Doctoral theses at NTNU, 2015:265
Doctoral theses at NTNU, 2015:265
Nils Melvær Nornes
Nils Melvær Nornes
Degeneration and related partial orders
in representation theory.
ISBN 978-82-326-1186-7 (printed version) ISBN 978-82-326-1187-4 (electronic version) ISSN 1503-8181
NTNU Norwegian University of Science and Technology Faculty of Information Technology, Mathematics and Electrical Engineering Department of Mathematical Sciences
Norwegian University of Science and Technology Thesis for the degree of Philosophiae Doctor
Nils Melvær Nornes
Degeneration and related partial orders
in representation theory.
Trondheim, October 2015
NTNU
Norwegian University of Science and Technology Thesis for the degree of Philosophiae Doctor
ISBN 978-82-326-1186-7 (printed version) ISBN 978-82-326-1187-4 (electronic version) ISSN 1503-8181
Doctoral theses at NTNU, 2015:265
© Nils Melvær Nornes
Printed by Skipnes Kommunikasjon as
CONTENTS
Acknowledgements.
Introduction.
I. Partial Orders on Representations of Algebras.
Published in Journal of Algebra 323 (2010), no. 7, 2058-2062.
II. Degenerations of submodules and composition series.
Submitted to Algebras and Representation Theory.
III. Module Degenerations and Finite Field Extensions.
ACKNOWLEDGEMENTS
Wait, I’m actually done now? Huh. Time to thank people then.
Thanks to my advisor, Professor Sverre O. Smal, for advising. And to my other coauthors, Tore Forbregd and Professor Steffen Oppermann, for um, coauthing. It has been a pleasure working with you all.
I am also grateful to the rest of the algebra group at NTNU for many fine discussions over coffee and other beverages, to the department staff for making everything run smoothly, and to the Norwegian tax payers for picking up the tab.
And finally, many thanks to my parents for all sorts of support over the years.
Nils Melvær Nornes Nornes, September 2015
INTRODUCTION
The general idea behind representation theory is, as the name sug- gests, to represent complicated objects by something we know bet- ter, namely vector spaces and linear transformations. It started out with linear representations of groups, which are homomorphisms from a group to a group of automorphisms on a vector space. Similar con- structions were made for Lie groups and Lie algebras, and eventually this was generalized to representations of associative algebras.
Let k be a field and Λ a finitely generated k-algebra. A finite- dimensional representation of Λ is an algebra homomorphism from Λ to a full matrix ring overk. Letdbe a natural number andMd(k) the ring of d×d-matrices overk. We denote by moddΛ the set of all d- dimensional representations of Λ, i.e. all algebra homomorphisms from Λ toMd(k). The representations in moddΛ correspond bijectively to the Λ-module structures on the vector space kd. The general linear group GLd(k) acts on moddΛ by conjugation, and the orbits of this action corresponds to isomorphism classes of modules.
Whenkis algebraically closed, the set moddΛ also has the structure of an affine variety. Then the closures of the GLd(k)-orbits are partially ordered by inclusion, and this gives a partial order calleddegeneration on the set ofd-dimensional Λ-modules. There are several other partial orders related to this, and we will study some of them.
The thesis consists of, in addition to this introduction, three papers:
Partial Orders on Representations of Algebras [7] (cowritten with Tore A. Forbregd and Sverre O. Smalø), Degenerations of Submodules and Composition Series [9] (cowritten with Steffen Oppermann) andMod- ule Degenerations and Finite Field Extensions [10].
1. Background
Let ρbe a representation in moddΛ. It defines a module structure on the vector space kd in the following way. For aλ∈Λ and x ∈kd let λ·x=ρ(λ)·x(where the multiplication on the right hand side is just matrix multiplication). Conversely, every module structure on kd gives us a representation. Given a Λ-moduleM with underlying vector space kd, every λ∈Λ defines a linear transformation fλ:kd →kd by fλ(x) =λ·x. The functionρM : Λ→ Md(k) given by ρM(λ) =fλ is
1
2 INTRODUCTION
an algebra homomorphism, and thus a representation. This gives us a bijection between the set of Λ-module structures on kd and moddΛ.
Since Λ is finitely generated, a representation is completely deter- mined by its value on a finite set of elements. Let{λ1, . . . , λn}be a set of generators for Λ, and letρ∈moddΛ. If we know the matricesρ(λi) for 0< i≤n, then we can reconstruct all ofρ. This means that we can identify moddΛ with a subset ofMd(k)n. This subset is closed in the Zariski topology. To show this we need to construct some polynomials.
An element (A1, . . . , An)∈ Md(k)n is in moddΛ if and only if for every noncommutative polynomial f such that f(λ1, . . . , λn) = 0 we have f(A1, . . . , An) = 0. Let k[{Xabc}0<a≤n,0<b≤d,0<c≤d] be the coordi- nate ring of Md(k)n. For every f with f(λ1, . . . , λn) = 0, we have a matrix
f
X111 · · · X11d ... . .. ... X1d1 · · · X1dd
,· · ·,
Xn11 · · · Xn1d ... . .. ... Xnd1 · · · Xndd
.
Each entry in this matrix is a polynomial that is 0 on moddΛ. LetS be the set of all these polynomials for allf such thatf(λ1, . . . , λn) = 0.
Then moddΛ is the zero set of S, and thus an affine variety.
The group variety GLd(k) acts on moddΛ by conjugation, that is, forg∈GLd(k) and (A1, . . . , An)∈moddΛ we haveg ?(A1, . . . , An) = (gA1g−1, . . . , gAng−1). This induces an isomorphism between the mod- ule represented byρand the module represented byg ? ρ, and thus we get a one-to-one correspondence between GLd(k)-orbits in moddΛ and isomorphism classes of d-dimensional Λ-modules. We are now ready for the definition of degeneration.
Definition. If the orbit corresponding to a moduleN is contained in the closure of the orbit corresponding to the module M, we say that M degenerates toN, and denote this byM ≤degN.
The simplest examples of module varieties occur when Λ =k[X], the polynomial ring in one variable. Here we have modd(k[X]) =Md(k), and the orbits are just similarity classes of matrices. That makes it easy to decide if two modules are isomorphic, we only have to compare the Jordan forms of their representations. If they have the same eigenvalues and block sizes, they are isomorphic.
It is also easy to decide if onek[X]-module degenerates to another.
M. Gerstenhaber showed in [8] that for A, B ∈ moddk[X] we have A≤degB if and only if rank(A−λ)i≥rank(B−λ)ifor alli∈Nand all eigenvaluesλofA.
INTRODUCTION 3
Any minimal degeneration of a k[X]-module A can be constructed by reducing the size of a Jordan block by one and increasing the size of a smaller block with the same eigenvalue (or creating a new block of size one).
In 1969, M. Artin [4] observed that for an exact sequence 0 //A //B //C //0
we haveB≤degA⊕C. An immediate consequence of this result is that an orbit is closed if and only if the corresponding module is semisimple.
For some algebras this tells the whole story, every minimal degener- ation is of this form. In particular this is true fork[X].
The existence of such sequences gives rise to a coarser order called
≤ext, which was first considered by S. Abeasis and A. Del Fra in [1].
Definition. LetM andN be Λ-modules. M ≤extN if for somen∈N there existnshort exact sequences
0 //Ai //Bi //Ci //0
such thatM 'B1,N 'An⊕Cn andBi'Ai−1⊕Ci−1for 2≤i≤n.
Abeasis and Del Fra also introduced a third order≤r based on the ranks of certain matrices.
They showed in [1], [2] and [3] that ≤ext,≤degand≤r are the same for all path algebras overAn-quivers, and over someDn-quivers. Later K. Bongartz showed in [5] that ≤deg and ≤ext are the same for all representation-directed algebras. This includes all path algebras over Dynkin quivers. He also showed this for the Kronecker algebra. As mentioned above, ≤ext and≤degare also the same fork[X].
In [11], C. Riedtmann proved the following.
Proposition 1. Let
0 //X //X⊕M //N //0 be an exact sequence inmod Λ. Then M ≤degN.
Using this she gave the first example of a proper degenerationM ≤deg
N whereN is indecomposable. WhenM ≤ext N, N is clearly decom- posable, so this shows that degeneration is strictly finer than the ext- order. In the same paper she presented an example due to J. Carlson which shows that one cannot cancel common summands in a degener- ation. This led her to introduce two new partial orders.
Definition. M virtually degenerates to N, denoted M ≤vdeg N, if there existsY ∈mod Λ such thatM⊕Y ≤degN⊕Y.
4 INTRODUCTION
Definition. M ≤HomN if for allX ∈mod Λ we have [X, M]≤[X, N].
She then showed that≤vdegand≤Homare the same for representation- finite algebras.
In 2000 G. Zwara proved the converse of Proposition 1 in [15].
Theorem 2.LetM andN beΛ-modules. The following are equivalent.
(1) M ≤degN.
(2) There exists a short exact sequence ofΛ-modules 0 //X //X⊕M //N //0. (3) There exists a short exact sequence ofΛ-modules
0 //N //Y ⊕M //Y //0.
This gives a completely module theoretic description of degenera- tions. We can use this as an alternative definition of degeneration, and since it does not involve geometry, we can relax the conditions on k.
It does not have to be algebraically closed any more, it does not even have to be a field. All we need is a commutative artin ring.
Without geometry it is no longer obvious that degeneration is a partial order, but this was proved by G. Zwara in [13].
Obviously we still have that M ≤ext N implies M ≤deg N, and M ≤degN impliesM ≤vdegN. From the new definition it is also easy to see that M ≤vdegN implies M ≤Hom N: IfM ≤vdeg N we have a Riedtmann sequence
0 //X //X⊕M ⊕Y //N⊕Y //0.
For anyZ ∈mod Λ we apply HomΛ(Z,−) to the Riedtmann sequence and get an exact sequence
0 //HomΛ(Z, X) //HomΛ(Z, X)⊕HomΛ(Z, M)⊕HomΛ(Z, Y)
//HomΛ(Z, N)⊕HomΛ(Z, Y), and summing up the lengths we see that [Z, M]≤[Z, N].
K. Bongartz has shown in [6] that for tame hereditary algebras over algebraically closed fields ≤deg and ≤Hom are the same. G. Zwara showed the same for representation-finite algebras over algebraically closed fields in [14]. Zwara’s result was later generalized to any representation- finite artin algebra by S. O. Smalø in [12].
The examples from Riedtmann and Carlson show that for arbitrary algebras ≤ext is strictly coarser than≤deg, which is again coarser than
≤vdeg. It is still not known if ≤Hom is different from≤vdeg.
INTRODUCTION 5
2. Overview of the thesis
In [12], S. O. Smalø introduced the following family of quasiorders.
Let M be a Λ-module and n a natural number. An n×n-matrix A with entries in Λ induces ak-linear map fromMnto itself. Denote the image of this map byAMn.
Definition. M ≤n N if `(Mn/AMn) ≤ `(Nn/ANn) for all n×n- matricesAwith entries in Λ.
Equivalently,M ≤nN if each matrix has greater or equal rank as a Mn-endomorphism than as aNn-endomorphism. This generalizes the rank order of Abeasis and Del Fra.
Clearly the relation≤n is reflexive and transitive, but for smallnit is not always antisymmetric.
In [7], cowritten with my fellow student Tore A. Forbregd and our adviser Professor Sverre O. Smalø, we show that≤d3always is a partial order on moddΛ. It seems like for large enoughn,≤nis equivalent to
≤Hom. In the paper we claimed this as a fact, but we did not give a proof. When the reviewer requested a proof, we realized that the proof we had in mind was incomplete. We decided to remove the statement, but unfortunately we wrote it twice and deleted it once, so it still appears in the published version.
While we have not found a proof, we have not found any counterex- ample either. In fact, in all examples we have looked at,≤nis either not a partial order or equivalent to≤Hom. We still have no examples where
≤n is a partial order but is different from ≤Hom. However, we have such an example for the closely related quasiorders ≤Hom−n obtained by loosening the conditions of the Hom-order.
Definition. Let M and N be Λ-modules and n a natural number.
M ≤Hom−nN if [X, M]≤[X, N] for all Λ-modulesX with`X ≤n.
Example 1. Let Qbe the Kronecker quiver, Q: 1
α //
β //2,
and let Λ =kQbe the path algebra. The representations
P1: k (10)
//
(01)
//k2, P2: 0 ////k
are the indecomposable projective modules, and
I1: k ////0, k2
( 1 0 )//
( 0 1 )
//k
6 INTRODUCTION
are the indecomposable injective modules. Let M =P2⊕I2 andN = P1⊕I1. We haveM ≤Hom−4 N, but [M, P2]> [N, P2], so M and N are not comparable in the Hom-order. We also have that`(EndΛM) =
`(EndΛN) = 3, which is something that cannot happen with a proper degeneration.
In [9], cowritten with Professor Steffen Oppermann, we show that a degenerationM ≤degN induces degenerations from submodules ofM to submodules of N. Given a submodule M0 ⊆M and a Riedtmann sequence, we construct a submodule N0 ⊆ N such that M0 ≤deg N0. This construction gives rise to a function from the set of submodules of M to the set of submodules of N, but this function does not seem to have any nice properties. We give examples that show, among other things, that it is neither injective nor surjective.
Since submodules degenerate to submodules we also have that com- position series in some sense degenerate to composition series. We give a geometric interpretation of this degeneration order using the subset of modΛ consisting of those homomorphisms whose images are con- tained in the ring of upper triangular matrices. Such a representation can be viewed as a representation of a composition series. With the right group action we get a correspondence between orbits and iso- morphism classes of composition series, and orbit closures give rise to degenerations.
In [10] we study the degeneration order for some algebras over fields that are not algebraically closed. In particular we look at modules over K⊗kΛ, whereKis a finite extension of the base field. These modules can also be viewed as Λ-modules, and we try to show how isomorphism classes and degenerations differ depending on which algebra we work over. The Λ-isomorphism class of a module may contain several differ- entK⊗kΛ-isomorphism classes, and in the case whereKis a normal extension we give a complete description of these. We show several examples where modules degenerate as Λ-modules but not asK⊗kΛ- modules. We also find some examples where M⊕M ≤degN⊕N but M does not degenerate toN.
References
[1] Abeasis, S. and Del Fra, A. Degenerations for the representations of an equior- iented quiver of typeAm. Boll. Un. Mat. Ital. Suppl. (1980), no. 2, 157–171 [2] Abeasis, S. and Del Fra, A. Degenerations for the representations of an equior-
iented quiver of typeDm. Adv in Math 52 (1980), no. 2, 157–171
[3] Abeasis, S. and Del Fra, A. Degenerations for the representations of a quiver of typeAm. J. Algebra 93 (1985), no. 2, 376-412
INTRODUCTION 7
[4] Artin, M. On Azumaya Algebras and Finite Dimensional Representations of Rings. J. Algebra 11 (1969), 532–563
[5] Bongartz, K. On Degenerations and Extensions of Finite Dimensional Modules.
Adv. in Math. 121 (1996), 245–287.
[6] Bongartz, K. Degenerations for representations of tame quivers. Ann. scient.
Ec. Norm. Sup., 4. serie, t. 28, (1995), 647–668.
[7] Forbregd, T. A., Nornes, N. M., Smalø, S. O. Partial Orders on Representations of Algebras. J. Algebra 323 (2010), no. 7, 2058–2062.
[8] Gerstenhaber, M. On nilalgebras and linear varieties of nilpotent matrices.
Ann. of Math. 70 (1959), 167–205.
[9] Nornes, N. M., Oppermann, S. Degenerations of Submodules and Composition Series. Submitted to Algebr. Represent. Theory.
[10] Nornes, N. M. Module Degenerations and Finite Field Extensions.
[11] Riedtmann, C. Degenerations for representations of quivers with relations.
Ann. Sci. Ecole Nomale Sup. 4 (1986), 275–301.
[12] Smalø, S. O. Degenerations of representations of associative algebras. Milan Journal of Math. 76, 1 (2008), 135–164.
[13] Zwara, G. A degeneration-like order for modules. Arch. Math. 71 (1998), 437- 444
[14] Zwara, G. Degenerations for modules over representation-finite algebras. Proc.
Amer. Math. Soc. 127, 5 (1999), 1313–1322.
[15] Zwara, G. Degenerations of finite-dimensional modules are given by extensions.
Composito Math. 121 (2000), 205–218.
I
PARTIAL ORDERS ON REPRESENTATIONS OF ALGEBRAS
TORE A. FORBREGD, NILS M. NORNES, AND SVERRE O. SMALØPublished in Journal of Algebra 323 (2010), no. 7, 2058-2062.
PARTIAL ORDERS ON REPRESENTATIONS OF ALGEBRAS
TORE A. FORBREGD, NILS M. NORNES, AND SVERRE O. SMALØ
Abstract. Let kbe a commutative artin ring and let Λ be an artin k-algebra. For each natural numberdlet repdΛ be the set of isomorphism classes of Λ-modules withk-length equal tod. For each natural numbern, ann×n-matrix with entries in Λ, can be considered as a k-endomorphism of Mn, where Mn denotes the direct sum ofncopies of the Λ-moduleM. The quasiorder≤non repdΛ is defined byM ≤n N if for everyn×n-matrixϕ, with entries in Λ, we have that`k(Mn/ϕMn)≤`k(Nn/ϕNn).
We show that the quasiorder≤nis a partial order on repdΛ for n≥d3.
1. Introduction
For an artink-algebra Λ, where kis a commutative artin ring, and a natural number d, let repdΛ be the set of isomorphism classes of Λ-modulesX such that thek-length ofX,`k(X), isd. One can define several partial orders on repdΛ, such as the degeneration order ≤deg, the virtual degeneration order ≤vdeg and the Hom-order ≤hom. The two first of these orders come from geometry whenkis an algebraically closed field. However, due to a result of C. Riedtmann combined with a result of G. Zwara (see [6] and [8]) these orders have a purely mod- ule theoretical interpretation. Namely, M≤degN is equivalent to the existence of a short exact sequence of Λ-modules of the form
0 //X //X⊕M //N //0.
and thus this can be taken as the definition for the relation ≤deg on repdΛ for alld. Furthermore, Zwara also showed that this is equivalent to the existence of a short exact sequence of the form
0 //N //M⊕X //X //0.
The relation≤vdegis defined by M≤vdegN if there exists a Λ-module Y such that Y ⊕M≤degY ⊕N. Finally, the Hom-relation is de- fined by M≤homN if `k(HomΛ(X, M)) ≤`k(HomΛ(X, N)) for all Λ- modules X. The fact that ≤hom is a partial order is due to a re- sult of M. Auslander. In [1] he showed that M ' N if and only if
11
12 TORE A. FORBREGD, NILS M. NORNES, AND SVERRE O. SMALØ
`(HomΛ(X, M)) =`(HomΛ(X, N)) for all finitely generated Λ-modules X. From Proposition 5 in the main section of this paper it follows that one does not need to consider allX to show thatM 'N, it is enough to look at a certain set of submodules of M and N. This is proved using a construction due to O. Iyama [4].
It is known thatM≤degNimpliesM≤vdegNwhich impliesM≤homN.
However, due to an example of J. Carlson we do not have that≤vdegis equivalent to ≤deg in general. It is still open whether≤vdegis equiva- lent to≤hom. For some classes of algebras it is known that these three orders coincide, e.g. algebras of finite representation type and tame hereditary algebras. For a quick overview the reader is referred to [7].
Here we will look at the Hom-order and the quasiorders ≤n. If for everyn×n-matrix,ϕ, with entries in Λ, we have that`k(Mn/ϕMn)≤
`k(Nn/ϕNn), then we write M ≤n N. Ifk is a field there is a strong link between these quasiorders and the Hom-order, in the sense that there exists a natural number nd such that ≤nd is a partial order on repdΛ and this partial order coincides with≤hom.1 In the case where Λ is of finite representation type, it is known that there is a universaln such that≤n is a partial order on repdΛ for anyd(see [5]). The main result, Theorem 6, states that ≤d3is a partial order.
2. Preliminaries
Let kbe a commutative artin ring and let Λ be an artink-algebra.
For a ring R we have that if M is a right R-module, then it is in a natural way a left Rop-module. Therefore, throughout this article all modules will be unital left modules. Denote by mod Λ the category of finitely generated Λ-modules and forM in mod Λ let addM be the additive closure ofM in mod Λ. If not specified otherwise, the length,
`(M), of a Λ-moduleM will mean its length as a k-module. We will write R(−,−) instead of HomR(−,−). For a Λ-module M let radM be the Jacobson radical ofM,that is the submodule ofM given by the intersection of all maximal submodules of M. For artin algebras it is known that radM = (rad Λ)·M. Moreover, the socle ofM, socM, is the sum of all simple submodules of M, i.e. the largest semisimple submodule ofM.
Let K0(mod Λ) = F(mod Λ)/R(mod Λ) denote the Grothendieck group of mod Λ, where F(mod Λ) is the free abelian group on the isomorphism classes of Λ-modules andR(mod Λ) is the subgroup gen- erated by all short exact sequences of Λ-modules. Let [M] denote the
1We don’t have a proof of this. See introduction.
PARTIAL ORDERS ON REPRESENTATIONS OF ALGEBRAS 13
element inK0(mod Λ) corresponding to the moduleM in mod Λ. Fur- thermore, for a ringR, denote byP(R) the full subcategory of modR consisting of projectiveR-modules.
For a module M, Mn denotes the direct sum of n copies of M. Let Mn(Λ) be the set ofn×n-matrices with entries in Λ. A matrix ϕ∈Mn(Λ) induces ak-endomorphism onMnby matrix multiplication from the left, and we denote the image of this homomorphism byϕMn. It also induces a Λ-endomorphism on Λnby matrix multiplication from the right, and we denote the image of this by Λnϕ.
Definition. The Hom-relation≤homon repdΛ is defined byM≤homN if`(HomΛ(X, M))≤`(HomΛ(X, N)) for all X in mod Λ.
It is obvious that the relation≤homis reflexive and transitive. In [1]
Auslander showed thatM 'Nif and only if`(Λ(X, M)) =`(Λ(X, N)) for all X in mod Λ, and thus that ≤hom is antisymmetric and hence a partial order. The result also holds in the more general setting of a commutative ring R and anR-linear abelian category where all mor- phism sets have finite lengths as R-modules. This generalization was proved by Bongartz in [3].
One can equivalently define a Hom-order≤0homby looking atΛ(M, X) and Λ(N, X), however this gives the same partial order as≤hom. For the convenience of the reader we will recall a proof of this fact.
Proposition 1. LetM and N be modules inrepdΛ. Then M≤homN if and only ifM ≤0homN.
Proof. We first consider the case where `(Λ(P, M) < `(Λ(P, N)) for an indecomposable projective Λ-module P. Since `(M) = `(N) and
`(M) = `(Λ(Λ, M)) there exists another indecomposable projective Λ-module P0 with `(Λ(P0, M)) > `(Λ(P0, N)), so M and N are in- comparable. Moreover, for an indecomposable projective Λ-moduleP,
`EndΛ(P)op(Λ(P, M)) is equal to the number of times the simple mod- ule P/radP occurs as a composition factor of M. Likewise, for an indecomposable injective Λ-moduleI,`EndΛ(I)(Λ(M, I)) is equal to the number of times the simple module socIoccurs as a composition factor of M. Since P andI are indecomposable, EndΛ(I) and EndΛ(P) are local rings, thus we see that
`k(Λ(P, M)) =`EndΛ(P)op(Λ(P, M))·`k(EndΛ(P)/rad(EndΛ(P)))
`k(Λ(M, I)) =`EndΛ(I)(Λ(M, I))·`k(EndΛ(I)/rad(EndΛ(I))).
Let I be the indecomposable injective Λ-module corresponding to P, i.e.I is the injective envelope ofP/radP. By the above we have that
`(Λ(P, M)< `(Λ(P, N)) implies`(Λ(M, I))< `(Λ(N, I)). Hence we get
14 TORE A. FORBREGD, NILS M. NORNES, AND SVERRE O. SMALØ
that `(Λ(P, M)) = `(Λ(P, N)) for all projective Λ-modules P if and only if`(Λ(M, I)) =`(Λ(N, I)) for all injective Λ-modulesI.
Now if`(Λ(P, M)) =`(Λ(P, N)) for all projective Λ-modulesP, it fol- lows from Corollary IV.4.3 in [2] that`(Λ(M, DTrX))≥`(Λ(N, DTrX)) if `(Λ(X, M)) ≥ `(Λ(X, N)), where DTr : mod Λ → mod Λ is the Auslander-Reiten translate on mod Λ. Hence, we get that`(Λ(X, M))≥
`(Λ(X, N)) for allX in mod Λ, if and only if`(Λ(M, X))≥`(Λ(N, X))
for allX in mod Λ.
Definition. Let M, N ∈ repdΛ. We say that M ≤n N if for every ϕ∈Mn(Λ) we have that`(Mn/ϕMn)≤`(Nn/ϕNn).
In general this gives a quasi-ordering on repdΛ, however it is not always antisymmetric. It is known that≤d5is a partial order on repdΛ (see [5]).
We now give some basic facts about these quasi-orderings.
Proposition 2. Let M andN be modules inrepdΛ, and let mandn be natural numbers. Then
(1) M ≤n N impliesM ≤m N whenever m≤n. In particular, if
≤m is a partial order, then so is≤n. (2) M≤homN impliesM ≤nN
Proof. Part 1 follows from the fact that ifm≤n, everym×m-matrix can be expanded to an×n-matrix simply by filling in enough zeros.
To show part 2, we consider the following exact sequence Λn−·ϕ//Λn //Λn/Λnϕ //0
withϕ∈Mn(Λ). By applying Λ(−, M) to the sequence above we get the following exact commutative diagram.
0 //Λ(Λn/Λnϕ, M) // Λ(Λn, M)Λ(−·ϕ,M// )
O
Λ(Λn, M)
O
Mn ϕ·− // Mn //Mn/ϕMn //0 Since the alternating sum of the lengths of the modules in an exact sequence equals zero, this yields that`(Mn/ϕMn) =`(Λ(Λn/Λnϕ, M)), and henceM≤homN implies thatM ≤nN for alln.
3. The Main Result
We begin by stating and proving the following lemma.
Lemma 3. Let M, N ∈ repdΛ and X ∈ repsΛ. If M ≤d2s N and N ≤d2sM, thenl(Λ(X, M)) =l(Λ(X, N)).
PARTIAL ORDERS ON REPRESENTATIONS OF ALGEBRAS 15
Proof. Note first that without loss of generality we may assume that annM = annN, since otherwise there is aλ∈Λ such that`(M/λM)6=
`(N/λN). Let Γ = Λ/annM. We then have that Γ⊆Endk(M) and hence`(Γ)≤d2, andΓ(X/annM·X, M)'Λ(X, M) andΓ(X/annM· X, N)'Λ(X, N). Then we may find a free resolution ofXover Γ which is of the form
Γd2s //Γs //X/annM·X //0.
By addingd2s−scopies of the algebra Γ to the second and third terms, we get the exact sequence Γd2s ϕ//Γd2s //X/annM·X⊕Γd2s−s //0, whereϕcan be described by a matrix inMd2s(Λ). By applyingΓ(−, M) and Γ(−, N) and counting lengths in the resulting sequences, we get that
`(Γ(X/annM·X, M) +`(ϕMd2s) =s·`(M)
`(Γ(X/annM·X, N) +`(ϕNd2s) =s·`(N).
Since M and N are both i repdΛ, we have `(M) = `(N) = d, and M ≤d2sN andN ≤d2sM implies that`(ϕMd2s) =`(ϕNd2s). Hence,
`(Λ(X, M)) =`(Γ(X/annM·X, M)) =`(Γ(X/annM·X, N)) =`(Λ(X, N)).
Note that if one considers all matrices, rather than just square ma- trices, adding Γd2s−sis not necessary. In other words, it is sufficient to look atd2s×s-matrices.
In [4] O. Iyama showed that for each L in mod Λ we can find sub- modules Li ⊂ L fori = 1, . . . , r such that gl.dim.EndΛ(Lr
i=0Li) <
∞ where L0 = L. In [4] these submodules are given by Li+1 = rad(EndΛ(Li))·Li, i.e. Li+1 is the submodule generated by the im- ages of all maps in the Jacobson radical of EndΛ(Li).
Remark 4. This construction behaves nicely with respect to direct sums, that is if L= M ⊕N then rad(EndΛ(L))·L'M1⊕N1 with M1 ⊂M andN1⊂N.
Proposition 5. LetM andN be in repdΛ and let L0=M ⊕N and Li+1= rad(EndΛ(Li))·Lifori= 1, . . . , rwithLr+1= 0. ThenM 'N if and only if`(Λ(X, M)) =`(Λ(X, N)) for all X inaddLr
i=0Li. Proof. ClearlyM 'N implies`(Λ(X, M)) =`(Λ(X, N)) for allX. So let C=Lr
i=0Liand Γ = EndΛ(C)op, and suppose that`(Λ(X, M)) =
`(Λ(X, N)) for all X in addC. It is sufficient to consider the inde- composable objects in addC. Let C 'Lt
j=1Cj be a decomposition of C into indecomposable Λ-modules. We then have an equivalence
16 TORE A. FORBREGD, NILS M. NORNES, AND SVERRE O. SMALØ
of subcategories Λ(C,−) : addC //P(Γ). For each Cj we have a k- isomorphismΓ(Λ(C, Cj),Λ(C, M))'Λ(Cj, M). SinceCj is in addC, by assumption we get that`(Γ(Λ(C, Cj),Λ(C, M))) =`(Γ(Λ(C, Cj),Λ(C, N))).
AsCj is indecomposable, EndΛ(Cj) is a local ring. Therefore we have that
`(Γ(Λ(C, Cj),Λ(C, M))) =`(Γ(Λ(C, Cj),Λ(C, N)))
also when we consider lenghts over EndΛ(Cj)op. This common num- ber, denoted mj, is the multiplicity of the simple Γ-module Sj =
Λ(C, Cj)/rad(Λ(C, Cj)) as a composition factor of Λ(C, M), and as a composition factor of Λ(C, N), for j = 1, . . . , t. We then have that [Λ(C, M)] =Pt
j=1mj[Sj] = [Λ(C, N)] as elements in the Grothendieck groupK0(mod Γ) of Γ. By [4], Γ has finite global dimension, and thus we have that the indecomposable projective Γ-modules constitute a ba- sis forK0(mod Γ). This implies that every projective Γ-module is de- termined by its composition factors, and thereforeΛ(C, M)'Λ(C, N).
Through the equivalence Λ(C,−) : addC //P(Γ) we get that M '
N.
By combining Lemma 3 and Proposition 5 we get the following result.
Theorem 6. The relation≤d3 is a partial order onrepdΛ.
Proof. It is enough to prove that the relation ≤d3 is antisymmetric.
Suppose thatM ≤d3N andN ≤d3 M. LetL0 =M ⊕N and let C= Lr
i=0LiwhereLi+1= rad(EndΛ(Li))·Lifori= 1, . . . , rwithLr+1= 0.
Let C = Lt
j=1Cj be a decomposition of C into indecomposable Λ- modules. By Remark 4 we have thatCj ⊂M or Cj ⊂N for all 1≤ j ≤t, and therefore sj =`(Cj)≤d. Sinced3 ≥sjd2, Lemma 3 yields that`(Λ(Cj, M)) =`(Λ(Cj, N)) for each indecomposable summandCj ofC, and hence`(Λ(X, M)) =`(Λ(X, N)) for allXin addC. Thus the conditions of Proposition 5 are satisfied and we have thatM 'N and the relation ≤d3is a partial order on repdΛ.
4. Closing Comment
For anM in repdΛ, the length of the first syzygy ofM as a module over Λ/annM is bounded byd3−d. Therefore, ≤n will be a partial order on repdΛ forn≥max{d3−d, d}.
References
[1] Auslander, M. Representation theory of finite dimensional algebras. Contemp.
Math. 13 (1982), 27–39.
PARTIAL ORDERS ON REPRESENTATIONS OF ALGEBRAS 17
[2] Auslander, M., Reiten, I. and Smalø, S. O. Representation Theory of Artin Algebras. Cambridge University Press, 1994.
[3] Bongartz, K. A generalization of a theorem of M. Auslander. Bull. London Math. Soc. 21, 3 (1989), 255–256.
[4] Iyama, O. Finiteness of representation dimension. Proceedings of the American Mathematical Society 131, 4 (2002), 1011–1014.
[5] Nornes, N. Partial orders related to the hom-order and degenerations. S˜ao Paulo Journal of Math. Sci. (To appear).
[6] Riedtmann, C. Degenerations for representations of quivers with relations.
Ann. Sci. Ecole Nomale Sup. 4 (1986), 275–301.
[7] Smalø, S. O. Degenerations of representations of associative algebras. Milan Journal of Math. 76, 1 (2008), 135–164.
[8] Zwara, G. Degenerations of finite-dimensional modules are given by extensions.
Composito Math. 121 (2000), 205–218.
II
DEGENERATIONS OF SUBMODULES AND COMPOSITION SERIES
NILS M. NORNES AND STEFFEN OPPERMANNSubmitted to Algebras and Representation Theory.
DEGENERATIONS OF SUBMODULES AND COMPOSITION SERIES
NILS M. NORNES AND STEFFEN OPPERMANN
Abstract.LetM andN be modules over an artin algebra such that M degenerates toN. We show that any submodule ofM degenerates to a submodule ofN. This suggests that a composition series ofM will in some sense degenerate to a composition series ofN.
We then study a subvariety of the module variety, consisting of those representations where all matrices are upper triangular. We show that these representations can be seen as representations of composition series, and that the orbit closures describe the above mentioned degeneration of composition series.
1. Introduction
Letkbe an algebraically closed field, and let Λ be a finite dimensional associativek-algebra with unity. We denote by mod Λ the category of finite dimensional unital left modules over Λ. For natural numbersm andn, letMm×n(k) denote the set ofm×n-matrices with entries ink, letMn(k) denote thek-algebra ofn×n-matrices andUn(k)⊆ Mn(k) the subalgebra of upper triangular matrices. GLn(k)⊆ Mn(k) denotes the general linear group, andUd(k)⊆GLd(k) denotes the subgroup of upper triangular matrices.
Fix a natural numberd. We want to study the set of left Λ-module structures on the vector spacekd. We have a one-to-one correspondence between this set and the set of k-algebra homomorphisms from Λ to Md(k). Iff is such a homomorphism, we obtain a module structure by settingλ·v:=f(λ)vforλ∈Λ andv∈kd. Conversely, if we have a module structure, we get a k-algebra homomorphism g by setting g(λ) := λ·u1 . . . λ·ud
, where ui is the ith unit column vector.
Such a homomorphism is called a d-dimensional representation of Λ, and we denote the set of all d-dimensional representations of Λ by moddΛ.
Let {λ1, . . . , λn} be a generating set of Λ. Then a representation ρ ∈ moddΛ is completely determined by its values on λi, so we can view moddΛ as a subset of Md(k)n. This subset is Zariski closed, so
21
22 NILS M. NORNES AND STEFFEN OPPERMANN
moddΛ has the structure of an affine variety. The group variety GLd(k) acts on moddΛ by conjugation, and its orbits correspond bijectively to the isomorphism classes of modules. We can now give the definition of degeneration of modules.
Definition.LetM andN be Λ-modules with representationsµandν in moddΛ. M degenerates to N ifν lies in the closure of the GLd(k)- orbit ofµ. This is denoted byM ≤degN.
Degeneration is a partial order on the set of isomorphism classes of d-dimensional modules. Thecodimensionof a degenerationM ≤degN, denoted codim(M, N), is the codimension of the orbit corresponding toN in the closure of the orbit corresponding toM. The dimension of an orbit GLd(k)∗µcan be computed by the formula dim GLd(k)∗µ= d2−[M, M], where [M, M] denotes thek-dimension of HomΛ(M, M).
From that we get codim(M, N) = [N, N]−[M, M].
In [8] G. Zwara, building on earlier work of C. Riedtmann in [4], gave a nice module-theoretic description of this partial order:
Theorem 1. Let M and N be Λ-modules. Then the following are equivalent:
(1) M≤degN
(2) There exists a short exact sequence0→N →M⊕Z→Z→0 in mod Λfor someZ∈mod Λ.
(3) There exists a short exact sequence0→X→M⊕X→N →0 in mod Λfor someX∈mod Λ.
The short exact sequences in Theorem 1 are calledRiedtmann-sequences.
In this paper we will use Riedtmann-sequences of the form 0→X → M⊕X→N →0, but all our results work equally well for sequences of the other form.
Now one can extend the notion of degeneration to algebras over arbitrary fields, and even over commutative artin rings, by using the existence of Riedtmann-sequences as the definition. G. Zwara showed in [7] that degeneration is a partial order also in this case. Here we define the codimension of M ≤deg N to be [N, N]−[M, M] (where [X, X] denotes length of HomΛ(X, X) as ak-module.)
One problem with the degeneration order is that in general one can- not cancel common summands, that is X⊕M ≤deg X⊕N does not imply M ≤degN. This led to the introduction of a new partial order calledvirtual degeneration in [4].
Definition. LetM andN be Λ-modules. M virtually degenerates to N if there exists a moduleX ∈mod Λ such thatX⊕M ≤degX⊕N. This is denoted byM≤vdegN.
DEGENERATIONS OF SUBMODULES AND COMPOSITION SERIES 23
The following proposition gives an alternative way of describing vir- tual degenerations. For a proof of the proposition see [6], section 2.
Proposition 2. LetM andN beΛ-modules. ThenM≤vdegN if and only if there is some finitely presented functorδ: mod Λ→modksuch that`(δ(X)) = [X, N]−[X, M]for allX ∈mod Λ.
Ifδis such a functor, we say that the degeneration isgiven by δ.
In section 2 we will prove the following:
Theorem 3. LetM andN be Λ-modules and letM0⊆ M be a sub- module.
(1) IfM ≤degN, then there exists a submoduleN0⊆N such that M0≤degN0.
(2) IfM≤vdegN, then there exists a submoduleN0⊆N such that M0≤vdegN0.
In section 3 we look at representations whose images are contained inUd(k), which we calltriangular representations. We show that these can be viewed as representations of composition series, and then we prove the following analogue of Theorem 1.
Theorem 4. Letµandν be triangularΛ-representations, and let re- spectively M1 i1 //. . . id−1//Md and N1 j1 //. . . jd−1//Nd be the cor- responding composition series. Thenν∈Ud(k)∗µif and only if there exists a commutative diagram
0
0
0
X1 h1 //
X2 h2 //
· · · hd−1 //Xd
X1⊕M1
h1 0 0 i1
//
X2⊕M2
h2 0 0 i2
//
· · ·
hd−1 0 0 id−1
//Xd⊕Md
N1
j1 //N2
j2 //· · · jd−1 //Nd
0 0 0
with exact columns.
To study degenerations of modules, one can look at the variety of quiver representations, repd(Q, ρ), instead of moddΛ. Let Q be a quiver with vertices Q0 = {1, . . . , n} and arrows Q1, and let d =
24 NILS M. NORNES AND STEFFEN OPPERMANN
(d1, . . . , dn)∈ Nn. Then repdQ = `
α∈Q1Mde(α)×ds(α)(k), wheres(α) and e(α) are respectively the start and end points of the arrow α, consists of all representations with dimension vectord. The group va- riety Gd = GLd1(k)×. . .×GLdn(k) acts on repdQ, and the orbits correspond to isomorphism classes. Given a set of relations ρ on Q, repd(Q, ρ) is the subvariety of repdQconsisting of all representations that satisfy the relations in ρ. K. Bongartz showed in [2] that the degeneration order we get from repd(Q, ρ) is the same as the one we get from moddkQ/hρi. He also showed a deeper geometric connection between these varieties, but we will not go into that in this paper.
Usually repd(Q, ρ) is much smaller than moddkQ/hρi, which makes it easier to perform computations.
In section 4 we introduce a similar smaller variety that can be used to study degenerations of composition series.
For general background on representation theory of algebras we refer the reader to [1]. For an introduction to the topic of module degener- ations, see [5].
2. Degenerations of submodules
In this section, letk be a commutative artin ring and let Λ be an artink-algebra. All modules considered in this paper have finite length.
We first prove part 1 of Theorem 3.
Proposition 5. Let M and N be Λ-modules and let M0 ⊆ M be a submodule. If M ≤degN, then there exists a submoduleN0 ⊆N such thatM0≤degN0.
Proof. Assume thatM ≤degN and letM0⊆M be a submodule. Then there exists an exact sequence
η: 0 //X
f g
//X⊕M //N //0.
Let X0 = {x ∈ X | gfn(x) ∈ M0 ∀n ≥ 0}, let iX : X0 → X and iM : M0 → M be the submodule inclusions. From the definition of X0, we see that f(X0) ⊆ X0 and g(X0) ⊆ M0. Thus, by restricting
fg
toX0, we get a homomorphism fg
X0⊕M0
X0 :X0→X0⊕M0. Let
DEGENERATIONS OF SUBMODULES AND COMPOSITION SERIES 25
N0= coker fg
X0⊕M0
X0 . We then have the commutative diagram 0
0
0 //X0
f g
X0⊕M0 X0
//
iX
X0⊕M0 //
iX 0 0 iM
N0
α //0
0 //X
f g
//
X⊕M //
N //0
X/X0
f g
//
X/X0⊕M/M0
0 0
with exact rows and columns. Since the top row is exact we have M0 ≤deg N0, so it remains to show thatα is a monomorphism. We have
kerf ={(x+X0)∈X/X0|f(x)∈X0}
={(x+X0)∈X/X0|gfn(x)∈M0∀n≥1}.
If (x+X0) is a non-zero element in kerf then x 6∈ X0 = {x ∈ X | gfn(x)∈M0∀n≥0}, so we must haveg(x)6∈M0and hence (x+X0)6∈
kerg. This means that ker
f g
= kerf ∩kerg = (0). Then by the
Snake Lemma we get that kerα= (0).
To prove the same result for virtual degenerations, we will need the following simple lemma.
Lemma 6. Let X and Y be Λ-modules, and let M ⊆ X⊕Y be a submodule. Then there exist submodules X0 ⊆ X and Y0 ⊆ Y such thatM ≤degX0⊕Y0.
Proof. Leti:M →X⊕Y be the inclusion andp:X⊕Y →X the projection on the first summand. We have a commutative diagram
0 //Y //X⊕Y p //X //0
0 //ker? pi
OO //M?
i
OO //impi?
OO //0
with exact rows. From the bottom row we make an exact sequence 0→kerpi→kerpi⊕M →kerpi⊕impi→0,
which shows thatM≤degimpi⊕kerpi.
26 NILS M. NORNES AND STEFFEN OPPERMANN
We can now complete the proof of Theorem 3.
Theorem 3. LetM andN be Λ-modules and letM0⊆ M be a sub- module.
(1) IfM ≤degN, then there exists a submoduleN0⊆N such that M0≤degN0.
(2) IfM≤vdegN, then there exists a submoduleN0⊆N such that M0≤vdegN0.
Proof. Part 1 was proved in Proposition 5, so it remains to prove part 2.
Assume thatM≤vdegN. Then there exists someY ∈mod Λ so that M⊕Y ≤degN ⊕Y. We have a submoduleM0⊆M, and we want to find submodulesN0⊆N andY0⊆Y such thatM0⊕Y0≤degN0⊕Y0. To do so we construct two descending chains of submodulesY =Y1⊇ Y2⊇. . .andN =N1⊇N2⊇. . ., whereM0⊕Yi≤degNi+1⊕Yi+1for alli.
We have thatM0⊕Y ⊆M⊕Y, so by Proposition 5, there exists a submoduleZ1⊆N⊕Y such thatM0⊕Y ≤degZ1. Then by Lemma 6, there exist submodulesN2⊆N andY2⊆Y such thatZ1≤degN2⊕Y2, so we haveM0⊕Y1≤degN2⊕Y2.
Fori >1, assume that we haveM0⊕Yi−1≤degNi⊕YiandYi⊆Yi−1. ThenM0⊕Yi⊆M0⊕Yi−1, and we can again apply Proposition 5 and Lemma 6 to find Ni+1 ⊆ Ni and Yi+1 ⊆ Yi such that M0⊕Yi ≤deg Ni+1⊕Yi+1.
SinceY is artin there is some j such that Yj = Yj−1, so we have M0⊕Yj≤degNj⊕Yj and thusM0≤vdegNj.
For a module M, let SubM denote the set of submodules of M.
The construction in the proof of Proposition 5 induces a functionφη: SubM → SubN. Note that if θ is a different Riedtmann-sequence for the same degeneration, the functionsφη and φθ may be different.
There are several questions that are natural to ask here, for example
• Isφη surjective?
• Is it injective?
• Is the codimension ofM0≤degN0bounded by the codimension ofM ≤degN?
• IfM≤degNis given by a finitely presented functorδ, isM0≤deg N0given by a subfunctor ofδ?
As the following examples show, the answer to each of these questions is in general no.
DEGENERATIONS OF SUBMODULES AND COMPOSITION SERIES 27
Example 7. Letk be a field,Q the Kronecker quiver, Q: 1
α //
β //2,
and consider the path algebra kQ and the kQ-modules given by the quiver representations
I2= k2
( 1 0 )
//
( 0 1 )
//k , S1= k
( 0 )
//
( 0 )
//0,
S2= 0
( 0 )
//
( 0 )//k , R= k
( 1 )
//
( 0 ) //k
DTrS1= k3 (1 0 00 1 0)
//
(0 1 00 0 1)//k2.
We have a degeneration I2 ≤deg R⊕S1 given by a Riedtmann- sequence
η: 0 //R //R⊕I2 //R⊕S1 //0.
Any(1,1)-dimensional regular moduleR0is isomorphic to a submodule ofI2, but whenR06'Rthe only submodule ofR⊕S1 it can degenerate to is the socle. Thus we see thatφη is not injective. On the other hand, there is ak-family of submodules ofR⊕S1 that are isomorphic to R.
But there is only one submodule of I2 that can degenerate to any of these, soφη is not surjective either.
Note also that we have[DTrS1, R⊕S1]−[DTrS1, I2] = 1≤[DTrS1, S1⊕ S2]−[DTrS1, R0] = 3, so if R0≤degS1⊕S2 is given by a functorδ, thenδ can not be a subfunctor of any functor giving the degeneration I2≤degR⊕S1.
In the above example the codimension of the degeneration decreases when we go to the submodules, that is, for modules M ≤deg N and submodulesM0 ≤deg N0 we have codim(M0, N0) ≤codim(M, N). As the next example shows, this does not hold in general.
Example 8. Let k be a field and Λ =k[X]/(X2), let S be the simple Λ-module and letp : ΛS and i:S ,→Λ be the natural projection and inclusion. From the Riedtmann-sequence
η: 0 //S
0 i0
//S⊕Λ2
0 0 1 0p0 1 0 0
//Λ⊕S2 //0
