Moduli of Representations, Quiver Grassmannians, and Hilbert Schemes
aa r X i v : . [ m a t h . R T ] M a y MODULI OF REPRESENTATIONS, QUIVER GRASSMANNIANS,AND HILBERT SCHEMES
LUTZ HILLE
Abstract.
It is a well established fact, that any projective algebraic varietyis a moduli space of representations over some finite dimensional algebra. Thisalgebra can be chosen in several ways. The counterpart in algebraic geometryis tautological: every variety is its own Hilber scheme of sheaves of length one.This holds even scheme theoretic. We use Beilinson’s equivalence to get similarresults for finite dimensional algebras, including moduli spaces and quivergrassmannians. Moreover, we show that several already known results can betraced back to the Hilbert scheme construction and Beilinson’s equivalence. Introduction
Assume k is an algebraically closed field and X is a projective subscheme of P n defined by some homogeneous equations f , . . . , f r in k [ X , . . . , X n ]. We want torealize X as a moduli space of quiver representations and as a quiver grassmannianin a natural way. Moreover, we also like to have a construction making the quiver assmall as possible. Let A be a bounded path algebra kQ/J , where Q is a finite quiverand J is an ideal of admissible relations in the path algebra kQ . Moduli spaces forquiver representations have been defined by King in [11], a quiver grassmannian isjust the variety of all submodules of a given module M of a fixed dimension vector.We note that we can consider moduli spaces and also quiver grassmannians withits natural scheme structure. Moreover, any quiver grassmannian is a moduli space(just a moduli space of submodules of a given module), and there are natural mor-phisms between quiver grassmannians and moduli spaces. Under certain additionalconditions these morphisms are even isomorphisms. Those isomorphisms are alwayshidden in our construction. Since these morphisms can be seen explicitely in ourconstruction we do not need any general result for those morphisms. This is themain reason why we use line bundles in our construction, for arbitrary vector bun-dles all constructions become much more technical. The other advantage of usingline bundles is that we can always use modules of dimension vector (1 , , . . . , , Theorem 1.1.
Let X be any projective scheme defined by equations f , . . . , f r in aprojective n -space P n . Then there exists a quiver Q , an ideal J in the path algebra kQ and a kQ –module M so that (1) X is isomorphic to the moduli space of all indecomposable kQ/J –modulesof dimension vector (1 , . . . , , and (2) X is isomorphic to the quiver grassmannian of all submodules of M ofdimension vector (1 , . . . , for the quiver Q . Note that Q can be chosen to be the Beilinson quiver, J is an ideal just defined bythe f i and M is the unique sincere injective cover of one simple module, in partic-ular M is indecomposable. For more details we refer to section 2. The result on quiver grassmannians recently attracted attention in connection with cluster alge-bras (see [10]) and in connection with Auslanders theory on morphisms determinedby objects. Ringel has already pointed out that the result above has been studiedby several authors ([16]), however ’can be traced back to Beilinson’ ([17]). Theprincipal aim of this note is to show how, we can use Beilinson, and even better,how we can even improve it. Eventually, we show that all the constructions at theend can be traced back to a tautological construction in algebraic geometry. Anyscheme X is its own Hilbert scheme of sheaves of lenght one: Hilb ( X ) = X .We note that the second result, for a variety X , was already stated in [9] and provenagain with different methods in [14]. It can certainly be traced back to the workin [4, 5]. An affine version was already proven in [8]. However, the first publishedresult in this direction was just an example in [7]. We will give a common framefor all those examples, in fact all are variants of the Hilbert scheme construction inalgebraic geometry and a variant of Beilinsons equvivalence. Ringel already noticedthat we can even work with the Kronecker quiver, thus, two vertices are sufficientfor Q . Improving this construction slightly, we can even realize any projectivesubscheme of the n –dimensional projective space as a quiver grassmannian for the( n + 1)–Kronecker quiver. This construction is again explicit. We denote the the m th homogenous component of the ideal I generated by the polynomials f i by I m .Thus the homogeneous coordinate ring of X is just ⊕ S m V /I m for some ( n + 1)–dimensional vector space V . We also denote by d a natural number greater or equalto the maximal degree of the the polynomials f i . Theorem 1.2.
Let X be any projective subscheme of the n –dimensional projectivespace P n . There is a module M = ( S d − V /I d − , S d V /I d ) over the Kronecker algebradefined by the natural map S d − V /I d − ⊗ V −→ S d V /I d . Then X is isomorphic tothe quiver grassmannian of submodules of M of dimension vector (1 , . The principal part of the note consists of a five step construction that we will useto get a realisation of X as such a moduli space. In addition we also add somemodifications of these steps allowing to simplify the quiver or the relations. Weexplain these five steps briefly. First note, that any scheme X is its own Hilbertscheme of sheaves of length one. So any projective scheme is a moduli space ofsheaves (in a rather trivial way). In a second step we use Beilinson’s equivalence toconstruct for X an algebra A = kQ/J . Roughly, we can take any tilting bundle T on P , extend it by any other vector bundle T ′ to R = T ⊕ T ′ and apply Hom( R, − )to the universal family of the Hilbert scheme X . In the particular case when T is the direct sum of the line bundles O ( i ), for i = 0 , . . . , n , we can just extendit by the line bundles O ( i ) for i = n + 1 , . . . , d . In this way, we get a family ofmodules over the Beilinson algebra for T and a family of modules over the ’enlarged’Beilinson algebra for R . If X is given by polynomials as above, all modules of thefamily also satisfy the equations f i , however now in the Beilinson algebra. Thus wedefine J to be the ideal generated by the f i in the enlarged Beilinson algebra. Notethat we have several choices for such realization, depending on where the relationstarts. However, one can check directly, that family of all modules of dimensionvector (1 , . . . ,
1) over A , the enlarged Beilinson algebra with relations J , coincideswith X , independent of the this realization. Consequently, the moduli space of allmodules of dimension vector (1 , . . . ,
1) over A is X as a scheme. In a final step, werealize X as a quiver grassmannian by using an injectice hull in A .We already mentioned that this construction is more general in the way that wecan take any tilting bundle T and any vector bundle T ′ , however the direct com-putation seems to be more sophisticated. So we use line bundles just for simplicity. ODULI OF REPRESENTATIONS, QUIVER GRASSMANNIANS, AND HILBERT SCHEMES 3
Even stronger, in general we do not need T to be a tilting bundle. For example, theconstruction also would work if we only take O ⊕ O ( d ) where d is at least the max-imal degree of the f i . Then we get a realization similar to Reinekes construction,that is in fact a variant of the realization for the Kronecker quiver.The paper is organized as follows. In the second section we present the five steps ofour construction together with a final note on framed moduli spaces. In the thirdsection we reprove the already known results using the construction in section 2.We conclude in the last section with some open problems and a proof of the secondtheorem.2. Hilbert schemes, Beilinson’s equivalence and Serre’s construction
We construct, using some elementary results from algebraic geometry, for any al-gebraic variety a moduli space of quiver representations, a quiver grassmannianand also further examples in five steps. We also note, that this even holds for anyscheme that is quasi-projective. So we obtain the authors example from 1996 [7],Huisgen-Zimmermann’s examples in her work on uniserial modules (see for example[8], and her work with Bongartz [4, 5] on Grassmannians, we apologize for beingnot complete), a variant of Reineke’s result for quiver-grassmannians from 2012[14] and last not least Michel Van den Bergh’s example, that appeared in a blog ofLieven le Bruyn. In fact, all the results at the end of this note can be proven usingthe following constructions.2.1.
Hilbert schemes. ([12])We take an algebraic variety X and consider sheaves of length one on X . There isa bijection between those sheaves and points of X . In a more sophisticated way wecan say X = Hilb ( X ) the Hilbert scheme of length one sheaves on X . Or, we canconsider any line bundle L on X as a fine moduli space of skyscraper sheaves bytaking the push forward to the diagonal in X × X . Each fiber of some point x forthe first projection is just the skyscraper sheaf in x .2.2. Beilinson’s tilting bundle. ([2])In the second step we transform the construction above to the representation the-oretic side using tilting. To keep the construction easy, we consider the Beilinsontilting bundle T = O ⊕ O (1) ⊕ ... ⊕ O ( n ) on the n –dimensional projective space P n .We denote by A the Beilinson algebra, it is the opposite of End( T ). Take k x to be askyscraper sheaf and let us compute Hom( T, k x ). Note that dim Hom( O ( i ) , k x ) = k ,thus we get thin sincere representations of A , that is each simple occurs with mul-tiplicity one. Now we can see by direct computations that the moduli space of thinsincere representations of A is just the projective space P n .This example can be easily generalized to any tilting bundle T , however, usingsheaves (that are not vector bundles) we do not get a flat family (the dimensionabove jumps at certain points).2.3. Relations of length at most n . ([20])Now we consider Y , any subvariety defined by equations f , . . . , f r in a projective n –space P n . Assume first deg f i ≤ n for all i . Note that the quiver of the Beilinsonalgebra above has n + 1 arrows from i to i + 1, we denote by x i , . . . , x in , and pathsthe monomials x ai (1) x a +1 i (2) ...x d − i ( d − x di ( d ) of degree d − a + 1 at most n . We define analgebra B as the quotient of A by the ideal J = ( f , f , . . . , f r ), where f is anylinear combination of path representing f in A . Note that any representative works,since the arrows in A commute, whenever this makes sense: x ai x a +1 j = x aj x a +1 i . LUTZ HILLE
This solves the problem we address to the next step already, since P n ⊂ P m for any n < m . Moreover, we could also use the fact, that any projective algebraic varietyis already defined by quadratic relations. However, using Serre’s theorem we canhandle also relations of any degree in P n . Figure 1.
Beilinson quiver2.4.
Relations of arbitrary length.
To obtain Y , as in the previous step, where deg f i ≤ d for any d > n , we considerthe sequence of line bundles O , . . . , O ( d ). The direct sum of these line bundles isno longer a tilting bundle, however the same computation as above shows that themoduli space of all thin sincere representations of A is still a projective n –space.Now the representatives f i of the polynomials f i live in A = End( ⊕ di =0 O ( i )) andthe moduli space of thin sincere representations of B = A/ ( f , . . . , f r ) is Y (evenscheme theoretic). The reader familiar with Serre’s construction will notice thatthis step is just inspired by this construction ([20]).2.5. Quiver Grassmannians.
In a final step, we use the embedding of the thin sincere representations of B inits minimal injective hull M . Note that I is the indecomposable injective modulewith the unique simple socle, that is the socle of any thin sincere representationof B . In case B = A (all f i are zero) we obtain the projective space as quivergrassmannian of thin sincere subrepresentations of I . In a similar way, also Y coincides with the quiver grassmannian of thin sincere subrepresentations of thelarge indecomposable injective B –module M . This final step goes back to Schofield[18] and was mentioned later also by Van den Bergh and Ringel ([17]).2.6. Framed moduli spaces.
We note that quiver grassmannians can be obtained also directly from a correspond-ing construction in algebraic geometry. Any skyscraper sheaf is the quotient of aline bundle
L −→ k x . If we consider the moduli space of all those quotients of L with fixed Hilbert series of the quotient sheaf, we get the (framed) Hilbert scheme,that coincides with the original one. If we apply Beilinson’s tilting again, we getfor L = O ( d ) a projective A –module. Thus P n is the quotient grassmannian for thelarge (that is sincere) indecomposable projective A –module. The same constructionworks with Y instead of P n .2.7. Reduction of the quiver.
Note that any vector bundle T = ⊕ i ∈ L O ( i ) for any L with at least two elements on P n defines a morphism from P n to the moduli space of modules of dimension vector(1 , , . . . , ,
1) over A = End( T ) (or even over the path algebra of A ) and also to thecorresponding quiver grassmannian for the injective hull M of an indecomposableof dimension vector (1 , , . . . , , n –space even two linebundles are sufficient. In the last section we modify this construction slightly andconsider L consisting of three, repectively even two, elements so that we still getan isomorphism for any subscheme X in P n .2.8. Proof.
We prove the Theorem 1.1 using the five steps above. In explicit termsthe module M is just defined by vector spaces M m = S m V /I m the m th graded partof the homogeneous coordinate ring. This becomes a module over the Beilinsonquiver using the natural map V ⊗ S m V −→ S m +1 V as a multiplication map asfollows. Take a basis v , . . . , v n of V and define the linear map of the i th arrow ODULI OF REPRESENTATIONS, QUIVER GRASSMANNIANS, AND HILBERT SCHEMES 5 just by tensoring with v i : S m V −→ S m +1 V . The commutative relations force thatthe moduli space (or the corresponding quiver grassmannian in M ) of thin sinceremodules is just a subscheme in P n defined by some of the polynomials f i . If weconsider sufficiently many degrees m , then any f i is realized in some homogeneouspart I m of the ideal I . For example the two degrees m = 0 , d are sufficient to seeany f i . Thus, if we take the Beilinson quiver with vertices 0 , , , . . . , d − , d wecan certainly realize the variety X .Conversely, we may ask how many degrees we need to realize X in the module M . A similar consideration as above shows that the three degrees m = 0 , e, d aresufficient, provided d is at least the maximum of all degrees of the polynomials f i ,and e can be any natural number with 0 < e < d . In particular, we can take e = 1or e = d −
1. This leads to the proof of the second theorem proven in the lastsection. The lemma below reduces than even to the Kronecker quiver.3.
Overview on results and some consequences
Some variatians.
Now we can use the construction above to get many varia-tions, we can not list all, however we should collect some. First we construct affineexamples. One way is to take open subvarieties, however we would like to char-acterize open subsets module theoretic. Huisgen-Zimmermann started to consideruniserial modules in [8]. To obtain affine varieties as moduli spaces of uniserialmodules, we consider a variant of the Beilinson quiver, we replace the first arrow x just by a path y z of length two. Then a thin sincere module is uniserial pre-cisely when its map y z is not zero. Thus it is the open subvariety (subscheme)defined by x = 1, that is an affine chart. Figure 2. modified Beilinson quiver3.2.
Consequences.
In our opinion there are two kind of consequences. First one might think that we cannow obtain results in algebraic geometry using representation theory. This seems tobe impossible, as far we consider any algebraic variety. However, restricting to somesubclasses this might be fruitful, we mention some open problems at the end of thisnote. Moreover, for our construction, using the Beilinson algebra, the relations aredirectly given by the defining polynomials. Thus we do not get any deeper insightby considering an algebraic variety as a moduli space of quiver representations.The second consequence concerns the realisation of a variety as a particular modulispace, that is more restrictive. This is often very useful and is already used quiteoften. The main open problem here seems to be to construct all moduli spaces ofquiver representations for a particular quiver. In general, for all dimension vectors,this is even open for the 3–arrow Kronecker quiver.3.3.
Results.
We use the construction in the previous section to prove some of thealready known results just by applying the five steps. We start with any projectivealgebraic variety and proceed with affine ones. As we already explained, we consider X as the scheme of length one sheaves on itself and apply the Beilinson tiltingbundle. LUTZ HILLE
Theorem 3.1. [7]
Any projective algebraic scheme of finite type is a fine modulispace of modules over some finite dimensional algebra (a bounded path algebra).Moreover, we can obtain it already for the thin sincere representations, that is theJordan-H¨older series contains each simple module just once in its composition seriesup to isomorphism.
Taking open parts we recover the result of Huisgen-Zimmermann, that was obtainedusing uniserial modules (Theorem G in [8]). Note, the result was stated in [8] in adifferent language, the notion of a moduli space was adapted by her only later.
Theorem 3.2. [8]
Any affine algebraic variety is a fine moduli space of uniserialmodules over some finite dimensional algebra (a bounded path algebra).
Then Grassmannians also appeared in Huisgen-Zimmermann’s work, however theidea was already introduced by Schofield [18] and then intensively used by Nakajima[13]. However, a similar result could be read of from the work of Bongartz andHuisgen-Zimmermann and was later explicitely stated in [9]. Here again we canuse thin sincere submodules of a module M or just uniserial modules. Theorem 3.3. [9]
Any projective algebraic variety is a quiver grassmannian.
Kronecker quiver.
Using the Beilinson construction with a rather small vec-tor bundle we can reduce the quiver even to the Kronecker quiver. This is almostthe same construction as in Reinekes work and based on the following geometricconstruction. Consider P n = P ( V ) embedded into P ( S m V ) with the m –uple em-bedding. Assume X is a subscheme in P ( V ) and consider its image in P ( S m V ). If m is larger than the maximal degree of the polynomials f i defining X , the equa-tions of X in P ( S m V ) are just linear and the defining equations of the embedding P ( V ) −→ P ( S m V ) (that are quadratic). Just modilfying the Beilinson construc-tion we can use the bundle O ⊕ O ( e ) ⊕ O ( d ). This reduces the construction to aquiver with three vertices. For e = d/ d sufficiently large, this correspondsto realizing X using quadratic equations. The corresponding module M consid-ered as a representation of a three vertex quiver ( M , M , M ) has a simple socle M = k = S V with M = S e V and M = S d V . Now we use Ringels idea toreduce to the Kronecker quiver S d − e V . Lemma 3.4.
With notation above and any d > e > we have an isomorphism ofquiver grassmannians as follows. The quiver grassmannian of submodules of M =( M , M , M ) of dimension vector (1 , , is isomorphic to the quiver grassmannianof submodules of ( M , M ) of dimension vector (1 , . Proof.
Note that the restriction of M to ( M , M ) defines a morphism of quivergrassmannians. Since M is just one–dimensional, any submodule ( M , M ) overthe Kronecker algebra of dimension vector (1 ,
1) extends uniquely to a submoduleof M of dimension vector (1 , , X being the projective space, this morphism is an isomorphism.Going back to M we just restrict this isomorphism to the subscheme defined by thepolynomials f i , consequently, both quiver grassmannians are also isomorphic. ✷ Taking d at least the degree of the defining equations f i and d = e +1 we realize X asa quiver grassmannian over the Kronecker algebra with ( n +1) vertices. This provesTheorem 1.2. Note that Reineke realized the linear subspace using an additionalarrow, however, this is not necessary.Obviously, we can not reduce to just one vertex, thus two vertices is the minimumwe can achieve. However, it is not clear whether we can still reduce the number of ODULI OF REPRESENTATIONS, QUIVER GRASSMANNIANS, AND HILBERT SCHEMES 7 arrows. Such a reduction would be more complicated and certainly independent ofBeilinsons result.3.5.
Affine versus projective.
At the end we discuss the problem how to obtainprojective examples from affine ones and vice versa. As we have mentioned above,one can take a projective variety, that is a moduli space, and obtain an affine coveras moduli spaces of uniserials by modifying the Beilinson quiver slightly.The converse, to obtain complete examples by glueing, is an open problem. Inparticular, let X be a complete variety that is not projective (see Hartshorne foran example [6], Ex 3.4.1 in appendix B) then to our knowledge there is no way sofar, to get X as a moduli space of representations. Moreover, it is clear that X can not be a quiver grassmannian, since the latter one is projective by definition.One might think that also moduli spaces are always projective, however, we shouldmention that moduli spaces as constructed in King’s paper [11] are, but there mightbe other constructions as well.3.6. Further open problems.
Since already Kronecker quivers are very compli-cated with respect to the geometry of quiver grassmannians it would be natural torestrict to particular classes of modules or quivers. As far we know, the problem todescribe all quiver grassmannians is open for Dynkin quivers and also tame quiv-ers. It also would be desirable to understand quiver grassmannians for the 3–arrowKronecker quiver. Moreover, inspired by cluster algebras, the main open problemseems to be to understand quiver grassmannians for exceptional modules over pathalgebras.If we use the explicit construction of the module M with M m = S m V /I m one cansee, that everything is even defined over any base field. For polynomials over theintegers everything is defined even over Z . Thus the construction also works in thesame fashion over an commutative ring. References [1] Altmann, Klaus; Hille, Lutz:
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