Geometry of noncommutative algebras

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Geometry of noncommutative algebras

Eivind Eriksen

BI Norwegian Business School

Arvid Siqveland

Buskerud University College

This is the author’s final, accepted and refereed manuscript to the article published in

Banach Center Publications, 93(2011): 69-82

D

The publisher, Institute of Mathematics of the Polish Academy of Sciences, allows the author to retain rights to “upload his last version of the article

on the BI Brage archive”. (Publisher’s policy 2012).

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 2011

GEOMETRY OF NONCOMMUTATIVE ALGEBRAS

EIVIND ERIKSEN

BI Norwegian School of Management, N-0442 Oslo, Norway E-mail: [email protected]

ARVID SIQVELAND

Buskerud University College, P.O. Box 235, N-3603 Kongsberg, Norway E-mail: [email protected]

Abstract. There has been several attempts to generalize commutative algebraic geometry to the noncommutative situation. Localizations with good properties rarely exist for noncommutative algebras, and this makes a direct generalization difficult. Our point of view, following Laudal, is that the points of the noncommutative geometry should be represented as simple modules, and that noncommutative deformations should be used to obtain a suitable localization in the noncommutative situation.

Let A be an algebra over an algebraically closed field k. IfA is commutative and finitely generated over k, then any simple A-module has the form M = A/m, the residue field, for a maximal ideal m ⊆ A, and the commutative deformation functor DefM has formal moduli Aˆm. In the general case, we may replace theA-moduleA/mwith the simpleA-moduleM, and use the formal moduli of the commutative deformation functorDefM as a replacement for the complete local ring Aˆm. We recall the construction of the commutative scheme simp(A), with points in bijective correspondence with the simpleA-modules of finite dimension overk, and with complete local ring at a point M isomorphic to the formal moduli of the corresponding simple moduleM.

The schemesimp(A)has good properties, in particular when there are no infinitesimal rela- tions between different points, i.e. whenExt¹_A(M, M⁰) = 0for all pairs of non-isomorphic simple A-modules M, M⁰. It does not, however, characterizeA. We use noncommutative deformation theory to define localizations, in general.

We consider the quantum plane, given byA=khx, yi/(xy−qyx), as an example. This is an Artin-Schelter algebra of dimension two.

2010Mathematics Subject Classification: Primary 14A22; Secondary 16G20.

Key words and phrases: noncommutative algebraic geometry, simple modules.

The paper is in final form and no version of it will be published elsewhere.

[1]

1. Noncommutative deformations of modules. Let k be a field. For any integer r≥ 1, we consider the category ar of r-pointed Artinian k-algebras. We recall that an object ina_ris an Artinian ringR, together with a pair of structural ring homomorphisms f : k^r → R and g : R → k^r with g◦f = id, such that the radical I(R) = ker(g) is nilpotent. The morphisms of ar are the ring homomorphisms that commute with the structural morphisms. It follows from this definition thatI(R)is the Jacobson radical of R, and therefore that the simple rightR-modules are the projections{k1, . . . , kr}ofk^r. Let A be an associative k-algebra. For any family M = {M1, . . . , Mr} of right A- modules, there is a noncommutative deformation functorDef_M:ar→Sets, introduced in Laudal [4]; see also Eriksen [2]. For an algebraR in a_r, we recall that a deformation ofMoverRis a pair (MR,{τi}_1≤i≤r), where MR is anR-Abimodule (on whichkacts centrally) that isR-flat, andτ_i :k_i⊗_RM_R→M_i is an isomorphism of rightA-modules for1≤i≤r. Moreover,(MR,{τi})and(M_R⁰,{τ_i⁰}) are equivalent deformations overR if there is an isomorphismη : M_R →M_R⁰ of R-A bimodules such that τ_i =τ_i⁰◦(1⊗η) for1≤i≤r. One may prove thatMR isR-flat if and only if

M_R∼= (R_ij⊗kM_j) =











R11⊗kM1 R12⊗kM2 . . . R1r⊗kMr

R₂₁⊗kM₁ R₂₂⊗kM₂ . . . R_2r⊗kM_r ... ... . .. . . . R_r1⊗_kM₁ R_r2⊗_kM₂ . . . R_rr⊗_kM_r











considered as a left R-module, and a deformation in Def_M(R) may be thought of as a right multiplication A → EndR(MR) of A on the left R-module MR that lifts the multiplicationρ:A→ ⊕i Endk(Mi)ofAon the familyM.

Let us assume thatM is aswarm, i.e. that Ext¹_A(Mi, Mj) has finite dimension over kfor1 ≤i, j≤r. ThenDef_M has a pro-representing hull or a formal moduli(H, MH), see Laudal [4], Theorem 3.1. This means thatH is a completer-pointedk-algebra in the pro-categoryˆar, and thatMH∈Def_M(H)is a family defined overH with the following versal property: For any algebraRina_rand any deformationM_R∈Def_M(R), there is a morphismφ:H →Rinˆarsuch thatDef_M(φ)(MH) =MR. The formal moduli(H, MH) is unique up to non-canonical isomorphism. However, the morphismφ is not uniquely determined by(R, MR).

WhenM is a swarm with formal moduli(H, M_H), right multiplication on theH-A bimoduleMH by elements inAdetermines an algebra homomorphism

η:A→End_H(M_H)

We write O^A(M) = End_H(M_H) and call it the algebra of observables. Since M_H is H-flat, we have that EndH(MH)∼= (Hij⊗bkHomk(Mi, Mj)), and it follows that O^A(M) is explicitly given as the matrix algebra











H11⊗bkEndk(M1) H12⊗bkHomk(M1, M2) . . . H1r⊗bkHomk(M1, Mr) H21⊗bkHomk(M2, M1) H22⊗bkEndk(M2) . . . H2r⊗bkHomk(M2, Mr)

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

Hr1⊗bkHomk(Mr, M1) Hr2⊗bkHomk(Mr, M2) . . . Hrr⊗bkEndk(Mr)











Let us writeρi :A →Endk(Mi)for the structural algebra homomorphism defining the

rightA-module structure onM_i for1≤i≤r, and ρ:A→ ⊕

1≤i≤r

End_k(M_i)

for their direct sum. Since H is a complete r-pointed algebra in ˆa_r, there is a natural morphismH →k^r, inducing an algebra homomorphism

π:O^A(M)→ ⊕

1≤i≤rEndk(Mi)

By construction, there is a right action ofO^A(M)on the family Mextending the right action ofA, in the sense that the diagram

A ^η //

ρJJJJJJ%%

JJ JJ

J O^A(M)

π

⊕

1≤i≤r

Endk(Mi)

commutes.

Lemma 1.1. Let f : A → B be an algebra homomorphism, and let M be a swarm of rightB-modules. IfMis a swarm of rightA-modules viaf, then then there is a natural algebra homomorphismO^A(M)→ O^B(M)such that the diagram

A ^ηA //

f

O^A(M)

B _η

B

//O^B(M)

commutes.

Proof. Let(H^A, M_HA)be the formal moduli ofDef^A_M, the noncommutative deformation functor ofMconsidered as a family of rightA-modules, and let(H^B, M_HB)be the formal moduli of Def^B_M, the noncommutative deformation functor ofMconsidered as a family of rightB-modules. SinceM_HB ∈Def^A_M(H^B)is also a lifting ofA-modules toH^B, there is a natural morphism H^A → H^B by the versal property of H^A, and hence a natural morphismO^A(M)→ O^B(M).

2. Laudal’s Generalized Burnside Theorem. LetAbe a finite-dimensional algebra over a fieldk. Then the simple right modules overAare the simple right modules over the semi-simple quotient algebra A/rad(A), where rad(A) is the Jacobson radical ofA. By the classification theory for semi-simple algebras, it follows that there are finitely many non-isomorphic simple rightA-modules.

We consider the noncommutative deformation functorDef_M:a_r→Setsof the family M={M1, M2, . . . , Mr}of simple rightA-modules. Clearly,Mis a swarm, henceDefM

has a formal moduli(H, M_H), and we consider the commutative diagram A ^η //

ρJJJJJJ%%

JJ JJ

J O^A(M)

π

⊕

1≤i≤r

End_k(M_i)

By a classical result, due to Burnside, the algebra homomorphismρis surjective whenk is algebraically closed. This result may be stated in the following form:

Theorem 2.1 (Burnside’s Theorem). If EndA(Mi) =kfor 1≤i≤r, then ρ is surjec- tive. In particular,ρis surjective when k is algebraically closed.

Proof. Consider the factorizationA→A/rad(A)→ ⊕i Endk(Mi)ofρ. IfEndA(Mi) =k for1 ≤ i ≤r, then A/rad(A)→ ⊕i End_k(M_i) is an isomorphism by the classification theory for semi-simple algebras. SinceEndA(Mi) is a division ring of finite dimension overk, it is clear thatEnd_A(M_i) =kwheneverk is algebraically closed.

Let us writeρ:A/radA→ ⊕iEndk(Mi)for the algebra homomorphism induced by ρ. We observe that ρ is surjective if and only if ρ is an isomorphism. Moreover, let us writeJ = rad(O^A(M))for the Jacobson radical ofO^A(M). Then we see that

J= (rad(H)ij⊗bkHomk(Mi, Mj)) = ker(π)

Sinceρ(radA) = 0 by definition, it follows thatη(radA)⊆J. Hence there are induced morphisms

gr(η)_q : rad(A)^q/rad(A)^q+1→J^q/J^q+1

for all q ≥ 0. We may identify gr(η)0 with ρ, since O^A(M)/J ∼= ⊕i Endk(Mi). The conclusion in Burnside’s Theorem is therefore equivalent to the statement thatgr(η)₀ is an isomorphism.

Theorem2.2 (Laudal’s Generalized Burnside Theorem). Let Abe a finite-dimensional algebra over a field k, and let M = {M1, M2, . . . , Mr} be the family of simple right A-modules. IfEndA(Mi) =k for1≤i≤r, thenη:A→ O^A(M)is an isomorphism. In particular,η is an isomorphism whenk is algebraically closed.

Proof. Since A and O^A(M) are complete in the rad(A)-adic and J-adic topologies, it follows that η is surjective if A → O^A(M)/J² is surjective. It is therefore enough to prove that η is injective and that gr(η)q is an isomorphism for q = 0 and q = 1. By Burnside’s Theorem, we know thatgr(η)₀is an isomorphism. To prove thatηis injective, let us consider the kernel ker(η) ⊆ A. It is determined by the obstruction calculus of Def_M; see Laudal [4], Theorem 3.2 for details. When A is finite-dimensional, the right regularA-moduleAA has a decomposition series

0 =F0⊆F1⊆ · · · ⊆Fn =AA

withF_p/F_p−1a simple rightA-module for1≤p≤n. That is,A_Ais aniterated extension of the modules inM. This implies thatηis injective; see Laudal [4], Corollary 3.1. Finally, we must prove thatgr(η)₁: rad(A)/rad(A)²→J/J²is an isomorphism. This follows from the Wedderburn-Malcev Theorem; see Laudal [4], Theorem 3.4 for details.

3. Properties of the algebra of observables. LetA be a finitely generated algebra over a field k, and let M = {M1, . . . , Mr} be any family of right A-modules of finite dimension over k. Even though A may be non-Noetherian, and it may be difficult to computeExt¹_A(Mi, Mj)using free resolutions, we may show the following result:

Lemma3.1. IfAis a finitely generatedk-algebra andMis a family of finite-dimensional right A-modules, thenMis a swarm.

Proof. We have that Ext¹_A(Mi, Mj) ∼= HH¹(A,Homk(Mi, Mj)) for 1 ≤ i, j ≤ r. The Hochschild cohomology group is given by

HH¹(A,Homk(Mi, Mj)) = Derk(A,Homk(Mi, Mj))/Innerk(A,Homk(Mi, Mj)) where we write Innerk(A,Homk(Mi, Mj)) for the inner derivations ofA with values in Homk(Mi, Mj). Since a derivation is determined by its values on a set algebra generators anddim_kHom_k(M_i, M_j)<∞, it follows thatExt¹_A(M_i, M_j)has finite dimension overk for1≤i, j≤r.

Hence Def_M has a formal moduli (M, M_H), and hence we may consider the algebra B=O^A(M)of observables. It is clear that

B/rad(B)∼=⊕

i

End_k(M_i)

is semi-simple, with M as the set of simple modules, so M is the family of simple right B-modules. In fact, it follows from the proof of Lemma 3.1 thatM is a swarm of B-modules, since a derivation on a power series algebra in a finite number of variables {x1, . . . , xm}is determined by its values onxi for1≤i≤m.

Proposition 3.2. If k is an algebraically closed field, then the algebra homomorphism η_B :B→ O^B(M)is an isomorphism.

Proof. Since Mis a swarm of A-modules and ofB-modules, we may consider the com- mutative diagram

A ^η

A //

η_A

O^A(M) =B

η_B

B _ηB //

O^B(M) =C

Bn η_Bn //O^Bn(M)

for alln≥1, whereB_n =B/(radB)ⁿ. By Laudal’s Generalized Burnside Theorem,η_Bn is an isomorphism for alln≥1. Since any deformation of the familyM, as B-module is also a deformation asBn-module, for somen, it follows thatC is the projective limit of O^Bn(M), hence the algebra homomorphismηB is also an isomorphism.

In particular, the proposition implies that the assignment (A,M) 7→ (B,M) is a closure operation whenkis algebraically closed. In other words, the algebraB=O^A(M) has the following properties:

1. The familyMis the family of simpleB-modules.

2. The family M has the same module-theoretic properties, in terms of extensions, higher extensions and Massey products, considered as a family of modules overB as overA.

Moreover, these properties characterize the algebraB=O^A(M), and make it natural to call it the algebra of observables.

4. Finite dimensional simple representations. Letkbe an algebraically closed field, and letAbe a finitely generatedk-algebra. We denote bysimp_n(A)the set of isomorphism classes of simple rightA-modulesM of dimensiondimkM =n, and by

simp(A) = [

n≥1

simp_n(A)

the set of isomorphism classes of simple rightA-modules of finite dimension over k.

Letρ:A→End_k(M)be the structure morphism of a simple moduleM ∈simp_n(A), and let mM = ker(ρ) be the corresponding primitive ideal. It follows from Burnside’s Theorem thatm_M ⊆Ais a maximal ideal. We define the radical

rad(A)^∞= \

M∈simp(A) m≥1

m^m_M

and say thatAisgeometric ifrad(A)^∞= 0.

Example4.1. The free associativek-algebraA=khx1, x2, . . . , xdiis geometric. On the other hand, a simplek-algebra A is geometric only if it has finite dimension overk. In particular, the first Weyl algebraA1(k)is not geometric.

5. The commutative scheme structure. Letk be an algebraically closed field, and let A be a finitely generated geometric k-algebra. In this section, we shall discuss the commutative scheme structure ofsimp_n(A).

For any integer n ≥ 1, let I(n) ⊆A be the ideal generated by the n-commutators {[a1, a₂, . . . , a_2n] :a₁, a₂, . . . , a_2n∈A}. We recall that thatn-commutators are given by

[a1, a2, . . . , a2n] = X

σ∈S2n

sgn(σ)·aσ(1)aσ(2)· · ·aσ(2n)

for any sequencea1, a2, . . . , a2n ∈A. We defineA(n) =A/I(n)to be the corresponding factor algebra. For anyM ∈simp_n(A), there is a factorizationA→A(n)→Endk(M)of the structure morphism; see Formanek [3]. Hence anyM ∈simp_n(A)can be considered as a simple rightA(n)-module in a natural way.

Lemma5.1. LetMbe a finite subset ofsimpA, letr=∩M∈MmM, and writeJ for the Jacobson radical ofO^A(M). Then the algebra homomorphism

A/r^m→ O^A(M)/J^m induced byη:A→ O^A(M)is surjective for allm≥2.

Proof. SinceMis a swarm, we may consider the algebra homomorphismη:A→ O^A(M) and the induced homomorphismη:A/r^m→ O^A(M)/J^m. We letB=A/r^m, which is a

finite dimensionalk-algebra, and consider the natural commutative diagram A

η//O^A(M)

&&MMMMMMMMMMM

B ^' //

η

77O^B(M) ^α //O^A(M)/J^m

Since M is the family of simple rightB-modules, it follows from Laudal’s Generalized Burnside Theorem that ηB : B → O^B(M) is an isomorphism. The obvious homomor- phism,η induces an algebra morphismα:O^B(M)→ O^A(M)/J^mthat commutes with O^A(M)→ O^A(M)/J^m, and it follows thatαis surjective.

Proposition 5.2. Let M, N ∈simp(A) be non-isomorphic simple left modules, and let r=mM ∩mN. Then we have

1. Ext¹_A(M, N)∼= Ext¹_A/rm(M, N)for allm≥1, 2. Ext¹_A(M, M)∼= Ext¹_A(n)(M, M)withn= dimkM.

Proof. The first part follows from Lemma 5.1 applied to M={M, N} for m= 2. For the second part, notice that any derivation maps a standard n-commutator into a sum of standardn-commutators. Hence any derivation of Awith values in End_k(M)factors throughA(n).

Example5.3. We remark that we may well have thatExt¹_A(M, N)andExt¹_A(n)(M, N) are non-isomorphic whenM, N insimp_n(A)are non-isomorphic simple modules. Consider for example the algebraA with quotientA(1), given by

A=

k[x] k[x]

0 k[x]

and A(1) =

k[x] 0 0 k[x]

We see that Ext¹_A(M, N) ∼= k and Ext¹_A(1)(M, N) = 0 when M = k[x]/(x)⊕0 and N = 0⊕k[x]/(x). On the other hand, we have thatExt¹_A(M, M)∼= Ext¹_A(1)(M, M)∼=k andExt¹_A(N, N)∼= Ext¹_A(1)(N, N)∼=k.

Lemma5.4. LetRbe anyk-algebra. IfR⊗kEnd_k(V)satisfy the standardn-commutator relations for a vector space V of dimensionn, then Ris commutative.

Proof. Letr₁, r₂∈R, and consider the element given by ([r1, r2]e11)e12e22e23e33. . . e_n−1,nen,n

Since this is a standardn-commutator inR⊗kEndk(V), it follows thatRis commutative if alln-commutators vanish.

Lemma 5.5. Let M ={M1, . . . , Mr} be a finite subset of simp_n(A). We may consider Mas a family of simple A(n)-modules, and we have that

1. Ext¹_A(n)(Mi, Mj) = 0 for1≤i6=j≤r

2. H^A(n)(Mi)∼=H^A(Mi)^comm=H^A(Mi)(1)for1≤i≤r

In particular, the pro-representing hull

H^A(n)(M) =











H^A(M₁)^comm 0 . . . 0 0 H^A(M2)^comm . . . 0

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

0 0 . . . H^A(M_r)^comm











is commutative.

Proof. Letsnbe a standardn-commutator inO^A(n)(M)∼=Mn(H^A(n)(M)). The algebra homomorphismη_m : A(n)/r^m → O^A(n)(M)/J^m induced by η : A(n) → O^A(n)(M) is surjective for allm≥2by Lemma 5.1. This implies that sn= 0 modJ^mfor allm≥2, hence sn = 0, and it follows from Lemma 5.4 that H^A(n) is commutative. To prove that the commutativization H^A(M_i)^comm ∼= H^A(n)(M_i), we consider the commutative diagram

A

**VVVVVVVVVVVVVVVVVVVVV

Z(A(n)) //

A(n)

α

H^A(M)⊗kEndk(M)

ttiiiiiiiiiiiiiiiii

H^A(M)^comm //H^A(M)^comm⊗kEndk(M)

whereM =MiandZ(A(n))is the center ofA(n). The existence ofαis a consequence of the fact thatI(n)⊆A maps to zero inH^A(M)^comm⊗_kEnd_k(M)∼= Mn(H^A(M)^comm).

We note that there are natural morphisms of formal moduli

H^A(M)→H^A(n)(M)→H^A(M)^comm→H^A(n)(M)^comm

SinceH^A(n)(M)is commutative, the composition

H^A(n)(M)→H^A(M)^comm→H^A(n)(M)^comm

must be an isomorphism. By Proposition 5.2, the tangent spaces of H^A(n)(M) and H^A(M)are isomorphic, and this proves thatH^A(M)^comm∼=H^A(n)(M)

Corollary5.6. Let A=khx1, x2, . . . , xdi, and letM ∈simp_n(A).Then H^A(M)^comm∼=H^A(n)(M)∼=k[[t1, t2, . . . , t_(d−1)n2+1]]

Proof. See Corollary 5.11 in Procesi [6].

In general, the familysimp_n(A)is, of course, not finite, and we consider the projec- tive limitη(n) of the algebra homomorphisms η(M) :A(n) → O^A(n)(M)for all finite subfamiliesM ⊆simp_n(A). By Lemma 5.5,η(n)is given by

η(n) :A(n)→ Y

M∈simp_n(A)

H^A(n)(M)⊗kEndk(M)

We shall writeO(n) = Imη(n)for the image ofη(n). We remark thatη(n)is not injective in general; see Example 11 in Laudal [5] for a counter-example.

Let us choose ak-linear base ofM for anyM ∈simp_n(A), and use this base to obtain an identificationEndk(M)∼= Mn(k). The codomain ofη(n)is given by

Y

M∈simp_n(A)

H^A(n)(M)⊗kEndk(M)∼= Mn(B(n))

whereB(n)is the commutativek-algebra B(n) = Y

M∈simp_n(A)

H^A(n)(M)

Let{x1, x2, . . . , xd}be a set of generators ofAas ak-algebra, and consider their images η(n)(x_i) = (x^p,q_i )∈M_n(B(n))for1≤i≤d. SinceB(n)is commutative, the subalgebra C(n) ⊆ B(n) generated by the elements {x^p,q_i : 1 ≤ i ≤ d, 1 ≤ p, q ≤ n} ⊆ B(n) is commutative. We notice that there are natural inclusionsO(n)⊆Mn(C(n))⊆Mn(B(n)).

Hence there is a natural composition ofk-algebra homomorphisms αM : Mn(C(n))→Mn(H^A(n)(M))→Mn(k)

for anyM ∈simp_n(A), and therefore an induced composition of algebra homomorphisms of the centers

Z(α_M) :C(n)→H^A(n)(M)→k

It follows that there is a natural set-theoretic injective mapt: simp_n(A)→Max(C(n)), defined byt(M) = ker(Z(αM)), whereMax(C(n)) denotes the set of maximal ideals of C(n).

Proposition 5.7. For allM ∈simp_n(A), there is a natural isomorphism C(n)b _t(M)∼=H^A(n)(M)

Proof. The algebra homomorphismηM :A(n)→H^A(n)(M)⊗kEndk(M)is topologically surjective for any M ∈ simp_n(A)by Lemma 5.1. This means that we have a surjective homomorphism

C(n)b _t(M)→H^A(n)(M)

By the versal property of H^A(n), there is a homomorphismH^A(n)(M)→C(n)b t(M)that composed with the former gives an automorphism of H^A(n)(M), and this implies that H^A(n)(M)→C(n)b _t(M)is injective. Let mM =t(M)∈Max(C(n)). Since

Mn(C(n))⊆ Y

M∈simp_n(A)

H^A(n)(M)⊗kEndk(V)

it follows that the finite dimensionalk-algebraM_n(C(n)/m²_V)sits in a finite dimensional quotient of some

Y

M∈M

H^A(n)(M)⊗kEnd_k(M)

whereM ⊆simp_n(A)is a finite subset. However, the homomorphism A(n)→ Y

M∈M

H^A(n)(M)⊗_kEnd_k(M)

is topologically surjective by Lemma 5.1. Hence the morphismA(n)→Mn(C(n)/m²_M)is surjective, and this implies thatH^A(n)(M)→C(n)b _t(M) is surjective.

Theorem5.8. For anyM ∈simp_n(A), there exists a Zariski-open neighbourhoodU_M of t(M)in Max(C(n))such that any maximal ideal m∈UM is the image m=t(N) =mN

of a unique simple moduleN∈simp_n(A). LetU(n)⊆Max(C(n))be the open subscheme U(n) = [

M∈simp_n(A)

UM

Then O(n) defines a noncommutative structure sheaf O(n) = O_simp

n(A) of Azumaya algebras on the topological spacesimp_n(A)with the Jacobson topology. The center S(n) ofO(n)defines a scheme structure onsimp_n(A), and there is a morphism of schemes

κ:U(n)→simp_n(A) such thatS(n)b _κ(M)∼=H^A(n)(M) for allM ∈simp_n(A).

Proof. Letρ:A→Mn(k)be the structure homomorphism ofM ∈simp_n(A). We write e_ij ∈ M_n(k) for the elementary matrices, and pick y_ij ∈ A such that ρ(y_ij) = e_ij for 1≤i, j ≤n. Letσ denote a cyclic permutation of the integers{1,2, . . . , n} of ordern, and define

s_k = [y_σk(1),σ^k(2), y_σk(2),σ^k(2), y_σk(2),σ^k(3), . . . , y_σk(n),σ^k(n)]

for0≤k≤n−1. Moreover, lets=s₀+s₁+· · ·+s_n−1 ∈I(n−1)⊆A. We see that ρ(s)∈Mn(k) is a matrix with entry1 in position (σ^k(1), σ^k(n))for k= 0,1, . . . , n−1 and0in all other positions. In particular,detρ(s) =±1, hencedet(s)∈C(n)is non-zero at the pointmM ∈Max(C(n))corresponding toM. PutUM =D(det(s))⊆Max(C(n)), and consider the localization O(n)_{s} ⊆ Mn(C(n)_{det(s)}), where the inclusion follows from general properties of localization. Any closed point m⁰_M ∈U_M corresponds to an n-dimensional representation ofA for which the elements∈I(n−1)is invertible. This representation can not have am-dimensional quotient withm < n, so it must be simple.

Sinces∈I(n−1),the localizedk-algebraO(n)_{s}does not have any simple modules of dimension other than n. In fact, for any finite dimensional O(n)_{s}-module M of dimension m, the image ˆs of s in Endk(M) must be invertible. However, the inverse ˆ

s⁻¹ must be the image of a polynomial of degree m−1 in s. Therefore, ifM is simple overO(n)_{s}, that is if the homomorphism O(n)_{s} →Endk(M) is surjective,M must also be simple over A. Since s ∈ I(n−1) it follows that m ≥ n. If m > n, we may construct in the same way as above an element inI(n)mapping into a nonzero element ofEndk(M). Since0 =I(n)⊆A(n), and thereforeI(n) = 0in O(n)_{s}, we have proved thatm=n. By a theorem of M. Artin,O(n)_{s}must be an Azumaya algebra with center S(n)_{s}=Z(O(n)_{s}); see Artin [1]. Therefore,O(n)defines a presheafO(n)of Azumaya algebras onsimp_n(A), with centerS(n) = Z(O(n)). AnyM ∈simp_n(A)corresponding tomM ∈Max(C(n))maps to a pointκ(M)∈simp_n(A). Letm_κ(M)be the corresponding maximal ideal ofS(n). SinceO(n)is locally Azumaya, it follows that

S(n)b m_κ(M) ∼=H^O(n)(M)∼=H^A(n)(M)

6. The noncommutative scheme structure. In this section, we shall use noncom- mutative deformations of modules to define localizations, and use this to construct a

“structure” presheaf of noncommutative algebras on the Jacobson topology defined on the

spacesimp(A)of finitely dimensional simple modules. Recall that the Jacobson topology is the topology defined, just as the Zariski topology in ordinary algebraic geometry, by considering for anya∈Athe subsetD(a) :={V ∈simp(A)|det(ρ(a))6= 0}, and showing that this set forms a basis for a topology.

6.1. Geometric localizations. The localization Lof a commutativek-algebraAin a maximal idealmis given by the following universal property: It is ak-algebraLtogether with a diagram

A ^ρL //

κC^ACCCCCC!!

C L

κ^L

A/m

such thatρL(a)is a unit inLwheneverκA(a)is a unit inA/mand such that for any other L⁰ with this property, there exists a unique morphismφ:L→L⁰ such thatρ_L⁰ =ρ_L◦φ.

In the following,Ais a not necessarily commutative, associative k-algebra.

Definition 6.1. Let A be any k-algebra and M = {M1, . . . , Mn} a family of right A-modules. ThenLis called ageometric localization ofAinMif there exists a diagram

A ^ρL //

κM^AMMMMM&&

MM MM

M L

κ^L

Qn

i=1Endk(Mi) such thatρL(a)is a unit in Lwheneverκ^A(a)is a unit in Qn

i=1Endk(Mi), and if O^A(M)' O^L(M).

We writeL=A_M,and notice that geometric localizations might not be unique.

Lemma 6.2. Assume that A is a geometric k-algebra, and that M = {M1, . . . , Mn} is a family of finite dimensional, simple right A-modules. Then the geometric localization A_M of Ain M={M1, . . . , Mn} exists, andsimp(A_M) =M.

Proof. We consider the structural morphism for the familyM;

A ^η //

κLLLLL%%

LL LL

LL O^A(M)

π

Qn

i=1Endk(Mi)

The subalgebraA_MofO^A(M)generated by the imageη(A)together with all inverses of elements inη(A)mapping to units inQn

i=1End_k(M_i)is a geometric localization ofAin M. Ifa∈Amaps to a unit inQn

i=1Endk(Mi), there exists an element b∈Asuch that abmaps to 1. Thenr:= 1−absits in the radical, andab= 1−ris invertible inO^A(M), sinceO^A(M) = lim

←O^A(M)/radⁿ, and soais invertible, with a unique left=right inverse.

To prove thatO^A(M) =O^AM(M), we need only see that any deformation ofMas an A-module is also a deformation asAM-module.

ForU any open set in the Jacobson topology, wedefine O(U) = lim

←

c⊆U

Ac⊆lim

←

c⊆U

O^A(c)

It is clear that this is a presheaf, thestructure presheaf,Osimp(A), onX= simp(A).

Proposition 6.3. IfA is geometric, we have an injective homomorphism, A⊂Γ(simpA,O_simp(A)) :=A.

If A is commutative, then (simp(A),Osimp(A)) ' (Spec(A),OSpec(A)), restricted to the closed points.

Proof. The first statement follows by definition of geometric and the rest is clear. See also [5].

Definition6.4. IfA=Awe callsimp_ncA= (simp(A),O_simp(A))an affine prescheme, and we say that the set of simple A-modules |simp(A)| is a prescheme for A. A not necessarily commutative prescheme is a topological space with a presheaf of rings that can be covered by affine preschemes.

Notice that ifAhas finite nilpotency, i.e. if the kernel of the morphism,

A→ Y

V∈simp(A)

End_k(V)

is nilpotent, thenA=A,therefore|simp(A)|is a prescheme forA.

7. Example: The quantum plane Aq = khx, yi/(xy−qyx). For |q| 6= 1 the finite dimensional simple modules, are the points of the two coordinate axes.

Let forαβ= 0,

Vα,β =Aq/(x−α, y−β) =khx, yi/(x−α, y−β)∼=k.

Recall that for a generalk-algebraA, andA-modulesM,N, Extⁱ_A(M, N)∼=HHⁱ(A,Homk(M, N))

whereHH^·is the Hochshild cohomology. Also recall thatHom_k(M, N)is anA-bimodule by the action onφ∈Homk(M, N)bya∈A given by(aφ)(m) =φ(ma)and(φa)(m) = φ(m)a.Computing the Hochshild cohomology leads to,

Ext¹_A

q(V_α,β, V_α0,β⁰)∼= Derk(A_q,Hom_k(V_α,β, V_α0,β⁰))/Inner,

where Derk(Aq,Homk(Vα,β, Vα⁰,β⁰)) denotes thek-derivations from Aq to the bimodule Hom_k(V_α,β, V_α⁰_,β⁰),and whereInnerdenotes the derivations on the formδ= ad(φ).

First of all, ifq= 1 we have the ordinary commutative affine plane, and everything is known. The points are exactly the simple one-dimensional modulesV_α,β,

dimkExt¹_A1(Vα,β, Vα⁰,β⁰) = 1if(α, β) = (α⁰, β⁰),otherwise0,and the geometric localiza- tion at a point is the ordinary localization. In the rest of this section, we assume|q| 6= 1.

The simple modules are then reduced to the modules,X_α :=V_α,0 and Y_β :=V_0,β, and to,X0=Y0=V0,0.

An easy computation leads to the results,

• α=β = 0⇒Ext¹_Aq(Xα, Xβ) =hdx, dyi,

• α=β6= 0⇒Ext¹_A

q(X_α, X_β) =hdxi,

• α=qβ6= 0⇒Ext¹_Aq(Xα, Xβ) =hdyi,

• otherwiseExt¹_A

q(X_α, X_β) = 0,

where d_x (resp.d_y) is the class of the derivation for which, d_x(x) = 1, d_x(y) = 0, (resp.

dy(x) =o, dy(y) = 1).

To computeExt¹_A

q(X_α, Y_β), forα·β6= 0we notice that the trivial derivations are given byadφ(x) =φx−xφ=φα,adφ(y) =φy−yφ=−βφ,or simply by,adφ=φ(αdx−βdy).

A general derivation δ : A_q → Hom_k(X_α, Y_β) = k is given by the values, δ(x) = η, δ(y) =ξ. The relation inAq,xy−qyx= 0implies the equation,

0 =δ(xy−qyx) =xδ(y) +δ(x)y−q(yδ(x) +δ(y)x) =αξ+ηβ.

So, ifα6= 0, ξ=−^β_αη andδ=ηdx−^β_αηdy=_α^η(αdx−βdy).Similarly, ifβ 6= 0, η=−^α_βξ andδ=−^α_βξd_x+ξd_y=−_β^ξ(αd_x−βd_y).So, in both cases we have,

Ext¹_Aq(Xα, Yβ) = 0.

We notice that the remaining cases (Yα, Yβ), (Yα, Xβ) follows by symmetry, except for the caseV :=X0=Y0. NowV =Aq/(x, y)∼=k, and we easily prove that,

Ext¹_Aq(V, V)∼=HH¹(Aq,Homk(V, V))∼= Derk(Aq,Endk(V))/Inner =hdx, dyi.

Following the algorithm of computing generalized Massey Products in e.g. [7], we find, Hˆ(V)∼=khhx, yii/(xy−qyx)

with the obvious algebraizationAq =H(V) =khx, yi/(xy−qyx), and a canonical injec- tion, ρ_c : A_q ,→ Hˆ(V). Hence also an injection ρ :A_q ,→Γ(simp(A_q),O_simp(Aq) = A).

However the element1 +xyin Ais a unit at all points ofsimp(A), therefore the inverse is an element of A, but not ofA. This proves that Aq does not map isomorphically to Γ(simpAq,OsimpAq)and so|simpAq|is not a scheme forAq.

Acknowledgements. The authors would like to thank the referee for several suggestions improving the paper. We would also like to thank the organizers of the conferenceAlgebra, Geometry and Mathematical Physics ’09 in Będlewo.

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Received January 30, 2010; Revised January 21, 2011