Noncommutative geometry (web draft, version 2)

310 Pages·1.727 MB·English

by  Le Bruyn L.

noncommutative geometry@n
Lieven Le Bruyn
Universiteit Antwerpen (UIA), B-2610 Antwerp (Belgium)
E-mail address: [e-mail]
URL: http://win-www.uia.ac.be/u/lebruyn/
2000 Mathematics Subject Classification. Primary 14A22,16A30 ; Secondary
13A50, 14D20, 14L24, 16G20, 53D30
Abstract. This is yet another version of the book
noncommutative geometry@n by Lieven Le Bruyn. Constructive criticism is,
asalways,[e-mail].
Introduction
”... La suite est trop confuse dans les notes pourˆetre exploitable
telle quelle.”
Bellaiche, Dat, Marin, Racinet, Randriambololona in [33].
Rather than adding to the plethora of pet-proposals for a noncommutative
geometry, we will focus in this book on some methods that are likely to prove
useful in the ’final theory’. Whereas the details of this theory are unclear at the
time of writing, the rough outline is slowly emerging.
The starting point is that a lot of interesting (families of) moduli spaces in
algebraicgeometryarespecialcasesoftheisomorphismprobleminsuitableAbelian
categories ab
-
moduli ⊂ iso(ab)
In recent years one has come to realize that many of these naturally occurring
Abelian categories are locally controlled by noncommutative algebras
ab=∪ rep A
i i
whererepA istheAbeliancategoryofallfinitedimensionalrepresentationsofthe
i
affine noncommutative algebra A and where the covering is compatible with the
i
naturalnotionsofisomorphismonbothsides. However,oneshouldnotviewrepA
as an affine noncommutative scheme. This is only justified under extra conditions
on A.
Among these noncommutative schemes rep A one singles out the smooth vari-
eties by imposing a noncommutative regularity condition on the algebra A. There
are several characterizations of commutative regular algebras. Generalizing these
to the world of noncommutative algebras leads to quite different notions of non-
commutative smoothness. We choose Grothendieck’s characterization in terms of
algebra lifts through nilpotent ideals. This approach has the advantage that the
resultingalg-smoothalgebrasbehavewellwithrespecttononcommutativedifferen-
tial forms and connections. An obvious disadvantage is that examples quickly lead
usawayfromthecosysettingofNoetherianalgebrasandintotheexoticwilderness
of universal algebra constructions.
Basicexamplesofalg-smoothalgebrasincludecoordinateringsofsmoothaffine
curves as well as path algebras of finite quivers. More intricate examples are con-
structed from these by applying universal algebra constructions such as free prod-
ucts, universal localization, passing to a Morita equivalent algebra, taking the n-th
root, and so on.
1
2 INTRODUCTION
In this book we will present methods to tackle the isomorphism problem for
smooth noncommutative varieties, that is, we want to describe
iso(rep A)
for A an alg-smooth algebra. Clearly, this is a wild problem so sooner or later we
willhitthewall. Allwecandoistotrytopushthewallabitfurther. Themethods
we will use are drawn from two classical sources : geometric invariant theory and
the theory of orders in central simple algebras.
We can partition rep A with respect to the dimension of the representation
G
rep A= rep A
n
n
where rep A is the affine scheme of n-dimensional representations of A. If A is
n
alg-smooth, each rep A is a smooth affine scheme (in particular, it is reduced).
n
The direct sum ⊕ on A-representation induces sum-morphisms
-
rep A×rep B rep A
n m m+n
Whereas rep A is reduced, it usually decomposes into several irreducible compo-
n
nents
G
rep A= rep A
n α
|α|=n
The component semigroup compA is the set of all occurring α, the addition is
induced by the sum morphisms and the dimension |α| defines an augmentation
compA - N.
Consider the subset simpA (resp. schurA) of all components containing a
simple (resp. a Schur) representation. The empire of the algebra A is the (infinite)
quiverEmpAwithavertexv foreveryα∈schurA. Thenumberofdirectedarrows
α
from the vertex v to v is equal to ext(α,β) which is the minimal dimension
α β
of Ext1(V,W) for V ∈ rep A and W ∈ rep A. With empA we denote the full
A α β
subquiverontheverticesv forα∈simpA. Thesequiverscontainstheinformation
α
about the noncommutative ´etale structure of repA.
The wall of the alg-smooth algebra A is the (usually finite) full subquiver
wallA of EmpA on the vertices corresponding to semigroup generators of compA.
Without being too dogmatic about it, let us define repA to be affine if and only
if wallA is strongly connected, that is, every pair of vertices v ,v belongs to an
α β
orientedcycleinwallA. Thismeansthatthereare’enough’simplerepresentations
to allow a meaningful reduction. In general, one can reduce to an affine setting by
taking suitable universal localizations of A.
We assume that repA is affine and that wallA is a finite quiver on the semi-
group generators {α ,...,α }. Then, A is said to be isomorphic in the noncom-
1 k
mutative ´etale topology to
B GhwallAi
where the algebra B is Morita equivalent to the path algebra hwallAi of the finite
quiver wallA. The Morita equivalence is determined by the dimensions |α| of
the semigroup generators α. We claim that path algebras of quivers (or Morita
equivalent algebras) play the role of affine spaces as being the only analytic local
structure for manifolds.
INTRODUCTION 3
The structure of empA is determined by the Euler form χ of wallA. If β ∈
A
compA it can be written as
β =e α +...+e α with e ∈N
1 1 k k i
and β ∈simpA if and only if
χ ((cid:15),δ )≤0 and chi (δ ,(cid:15))≤0
A i A i
for all 1 ≤ i ≤ k with (cid:15) = (e ,...,e ) (unless, wallA is just one oriented cycle in
1 k
which case (cid:15) must be (1,...,1)). Further, the arrows from v to v in empA are
β γ
determined by the wall as
ext(β,γ)=δ −χ ((cid:15),η)
βγ A
if η =(f ,...,f ) and γ =f α +...+f α . In particular, if A and A0 are in the
1 k 1 1 k k
same ´etale isomorphismclass, then simpA=simpA0. We next define when such A
and A0 are birational in the noncommutative Zariski topology.
Let α ∈ compA∩compA0 with |α| = n, and consider the natural basechange
actionofGL onrep Aandonrep A0 ofwhichtheorbitsarepreciselytheisomor-
n α α
phism classes of representations. From invariant theory we recall that the closed
orbits can be classified by the affine scheme corresponding to the ring of polyno-
mial GL -invariants. Michael Artin proved that the closed orbits determine the
n
isoclasses of semisimple n-dimensional representations and Claudio Procesi proved
that the ring of polynomial invariants is generated by traces of monomials in the
algebra generators. The corresponding quotient maps
rep -π- iss A rep A0 π-0 iss A0
α α α α
send the n-dimensional representation to the isomorphism class of the direct sum
ofitsJordan-H¨oldercomponents. Ifα∈simpA=simpA0 thentheinducedPGL -
n
action is generically free, whence there is an open subset in the quotient varieties
overwhichtherepresentationvarietyisaprincipalPGL -fibration. Hencetheyde-
n
fine two centralsimple algebras Σ resp. Σ0 ofdimension n2 overthe functionfield
α α
C(iss A) resp. C(iss A0). We call A and A0 birational in the noncommutative
α α
Zariski topology if and only if for all α ∈ simpA = simpA0 we have that iss A is
α
birational to iss A0 (hence they have the isomorphic functionfields andrespecting
α
this isomorphism we have that
Σ ’Σ0
α α
R
We can express this condition without reference to geometry. Define A to be
α
the algebra obtained from A by first adjoining formally all traces of monomials
in the algebra generators, then modding out all relations coming from the Cayley-
Hamiltonidentityforn×nmatrices(|α|=n)andfinallytakingthedirectsummand
corresponding to the irreducible component rep A. These algebras natural come
α R
equipped with a trace map and we define its image tr A to be the commutative
algebraH A. Reformulatingtheabove,wehavethatAaαndA0 arebirationalifand
α
only if
Z Z
A and A0
α α
are orders in the same central simple algebra for all α∈simpA=simpA0.
The solution to the isomorphism problem for finite dimensional representation
of the alg-smooth algebra A combines the ´etale and the Zariski invariants of A.
4 INTRODUCTION
The main result asserts that
Z Z
G
iso(repA)= iso(null Q)×azu A×...×azu A
α
(Q,α) | {z } β1 βl
´etale | {z }
Zariski
where the disjoint union is taken over all quiver settings (Q,α) with Q a finite
R
subquiverofempAonthevertices{v ,...,v }⊂simpAandwhereazu Aisthe
R β1 βl βi
Azumaya locus of A. The correspondence is given as follows. Let M ∈ rep A,
βi β
then its image ξ =π (M) in iss A is given by the semi-simplification
β β
Mss =S⊕e1 ⊕...⊕S⊕el
1 l
wheretheS arenon-isomorphicsimpleslyinginrep Aandoccurringwithmulti-
i βi
plicitye inMss. ThisalreadyaccountsfortheZariskipart. Letmbethemaximal
i
idealofC[iss A]correspondingtoπ (M),thenthem-adiccompletionoftheorder
β β
R
A is fully determined by a quiver-setting
β
mˆ ˆ0
Z Z
AG hQi
β α
where Q is a quiver on l vertices (corresponding to the distinct simple components
of Mss) and α = (e ,...,e ) (the multiplicities of the simple components). In Q,
1 l
the number of arrows from the vertex corresponding to S to that of S is given by
i j
# { joo i }=dim Ext1A(Si,Sj)
and one verifies that this number is ext(β ,β ) whence Q is the full subquiver of
i j
empAonthevertices{v ,.(cid:24)(cid:31)(cid:25)(cid:30).(cid:26)(cid:29).(cid:27)(cid:28),v }. In(cid:24)(cid:31)(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)geometricterms,thelocaldescriptionimplies
β1 bl
that the fiber of the quotient map in ξ is isomorphic as GL -variety to
n
π−1(ξ)’GL ×GL(α)null Q
β n α
wherenull QisthenullconeforthebasechangeactiongroupGL(α)onthespace
α
ofα-dimensionalrepresentationsofQ. Inparticular,GL -orbitsinthefiberπ−1(ξ)
n
correspondone-to-onetoGL(α)-orbitsinthenullcone,whichaccountsforthe´etale
part in the above description.
If there are not enough simple representations, that is if repA is not affine, it
is better to consider the bigger empire EmpA on the Schur roots of A. One replaces
the process of semisimplification by that of taking the Jordan-H¨older components
with respect to a suitable stability structure on repA. Using Schofield’s theory of
universal localization at Sylvester rank function one can usually reduce to the case
treated before, that is,
ressA=∪ repA
i Σi
where the universal localizations are taken such that repA is affine and where
Σ
ressA are the finite dimensional representations of A which are semistable for
some stability structure on repA. If one fixes a stability structure θ, then the
substitute for iss A is the moduli space moss (A,θ) whose points parametrize
α α
direct sums of θ-stable representations of A of total dimension α. The basis idea
of the above local description of ressA is that a θ-stable representation becomes
simple in a suitable universal localization A .
Σ
In the special, but important, case of path algebras of quivers, these moduli
spacesalsoplayacrucialroleintheremainingcombinatorialproblemofdescribing
INTRODUCTION 5
theorbitsinnull Q. WegivearepresentationtheoreticinterpretationoftheHes-
α
selink stratification of these nullcones in terms of associated string quiver settings
where the underlying quiver is directed. Non-emptiness of a potential stratum is
decided by the non-emptiness of the corresponding moduli space where the stabil-
ity structure is determined by the coweight. Unfortunately, there is a twist in the
tail. Whereas the moduli spaces account for most of the orbits in a given stratum,
to describe all orbits one has to enlarge the quiver and study the orbits under a
parabolic group. Here, we hit the wall with the methods presented in this book.
After all, describing repA, even in the special case of the free algebra, is a hopeless
problem.
This book is organized as follows. The first two chapters set the main stage,
we define alg-smooth algebras, give examples of them and show that they are the
natural class of noncommutative smooth algebras to consider from a noncommuta-
tive differential geometric perspective. Then, we introduce representation schemes
of affine algebras as the main tool to study these alg-smooth algebras. Whereas
for alg-smooth algebras one often gets by using only the reduced structure, for ar-
bitraryalgebrastheschemestructureisneeded. Thisschemestructurecontainsall
-
information about algebra morphisms A M (C) where C is a commutative
n
algebra. In fact, one can even equip these schemes with a thickening structure,
-
inspiredbytheworkofKapranov,toincludeallalgebramorphismsA M (B)
n
where B is a noncommutative infinitesimal extension of a commutative ring.
In the third and fourth chapter we show that one can develop a geometry at
level n having all the sophistication of ordinary commutative geometry (which is
level 1). More precisely, if GL(n)-aff is the category of all commutative affine
schemes equipped with a linear GL -action, then there is a triangle
n
alg@n
(cid:0)(cid:18) @
R (cid:0) @
n(cid:0) @trepn
(cid:0) @
(cid:0) @R
rep -
alg n GL(n)-aff
-
The fundamental anti-equivalence spec : commalg aff of commutative
algebraic geometry extends to a left inverse ⇑n assigning to an affine GL -scheme
n
afX its witness algebra which is the algebra of GL -equivariant polynomial maps
n
afX - M (C). There is the commuting diagram of functors
n
trep -
alg@n (cid:27) n GL(n)-aff
⇑n
tr quot
? ?
-
commalg aff
spec
where quot is the quotient functor which assigns to an affine scheme with GL -
n
action afX the affine scheme determined by the ring of polynomial invariants
C[afX]GLn.
The fifth chapter is pivotal in our approach to iso(repA). We recall enough
of´etale cohomology to describe the Brauer group of functionfields by the coniveau
6 INTRODUCTION
spectral sequence and to describe orders by cohomology pointed sets of automor-
phism schemes provided we know their ´etale local description. The latter is given
by applying the Luna-Knop theory of ´etale slices to the setting of representation
schemes. As an illustration of the force of these two methods we characterize all
central simple algebras over a projective smooth surface having a noncommutative
smooth model.
Thelasttwochaptersapplythemachinerydevelopedsofartotheisomorphism
problem of finite dimensional representations of alg-smooth algebras and their
contents was described above.
A major conceptual problem in writing this book was that it assumes some
familiarity with quite different topics : commutative algebraic geometry, invariant
theory, representation theory, ´etale cohomology, Brauer groups, universal algebra,
Azumaya algebras and p.i.-theory to name of few. Whereas I tried to include as
many details as feasible, the reader may want to consult some standard texts for
more details. I recommend, respectively, the books by Robin Hartshorne [22],
Hanspeter Kraft [36], Peter Gabriel and Andrei Roiter [19], J.S. Milne [47], Ina
Kersten [29], Aidan Schofield [60], Maxim Knus and Manuel Ojanguren [32] and
Claudio Procesi [52].
Notation
Machines.
• commalg the category of commutative C-algebras.
• alg the category of all C-algebras.
• hmi=Chx ,...,x i the free algebra in m variables.
1 m
• h∞i=Chx ,x ,...i the free algebra in infinitely many variables.
1 2
• hQi=CQ the path algebra of a finite quiver Q.
• AGA0 : A is Morita-equivalent to A0.
• A∗A0 the algebra free product of A and A0.
• mod A the category of left A-modules.
• projnod A the finitely generated projective left A-modules.
• A the universal localization of A at a set Σ of maps in projmod A.
Σ
• u(Σ) : the upper envelope of a set Σ of maps in projmod A.
• Brat A : the Bratelli diagram of an inductive limit A of semi-simple
algebras.
• dgalg the category of differential graded C-algebras.
• Ω A the ring of noncommutative differential forms of A.
• Ωev A the ring of even noncommutative differential forms of A.
• T(A) the tensor algebra of A.
• ⊥ the universal algebra for based linear maps from A.
A
• ∇ (resp. ∇ ) a right (resp. left) connection.
r l
• DerC A the Lie algebra of C-derivations of A.
• Ω A the ring of B-relative noncommutative differential forms of A.
B
Thickenings.
√
• nA the n-th root algebra of A.
• C[F] the coordinate ring of the affine scheme F.
• rep A the n-dimensional representation functor of A.
n √
-
• i the universal map A M (nA).
A n
• poisson the category of commutative Poisson algebras.
• A the Abelianization A of A.
ab [A,A]
• ALie the Lie algebra structure on A given by commutators.
• Fk A the k-th part of the commutator filtration on A.
• gr A the associated graded algebra for the commutator filtration on A.
• Qµ(A)themicro-localizationofAatS wrt. thecommutatorfiltrationon
S
A.
• Oµ the formal structure defined by A on spec A .
A ab
• hdi , the formal structure on Ad determined by hdi.
[[ab]]
7
8 NOTATION
• f the free Lie algebra on d variables.
d
• thick the category of thickenings of commutative algebras.
• thick.d the category of d-thickenings of commutative algebras.
Rd -
• : alg thick.d, the d-th thickening functor.
R1∞ -
• : alg thick, the thickening functor.
1
Necklaces.
• vect the category of C-vectorspaces.
• [A,A] the subspace spanned by all commutators of A.
v
• nx the necklace associated to the word w.
w
• s the i-th Newton symmetric function.
i
• neck the space spanned by all necklaces in X ={x ,...,x }.
d d 1 d
• {−,−} the Kontsevich bracket on necklaces.
K
H -
• : alg commalg the necklace functor.
• alg@ the category of C-algebras with trace.
R -
• : alg alg@ the trace functor.
• σ the i-th elementary symmetric function.
i
• χ(n)(t) the formal Cayley-Hamilton polynomial of degree n.
a
• alg@n the category of Cayley-Hamilton algebras of degree n.
R -
• : alg alg@n the Cayley-Hamilton functor of degree n.
Hn -
• : alg commalg the necklace functor of degree n.
n -
• ↓ : alg commalg the n-th invariant functor.
n
• DR∗ the Karoubi complex.
• H∗ noncommutative de Rham cohomology.
dR
• DR∗ the B-relative Karoubi complex.
B
• H∗ noncommutative B-relative de Rham cohomology.
B,dR
• neck the necklace Lie algebra of a symmetric quiver.
Q
Witnesses.
• S the symmetric group on d letters.
d
• λ a partition (or conjugacy class of S ).
d
• λ∗ the dual partition.
• c the Young symmetrizer.
λ
• fund fundamental n-th necklace relation.
n
-
• ↑ : alg alg@n the n-th equivariant functor.
n
• cha fundamental n-th trace relation.
n
• trep A scheme of trace preserving n-dimensional representations.
n
• GL(n)-aff the category of affine schemes with GL -action.
n
-
• ⇑n : GL(n)-aff alg@n the witness algebra functor.
• simpGL the isomorphism classes of irreducible GL -representations.
n n
• V the isotypical component of V of type s∈simpGL .
(s) n
• iss A the quotient scheme of rep A under the action of GL .
n n n
• rrep A the reduced variety of rep A.
n n
• riss A the reduced variety of iss A.
n n
• O(M) the GL -orbit of M ∈rep A.
n n
• O(M) the Zariski closure of the orbit O(M).