Formal structures and representation spaces

14 Pages·0.24 MB·English

by  Lieven Le Bruyn

FORMAL STRUCTURES AND REPRESENTATION SPACES
LIEVEN LE BRUYN AND GEERT VAN DE WEYER
Abstract. M. Kapranov introduced and studied in math.AG/9802041 the
1 noncommutative formal structure of a smooth affine variety. In this note we
0 showthathisconstructionisaparticularcaseofmicro-localizationandextend
0 the construction functoriallyto representation schemes ofaffine algebras. We
2 describeexplicitlytheformalcompletionsinthecaseofpathalgebrasofquivers
andinitiatethestudyoftheirfinitedimensionalrepresentations.
n
a
J
1. Introduction.
9
2 Let R be an associative C-algebra,RLie its Lie structure and RLie the subspace
m
spannedbythe expressions[r1,[r2,...,[rm 1,rm]...] containingm 1 instancesof
A] Lie brackets. The commutator filtration of−R is the (increasing) filt−ration by ideals
R (Fk R)k∈Z with Fk R=R for d∈N and
h. F−k R= RRiL1ieR...RRiLmieR
at Xm i1+..X.+im=k
m Observe that all C-algebra morphisms preserve the commutator filtration. The
[ associatedgradedgrF Risa(negatively)gradedcommutativePoissonalgebrawith
1 part of degree zero Rab = [RR,R].
v Denote with nilk the category of associative C-algebras R such that F−kR=0 (in
1 particular, nil =commalg the category of commutative C-algebras). An algebra
1
3 A Ob(nil ) is said to be k-smooth if and only if for all T Ob(nil ), all nilpotent
k k
12 tw∈osided ideals I ⊳T and all C-algebra morphisms A φ-∈ TI there exist a lifted
0 C-algebra morphism
1
T
0 T --
math/ I....∃.φ.˜........φI6
:
v A
i
X making the diagram commutative.
r Kapranovproves[4,Thm1.6.1]thatanaffinecommutativesmoothalgebraC hasa
a unique(upto C-algebraisomorphismidenticalonC)k-smooththickeningC(k) with
C(k) C. The inverse limit (connecting morphisms are given by the uniqueness
ab ≃
result)
Cf =lim C(k)
←
is then called the formal completion of C. Clearly, one has Cf =C.
ab
Example 1.1. ConsidertheaffinespaceAd withcoordinateringC[x ,...,x ]and
1 d
orderthecoordinatefunctionsx <x <...<x . Letf bethefreeLiealgebraon
1 2 d d
Cx ... Cx which has an ordered basis B = B defined as follows. B is
1 d k 1 k 1
⊕ ⊕ ∪ ≥
the orderedset x ,...,x and B = [x ,x ] j <i ,orderedsuch that B <B
1 d 2 i j 1 2
{ } { | }
1
2 LIEVEN LE BRUYN AND GEERT VAN DE WEYER
and [x ,x ] < [x ,x ] iff j < l or j = l and i < k. Having constructed the ordered
i j k l
sets B for l<k we define
l
B = [t,w] t=[u,v] B,w B such that v w<t for l <k .
k l k l
{ | ∈ ∈ − ≤ }
For l <k we let B <B and B is ordered by [t,w]<[t.w ] iff w <w or w =w
l k k ′ ′ ′ ′
and t<t. It is well known (see for example [10, Ex. 5.6.10]) that B is an ordered
′
C-basis of the Lie algebra f and that its enveloping algebra
d
U(f )=C x ,...,x
d 1 d
h i
is the free associative algebra on the x . We number the elements of B ac-
i k 2 k
∪ ≥
cording to the order b ,b ,... and for b B we define ord(b ) = k 1 (the
1 2 i k i
{ } ∈ −
number of brackets needed to define b ). Let Λ be the set of all functions with
i
finite support λ : B - N and define ord(λ) = λ(b )ord(b ). Rephras-
k 2 k i i
∪ ≥
ing the Poincar´e-Birkhoff-Witt result for U(f ) we have that any noncommutative
d
polynomial p C x ,...,x can be written uniquely asPa finite sum
1 d
∈ h i
p= [[f ]]M
λ λ
λ Λ
X∈
where[[f ]] C[x ,...,x ] = S(B ) and M = bλ(bi). In particular, for every
λ ∈ 1 d 1 λ i i
λ,µ,ν Λ, there is a unique bilinear differential operator with polynomial coeffi-
∈ Q
cients
Cλνµ :C[x1,...,xd]⊗CC[x1,...,xd] - C[x1,...,xd]
defined by expressing the product [[f]] M . [[g]] M in C x ,...,x uniquely as
λ µ 1 d
[[Cν (f,g)]] M , see [4, Prop. 3.4.3]. By associativhity of C xi,...,x the
ν Λ λµ ν h 1 di
Cν ∈satisfy the associativity constraint, that is, we have equality of the trilinear
Pλµ
differential operators
Cν (Cµ1 id)= Cν (id Cµ2 )
µ1λ3 ◦ λ1λ2 ⊗ λ1µ2 ◦ ⊗ λ2λ3
Xµ1 Xµ2
for all λ ,λ ,λ ,ν Λ. Kapranov defines the algebra C x ,...,x to be the
C-vector1spa2ce 3of p∈ossibly infinite formal sums [[fh]] 1M witdhi[[mabu]l]tiplication
λ Λ λ λ
defined by the operators Cν . ∈
To provethat this algebrλaµis the formalcomplPetion ofC[x ,...,x ] observethat
1 d
the commutator filtration on C x ,...,x has components
1 d
h i
F−k C x1,...,xd = [[fλ]]Mλ, λ:ord(λ) k
h i { ∀ ≥ }
λ
X
Malgoerbeoravsera,tnhdetqhueotfaiecnttthFa−CtkhxaC1lhg,x.e.1.b,,x..rd.a,ixdmioirspkh-issmmosopthresuesrivnegtthheeclioftminmguptarotoprerfityltroaftiforene.
Therefore,
C x ,...,x
C[x ,...,x ]f =lim h 1 di C x ,...,x .
1 d F k C x ,...,x ≃ h 1 di[[ab]]
← − h 1 di
Let X be a (commutative) smooth affine variety (both assumptions are crucial !),
then Kapranov uses the formal completion of the algebra of functions, C[X]f, to
defineasheafofnoncommutativealgebrasonX, f theKapranovformalstructure
OX
on X.
From[4, 4]itfollowsthatitisnotpossibleingeneraltoextendaKapranovformal
§
structure on an arbitrarysmooth variety X from that on the affine open pieces. In
fact, the obstruction gives important new invariants of smooth varieties, related to
Atiyah classes.
When X is affine, smoothness is essential to construct and prove uniqueness of the
Kapranovformalstructure f . At present there is no naturalfunctorial extension
OX
FORMAL STRUCTURES AND REPRESENTATION SPACES 3
offormalstructurestoarbitraryaffinevarieties. Oneofthemajorgoalsofthisnote
istoconstructsuchanextensionforrepresentationspacesofaffinenon-commutative
algebras.
Example 1.2. Letusrecalltheconstructionof f from[4]. LetA (C)bethed-th
OAd d
Weyl algebra,that is, the ring of differential operators with polynomialcoefficients
on Ad. Let Ad be the structure sheaf on Ad then it is well-known that the ring
of sections OAd(U) on any Zariski open subset U ⊂ - Ad is a left Ad(C)-module.
O
Kapranovdefinesin[4,Thm. 3.5.3]asheaf f ofnoncommutativealgebrasonAd
OAd
by taking as its sections over U the algebra
OAfd(U)=Chx1,...,xdi[[ab]]C[x1⊗,...,xd]OAd(U)
that is the C-vectorspace of possibly infinite formal sums [[f ]] M with
λ Λ λ λ
fλ Ad(U) and the multiplication is given as before by the a∈ction of the bi-
line∈arOdifferential operators Cλνµ on the left Ad(C)-module OAdP(U), that is, for all
f,g Ad(U) we have
∈O
[[f]]M .[[g]]M = [[Cν (f,g)]]M
λ µ λµ ν
ν
X
The sheaf of noncommutative algebras f is called Kapranov’s formal structure
OAd
on Ad.
In the next section we will prove that the construction of Kapranov’sformal struc-
ture f is a special case of micro-localization. In section three we will use this
OX
strategytoextendtheconstructionofformalstructuresinafunctorialwaytoaffine
commutativeschemesX =rep A,theschemeofalln-dimensionalrepresentations
n
of an affine C-algebra A. In section four we will work out the special important
case when A is the path algebra of a quiver and in the final section we will make
some comments on the representationtheory of these formal completions.
Acknowledgement. We like to thank F. Van Oystaeyen and A. Schofield for
helpful discussions on the contents of section 2, respectively 4.
2. Microlocal interpretation.
Werecallbrieflythealgebraicconstructionofmicrolocalization. Formoredetails
werefertothemonographs[9]and[12]. LetRbeafilteredalgebrawithaseparated
filtration Fn n Z and let S be a multiplicatively closed subset of R containing 1
{ } ∈
but not 0. For any r F F we denote its principal character σ(r) to be the
n n 1
∈ − −
image of r in the associated graded algebra gr(R). We assume that the set σ(S)
is a multiplicatively closed subset of gr(R) (always not containing 0 whence σ is
multiplicative on S). We define the Rees ring R˜ to be the graded algebra
R˜ = n ZFntn ⊂ - R[t,t−1]
⊕ ∈
where t is an extra central variable. If σ(s) gr(R) then we define the element
n
∈
s˜ = stn R˜ . The set S˜ = s˜,s S is a multiplicatively closed subset of
n
∈ { ∈ }
homogeneous elements in R˜.
Assume that σ(S) is an Ore set in gr(R) = R˜ , then for every n N the image
(t) ∈ 0
π (S˜) is an Oresetin R˜ where R˜ -- R˜ is the quotientmorphism. Hence, we
n (tn) (tn)
haveaninversesystemofgradedlocalizationsandcanformthe inverselimitin the
graded sense
R˜
Qµ(R˜)=limg π (S˜) 1
S˜ n − (tn)
←
4 LIEVEN LE BRUYN AND GEERT VAN DE WEYER
The element t acts torsionfree on this limit and hence we can form the filtered
algebra
Qµ(R˜)
Qµ(R)= S˜
S (t 1)Qµ(R˜)
− S˜
which is the micro-localization of R at the multiplicatively closed subset S. We
recall that the associated graded algebra of the microlocalization can be identified
as
gr(Qµ(R))=σ(S) 1gr(R).
S −
Let R be a C-algebra with R = R = C. We assume that the commutator
ab [R,R]
filtration (Fk)k Z introduced in (1.3) is a separated filtration on R. Observe that
∈
thisisnotalwaysthecase(forexampleconsiderU(g)forgasemi-simpleLiealgebra)
but often one can repeat the argument below replacing R with R .
Fn
Observethatgr(R)isanegativelygradedcommutativealgebraw∩ithpartofdegree
zero C. Take a multiplicatively closed subset S of C, then S = S +[R,R] is a
c c
multiplicatively closedsubsetofRwith the propertythatσ(S)=S andclearlyS
c c
isanOresetingr(R). Therefore,S˜isamultiplicativelyclosedsetoftheReesringR˜
consisting of homogeneous elements of degree zero. Observing that (tn) =F ntn
0 −
for all n N we see that
0
∈
R
Qµ(R)=lim π (S) 1
S n − F n
← −
where R π-n- R is the quotient morphism and Qµ is filtered again by the com-
F−n S
mutator filtration and has as associated graded algebra
gr(Qµ(R))=S 1gr(R).
S c−
That is, the rings constructed in [4, 2] are just microlocalizations.
As explained in more detail in [12§] one can define a microstructure sheaf µ on
OR
the affine scheme X of C by taking as its sections over the affine Zariski open set
X(f)
Γ(X(f), µ)=Qµ (R)
OR Sf
where S = 1,f,f2,... +[R,R]. Comparing with [4] we have proved.
{ }
Theorem 2.1. Let X = spec R be smooth, then the microstructure sheaf µ
ab OR
coincides with Kapranov’s formal structure f .
OX
An important remark to make is that one really needs microlocalization to con-
struct a sheaf of noncommutative algebras on X. If by some fluke we would have
thatalltheS arealreadyOresetsinR,wemightoptimisticallyassumethattaking
f
as sections over X(f) the Ore localization Sf−1R we would define a sheaf OR over
X. This is in general not the case as the Ore set S need no longer be Ore in a
g
localization Sf−1R !
3. Representation schemes.
We will first show that representation schemes of noncommutative formally
smooth algebras give smooth affine varieties and are therefore endowed with a
Kapranov formal structure. A C-algebra A is said to be formally smooth if and
only if it has the lifting property with respect to test-objects in alg. That is, let
I⊳T beanilpotenttwosidedideal,thenonecancomplete anyC-algebramorphism
FORMAL STRUCTURES AND REPRESENTATION SPACES 5
diagram
T
--
T
I....∃.φ.˜........ I6φ
A
IfwerestrictbothAandthetestobjects(T,I)tocommalgwegetGrothendieck’s
notion of formally smooth commutative algebras. Grothendieck showed that if A
is essentially of finite type, then this formal smoothness notion coincides with the
usual geometric notion of smoothness.
Motivated by this analogy, Quillen and Cunz [3] have suggested to take formally
smooth algebras as coordinate rings of noncommutative smooth varieties.
Let A= Chx1I,.A..,xdi be an affine C-algebra. The representation space repn A is the
affine scheme representing the functor
commalg Homalg(A,Mn(−)) - sets.
Its coordinate ring C[rep A] can be described as follows. For each x consider an
n k
n nmatrixofindeterminatesX =(x ) . Foreveryrelationf(x ,...,x ) I
k ij,k i,j 1 d A
co×nsiderthe n n matrixinM (C[x , i,j,k])givenby f(X ,...,X )=(f∈) .
n ij,k 1 d ij i,j
Each of the f ×is a polynomial in C[x ∀, i,j,k]. Then we have
ij ij,k
∀
C[x , i,j,k]
C[rep A]= ij,k ∀
n (f , f I )
ij A
∀ ∈
RepresentabilityoftheabovefunctorequipsuswithauniversalC-algebramorphism
A jA- M (C[rep A]) such that for every C-algebra morphism A φ- M (C)
n n n
with C commutative
A jA - M (C[rep A])
n n
φ ?(cid:9)........∃..!M...n.(ψ)
M (C)
n
thereisauniquemorphismC[rep A] ψ- C,makingtheabovediagramcommute.
n
Fore example, if I = 0, that is A C x ,...,x , then C[rep A] =
C[x , i,j,k], whenAce ≃ h 1 di n
ij,k
∀
rep C x ,...,x =Adn2 =M (C) ... M (C).
n h 1 di n ⊕ ⊕ n
d
The universal map j maps x to the generic matrix X = (x ) and the mor-
A k | {zk ij,k}i,j
phismψ isdeterminedbysending the variablex tothe (i,j)-entryofthe matrix
ij,k
φ(x ).
i
Moregenerally,ifAisanaffineformallysmoothalgebra,thenalltherepresentation
spaces rep A are affine smooth (commutative) varieties. Indeed, as C[rep A] is
n n
affine, it suffices to verify Grothendieck’s formal smoothness property
T
-- c
T
c
I....?.φ.˜........ φI6
C[rep A]
n
6 LIEVEN LE BRUYN AND GEERT VAN DE WEYER
for T a commutative algebrawith nilpotent ideal I. Using formalsmoothness of A
c
we have the existence of ψ
T
-- c
M (T ) M ( )
n c n
∃ψ................6I........∃..M..n..(φ˜) 6MIn(φ)
A - M (C[rep A])
jA n n
and universality of j provides us with the required lift φ˜, proving
A
Proposition 3.1. If A is a formally smooth algebra then rep A is an affine
n
smooth variety for all n.
ForanarbitraryalgebraA,however,therepresentationspacerep Aisingeneral
n
notsmoothnorevenreducedsoasmentionedbeforeitisnotimmediatelyclearhow
to define a canonicaland sufficiently functorialformalstructure on it. We will now
show how this can be done.
The starting point is that for every associative algebra A the functor
alg Homalg(A,Mn(−)) - sets
isrepresentableinalg. Thatis,there existsanassociativeC-algebra √nAsuchthat
there is a natural equivalence between the functors
Hom (A,M ( )) Hom (√nA, ).
alg n alg
− n∼.e. −
In other words, for every associative C-algebra B, there is a functorial one-to-one
correspondence between the sets
-
algebra maps A M (B)
n
(algebra maps √nA - B
Example 3.2. If A = C x ,...,x , then it is easy to see that √nA =
1 d
h i
C x ,...,x . For, given an algebra map A φ- M (B) we obtain an al-
11,1 nn,d n
gehbra map √nA i- B by sending the free variable x to the (i,j)-entry of the
ij,k
matrix φ(x ) M (B). Conversely, to an algebra map √nA ψ- B we assign the
k n
∈ -
algebra map A M (B) by sending x to the matrix (ψ(x )) M (B).
n k ij,k i,j n
∈
Clearly, these operations are each others inverses.
Todefine √nAingeneral,considerthe freealgebraproductA M (C) andconsider
n
∗
the subalgebra
√nA=A M (C)Mn(C) = p A M (C) p.(1 m)=(1 m).p m M (C)
n n n
∗ { ∈ ∗ | ∗ ∗ ∀ ∈ }
Before we can prove the universal property of √nA we need to recall a property
that M (C) shares with any Azumaya algebra : if M (C) φ- R is an algebra
n n
morphism and if RMn(C) = r R r.φ(m)=φ(m).r m Mn(C) , then we have
R Mn(C) CRMn(C). { ∈ | ∀ ∈ }
In≃particula⊗r, if we apply this to R = A M (C) and the canonical map
n
∗
M (C) φ- A M (C) where φ(m) = 1 m we obtain that M (√nA) =
n n n
∗ ∗
Mn(C) C √nA=A Mn(C).
⊗ ∗
Hence, if √nA f- B is an algebra map we can consider the composition
A idA-∗1 A Mn(C) Mn(√nA) Mn(-f) Mn(B)
∗ ≃
FORMAL STRUCTURES AND REPRESENTATION SPACES 7
-
to obtain an algebra map A M (B). Conversely, consider an algebra map
n
A g- M (B) and the canonical map M (C) i- M (B) which centralizes B
n n n
in M (B). Then, by the universal property of free algebra products we have an
n
algebra map A Mn(C) g∗-i Mn(B) and restricting to √nA we see that this maps
∗
factors
A Mn(C) g∗-i Mn(B)
∗
6 6
∪ ∪
√nA ................-... B
and one verifies that these two operations are each others inverses.
It follows from the functoriality of the √n. construction that C x1,...,xd -- A
h i
implies that n C x ,...,x -- √nA. Therefore, if A is affine and generated by
1 d
h i
d elements, then √nA is also affine and generated by dn2 elements.
≤ Next, we dpefine a formal completion of C[rep A] i≤n a functorial way for any
n
associative algebra A. Equip √nA with the commutator filtration
... ⊂ - F √nA ⊂- F √nA ⊂ - √nA = √nA = ...
2 1
− −
Because algebra morphisms are commutator filtration preserving, it follows from
the universal property of √nA that √nA is the object in nil representing the
F−k√nA k
functor
nil Homalg(A,Mn(−)) - sets.
k
In particular, because the categories commalg and nil are naturally equivalent,
1
we deduce that
√nA √nA
√nA = = C[rep A]
ab [√nA, √nA] F √nA ≃ n
1
−
because both algebras represent the same functor. We now define
√nA
√nA =lim .
[[ab]] F √nA
← k
−
AssumenowthatAis formallysmooth,thensois √nAbecausewehaveseenbefore
that
M (√nA) A M (C)
n n
≃ ∗
and the class of formally smooth algebras is easily seen to be closed under free
products and matrix algebras. Alternatively, one can apply Bergman’s coproduct
theorems [2] or [11, Thm. 2.20] for a strong version.
As a consequence, we have for every k N that the quotient √nA is k-smooth.
∈ F−k√nA
Moreover,we have that
√nA √nA
( ) C[rep A].
F √nA ab ≃ [√nA, √nA] ≃ n
k
−
BecauseC[rep A]isanaffinecommutativesmoothalgebra,wededucefromKapra-
n
nov’s uniqueness result of k-smooth thickenings that
√nA
C[rep A](k)
n ≃ F √nA
k
−
and consequently that the formal completion of C[rep A] can be identified with
n
C[rep A]f √nA .
n ≃ [[ab]]
8 LIEVEN LE BRUYN AND GEERT VAN DE WEYER
Theorem 3.3. Defining for an arbitrary C-algebra A the formal completion of
C[rep A]tobe √nA givesacanonicalextensionofKapranov’sformalstructure
n [[ab]]
on affine smooth commutative algebras to the class the coordinate rings of represen-
tation spaces on which it is functorial in the algebras.
There is a natural action of GL by algebra automorphisms on √nA. Let u
n A
denote the universal morphism A uA- M (√nA) corresponding to the identity
n
map on √nA. For g GL we can consider the composed algebra map
n
∈
A u-A M (√nA)
n
@
@
@ g.g−1
ψg @
R@ ?
M (√nA)
n
Then g acts on √nA via the automorphism √nA φg- √nA corresponding the the
composition ψ . It is easy to verify that this defines indeed a GL -action on √nA.
g n
The formal structure sheaf f defined over rep A, constructed from √nA
Orepn A n
by microlocalization as in the foregoing section, will be denoted from now on by
f . We see that it actually has a GL -structure which is compatible with the
O√nA n
GL -action on rep A.
n n
Finally we should clarify what representation theoretic information is contained
in √nA . The reduced variety of rep A gives information about the C-algebra
[[ab]] n
maps A - M (C). The scheme structure of rep A gives us the C-algebra
n n
maps A - M (C) where C is a finite dimensional commutative C-algebra. The
n
formal structure now gives us the C-algebra maps A - M (B) where B is a
n
finite dimensional noncommutative but basic C-algebra. Recall that an algebra is
said to be basic if all its simple representations have dimension one.
4. Path algebras of quivers.
Even in the case when A is formally smooth it is by no means easy to describe
andmanipulatethen-throotalgebra √nAandits correspondingformalcompletion
√nA . Inthis andthe next sectionwe will discussthese facts in the special(but
[[ab]]
important) case of path algebras of quivers.
Let Q be a quiver, that is a directed graph on a finite set Q = v ,...,v
v 1 k
{ }
of vertices having a finite set Q = a ,...,a of arrows where we allow multiple
a 1 l
arrowsbetweenverticesandloops in{vertices. W} e willdepictvertexvi by (cid:31)(cid:24)(cid:25)(cid:30)i(cid:26)(cid:29)(cid:28)(cid:27) and
an arrow a from vertex vi to vj by (cid:31)(cid:24)(cid:25)(cid:30)j(cid:26)(cid:29)(cid:28)(cid:27)oo a (cid:31)(cid:24)(cid:25)(cid:30)i(cid:26)(cid:29)(cid:28)(cid:27).
The path algebra CQ has as underlying C-vectorspace basis the set of all ori-
ented paths in Q, including those of length zero corresponding to the vertices v .
i
Multiplication in CQ is induced by (left) concatenation of paths. More precisely,
1=v +...+v is a decompositionof1 into mutually orthogonalidempotents and
1 k
further we define
vj.a is always zero unless (cid:31)(cid:24)(cid:25)(cid:30)j(cid:26)(cid:29)(cid:28)(cid:27)oo a (cid:31)(cid:24)(cid:25)(cid:30)(cid:26)(cid:29)(cid:28)(cid:27) in which case it is the path a,
•
a.vi is always zero unless (cid:31)(cid:24)(cid:25)(cid:30)(cid:26)(cid:29)(cid:28)(cid:27)oo a (cid:31)(cid:24)(cid:25)(cid:30)i(cid:26)(cid:29)(cid:28)(cid:27) in which case it is the path a,
•
a .a isalwayszerounless (cid:31)(cid:24)(cid:25)(cid:30)(cid:26)(cid:29)(cid:28)(cid:27)oo ai (cid:31)(cid:24)(cid:25)(cid:30)(cid:26)(cid:29)(cid:28)(cid:27)oo aj (cid:31)(cid:24)(cid:25)(cid:30)(cid:26)(cid:29)(cid:28)(cid:27) inwhichcaseitisthepath
i j
•
a a .
i j
FORMAL STRUCTURES AND REPRESENTATION SPACES 9
In order to see that CQ is formally smooth, take an algebra T with a nilpotent
twosided ideal I ⊳T and consider
T
--
T
I....?.φ.˜........ I6φ
CQ
Thedecomposition1=φ(v )+...+φ(v )intomutuallyorthogonalidempotentsin
1 k
T can be lifted up the nilpotent ideal I to a decomposition 1=φ˜(v )+...+φ˜(v )
I 1 k
into mutually orthogonalidempotents in T. But then, taking for every arrow a
(cid:31)(cid:24)(cid:25)(cid:30)j(cid:26)(cid:29)(cid:28)(cid:27)oo a (cid:31)(cid:24)(cid:25)(cid:30)i(cid:26)(cid:29)(cid:28)(cid:27) an arbitrary element φ˜(a)∈φ˜(vj)(φ(a)+I)φ˜(vi)
gives a required lifted algebra morphism CQ φ˜- T.
Next,wewilldescribethesmoothaffineschemesrep CQ. Considerthesemisim-
n
ple subalgebra V = C ... C generated by the vertex-idempotents v ,...,v .
1 k
× × { }
k
Everyn-dimensionalrepresentationofV issemi-simpleanddeterminedbythemul-
| {z }
tiplicities by which the factors occur. That is, we have a decomposition
rep V = GL /(GL ... GL )= rep V
n n a1 × × ak α
Gai=n Gα
into homogeneous spaPces where α runs over the dimension vectors α =
(a ,...,a ) such that a = n. The inclusion V ⊂ - CQ induces a map
1 k i i
rep CQ ψ- rep V and we have the decompositionofrep CQ into associated
n n P n
fiber bundles
ψ 1(rep V)=GL GL(α)rep Q.
− α n× α
Here,GL(α)=GL ... GL embeddedalongthediagonalinGL andrep Q
a1× × ak n α
is the affine space of α-dimensional representations of the quiver Q. That is,
rep Q= M (C)
α aj×ai
(cid:31)(cid:24)(cid:25)(cid:30)j(cid:26)(cid:29)(cid:28)(cid:27)oo Ma (cid:31)(cid:24)(cid:25)(cid:30)i(cid:26)(cid:29)(cid:28)(cid:27)
andGL(α)actsonthisspaceviabase-changeinthevertex-spaces. Thatis,rep CQ
n
is the disjoint union of smooth affine components depending on the dimension vec-
tors α=(a ,...,a ) such that a =n.
1 k i
P
The importance of the class of path algebras comes from the fact that if A is
an arbitrary affine formally smooth algebra, the ´etale local GL -structure of the
n
representationspace rep A is determined by a rep Q for a certain quiver Q and
n α
dimension vector α. We refer to [6] or [7] for more details.
Wehaveseenbeforethat √nCQisformallysmooth,hencewewouldliketodetermine
the quiver settings relevant for the study of the representation space rep √nCQ
m
forarbitrarym N. Beforewecandothisweneedtorecallthe notionofuniversal
∈
localization. We refer to [11, Chp. 4] for full details.
Let A be a C-algebra and projmod A the category of finitely generated projective
left A-modules. Let Σ be some class of maps in this category (that is some left
A-module morphisms between certain projective modules). In [11, Chp. 4] it is
shown that there exists an algebra map A jΣ- A with the universal property
Σ
that the maps A σ have an inverse for all σ Σ. A is called the universal
Σ A Σ
⊗ ∈
localization of A with respect to the set of maps Σ.
10 LIEVEN LE BRUYN AND GEERT VAN DE WEYER
When A is formally smooth, so is A . Indeed, consider a test-object (T,I) in alg,
Σ
then we have the following diagram
T
--
T
.................6ψI.....φ.˜........φI6
-
A A
Σ
jΣ
where ψ exists by smoothnessof A. By Nakayama’slemma allmaps σ Σ become
isomorphisms under tensoring with ψ. Then, φ˜ exists by the universal∈property of
A .
Σ
Consider the special case when A is the path algebra CQ of a quiver on k ver-
tices. Then, we can identify the isomorphism classes in projmod CQ with Nk. To
each vertex v correspondsan indecomposable projective left CQ-ideal P having as
i i
C-vectorspacebasis all paths in Q starting at v (similarly, there is an indecompos-
i
able projective right CQ-ideal Pr with basis the paths ending in v ). We can also
i i
determine the space of homomorphisms
HomCQ(Pi,Pj)= Cp
(cid:31)(cid:24)(cid:25)(cid:30)i(cid:26)(cid:29)(cid:28)(cid:27)oo o/ Mo/po/ o/ (cid:31)(cid:24)(cid:25)(cid:30)j(cid:26)(cid:29)(cid:28)(cid:27)
where p is an oriented path in Q starting at v and ending at v . Therefore, any
j i
A-module morphism σ between two projective left modules
σ-
P ... P P ... P
i1 ⊕ ⊕ iu j1 ⊕ ⊕ jv
can be represented by an u v matrix M whose (p,q)-entry m is a linear com-
σ pq
×
bination of oriented paths in Q starting at v and ending at v .
jq ip
Now, form an v u matrix N of free variables y and consider the algebra CQ
σ pq σ
which is the quo×tient of the free product CQ C y ,...,y modulo the ideal of
11 uv
∗ h i
relations determined by the matrix equations
v 0 v 0
i1 j1
Mσ.Nσ = ...  Nσ.Mσ = ... 
0 v 0 v
 iu  jv
Equivalently,CQ is the pathalgebraofa quiver withrelations where the quiveris
σ
Q extendedwith arrowsy fromv to v for all1 p u and1 q v andthe
pq ip jq ≤ ≤ ≤ ≤
relations are the above matrix entry relations.
Repeatingthisprocedureforeveryσ ΣweobtaintheuniversallocalizationCQ .
Σ
Observe that if Σ is a finite set of ma∈ps, then the universal localization CQ is an
Σ
affine algebra.
It is easy to see that the representation space rep CQ is an affine Zariski open
n σ
subscheme (but possibly empty) of rep CQ. Indeed, if m = (m ) rep Q,
then m determines a point in rep CQn if and only if the matraiceas∈M (mα) in
n Σ σ
whichthearrowsareallreplacedbythematricesm areinvertibleforallσ Σ. In
a
∈
particular, this induces numerical conditions on the dimension vectors α such that
rep Q = . Let α=(a ,...,a ) be a dimension vector such that a =n then
α Σ 6 ∅ 1 k i
every σ Σ say with
∈ P
P1⊕e1 ⊕...⊕Pk⊕ek σ- P1⊕f1 ⊕...⊕Pk⊕fk
gives the numerical condition
e a +...+e a =f a +...+f a .
1 1 k k 1 1 k k