . , lz)

is an order with centerR = C[Cd/G]. In fact,A is just the skew group algebra

A = C[x1, . . . , xd]#G

Let us give one explicit example illustrating both approaches to the Kleinian singularityC

2/Z3.

Example 3.3 Consider the natural action ofZ3 onC2 via its embedding inSL2(C) send-ing the generator to the matrix

[ρ 00 ρ−1

]

whereρ is a primitive3-rd root of unity. Z3 has three one-dimensional simplesR1 =Ctriv, R2 = Cρ andR2 = Cρ2 . As V = C2 = R2 ⊕ R3 it follows that the McKay quiversetting(Q,α) is

1

y3

zz

x1

��1

x3

::

y2-- 1

y1

ZZ

y2

mm


Consider the matrices

X =

0 0 x3x1 0 00 x2 0

and Y =

0 y1 00 0 y2y3 0 0

then the variety of commuting matrices is determined by the matrix-entries of[X,Y ] thatis

I = (x3y3 − y1x1, x1y1 − y2x2, x2y2 − y3x3)

so the skew-group algebra is the quotient of the Smooth orderAαQ (which incidentally isone of our zero-settings for dimension4)

C[x, y]#Z3 ≃AαQ

(x3y3 − y1x1, x1y1 − y2x2, x2y2 − y3x3)

Takingyi = x∗i this coincides with the description via preprojective algebras as the momentelement is

m =

3∑

i=1

[xi, x∗i ] = (x3y3 − y1x1)e1 + (x1y1 − y2x2)e2 + (x2y2 − y3x3)e3

where theei are the vertex-idempotents.

jotter :

Many interesting examples of orders are of the following form :

A =AαQ∗

I

satisfying the following conditions :

• α = (e1, . . . , ek) is the dimension vector of a simple representation ofA, and

• the centerR = Z(A) is an integrally closed domain.

These requirements (which are often hard to verify!) imply thatA is an order overR inalg@n wheren is the total dimension of the simple representation, that is|α| =

∑

i ei.

Observe that such orders occur in the study of quotient singularities (see above) or as theétale local structure of (almost all) orders. From now on, this will be the setting we willwork in.

3.2 Stability structures

ForA = AαQ∗/I we define the affine variety ofα-dimensional representations

repα A = {V ∈ repα Q∗ | r(V ) = 0 ∀r ∈ I}

The action ofGL(α) =∏

iGLei by basechange onrepα Q∗ induces an action (actually

of PGL(α)) onrepα A. Usually,repα A will have singularities but it may be smooth onthe Zariski open subset ofθ-semistable representations which we will now define.

A characterof GL(α) is determined by an integralk-tupleθ = (t1, . . . , tk) ∈ Zk

χθ : GL(α) ✲ C∗ (g1, . . . , gk) 7→ det(g1)t1 . . . det(gk)tk


Characters definestability structureson A-representations but as the acting group onrepα A is reallyPGL(α) = GL(α)/C

∗(1e1 , . . . , 1ek) we only consider charactersθ sat-isfying θ.α =

∑

i tiei = 0.

If V ∈ repα A andV′ ⊂ V is anA-subrepresentation, that isV ′ ⊂ V as representations

of Q∗ and in additionI(V ′) = 0, we denote the dimension vector ofV ′ by dimV ′.

Definition 3.4 Forθ satisfyingθ.α = 0, a representationV ∈ repα A is said to be

• θ-semistableif and only if for every properA-subrepresentation0 6= V ′ ⊂ V wehaveθ.dimV ′ ≥ 0.

• θ-stableif and only if for every properA-subrepresentation0 6= V ′ ⊂ V we haveθ.dimV ′ > 0.

For any settingθ.α = 0 we have the following inclusions of Zariski openGL(α)-stablesubsets ofrepα A

repsimpleα A ⊂ repθ−stableα A ⊂ rep

θ−semistα A ⊂ repα A

but one should note that some of these open subsets may actually be empty!

Recall that a point of the algebraic quotient varietyissα A = repα//GL(α) representsthe orbit of anα-dimensional semi-simple representationV and such representations canbe separated by the valuesf(V ) wheref is a polynomial invariant onrepα A. This followsbecause the coordinate ring of the quotient variety

C[issα A] = C[repα A]GL(α)

and points correspond to maximal ideals of this ring. Recallfrom [31] that the invariant ringis generated by taking traces along oriented cycles in the marked quiver-setting(Q∗, α).

jotter :

For θ-stable andθ-semistable representations there are similar results andmorally oneshould viewθ-stable representations as corresponding to simple representations whereasθ-semistables are arbitrary representations.

For this reason we will only be able to classify direct sums ofθ-stable representations bycertain algebraic varieties which are called themoduli spacesof semistables representa-tions.

The notion corresponding to a polynomial invariant in this more general setting is that of apolynomial semi-invariant. A polynomial functionf ∈ C[repα A] is said to be aθ-semi-invariant ofweightl if for all g ∈ GL(α) we have

g.f = χθ(g)lf

whereχθ is the character ofGL(α) corresponding toθ. A representationV ∈ repα A is θ-semistable if and only if there is aθ-semi-invariantf of some weightl such thatf(V ) 6= 0.

Clearly,θ-semi-invariants of weight zero are just polynomial invariants and the multipli-cation ofθ-semi-invariants of weightl resp. l′ has weightl + l′. Hence, the ring of allθ-semi-invariants

C[repα A]GL(α),θ = ⊕∞l=0{f ∈ C[repα A] |∀g ∈ GL(α) : g.f = χθ(g)

lf }

is a graded algebra with part of degree zeroC[issα A]. But then we have aprojectivemorphism

proj C[repα A]GL(α),θ π✲✲ issα A


such that all fibers ofπ are projective varieties. The main properties ofπ can be deducedfrom [22]

Theorem 3.5 Points inproj C[repα A]GL(α),θ are in one-to-one correspondence with

isomorphism classes of direct sums ofθ-stable representations of total dimensionα.

If α is such that there areα-dimensional simpleA-representations, thenπ is a birationalmap.

Definition 3.6 We callproj C[repα A]GL(α),θ the moduli spaceof θ-semistable repre-

sentations ofA and denote it withmoduliθα A.

Example 3.7 In the case of Kleinian singularities, see example 3.1, if wetakeθ to be ageneric character such thatθ.α = 0, then the projective map

moduliθα A ✲✲ X = C2/G

is a minimal resolution of singularities. Note that the map is birational asα is the dimensionvector of a simple representation ofA = Π0, see [14].

Example 3.8 For general quotient singularities, see example 3.2, assume that the first ver-tex in the McKay quiver corresponds to the trivial representation. Take a characterθ ∈ Zk

such thatt1 < 0 and allti > 0 for i ≥ 2, for example take

θ = (−

k∑

i=2

dimRi, 1, . . . , 1)

Then, the corresponding moduli space is isomorphic to

moduliθα A ≃ G− Hilb Cd

the G-equivariant Hilbert schemewhich classifies all#G-codimensional idealsI ⊳C[x1, . . . , xd] where

C[x1, . . . , xd]

I≃ CG

asG-modules, hence in particularI must be stable under the action ofG. It is well knownthat the natural map

G− Hilb Cd ✲✲ X = Cd/G

is a minimal resolution ifd = 2 and if d = 3 it is often a crepant resolution, for examplewheneverG is Abelian. In non-Abelian cases it may have remaining singularities thoughwhich often are of conifold type. See [13] for more details.

jotter :

My motivation for this series of talks was to look for a non-commutative explanation forthe omnipresence of conifold singularities in partial resolutions of three dimensional quo-tient singularities as well as to have a conjectural list of possible remaining singularitiesfor higher dimensional quotient singularities.

Example 3.9 In theC2/Z3-example one can takeθ = (−2, 1, 1). The following represen-tations

1

1

yy

a

��1

0

99

1 ++1

0

XX

0kk

1

1

yy

1

��1

0

99

b ++1

0

XX

ckk

1

d

yy

1

��1

0

99

0 ++1

0

XX

1kk


are all nilpotent and areθ-stable. In fact ifbc = 0 they are representants of the exceptionalfiber of the desingularization

moduliθα A ✲✲ issα A = C2/Z3

3.3 Partial resolutions

It is about time we state the main re

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arXiv:math/0312221v1 [math.RA] 11 Dec 2003 · 2018. 11. 9. · lecture 1. non-commutative algebra 8 with qis a primitive n-th root of unity, is a central simpleK-algebra of dimensionn2