Matrix factorizations for quantum complete intersections

PETTER ANDREAS BERGH AND KARIN ERDMANN

ABSTRACT. We introduce twisted matrix factorizations for quantum complete intersec- tions of codimension two. For such an algebra, we show that in a given dimension, almost all the indecomposable modules with bounded minimal projective resolutions correspond to such factorizations.

1. INTRODUCTION

In this paper, we study twisted matrix factorizations for four dimensional quantum complete intersections of the form

A=k〈x,y〉/(x²,x y−q y x,y²),

wherekis a field andqis a nonzero element ofk. Namely, for the algebra B=k〈x,y〉/(x²,y²,x y x,y x y),

we consider the homotopy categoryHFact¡

F

^(B),Sν,ηw¢

of twisted matrix factoriza- tions of the elementw=x y−q y x, the twisting being with respect to the automorphism νdefined byx7→ −q⁻¹xandy7→ −q y. By [BJ2], this homotopy category is triangulated in a natural way. It is related to the categories studied in [CCKM], whose twisted ma- trix factorizations are in some sense the complements of the ones we define. Namely, in [CCKM] the factorizations are taken with respect to a regular element, whereas our elementwis actually a socle element in the algebraB. The resulting theories are quite different.

Similarly to the classical commutative case, reduction modulo the elementw in- duces a triangle functor

HFact¡

F

^(B),Sν,ηw¢

−→Kac(

F

^(A))

from this homotopy category of twisted matrix factorizations overBinto the homotopy category of acyclic complexes of free modules over the quantum complete intersection A. The latter is equivalent to the stable module category modA, and so we obtain a triangle functor

HFact¡

F

^(B),S_ν^,ηw

¢−→modA.

The image of a twisted matrix factorization is an A-module with a “twisted two- periodic” minimal free resolution. In particular, its minimal free resolution is bounded.

In our main result, we show that when the fieldkis algebraically closed, then almost all the indecomposable A-modules with bounded minimal projective resolutions are obtained this way, in the following sense. In a given dimension, all except two of the A-modules with such projective resolutions belong to the image of the triangle functor, and thus correspond to twisted matrix factorizations overB.

2010Mathematics Subject Classification. 16D50, 16E05, 16E35, 16G10, 16S80, 18E30.

Key words and phrases. Twisted matrix factorizations, quantum complete intersections.

The first author was partly supported by grant 221893 from the Research Council of Norway.

1

2. PRELIMINARIES

In this section, we recall some of the theory from [BJ2]. Fix an additive category

C

^, an additive automorphismS:

C

→

C

, and a natural transformation 1_C→Swith

ηSM=SηM

for allM∈

C

^{. A (}

C

,S)-factorizationofηis a sequence M ^f

//

_N ^g

//

_SM of objects and morphisms in

C

, satisfying

g◦f =ηM, (S f)◦g=ηN. We denote such a factorization by (M,N,f,g). A morphism

θ: (M1,N1,f1,g1)→(M2,N2,f2,g2)

between factorizations is a pairθ=(ψ,φ) of morphisms in

C

^{, with}ψ:M₁→M₂and φ:N₁→N₂, such that the diagram

M1 f₁

//

ψ

N1 g₁

//

φ

SM1 Sψ

M₂ ^f2

//

N₂ ^g2

//

SM₂

commutes. Such a morphism is an isomorphism if bothψandφare isomorphisms in

C

^.

The collection of all (

C

,S)-factorizations of η, together with the morphisms just described, becomes an additive category Fact(

C

^,S,η). Its homotopy category HFact(

C

^,S,η) has the same objects, but the morphisms are the homotopy equivalence classes [θ] of morphismsθinFact(

C

^,S,η). Here, the notion of homotopy is the standard one. Namely, two morphisms

θ=(ψ,φ) : (M₁,N₁,f₁,g₁)→(M₂,N₂,f₂,g₂) θ⁰=(ψ⁰,φ⁰) : (M₁,N₁,f₁,g₁)→(M₂,N₂,f₂,g₂) inFact(

C

^,S,η) are homotopic if there exist diagonal morphisms

M₁ ^f1

//

ψ⁰ ψ

N₁ ^g1

//

φ⁰ φ

s

xx

SM₁

Sψ⁰ Sψ

t

xx

M2

f₂

//

_N2 ^g2

//

_SM2

satisfying

ψ−ψ⁰ = s◦f₁+(S⁻¹g₂)◦(S⁻¹t) φ−φ⁰ = t◦g1+f2◦s.

Homotopies are compatible with all the operations inFact(

C

^,S,η), thus the homotopy categoryHFact(

C

^,S,η) is also additive.

Given a (

C

^,S)-factorization (M,N,f,g) of η, we obtain a new factorization (N,SM,g,S f) by rotating to the left. The suspensionΣ(M,N,f,g) of (M,N,f,g) is this new factorization with a sign change on the maps, i.e.Σ(M,N,f,g)=(N,SM,−g,−S f):

(M,N,f,g) : M ^f

//

N ^g

//

SM

Σ(M,N,f,g) : N ^−g

//

_SM ^{−S f}

//

_SN

This assignment induces an additive automorphismΣonHFact(

C

^,S,η). Next, consider a morphismθ

M₁ ^f1

//

ψ

N₁ ^g1

//

φ

SM₁

Sψ

M₂ ^f2

//

_N2 ^g2

//

_SM2

of (

C

,S)-factorizations ofη. Its mapping coneC_θis the (

C

,S)-factorization

N1⊕M2

µ−g₁ 0 φ f2

¶

//

_SM1⊕N2

µ−S f1 0 Sψ g2

¶

//

_SN1⊕SM2

ofη. There are natural morphisms

i_θ: (M2,N2,f2,g2)→C_θ, πθ:C_θ→Σ(M1,N1,f1,g1) inFact(

C

^,S,η), and these are used to endow the suspended category¡

HFact(

C

^,S,η),Σ¢ with the structure of a triangulated category. Namely, given a morphism

[θ] : (M₁,N₁,f₁,g₁)→(M₂,N₂,f₂,g₂) inHFact(

C

^,S,η), we declare the triangle

(M₁,N₁,f₁,g₁) ^[θ]

//

_{(M2,N2,f2,g2)} ^[iθ]

//

_Cθ ^[πθ]

//

_{Σ(M1,N1,f1,g1)} inHFact(

C

^,S,η) to be a standard triangle.

Theorem 2.1. [BJ2]Let(

C

^,S)be a suspended additive category,ηa central element, and

∆the collection of all triangles inHFact(

C

^,S,η)isomorphic to a standard triangle. Then

¡HFact(

C

^,S,η),Σ,∆¢

is a triangulated category.

3. MATRIX FACTORIZATIONS

In this section, we establish some general theory of generalized matrix factorizations over arbitrary rings, not necessarily commutative. We shall work with the following setup.

Notation. LetBbe a ring,w∈Ba central element, and denote the factor ringB/(w) by A. Furthermore, letν:B→Bbe an automorphism which fixes the elements of the ideal (w), that is,ν(bw)=bwfor allb∈B.

Note that the automorphismνfixes the elementw, and that it induces an automor- phism on A: we denote also this byν. Now for every leftB-moduleM, consider the twisted module_νM, on whichBacts viaν, i.e.b·m=ν(b)mforb∈Bandm∈νM. The assignmentM7→νMinduces an additive functorS_ν: ModB→ModBon the category of leftB-modules, a functor which acts as the identity on morphisms. This functor is an automorphism with inverseS_ν−1. Now for everyM∈ModB, letηM:M→S_νMbe the multiplication map given bym7→w m. Sinceν(b)w=ν(bw)=bwfor allb∈B, the equalities

ηM(bm)=w bm=ν(b)w m=b·(w m)=b·ηM(m)

hold, showing that the mapηMisB-linear. Moreover, for everyB-linear mapM−→^f Nin ModBthe diagram

M ^f

//

ηM

N

ηN

S_νM ^f=Sνf

//

_SνN

commutes, hence the collection

ηw={ηM|M∈ModB}

forms a natural transformationηw: 1_ModB→S_ν. It is easy to see that the coordinate maps commute with the functorS_ν, that is,ηS_νM=S_νηMfor allM∈ModB: this follows from the fact thatν(w)=w. Hence the natural transformationηwis a central element in the suspended additive category (ModB,S_ν).

We now follow the setup from Section 2. Consider the homotopy category HFact¡

ModB,S_ν,ηw

¢ of (ModB,S_ν)-factorizations of ηw, which is triangulated (in terms of standard triangles) by Theorem 2.1. By definition, its elements are sequences

M ^f

//

_N ^g

//

_νM of leftB-modules andB-linear maps, with

g◦f =ηM, f ◦g=ηN.

This is a noncommutative version of a category of matrix factorizations, and related to the categories studied in [CCKM], one important difference being that our elementwis not regular inB. The following result shows that reduction modulowinduces a triangle functor fromHFact¡

ModB,S_ν,ηw

¢to the homotopy categoryK(ModA) of complexes of leftA-modules.

Proposition 3.1. Let B be a ring, w ∈ B a central element, put B/(w)=A, and let ν:B → B be an automorphism with ν(bw)= bw for all b ∈ B . Furthermore, let S_ν: ModB→ModB be the twisting functor given byν, andηw: 1_ModB→S_νthe natural transformation with coordinatesηM:M→νM given byηM(m)=w m. Then reduction modulo w induces a triangle functor

HFact¡

ModB,S_ν,ηw

¢−→K(ModA) . The image of a factorization(M,N,f,g)is the complex

· · ·

//

_νn−1(N/w N) ^g

//

_νn(M/w M) ^f

//

_νn(N/w N) ^g

//

_νn+1(M/w M)

//

_{· · ·} of left A-modules and maps, with M/w M in degree zero.

Proof. Let (M,N,f,g) be a (ModB,S_ν)-factorization ofηw. Reduction modulowgives a sequence

M/w M ^f

//

N/w N ^g

//

_ν(M/w M)

of leftA-modules and maps. Sinceg◦f =w·1_Mandf◦g=w·1_N, the sequence

· · ·

//

_{νn−1(N/w N)} ^g

//

_νn(M/w M) ^f

//

_νn(N/w N) ^g

//

_{νn+1(M/w M)}

//

_{· · ·} is a complex. A homotopy between morphisms inFact¡

ModM,S_ν,ηw

¢becomes a ho- motopy between complexes after reduction, hence we obtain an additive functor

T:HFact¡

ModB,S_ν,ηw

¢−→K(ModA),

to the homotopy category of complexes of leftA-modules. The triangulated structures on the two categoriesHFact¡

ModB,S_ν,ηw

¢andK(ModA) are analogous: the distin- guished triangles are those that are isomorphic to the standard triangles. Consequently,

the functorTis a triangle functor.

In the next section, we shall mainly be dealing with free modules. The maps are then given by matrices, and the factorizations are noncommutative versions of classical matrix factorizations.

Let

F

(B) be the category of finitely generated free left B-modules. The twist_νF of a free module F is again a free module, hence the twisting functorS_ν: ModB → ModBrestricts to a functorS_ν:

F

^(B)→

F

(B). Similarly, the natural transformation ηw: 1_ModB→S_νrestricts to a transformationηw: 1_F(B)→S_ν, which becomes a central

element in the suspended additive category (

F

^(B),S_ν). As in Proposition 3.1, reduction modulowinduces a triangle functorHFact¡

F

^(B),S_ν^,ηw

¢→K(

F

(A)). We record this in the following result.

Proposition 3.2. Let B be a ring, w∈B a central element, put B/(w)=A, and letν:B→ B be an automorphism withν(bw)=bw for all b∈B . Furthermore, let S_ν:

F

^(B)→

F

^(B)be the twisting functor given byν, andηw: 1_F(B)→S_νthe natural transformation with coordinatesηF:F→νF given byηF(m)=w m. Then reduction modulo w induces a triangle functorHFact¡

F

^(B),S_ν^,ηw

¢→K(

F

^(A)).

4. QUANTUM COMPLETE INTERSECTIONS

In this section, we apply the theory from the previous section to a class of quantum complete intersections. We fix the following notation throughout.

Notation. Letkbe a field andBthe noncommutativek-algebra B=k〈x,y〉/(x²,y²,x y x,y x y).

Furthermore, letqbe a nonzero element ink,wthe quantum commutatorx y−q y xin B, andAthe algebraB/(w). ThusAis the four dimensional quantum complete inter- section

A=k〈x,y〉/(x²,x y−q y x,y²).

Our aim is to show that whenk is algebraically closed, then in a given dimension, almost all - in fact, all except two - of the finitely generated indecomposable left A- modules with bounded-rank minimal free resolutions are obtained from twisted matrix factorizations of the element w overB. By this, we mean matrix factorizations with respect to a functorS which is given by twisting with an algebra automorphism. The twisting automorphismν:B→Bof interest acts on the generators by

ν(x)= −q⁻¹x, ν(y)= −q y,

and is closely related to the Nakayama automorphism onA. Namely, by [Ber, Lemma 3.1], the latter mapsxtoq⁻¹xandytoq y. Note thatν(bw)=bwfor allb∈B, hence we are in the setting from Section 3.

Remark 4.1. The automorphismνwe have just defined is in some sense the only nat- ural one when studying twisted matrix factorizations overB, at least among automor- phismsµ:B→Bgiven byµ(x)=αxandµ(y)=α⁻¹yfor someα∈k. Namely, suppose that

B ^f

//

_B ^g

//

_µB is a rank one factorization inHFact¡

F

^(B),S_µ^,ηw

¢. Then the maps are given by right multiplication with elements inB, and these are of the formc₀+c₁x+c₂y+c₄x y+c₅y x.

Such an element is invertible inBif and only ifc₀is nonzero, and then the factorization is zero in the homotopy categoryHFact¡

F

^(B),S_µ^,ηw

¢. Moreover, the socle elementsx y andy xplay no role in the factorization. Consequently, we may assume that the mapsf andgare given by right multiplication with elementsβ1x+β2yandγ1x+γ2y, respec- tively. The compositionsg◦f andf ◦gboth equalηB, which is given by multiplication withw. It is not hard to see that this forces the scalarsβ1,β2,γ1andγ2to be nonzero, and thatαmust be−q⁻¹. Thusµ=ν.

The canonical way of producing an A-module from a twisted matrix factorization overBis to reduce modulow and then take the image of one of the maps. We shall now show that this assignment is functorial. By Proposition 3.2, reduction modulow induces a triangle functor

HFact¡

F

^(B),Sν,ηw

¢−→K(

F

^{(A)) .}

The following result shows that in our present setting, this functor maps the matrix fac- torizations, that is, the (

F

^(B),S_ν)-factorizations ofηw, toacycliccomplexes of freeA- modules.

Theorem 4.2. Let k be a field and B the k-algebra k〈x,y〉/(x²,y²,x y x,y x y).Take an element06=q∈k, put w=x y−q y x, and consider the quantum complete intersection

A=B/(w)'k〈x,y〉/(x²,x y−q y x,y²).

Furthermore, letνbe the automorphism on B defined byν(x)= −q⁻¹x andν(y)= −q y.

Finally, let

F

^(B)be the category of finitely generated free left B -modules, S_ν:

F

^(B)→

F

^(B)the twisting functor given byν, andηw: 1_F(B)→S_νthe natural transformation with coordinatesηF:F→νF given byηF(m)=w m. Then reduction modulo w induces a triangle functor

HFact¡

F

^(B),S_ν^,ηw

¢−→K_ac(

F

^{(A)) ,}

whereK_ac(

F

^(A))is the homotopy category of acyclic complexes of finitely generated free left A-modules.

Proof. Consider a factorization

F ^f

//

_G ^g

//

_νF inHFact¡

F

^(B),Sν,ηw¢

. Sinceg◦f =ηFandf◦g=ηG, the freeB-modulesFandGare of the same rank, sayr. We may therefore assume thatF=G=B^r, and that the maps are given by multiplying the standard generators ofB^r (that is, the row vectors with a single nonzero entry, the unit ofB) on the right byr×rmatricesC=(c_{i j}) andD=(d_{i j}).

Thus for arbitrary elements the maps are given by

f:u7→uC, g:u⁰7→ν(u⁰)D.

Hereν(u⁰) is the row vector inB^r obtained fromu⁰by applying the automorphismνto all its entries. Note that we may assume that none of the matrices contain a unit: such an entry would imply that the factorization (F,G,f,g) has a direct summand which is zero in the homotopy categoryHFact¡

F

^(B),S_ν^,ηw

¢.

Now take the standard generatorsb₁, . . . ,b_rofB^r. Applyingg◦f to these gives g◦f(b_i) = g(b_iC)

= g à r

X

j=1

ci jbj

!

=

r

X

j=1

ν(c_{i j})b_jD

=

r

X

j=1

ν(ci j) Ã _r

X

k=1

d_{j k}b_k

!

=

r

X

k=1

à r

X

j=1

ν(ci j)d_{j k}

! b_k

= b_iν(C)D,

whereν(C) is the matrix obtained fromC by applying the automorphismνto all its entries. Since the composite mapg◦f is given by multiplying the standard generators on the right by the matrixν(C)D, for an arbitrary element it is given by

g◦f:u7→ν(u)ν(C)D.

Similarly, the matrix for the composite mapS_ν(f)◦gisDC, so that for an arbitrary ele- ment it is given by

S_ν(f)◦g:u⁰7→ν(u⁰)DC.

Now look at the matricesCandD. Since none of their entries are units and the algebra Bhas a basis {1,x,y,x y,y x}, we may decompose the matrices as

C = xC₁+yC₂+x yC₃+y xC₄, D = xD₁+yD₂+x yD₃+y xD₄, where theC_i andD_iarer×rmatrices overk. This gives

ν(C)D = ¡

−q⁻¹xC₁−q yC₂+x yC₃+y xC₄¢ ¡

xD₁+yD₂+x yD₃+y xD₄¢

= −q⁻¹x yC₁D₂−q y xC₂D₁,

which must equal x y I−q y x I, sinceg◦f =ηF and the matrix for this map is (x y− q y x)I, whereIis ther×ridentlty matrix. Consequently, the matricesC₁,C₂,D₁,D₂are invertible.

Letu∈Gbe an element in Kerg. We may writeuasu=u₀+xu₁+yu₂+x yu₃+y xu₄, where theuiare row vectors ink^r. Then

g(u) = ν(u)D

= ¡

u₀−q⁻¹xu₁−q yu₂+x yu₃+y xu₄¢ ¡

xD₁+yD₂+x yD₃+y xD₄¢

= xu₀D₁+yu₀D₂+x yu₀D₃+y xu₀D₄−q⁻¹x yu₁D₂−q y xu₂D₁, and this is zero in_νF. Since every element in_νFcan be written uniquely as a sumu₀⁰+ xu⁰₁+yu⁰₂+x yu⁰₃+y xu₄⁰, where theu⁰_i are row vectors ink^r, we see immediately that u₀D₁=0. ButD₁is invertible, henceu₀=0, and so

0= −q⁻¹x yu₁D₂−q y xu₂D₁

in_νF. This forcesu₁andu₂to vanish as well (u₁because the matrixD₂is also in- vertible), givingu=x yu₃+y xu₄. But thenumust belong to Imf. Namely, since the matricesC₁andC₂are of rankr, there are row vectorsu⁰₁,u₂⁰ ∈k^r withu⁰₁C₁=u₄and u₂⁰C₂=u₃, giving

f(yu⁰₁+xu⁰₂) = ¡

yu₁⁰+xu⁰₂¢ ¡

xC₁+yC₂+x yC₃+y xC₄¢

= y xu₁⁰C₁+x yu⁰₂C₂

= y xu4+x yu3

= u.

This shows that Kerg⊆Imf.

Now take an elementu∈G, and suppose thatu+wGbelongs to Kergin the reduced sequence

F/w F ^f

//

_G/wG ^g

//

_{ν(F/w F)}

Theng(u)=w u=g◦f(u⁰) for someu⁰∈F, givingg(u−f(u⁰))=0. We showed above that Kerg⊆Imf, henceumust belong to Imf. It follows that the elementu+wGbelongs to Imf, so the reduced sequence is exact. Similarly, the sequence

· · ·

//

_νn−1(G/wG) ^g

//

_νn(F/w F) ^f

//

_νn(G/wG) ^g

//

_νn+1(F/w F)

//

_{· · ·} is acyclic, hence the image of the functor is contained inK_ac(

F

^(A)).

The following will be useful later.

Remark 4.3. As shown in the above proof, thexandy“components” of the two maps in a nonzero factorization inHFact¡

F

^(B),Sν,ηw¢

are invertible matrices. Namely, let F ^f

//

_G ^g

//

_νF

be a nonzero rankrfactorization, withf andgdefined in terms ofr×rmatricesCand D, respectively. These matrices decompose as

C = xC1+yC2+x yC3+y xC4, D = xD₁+yD₂+x yD₃+y xD₄,

where theC_i andD_i arer×r matrices overk. The four matricesC₁,C₂,D₁,D₂were shown to be invertible.

Conversely, letC₁andC₂be invertibler×r matrices over k, and consider theB- linear maps f:B^r →B^r andg:B^r →νB^r defined in terms of the matricesxC₁+yC₂ andxC₂⁻¹−q yC₁⁻¹, respectively. Then the matrix for the compositiong◦f is

ν¡

xC₁+yC₂¢ ¡

xC⁻₂¹−q yC₁⁻¹¢

=¡

−q⁻¹xC₁−q yC₂¢ ¡

xC⁻₂¹−q yC₁⁻¹¢

=w I, and for the compositionS_ν(f)◦git is

¡xC₂⁻¹−q yC₁⁻¹¢ ¡

xC₁+yC₂¢

=w I. Thusg◦f =ηB^r =S_ν(f)◦g, and so

B^{r f}

//

_B^{r g}

//

_νB^r is an element ofHFact¡

F

^(B),S_ν^,ηw

¢.

In the commutative case, when the element one factors out is a non-zerodivisor in the ring (andSis the identity), the triangle functor from the homotopy category of ma- trix factorizations to the homotopy category of acyclic complexes over the factor ring is always faithful, as shown in [BJ1, Proposition 3.3]. The following example shows that this is not the case here.

Example. Consider the diagram

B ^·(x+y)

//

_B ^{·(x−q y)}

//

_νB

As in the proof of Theorem 4.2, the composition of the two maps is given by 17→ν(x+y)(x−q y)=(−q⁻¹x−q y)(x−q y)=w.

Moreover, the composition of the two maps

B ^{·(x−q y)}

//

_νB ^Sν(·(x+y))

//

_νB is given by

17→(x−q y)(x+y)=x y−q y x=w, hence the top diagram is an object ofFact¡

F

^(B),S_ν^,ηw

¢. Now the diagram

B ^·(x+y)

//

0

B ^{·(x−q y)}

//

·w

νB

0

B ^·(x+y)

//

_B ^{·(x−q y)}

//

_νB

commutes, and therefore represents a morphism inFact¡

F

^(B),S_ν^,ηw¢

. It is easily seen that this morphism is not nullhomotopic, and so it represents a nonzero morphism in the homotopy categoryHFact¡

F

^(B),S_ν^,ηw

¢. However, this morphism trivially maps to zero by the functorHFact¡

F

^(B),S_ν^,ηw

¢→K_ac(

F

^(A)).

Returning to the general theory, we now show that we can assign modules to twisted matrix factorizations functorially. Namely, since Ais a local selfinjective algebra, the

homotopy category K_ac(

F

(A)) of acyclic complexes of finitely generated free left A- modules is equivalent to the stable module category modAof finitely generated left modules. The canonical equivalence

K_ac(

F

^(A))−→modA

maps an acyclic complex to the image of its degree zero map. The following is therefore an immediate corollary to Theorem 4.2.

Corollary 4.4. Let k be a field and B the k-algebra k〈x,y〉/(x²,y²,x y x,y x y).Take an element06=q∈k, put w=x y−q y x, and consider the quantum complete intersection

A=B/(w)'k〈x,y〉/(x²,x y−q y x,y²).

Furthermore, letνbe the automorphism on B defined byν(x)= −q⁻¹x andν(y)= −q y.

Finally, let

F

^(B)be the category of finitely generated free left B -modules, S_ν:

F

^(B)→

F

^(B)the twisting functor given byν, andηw: 1_F(B)→S_νthe natural transformation with coordinatesηF:F→νF given byηF(m)=w m. Then reduction modulo w induces a triangle functor

T:HFact¡

F

^(B),S_ν^,ηw

¢−→modA,

wheremodA is the stable module category of finitely generated left A-modules. The image of a factorization(F,G,f,g)inHFact¡

F

^(B),S_ν^,ηw

¢is the image of the map F/w F −→^f G/wG.

As mentioned, we shall show that whenk is algebraically closed, then in a given dimension, almost all the finitely generated indecomposable left A-modules with bounded minimal projective resolutions are obtained from twisted matrix factoriza- tions of the elementw overB. To do this, we use the classification of the indecom- posableAmodules. Recall first that thecomplexityof a finitely generated leftA-module Mwith minimal projective resolution

· · · →Q₂→Q₁→Q₀→M→0 is defined as

cx_AM^def=inf{t∈N∪{0}| ∃a∈Rsuch that dim_kQ_n≤an^t−1fornÀ0}.

The modules of complexity one are precisely the ones with bounded-rank minimal free resolutions. In [BGMS], a minimal projective bimodule resolution

PA: · · · →P₂→P₁→P₀→A→0

ofAwas constructed, with dim_kP_n=16(n+1). Now ifMis a finitely generated leftA- module, then the complexPA⊗AMis acyclic, and is therefore a projective resolution of M. It may no longer be minimal, but its rate of growth is at most that ofPA. Thus the complexity of anyA-module is at most two.

Note that if the parameterqis not a root of unity, then it was shown in [Sch, Propo- sition 4.1] that there exist non-periodicA-modules of complexity one: the acyclic com- plex

· · · →A ^·(x+(−q)ⁿ⁺¹y)

−−−−−−−−−−→A^·(^x+(−q)ⁿy)

−−−−−−−−→A^·(^x+(−q)ⁿ⁻¹y)

−−−−−−−−−−→A→ · · ·

gives rise to one class of such modules. However, ifq isa root of unity, then by [Sch, Proposition 4.1] again there are no suchA-modules: in this case, the indecomposable modules of complexity one are precisely the periodic ones.

The classification of the indecomposableA-modules goes back to Kronecker. What follows is based on [Sch, Section 4], which again is based on [Baš, Co1, Co2, HeR]. It classifies the modules according to their complexities.

Fact 4.5. (1) Since complexity zero is the same as finite projective dimension, there is only one such indecomposable, namelyAitself. The modules of complexity two, that is, the ones having unbounded (but linearly growing) minimal projective resolutions, are the cokernels of the maps in the minimal complete resolution of the modulekover the algebrak[x,y](x²,y²). These modules are all of odd dimension overk. To see this, take any indecomposableA-module which is not projective. AsAis injective, such a module is annihilated byx yand hence it is a module for the algebraA/(x y), which is indepen- dent ofq. As described in [Sch], for eachn≥1 there are precisely two indecomposable modules of dimension 2n+1 (and ifn=0, justk). Moreover, it is shown that when work- ing with the algebrak[x,y]/(x²,y²), all these modules are kernels in a minimal complete resolution ofk. When working with the algebraA, the same holds. One only needs to check that the syzygy with respect toAof an odd-dimensional indecomposable module is the same as the syzygy with respect tok[x,y]/(x²,y²).

(2) The modules we are interested in are the ones of complexity one, that is, the ones with bounded-rank minimal free resolutions. These are all of even dimension overk, andin what follows we assume that k is algebraically closed. For eachn≥1, the inde- composableA-modules of dimension 2nare

{Cn(λ)|λ∈k∪{∞}},

and these are described as follows. Forλ∈k, denote byJ_n(λ) then×nJordan matrix





















λ 1 0 · · · 0 0 0 λ 1 · · · 0 0 ... ... ... . .. ... ...

... ... ... . .. ... ...

0 0 0 · · · λ 1 0 0 0 · · · 0 λ





















with eigenvalueλ. The underlying vector space ofCn(λ) isk²ⁿ, with the action ofxand ygiven by the 2n×2nblock matrices

x 7→

µ 0 Jn(λ)

0 0

¶

y 7→

µ 0 I_n

0 0

¶

whereI_nis then×nidentity matrix. The moduleC_n(∞) has the same underlying vector space, with the action ofxandygiven by

x 7→

µ 0 I_n

0 0

¶

y 7→

µ 0 J_n(0)

0 0

¶

Note that every one of these modules hasnindependent generators.

We are now ready to state and prove the main result.

Theorem 4.6. Let k be an algebraically closed field, q∈k a nonzero element, and A the quantum complete intersection

A=k〈x,y〉/(x²,x y−q y x,y²).

Furthermore, let B,w,νand the triangle functor T:HFact¡

F

^(B),S_ν^,ηw

¢−→modA be as inCorollary 4.4, and

{C_n(λ)|n∈N,λ∈k∪{∞}}

the indecomposable left A-modules with bounded projective resolutions. Then C_n(λ) is in the image of T if and only if λ∉{0,∞}. Thus, for06=λ∈k there is a factoriza- tion(F,G,f,g)∈HFact¡

F

^(B),S_ν^,ηw

¢with C_n(λ)isomorphic to the image of the map F/w F−→^f G/wG, whereas for C_n(0)and C_n(∞)there are no such factorizations.

Proof. We start by proving the following. Let

B^{n f}

//

B^{n g}

//

_νBⁿ be a ranknfactorization inHFact¡

F

^(B),Sν,ηw¢

, and suppose that the matrix forf is of the formx I+yC+x yC⁰+y xC⁰⁰, withCinvertible (the matricesI,C,C⁰,C⁰⁰aren×n matrices overk, withIthe identity matrix). We claim that the image of the mapF/w F−→^f G/wGis isomorphic to the leftA-module with underlying vector spacek²ⁿ, and with the action ofxandygiven by

x 7→

µ 0 qC

0 0

¶

y 7→

µ 0 I 0 0

¶

To see this, note that the image of the map f is generated as aB-module by then rowsζ1, . . . ,ζn of the matrixx I+yC+x yC⁰+y xC⁰⁰. We describe the action ofxandy on these generators, in terms of the standard generatorsb₁, . . . ,b_nofBⁿ, whereb_iis the row vector with a single nonzero entry, the unit ofB. First, sincey²=0=y x yinB, we see thaty¡

x I+yC+x yC⁰+y xC⁰⁰¢

=y x I, and so yζi=y xb_i. Similarly, sincex²=0=x y x, we obtainx¡

x I+yC+x yC⁰+y xC⁰⁰¢

=x yC, hence xζi=x y

n

X

j=1

c_{i j}b_j,

whereC=(c_{i j}). Rewritingx yasq y x+w, we obtain xζi = (q y x+w)

n

X

j=1

c_{i j}b_j

=

n

X

j=1

qc_{i j}(y xb_j)+w

n

X

j=1

c_{i j}b_j

=

n

X

j=1

qci j(yζj)+w

n

X

j=1

ci jbj,

where we have used the equalityyζj=y xb_j from above. All this shows that when we factor outw, then the image of the mapf is spanned as a vector space by the cosets of

ζ1, . . . ,ζn,yζ1, . . . ,yζn.

The equalityyζi =y xb_i givesx(yζi)=0=y(yζi, and using it once more we see that the 2ncosets are linearly independent. The action ofxandyon the vector space they generate are given by the two matrices, hence the claim follows.

Having proved the claim, letλbe a nonzero element ink. Then then×n Jordan matrixJ_n(λ) is invertible, and so by Remark 4.3 there exists a factorization

B^{n f}

//

_B^{n g}

//

_νBⁿ

inHFact¡

F

^(B),S_ν^,ηw

¢withx I+y q⁻¹J_n(λ) as the matrix for the mapf. By the claim above, the image of the mapf has underllying vector spacek²ⁿ, and with the action of xandygiven by

x 7→

µ 0 J_n(λ)

0 0

¶

y 7→

µ 0 I 0 0

¶

This is precisely theA-moduleC_n(λ).

Finally, we show that the A-modulesCn(0) andCn(∞) cannot be obtained from twisted matrix factorizations overB, for anyn. Namely, take any nonzero rankn fac- torization

F ^f

//

_G ^g

//

_νF inHFact¡

F

^(B),S_ν,ηw¢

. By Remark 4.3, the matrix forf is of the formxC₁+yC₂+x yC₃+ y xC₄, where theC_i aren×nmatrices overk, and withC₁andC₂invertible. Consider the factorization (F,G,C₁⁻¹f,gC₁), in which the matrix for the mapC₁⁻¹f is that for f multiplied on the left withC₁⁻¹, whereas the matrix forgC₁is that forg multiplied on the right withC₁. The diagram

F ^f

//

·C1

G ^g

//

·I

νF

·C1

F ^C

−11 f

//

_G ^gC1

//

_νF

shows that the two factorizations are isomorphic inHFact¡

F

^(B),S_ν^,ηw

¢, and they are therefore mapped by the functorT to isomorphicA-modules. The matrix for the map C₁⁻¹f isx+yC⁻¹₁ C₂+x yC₁⁻¹C₃+y xC₁⁻¹C₄, hence from the claim we proved the image of these factorizations underT is isomorphic to the followingA-module: the underlying vector space isk²ⁿ, and the action ofxandyare given by

x 7→

µ 0 qC₁⁻¹C₂(λ)

0 0

¶

y 7→

µ 0 I 0 0

¶

Both these matrices have rankn. However, from Fact 4.5 we see that forC_n(0), the matrix that defines the action ofxhas rankn−1, as does the matrix that defines the action of y onC_n(∞). Namely, these actions are given by 2n×2nblock matrices with J_n(0) as the nonzeron×n block. This shows that theA-modulesC_n(0) andC_n(∞) cannot be

obtained from twisted matrix factorizations overB.

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PETTERANDREASBERGH, INSTITUTT FOR MATEMATISKE FAG, NTNU, N-7491 TRONDHEIM, NORWAY E-mail address:[email protected]

KARINERDMANN, MATHEMATICALINSTITUTE, UNIVERSITY OFOXFORD, ANDREWWILESBUILDING, RAD- CLIFFEOBSERVATORYQUARTER, WOODSTOCKROAD, OXFORD, OX2 6GG, UNITEDKINGDOM

E-mail address:[email protected]