The algebra of integro-differential operators on a polynomial algebra
aa r X i v : . [ m a t h . R A ] F e b The algebra of integro-differential operators on a polynomialalgebra
V. V. Bavula
Abstract
We prove that the algebra I n := K h x , . . . , x n , ∂∂x , . . . , ∂∂x n , R , . . . , R n i of integro-differentialoperators on a polynomial algebra is a prime, central, catenary, self-dual, non-Noetherian alge-bra of classical Krull dimension n and of Gelfand-Kirillov dimension 2 n . Its weak homologicaldimension is n , and n ≤ gldim( I n ) ≤ n . All the ideals of I n are found explicitly, there areonly finitely many of them ( ≤ n ), they commute ( ab = ba ) and are idempotent ideals( a = a ). The number of ideals of I n is equal to the Dedekind number d n . An analogue ofHilbert’s Syzygy Theorem is proved for I n . The group of units of the algebra I n is described(it is a huge group). A canonical form is found for each integro-differential operators (byproving that the algebra I n is a generalized Weyl algebra). All the mentioned results holdfor the Jacobian algebra A n (but GK ( A n ) = 3 n , note that I n ⊂ A n ). It is proved that thealgebras I n and A n are ideal equivalent. Key Words: the algebra of integro-differential operators on a polynomial algebra, catenaryalgebra, the classical Krull dimension, the global dimension, the weak homological dimension,the Gelfand-Kirillov dimension, the Weyl algebras, the Jacobian algebras, the prime spectrum.Mathematics subject classification 2000: 16E10, 16D25, 16S32, 16P90, 16U60, 16U70,16W50.
Contents
1. Introduction.2. Defining relations for the algebra I n .3. Ideals of the algebra I n .4. The Noetherian factor algebra of the algebra I n .5. The group of units of the algebra I n and its centre.6. The weak and the global dimensions of the algebra I n .7. The weak and the global dimensions of the Jacobian algebra A n . Throughout, ring means an associative ring with 1; module means a left module; N := { , , . . . } is the set of natural numbers; K is a field of characteristic zero and K ∗ is its group of units; P n := K [ x , . . . , x n ] is a polynomial algebra over K ; ∂ := ∂∂x , . . . , ∂ n := ∂∂x n are the partialderivatives ( K -linear derivations) of P n ; End K ( P n ) is the algebra of all K -linear maps from P n to P n ; the subalgebra A n := K h x , . . . , x n , ∂ , . . . , ∂ n i of End K ( P n ) is called the n ’th Weyl algebra.
Definition , [11]: The
Jacobian algebra A n is the subalgebra of End K ( P n ) generated by theWeyl algebra A n and the elements H − , . . . , H − n ∈ End K ( P n ) where H := ∂ x , . . . , H n := ∂ n x n . Clearly, A n = N ni =1 A ( i ) ≃ A ⊗ n where A ( i ) := K h x i , ∂ i , H − i i ≃ A . The algebra A n contains all the integrations R i : P n → P n , p R p dx i , since Z i = x i H − i : x α ( α i + 1) − x i x α .
1n particular, the algebra A n contains the algebra I n := K h x , . . . , x n , ∂ , . . . , ∂ n , R , . . . , R n i of polynomial integro-differential operators . Note that I n = N ni =1 I ( i ) ≃ I ⊗ n where I ( i ) := K h x i , ∂ i , R i i .The paper proceeds as follows. In Section 2, two sets of defining relations are given for thealgebra I n (Proposition 2.2); a canonical form is found for each element of I n by showing that thealgebra I n is a generalized Weyl algebra (Proposition 2.2.(2)); the Gelfand-Kirillov dimension ofthe algebra I n is 2 n (Theorem 2.3).In Section 3, a new equivalence relation, the ideal equivalence , on the class of algebras isintroduced: two algebras A and B are ideal equivalent if there exists a bijection f from the set J ( A ) of all the ideals of the algebra A to the set J ( B ) of all the ideals of the algebra B such that,for all a , b ∈ J ( A ), f ( a + b ) = f ( a ) + f ( b ) , f ( a ∩ b ) = f ( a ) ∩ f ( b ) , f ( ab ) = f ( a ) f ( b ) . The algebras I n and A n are ideal equivalent (Theorem 3.1). As a result, we have for free manyresults for the ideals of I n using similar known results for the ideals of A n of [11]. Name just afew: • I n is a prime, catenary algebra of classical Krull dimension n , and there is a unique maximalideal a n of the algebra I n . • ab = ba and a = a for all a , b ∈ J ( I n ). • The lattice J ( I n ) is distributive. • Classifications of all the ideals and the prime ideals of the algebra I n are given. • The set J ( I n ) is finite. Moreover, |J ( I n ) | = d n where d n is the Dedekind number, and2 − n + P ni =1 ni ) ≤ d n ≤ n . • P n is the only (up to isomorphism) faithful simple I n -module.The fact that certain rings of differential operators are catenary was proved by Brown, Goodearland Lenagan in [24].In Section 4, it is proved that the factor algebra I n / a is Noetherian iff the ideal a is maximal(Proposition 4.1); and GK ( I n / a ) = 2 n for all the ideals a of I n distinct from I n (Lemma 4.2).In Section 5, for the algebra I n an involution ∗ is introduced such that ∂ ∗ i = R i , R ∗ i = ∂ i , and H ∗ i = H i , see (18). This means that the algebra I n is ‘symmetrical’ with respect to derivationsand integrations. a ∗ = a for all ideals a of the algebra I n (Lemma 5.1.(1)). Each ideal of thealgebra I n is an essential left and right submodule of I n (Lemma 5.2.(2)). The group I ∗ n of unitsof the algebra I n is described: I ∗ n = K ∗ × (1 + a n ) ∗ ⊇ K ∗ × GL ∞ ( K ) ⋉ · · · ⋉ GL ∞ ( K ) | {z } n − and its centre is K ∗ (Theorem 5.6). For n = 1, the group I ∗ n is found explicitly, I ∗ ≃ K ∗ × GL ∞ ( K )(Corollary 5.7). The centre of the algebra I n is K (Lemma 5.4.(2)). It is proved that, for a K -algebra A , the algebra A ⊗ I n is prime iff the algebra A is prime (Corollary 5.3).In Section 6, we prove that the weak (w.dim) dimension of the algebra I n is n (Theorem 6.2).Moreover, wdim( I n / p ) = n for all the prime ideals p ∈ Spec( I n ) (Corollary 6.4). Recall that foreach Noetherian ring its weak dimension coincides with its global dimension (in general, this iswrong for non-Noetherian rings). In 1972, Roos proved that the global dimension of the Weylalgebra A n is n , [55]. This result was generalized by Chase [28] to the ring of differential operatorson a smooth affine variety. Goodearl obtained formulae for the global dimension of certain ringsof differential operators [36], [37] (see also Levasseur [48], and van den Bergh [61]). Holland and2tafford found the global dimension of the ring of differential operators on a rational projectivecurve [35] (see also Smith and Stafford [59]).Many classical algebras are tensor product of algebras (eg, P n = P ⊗ n , A n = A ⊗ n , A n = A ⊗ n , I n = I ⊗ n , etc). In general, it is difficult to find the dimension d ( A ⊗ B ) of the tensor productof two algebras (even to answer the question of when it is finite). In [33], it was pointed outby Eilenberg, Rosenberg and Zelinsky that ‘ the questions concerning the dimension of the tensorproduct of two algebras have turned out to be surprisingly difficult. ’ An answer is known if one ofthe algebras is a polynomial algebra: Hilbert ′ s Syzygy Theorem : d ( P n ⊗ B ) = d ( P n ) + d ( B ) = n + d ( B ) , where d = wdim , gldim. In [8], [9], an analogue of Hilbert’s Syzygy Theorem was established forcertain generalized Weyl algebras A (eg, A = A n , the Weyl algebra):l . gldim( A ⊗ B ) = l . gldim( A ) + l . gldim( B )for all left Noetherian finitely generated algebras B ( K is an algebraically closed uncountable fieldof characteristic zero). In this paper, a similar result is proved for the algebra I n and for all its primefactor algebras but for the weak dimension (Theorem 6.5). It is shown that n ≤ gldim( I n ) ≤ n (Proposition 6.7).In Section 7, we prove that the weak dimension of the Jacobian algebra A n is n (Theorem7.2), and wdim( A n / p ) = n for all the prime ideals p ∈ Spec( A n ) (Corollary 7.3). An analogueof Hilbert’s Syzygy Theorem is proved for the Jacobian algebra A n and its prime factor algebras(Theorem 7.4). It is shown that n ≤ gldim( A n ) ≤ n (Proposition 7.5).The algebra I = A h R i is an example of the Rota-Baxter algebra. The latter appeared inthe work of Baxter [21] and further explored by Rota [56, 57], Cartier [25], and Atkinson [4], andmore recently by many others: Aguiar, Moreira [1]; Cassidy, Guo, Keigher, Sit, Ebrahimi-Fard[26], [32]; Connes, Kreimer, Marcoli [29], [30], name just a few. From the angle of the Rota-Baxteralgebras the algebra I was studied by Regensburger, Rosenkranz and Middeke [54]. I n In this section defining relations are found for the algebra I n and it is proved that the algebra I n is a generalized Weyl algebra (Proposition 2.2) of Gelfand-Kirillov dimension 2 n (Theorem 2.3)which is neither left nor right Noetherian (Lemma 2.4). Generalized Weyl Algebras . Let D be a ring, σ = ( σ , ..., σ n ) be an n -tuple of commutingring endomorphisms of D , and a = ( a , ..., a n ) be an n -tuple of elements of D . The generalizedWeyl algebra A = D ( σ, a ) (briefly, GWA) of degree n is a ring generated by D and 2 n elements x , ..., x n , y , ..., y n subject to the defining relations [6], [7]: y i x i = a i , x i y i = σ i ( a i ) ,x i d = σ i ( d ) x i , dy i = y i σ i ( d ) , d ∈ D, [ x i , x j ] = [ y i , y j ] = [ x i , y j ] = 0 , i = j, where [ x, y ] = xy − yx . We say that a and σ are the sets of defining elements and endomorphismsof A respectively. For a vector k = ( k , ..., k n ) ∈ Z n , let v k = v k (1) · · · v k n ( n ) where, for 1 ≤ i ≤ n and m ≥ v m ( i ) = x mi , v − m ( i ) = y mi , v ( i ) = 1. It follows from the definition that A = L k ∈ Z n A k is a Z n -graded algebra ( A k A e ⊆ A k + e , for all k, e ∈ Z n ), where A k = v k, − Dv k, + ; v k, + := Q k i > v k i ( i ) and v k, − = Q k i < v k i ( i ). The tensor product (over the ground field) A ⊗ A ′ of generalized Weyl algebras of degree n and n ′ respectively is a GWA of degree n + n ′ : A ⊗ A ′ = D ⊗ D ′ (( τ, τ ′ ) , ( a, a ′ )) . P n be a polynomial algebra K [ H , . . . , H n ] in n indeterminates and let σ = ( σ , ..., σ n ) bethe n -tuple of commuting automorphisms of P n such that σ i ( H i ) = H i − σ i ( H j ) = H j , for i = j . The algebra homomorphism A n → P n (( σ , ..., σ n ) , ( H , . . . , H n )) , x i x i , ∂ i y i , i = 1 , . . . , n, (1)is an isomorphism. We identify the Weyl algebra A n with the GWA above via this isomorphism.Note that H i = ∂ i x i = x i ∂ i + 1.It is an experimental fact that many small quantum algebras/groups are GWAs. More aboutGWAs and their generalizations the interested reader can find in [2, 3, 12, 15, 16, 17, 18, 19, 20,22, 27, 34, 40, 42, 43, 44, 45, 50, 51, 52, 53, 58, 60].Suppose that A is a K -algebra that admits two elements x and y with yx = 1. The element xy ∈ A is an idempotent, ( xy ) = xy , and so the set xyAxy is a K -algebra where xy is its identityelement. Consider the linear maps σ = σ x,y , τ = τ x,y : A → A which are defined as follows σ ( a ) = xay, τ ( a ) = yax. (2)Then τ σ = id A and στ ( a ) = xy · a · xy , and so the map σ is an algebra monomorphism with σ (1) = xy and A = σ ( A ) M ker( τ ) . (3)In more details, σ ( A ) ∩ ker( τ ) = 0 since τ σ = id A . Since ( στ ) = στ , we have the equality A = im( στ ) L im(1 − στ ). Clearly, im( στ ) ⊆ im( σ ) and im(1 − στ ) ⊆ ker( τ ) as τ σ = id A . Then A = im( σ ) + ker( τ ), i.e. (3) holds. In general, the map τ is not an algebra endomorphism andits kernel is not an ideal of the algebra A . Suppose that the algebra A contains a subalgebra D such that σ ( D ) ⊆ D (and so xy = σ (1) ∈ D ), and that the algebra A is generated by D , x , and y . Since yx = 1, we have x i D D ≃ D and D Dy i ≃ D . It follows from the relations: yx = 1 , xy = σ (1) ,xd = σ ( d ) x, dy = yσ ( d ) , d ∈ D, that A = P i ≥ y i D + P i ≥ Dx i . Suppose, in addition, that the sum is a direct one. Then thealgebra A is the GWA D ( σ, Lemma 2.1 [14]
Keep the assumptions as above, i.e. A = D h x, y i = L i ≥ y i D L L i ≥ Dx i and σ ( D ) ⊆ D . Then A = D ( σ, . If, in addition, τ ( D ) ⊆ D and the element xy is central in D .Then Dx i = x i D and Dy i = y i D for all i ≥ .Definition , [13]. The algebra S n of one-sided inverses of P n is an algebra generated over a field K by 2 n elements x , . . . , x n , y n , . . . , y n that satisfy the defining relations: y x = · · · = y n x n = 1 , [ x i , y j ] = [ x i , x j ] = [ y i , y j ] = 0 for all i = j, where [ a, b ] := ab − ba is the algebra commutator of elements a and b .By the very definition, the algebra S n ≃ S ⊗ n is obtained from the polynomial algebra P n byadding commuting, left (but not two-sided) inverses of its canonical generators. The algebra S is a well-known primitive algebra [39], p. 35, Example 2. Over the field C of complex numbers,the completion of the algebra S is the Toeplitz algebra which is the C ∗ -algebra generated bya unilateral shift on the Hilbert space l ( N ) (note that y = x ∗ ). The Toeplitz algebra is theuniversal C ∗ -algebra generated by a proper isometry.The Jacobian algebra A n contains the algebra S n where y := H − ∂ , . . . , y n := H − n ∂ n . A n is the subalgebra of End K ( P n ) generated by the algebra S n and the 2 n invertible elements H ± , . . . , H ± n of End K ( P n ).The algebras S n and A n are much more better understood than the algebra I n . We will seethat the three classes of algebras have much in common. In particular, they are GWAs. Moreover,we will deduce many results for the algebra I n from known results for the algebras S n and A n in[11], [13], [14]. The algebra S n is a GWA . Clearly, S n = S (1) ⊗ · · · ⊗ S ( n ) ≃ S ⊗ n where S ( i ) := K h x i , y i | y i x i = 1 i ≃ S and S n = L α,β ∈ N n Kx α y β where x α := x α · · · x α n n , α = ( α , . . . , α n ), y β := y β · · · y β n n , β = ( β , . . . , β n ). In particular, the algebra S n contains two polynomial subalge-bras P n and Q n := K [ y , . . . , y n ] and is equal, as a vector space, to their tensor product P n ⊗ Q n .Note that also the Weyl algebra A n is a tensor product (as a vector space) P n ⊗ K [ ∂ , . . . , ∂ n ] ofits two polynomial subalgebras.When n = 1, we usually drop the subscript ‘1’ if this does not lead to confusion (we do thesame also for the algebras A , A and I ). So, S = K h x, y | yx = 1 i = L i,j ≥ Kx i y j . For eachnatural number d ≥
1, let M d ( K ) := L d − i,j =0 KE ij be the algebra of d -dimensional matrices where { E ij } are the matrix units, and M ∞ ( K ) := lim −→ M d ( K ) = M i,j ∈ N KE ij be the algebra (without 1) of infinite dimensional matrices. The algebra M ∞ ( K ) = L k ∈ Z M ∞ ( K ) k is Z -graded where M ∞ ( K ) k := L i − j = k KE ij ( M ∞ ( K ) k M ∞ ( K ) l ⊆ M ∞ ( K ) k + l for all k, l ∈ Z ).The algebra S contains the ideal F := L i,j ∈ N KE ij , where E ij := x i y j − x i +1 y j +1 , i, j ≥ . (4)Note that E ij = x i E y j and E = 1 − xy . For all natural numbers i , j , k , and l , E ij E kl = δ jk E il where δ jk is the Kronecker delta function. The ideal F is an algebra (without 1) isomorphic tothe algebra M ∞ ( K ) via E ij E ij . In particular, the algebra F = L k ∈ Z F ,k is Z -graded where F ,k := L i − j = k KE ij ( F ,k F ,l ⊆ F ,k + l for all k, l ∈ Z ). For all i, j ≥ xE ij = E i +1 ,j , yE ij = E i − ,j , E ij x = E i,j − , E ij y = E i,j +1 , (5)where E − ,j := 0 and E i, − := 0. xE i,j = E i +1 ,j +1 x, E ij y = yE i +1 ,j +1 . (6) S = K ⊕ xK [ x ] ⊕ yK [ y ] ⊕ F, (7)the direct sum of vector spaces. Then S /F ≃ K [ x, x − ] =: L , x x, y x − , (8)since yx = 1, xy = 1 − E and E ∈ F .The algebra S n = N ni =1 S ( i ) contains the ideal F n := F ⊗ n = M α,β ∈ N n KE αβ , where E αβ := n Y i =1 E α i β i ( i ) , E α i β i ( i ) := x α i i y β i i − x α i +1 i y β i +1 i . Note that E αβ E γρ = δ βγ E αρ for all elements α, β, γ, ρ ∈ N n where δ βγ is the Kronecker deltafunction.Using Lemma 2.1, we can show that the algebra S ( i ) is the GWA F , ( i )( σ i ,
1) where F , ( i ) := K L L k ≥ KE kk ( i ) and σ i ( a ) = x i ay i (moreover, σ i (1) = 1 − E ( i ) and σ i ( E kk ( i )) = E k +1 ,k +1 ( i )).Therefore, S n = N ni =1 F , ( i )( σ i ,
1) = F n, (( σ , . . . , σ n ) , (1 , . . . , F n, := N ni =1 F , ( i ). The algebra F n, is a commutative, non-finitely generated, non-Noetherian algebra, it contains the direct sum L α ∈ N n KE αα of ideals, hence F n, is not a primealgebra. The algebra S n = L α ∈ Z n S n,α is a Z n -graded algebra where S n,α = F n, v α = v α F n, forall α ∈ Z n where v α := Q ni =1 v α i ( i ) and v j ( i ) := ( x ji if j ≥ ,y − ji if j < . The map τ i : S n → S n , a y i ax i , is not an algebra endomorphism but its restrictionto the subalgebra F n, of S n is a K -algebra epimorphism, τ i ( F n, ) = F n, , with ker( τ i | F n, ) = KE ( i ) N N j = i F , ( j ). For all j ∈ N and d ∈ F n, , dx ji = x ji τ ji ( d ) and y ji d = τ ji ( d ) y ji .The algebra I n := K h ∂ , . . . , ∂ n , R , . . . , R n i of integro-differential operators with constant coef-ficients is canonically isomorphic to the algebra S n : S n → I n , x i Z i , y i ∂ i , i = 1 , . . . , n. (9)For n = 1 this is obvious since the map above is a well-defined epimorphism (since ∂ R = 1) whichmust be an isomorphism as the algebra I is non-commutative but any proper factor algebra of S is commutative [13]. Then the general case follows since S n ≃ S ⊗ n and I n ≃ I ⊗ n . The Jacobian algebra A n is a GWA . The Jacobian algebra A n = D n (( σ , . . . , σ n ) , (1 , . . . , D n := N ni =1 D ( i ), D ( i ) = L − ( i ) M L +1 ( i ) M F , ( i ) , F , ( i ) := M s ≥ KE ss ( i ) , L − ( i ) := M s,t ≥ K H i − s ) t , ( H i − s ) := H i − s + E s − ,s − ( i ) , L +1 ( i ) := K [ H ± i , ( H i + 1) − , ( H i + 2) − , . . . ] , and σ i ( a ) = x i ay i . In more detail, for all natural numbers s ≥ t ≥ σ i ( E ss ( i )) = E s +1 ,s +1 ( i ) , σ i (( H i − t ) ) = ( H i − t − σ i (1) ,σ i ( H i ) = H i − H i − σ i (1) = ( H i − σ i (1) . The algebra D n is a commutative, non-finitely generated, non-Noetherian algebra, it contains thedirect sum L α ∈ N n KE αα of ideals (and so D n is not a prime algebra). Note that H i E ss ( i ) = E ss ( i ) H i = ( s + 1) E ss ( i ). Clearly, A n = N ni =1 A ( i ) = N ni =1 D ( i )( σ i , A n = L α ∈ Z n A n,α is a Z n -graded algebra where A n,α = D n v α = v α D n , v α := Q ni =1 v α i ( i ) and v j ( i ) := ( x ji if j ≥ ,y − ji if j < . (10)The map τ i : A n → A n , a y i ax i , is not an algebra endomorphism but its restriction to thesubalgebra D n of A n is a K -algebra epimorphism, τ i ( D n ) = D n , ker( τ i | D n ) = KE ( i ) N N j = i D ( j ).In more detail, for a, b ∈ D , τ ( a ) τ ( b ) = ya (1 − E ) bx = τ ( ab ) − yaE bx = τ ( ab ) , since aE b ∈ KE and yE = 0. For all j ∈ N and d ∈ D n , dx ji = x ji τ ji ( d ) and y ji d = τ ji ( d ) y ji .Indeed, when n = 1, xτ ( d ) = (1 − E ) dx = dx − E dx = dx since E d ∈ KE and E x = 0. The algebra I n is a GWA . Since x i = R i H i , the algebra I n is generated by the elements { ∂ i , H i , R i | i = 1 , . . . , n } , and I n = L ni =1 I ( i ) where I ( i ) := K h ∂ i , H i , R i i = K h ∂ i , x i , R i i ≃ I .By (9), when n = 1 the following elements of the algebra I = K h ∂, H, R i , e ij := Z i ∂ j − Z i +1 ∂ j +1 , i, j ∈ N , (11)6atisfy the relations: e ij e kl = δ jk e il . Note that e ij = R i e ∂ j . The matrices of the linear maps e ij ∈ End K ( K [ x ]) with respect to the basis { x [ s ] := x s s ! } s ∈ N of the polynomial algebra K [ x ] arethe elementary matrices, i.e. e ij ∗ x [ s ] = ( x [ i ] if j = s, j = s. It follows that e ij = j ! i ! E ij , (12) Ke ij = KE ij , and F = L i,j ≥ Ke ij ≃ M ∞ ( K ). Moreover, F n = L α,β ∈ N n Ke αβ where e αβ := Q ni =1 e α i β i ( i ) and e α i β i ( i ) := R α i i ∂ β i i − R α i +1 i ∂ β i +1 i .The next proposition gives a finite set of defining relations for the algebra I n and shows thatthe algebra I n is a GWA (and so we have another set of defining relations for the algebra I n ). Proposition 2.2
1. The algebra I n is generated by the elements { ∂ i , R i , H i | i = 1 , . . . , n } thatsatisfy the defining relations: ∀ i : ∂ i Z i = 1 , [ H i , Z i ] = Z i , [ H i , ∂ i ] = − ∂ i , H i (1 − Z i ∂ i ) = (1 − Z i ∂ i ) H i = 1 − Z i ∂ i , ∀ i = j : a i a j = a j a i where a k ∈ { ∂ k , Z k , H k } .
2. The algebra I n = N ni =1 D ( i )( σ i ,
1) = D n (( σ , . . . , σ n ) , (1 , . . . , is a GWA ( R i ↔ x i , ∂ i ↔ y i , H i ↔ H i ) where D n := N ni =1 D ( i ) , D ( i ) := K [ H i ] L L j ≥ Ke jj ( i ) , H i e jj ( i ) = e jj ( i ) H i = ( j + 1) e jj ( i ) , and the K -algebra endomorphisms σ i of D n are given by the rule σ i ( a ) := R i a∂ i ( σ i ( H i ) = H i − , σ i e jj ( i )) = e j +1 ,j +1 ( i ) ). Moreover, the algebra I n = L α ∈ Z n I n,α is Z n -graded where I n,α = D n v α = v α D n for all α ∈ Z n where v α := Q ni =1 v α i ( i ) and v j ( i ) := R ji if j > , if j = 0 .∂ − ji if j < . (The canonical basis for the algebra I n ) I n = L α ∈ Z n I n,α and, for all α ∈ Z n , I n,α = v α, + D n v α, − ≃ D n ( v α, + dv α, − ↔ d ) where v α, + := Q α i > v α i ( i ) and v α, − := Q α i < v α i ( i ) .So, each element a ∈ I n is a unique finite sum a = P α ∈ Z n v α, + a α v α, − for unique elements a α ∈ D n .Proof . It suffices to prove the statements for n = 1 since I n = N ni =1 I ( i ). So, let n = 1 and I ′ be an algebra generated by symbols ∂ , R , and H that satisfy the defining relations of statement 1.The algebra I is generated by the elements ∂ , R , and H ; and they satisfy the defining relations ofstatement 1 as we can easily verify. Therefore, there is the natural algebra epimorphism I ′ → I given by the rule: ∂ ∂ , R R , H H . It follows from the relations of statement 1 and fromthe equalities e ij = (R i − j e jj if i ≥ j,e ii ∂ j − i if i < j, that I ′ = X i ≥ D ′ ∂ i + D ′ + X i ≥ Z i D ′ = X i ≥ ∂ i D ′ + D ′ + X i ≥ D ′ Z i where D ′ := K h H i + P i ≥ Ke ii . Since ∂ R = 1, the left D ′ -modules D ′ and D ′ ∂ i are isomorphic,and the right D ′ -modules R i D ′ and D ′ are isomorphic. Using the Z -grading of the Jacobianalgebra A and the fact that I ⊆ A , we have I = M i ≥ D ∂ i M D M M i ≥ Z i D = M i ≥ ∂ i D M D M M i ≥ D Z i D = K [ H ] L L i ≥ Ke ii = K [ H ] L L i ≥ KE ii since Ke ii = KE ii . Note that the left D -modules D and D ∂ i are isomorphic and the right D -modules D and R i D are isomorphicsince ∂ R = 1. This implies that the sum for I ′ above is a direct one. Therefore, I ′ ≃ I and therelations in statement 1 are defining relations for the algebra I and D ′ = D . The condition ofLemma 2.1 hold, and so I = D ( σ,
1) with D R i = R i D and D ∂ i = ∂ i D for all i ≥
1. Theproof of statements 1 and 2 of the proposition is complete.Statement 3 follows from statement 2 and the fact that, for all α ∈ Z n , the linear map I n,α → D n , b u α, − bu α, + , is a bijection since u α, − v α, + = 1 and v α, − u α, + = 1 where u α, − := Q α i > v − α i ( i ) and u α, + := Q α i < v − α i ( i ). (cid:3) Definition . For each element a ∈ I n , the unique sum for a in statement 3 of Proposition 2.2 iscalled the canonical form of a .The map τ i : I n → I n , a ∂ i a R i , is not an algebra endomorphism but its restriction to the sub-algebra D n of I n is a K -algebra epimorphism , τ i ( D n ) = D n with ker( τ i | D n ) = Ke ( i ) L L j = i D ( j ).In more detail, for n = 1 and a, b ∈ D , τ ( a ) τ ( b ) = ∂a (1 − e ) b Z = τ ( ab ) − ∂ae b Z = τ ( ab )since ae b ∈ Ke and ∂e = 0. For all j ∈ N and d ∈ D n , d R ji = R ji τ ji ( d ) and ∂ ji d = τ ji ( d ) ∂ ji .Indeed, for n = 1, R τ ( d ) = (1 − e ) d R = d R − e d R = d R since e d ∈ Ke and e R = 0.Note that τ i ( H j ) = H j + δ ij and τ i ( e st ( j )) = ( e s − ,t − ( i ) if i = j,e st ( j ) if i = j. (13)It follows that T ni =1 ker( τ i | D n ) = K Q ni =1 e ( i ) = Ke and T ni =1 ker( τ i | D n −
1) = K .For the definition and properties of the Gelfand-Kirillov dimension GK the reader is referredto [47] and [49]. Theorem 2.3
The Gelfand-Kirillov dimension
GK ( I n ) of the algebra I n is n .Proof . Since A n ⊆ I n , we have the inequality 2 n = GK ( A n ) ≤ GK ( I n ). To prove the reverseinequality let us consider the standard filtration { I n,i } i ∈ N of the algebra I n with respect to theset of generators { ∂ i , H i , R i | i = 1 , . . . , n } of the algebra I n . By Proposition 2.3, I n,i ⊆ I ′ n,i := L | α |≤ i v α D n,i where D n,i := N nj =1 D ,i ( j ) and D ,i ( j ) := L is =0 KH sj L L it =0 Ke tt ( j ). Thendim( I n,i ) ≤ dim( I ′ n,i ) ≤ (2 i + 1) n (2 i + 2) n , and so GK ( I n ) ≤ n , as required. (cid:3) Lemma 2.4
The algebra I n is neither left nor right Noetherian. Moreover, it contains infinitedirect sums of nonzero left (resp. right) ideals.Proof . Since I n ≃ I ⊗ n , it suffices to prove the lemma for n = 1. The ideal F = L i,j ≥ KE ij ofthe algebra I is the infinite direct sum L j ≥ ( L i ≥ KE ij ) (resp. L i ≥ ( L j ≥ KE ij )) of nonzeroleft (resp. right) ideals, and the statements follow. (cid:3) I n In this section, we prove that the restriction map (Theorem 3.1) from the set of ideals J ( A n ) ofthe algebra A n to the set of ideals J ( I n ) of the algebra I n is a bijection that respects the threeoperations on ideals: sum, intersection and product. As a consequence, we obtain many resultsfor the ideals of the algebra I n using similar results for the ideals of the algebra A n in [11], seeCorollary 3.3 and Corollary 3.4: a classification of all the ideals of I n (there are only finitely manyof them) and a classification of prime ideals of I n , etc.8 efinition . Let A and B be algebras, and let J ( A ) and J ( B ) be their lattices of ideals. Wesay that a bijection f : J ( A ) → J ( B ) is an isomorphism if f ( a ∗ b ) = f ( a ) ∗ f ( b ) for ∗ ∈ { + , · , ∩} ,and in this case we say that the algebras A and B are ideal equivalent . The ideal equivalence isan equivalence relation on the class of algebras.The next theorem shows that the algebras A n and I n are ideal equivalent. Theorem 3.1
The restriction map J ( A n ) → J ( I n ) , a a r := a ∩ I n , is an isomorphism (i.e. ( a ∗ a ) r = a r ∗ a r for ∗ ∈ { + , · , ∩} ) and its inverse is the extension map b b e := A n b A n .Proof . The theorem follows from Theorem 3.2. (cid:3) Recall that F n, ⊂ I n ⊂ A n ⊂ End K ( P n ). The subset of J ( F n, ), J ( F n, ) σ,τ := { b ∈J ( F n, ) | σ i ( b ) ⊆ b , τ i ( b ) ⊆ b for all i = 1 , . . . , n } , is closed under addition, multiplication andintersection of ideals where σ i ( a ) = R i a∂ i and τ i ( a ) = ∂ i a R i (recall that the maps σ i , τ i : F n, → F n, are K -algebra homomorphisms; τ i (1) = 1 but σ i (1) = 1 − e ( i )). Theorem 3.2
1. The restriction map J ( I n ) → J ( F n, ) σ,τ , a a r := a ∩ F n, , is an iso-morphism (i.e. ( a ∗ a ) r = a r ∗ a r for ∗ ∈ { + , · , ∩} ) and its inverse is the extension map b b e := I n b I n .2. The restriction map J ( A n ) → J ( F n, ) σ,τ , a a r := a ∩ F n, , is an isomorphism (i.e. ( a ∗ a ) r = a r ∗ a r for ∗ ∈ { + , · , ∩} ) and its inverse is the extension map b b e := A n b A n . The proof of Theorem 3.2 is given at the end of this section. Now, we obtain some consequencesof Theorem 3.1.The next corollary shows that the ideal theory of I n is ‘very arithmetic.’ Let B n be the set ofall functions f : { , , . . . , n } → { , } . For each function f ∈ B n , I f := I f (1) ⊗ · · · ⊗ I f ( n ) is theideal of I n where I := F and I := I . Let C n be the set of all subsets of B n all distinct elements ofwhich are incomparable (two distinct elements f and g of B n are incomparable if neither f ( i ) ≤ g ( i )nor f ( i ) ≥ g ( i ) for all i ). For each C ∈ C n , let I C := P f ∈ C I f , the ideal of I n . The number d n ofelements in the set C n is called the Dedekind number . It appeared in the paper of Dedekind [31].An asymptotic of the Dedekind numbers was found by Korshunov [46].
Corollary 3.3
1. The algebra I n is a prime algebra.2. The set of height one prime ideals of the algebra I n is { p := F ⊗ I n − , p := I ⊗ F ⊗ I n − , . . . , p n := I n − ⊗ F } .3. Each ideal of the algebra I n is an idempotent ideal ( a = a ).4. The ideals of the algebra I n commute ( ab = ba ).5. The lattice J ( I n ) of ideals of the algebra I n is distributive.6. The classical Krull dimension cl . Kdim( I n ) of the algebra I n is n .7. ab = a ∩ b for all ideals a and b of the algebra I n .8. The ideal a n := p + · · · + p n is the largest (hence, the only maximal) ideal of I n distinctfrom I n , and F n = F ⊗ n = T ni =1 p i is the smallest nonzero ideal of I n .9. (A classification of ideals of I n ) The map C n → J ( I n ) , C I C := P f ∈ C I f is a bijectionwhere I ∅ := 0 . The number of ideals of I n is the Dedekind number d n . Moreover, − n + P ni =1 ni ) ≤ d n ≤ n . For n = 1 , F is the unique proper ideal of the algebra I .10. (A classification of prime ideals of I n ) Let
Sub n be the set of all subsets of { , . . . , n } . Themap Sub n → Spec( I n ) , I p I := P i ∈ I p i , ∅ 7→ , is a bijection, i.e. any nonzero primeideal of I n is a unique sum of primes of height 1; | Spec( I n ) | = 2 n ; the height of p I is | I | ; and p I ⊂ p J iff I ⊂ J .Proof . By Theorem 3.1, the statements hold for the algebra I n since the same statements holdfor the algebra A n , and below references are given for their proofs in [11].1. Corollary 2.7.(5).2. Corollary 3.5.3. Theorem 3.1.(2).4. Corollary 3.10.(3).5. Theorem 3.11.6. Corollary 3.7.7. Corollary 3.10.(3).8. Corollary 2.7.(4,7).9. Theorem 3.1.10. Corollary 3.5.11. Corollary 3.6. (cid:3) For each ideal a of I n , Min( a ) denotes the set of minimal primes over a . Two distinct primeideals p and q are called incomparable if neither p ⊆ q nor p ⊇ q . The algebras I n have beautifulideal theory as the following unique factorization properties demonstrate. Corollary 3.4
1. Each ideal a of I n such that a = I n is a unique product of incompara-ble primes, i.e. if a = q · · · q s = r · · · r t are two such products then s = t and q = r σ (1) , . . . , q s = r σ ( s ) for a permutation σ of { , . . . , n } .2. Each ideal a of I n such that a = I n is a unique intersection of incomparable primes, i.e. if a = q ∩ · · · ∩ q s = r ∩ · · · ∩ r t are two such intersections then s = t and q = r σ (1) , . . . , q s = r σ ( s ) for a permutation σ of { , . . . , n } .3. For each ideal a of I n such that a = I n , the sets of incomparable primes in statements 1 and2 are the same, and so a = q · · · q s = q ∩ · · · ∩ q s .4. The ideals q , . . . , q s in statement 3 are the minimal primes of a , and so a = Q p ∈ Min( a ) p = ∩ p ∈ Min( a ) p .Proof . The same statements are true for the algebra A n (Theorem 3.8, [11]). Now, the corollaryfollows from Theorem 3.1. (cid:3) The next corollary gives all decompositions of an ideal as a product or intersection of ideals.
Corollary 3.5
Let a be an ideal of I n , and M be the minimal elements with respect to inclusionof the set of minimal primes of a set of ideals a , . . . , a k of I n . Then1. a = a · · · a k iff Min( a ) = M .2. a = a ∩ · · · ∩ a k iff Min( a ) = M .Proof . The same statements are true for the algebra A n (Theorem 3.12, [11]), and the corollaryfollows from Theorem 3.1. (cid:3) This is a rare example of a noncommutative algebra of classical Krull dimension > R of finite classicalKrull dimension is called catenary if, for each pair of prime ideals p and q with p ⊆ q , all maximalchains of prime ideals, p = p ⊂ p ⊂ · · · ⊂ p l = q , have the same length l . Corollary 3.6
The algebra I n is catenary.Proof . This follows from Corollary 3.3.(10, 11). (cid:3) orollary 3.7 The same statements (with obvious modifications) of Corollaries 3.3 and 3.4 holdfor the ideals J ( F n, ) σ,τ of the algebra F n, rather than I n (we leave it to the reader to formulatethem). Proposition 3.8
The polynomial algebra P n is the only (up to isomorphism) faithful simple I n -module.Proof . The I n -module P n is faithful (as I n ⊂ End K ( P n )) and simple since the A n -module P n is simple and A n ⊂ I n . Let M be a faithful simple I n -module. Then F n M = 0, i.e. e β m = 0for some elements β ∈ N n and m ∈ M . The I n -module P n ≃ I n / P ni =1 I n ∂ i is simple. Therefore,the I n -module epimorphism P n → M = I n e β m = P α ∈ N n Ke αβ m , 1 e β m , is an isomorphism.The proof of the proposition is complete. (cid:3) For a ring R and its element r ∈ R , ad( r ) : s [ r, s ] := rs − sr is the inner derivation of thering R associated with the element r . The proof of Theorem 3.2 . We split the proof of Theorem 3.2 into several statements(which are interesting on their own) to make the proof clearer.The algebra D n is the zero component of the Z n -graded algebra I n = D n (( σ , . . . , σ n ) , (1 , . . . , σ i ( D n ) ⊆ D n and τ i ( D n ) ⊆ D n for all i where σ i ( a ) = R i a∂ i and τ i ( a ) = ∂ i a R i . Let J ( D n ) σ,τ := { b ∈ J ( D n ) | σ i ( b ) ⊆ b , τ i ( b ) ⊆ b for all i = 1 , . . . , n } . Similarly, the algebra D n is thezero component of the Z n -graded algebra A n = D n (( σ , . . . , σ n ) , (1 , . . . , σ i ( D n ) ⊆ D n and τ i ( D n ) ⊆ D n for all i where σ i ( a ) = x i ay i and τ i ( a ) = y i ax i . Let J ( D n ) σ,τ := { b ∈J ( D n ) | σ i ( b ) ⊆ b , τ i ( b ) ⊆ b for all i = 1 , . . . , n } . Theorem 3.9
1. For each ideal a of the algebra I n , a = L α ∈ Z n v α, + a r v α, − where a r := a ∩ D n ∈ J ( D n ) σ,τ and, for each ideal b ∈ J ( D n ) σ,τ , b e := I n b I n = L α ∈ Z n v α, + b v α, − where v α, + := Q α i > v α i ( i ) , v α, − := Q α i < v α i ( i ) , and v j ( i ) is defined in Proposition 2.2.(2).2. For each ideal a of the algebra A n , a = L α ∈ Z n v α, + a r v α, − where a r := a ∩ D n ∈ J ( D n ) σ,τ and, for each ideal b ∈ J ( D n ) σ,τ , b e := A n b A n = L α ∈ Z n v α, + b v α, − where v α, ± are as abovebut the elements v j ( i ) are defined in (10).Proof . 1. Let a be an ideal of the algebra I n . The algebra I n = L α ∈ Z n I n,α is a Z n -gradedalgebra with I n,α := T ni =1 ker(ad( H i ) − α i ) for all α ∈ Z n . Then a is a homogeneous ideal,that is a = L α ∈ Z n a α where a α := a ∩ I n,α . The ideal a := a ∩ D n = a r of the algebra D n belongs to the set J ( D n ) σ,τ since σ i ( a ) = R i a ∂ i ⊆ a and τ i ( a ) = ∂ i a R i ⊆ a for all i = 1 , . . . , n . By Proposition 2.2.(2), a α = v α, + b α v α, − for some ideal b α of the algebra D n : a α = D n a α D n = D n v α, + b α v α, − D n = v α, + τ α, + ( D n ) b α τ α, − ( D n ) v α, − = v α, + D n b α D n v α, − since τ α, ± ( D n ) = D n where τ α, + := Q α i > τ i and τ α, − := Q α i < τ i . Let u α, − := Y α i > v − α i ( i ) , u α, + := Y α i < v − α i ( i ) . (14)The ideal b α is unique since v α, + b α v α, − = v α, + b ′ α v α, − implies b α = 1 · b α · u α, − v α, + b α v α, − u α, + = u α, − v α, + b ′ α v α, − u α, + = 1 · b ′ α · b ′ α . Moreover, b α = a for all α ∈ Z n since a ⊇ u α, − a n,α u α, + = u α, − v α, + b α v α, − u α, + = 1 · b α · b α .On the other hand, a n,α ⊇ v α, + a v α, − , and so a ⊆ b α .Let b ∈ J ( D n ) σ,τ . Then b er = b since b ⊆ b er = ( I n b I n ) r = X α ∈ Z n I n,α b I n, − α = X α ∈ Z n v α D n b D n v − α (Proposition 2.2 . (2))= X α ∈ Z n σ α, + τ α, − ( b ) v α v − α ⊆ b D n = b , σ α, + := Q α i > σ i . Therefore, b e = L α ∈ Z n v α, + b v α, − , by the first part of statement 1.2. Repeat the proof of statement 1 replacing ( I n , D n ) by ( A n , D n ) and making obvious modi-fications. (cid:3) For each function f ∈ B n , let f f := f f (1) ⊗ · · · ⊗ f f ( n ) where f := F , , f := F , ; d f := d f (1) ⊗ · · · ⊗ d f ( n ) where d := F , , d := D ; d ′ f := d ′ f (1) ⊗ · · · ⊗ d ′ f ( n ) where d ′ := F , , d ′ := D . Note that f f ∈ J ( F n, ) σ,τ , d f ∈ J ( D n ) σ,τ , and d ′ f ∈ J ( D n ) σ,τ . Lemma 3.10
1. The map C n → J ( F n, ) σ,τ , C f C := P f ∈ C f f , is a bijection where f ∅ := 0 .2. The map C n → J ( D n ) σ,τ , C d C := P f ∈ C d f , is a bijection where d ∅ := 0 .3. The map C n → J ( D n ) σ,τ , C d ′ C := P f ∈ C d ′ f , is a bijection where d ′∅ := 0 .Proof . 1. It follows from F n, = N ni =1 ( K + P j ∈ N Ke jj ( i )), σ i ( e jj ( i )) = e j +1 ,j +1 ( i ), τ i ( e jj ( i )) = e j − ,j − ( i ), e kk ( i ) e jj ( i ) = δ jk e jj ( i ) that any ideal b ∈ J ( F n, ) σ,τ is a sum P f ∈ C ′ f f . Then b = f C for a unique element C ∈ C n ( C is the set of all the maximal elements of C ′ , it does not dependon C ′ ), and so the map C f C is a bijection.2. Similarly, it follows from D n = N ni =1 ( K [ H i ] + P j ∈ N Ke jj ( i )), τ i ( H i ) = H i + 1 (hence K [ H i ] = S s ≥ ker( τ i − s ) and the actions of the endomorphisms σ i , τ i on the matrix units e jj ( i )that any ideal b ∈ J ( D n ) σ,τ is a sum P f ∈ C ′ d f . Then b = d C for a unique element C ∈ C n ( C isthe set of all the maximal elements of C ′ , it does not depend on C ′ ), and so the map C d C isa bijection.3. Statement 3 follows from statement 2 since the commutative algebra D n is a localization ofthe commutative algebra D n at the monoid generated by the set { ( H i − j ) | i = 1 , . . . , n ; 0 = j ∈ N } of nonzero divisors and d ′ C = D n d C for all C ∈ C n . (cid:3) Corollary 3.11
1. The restriction map J ( D n ) σ,τ → J ( F n, ) σ,τ , b b r := b ∩ F n, , is anisomorphism (i.e. ( b ∗ b ) r = b r ∗ b r for ∗ ∈ { + , · , ∩} ) and its inverse is the extension map c c e := D n c .2. The restriction map J ( D n ) σ,τ → J ( F n, ) σ,τ , b b r := b ∩ F n, , is an isomorphism (i.e. ( b ∗ b ) r = b r ∗ b r for ∗ ∈ { + , · , ∩} ) and its inverse is the extension map c c e := D n c .Proof . 1. Statement 1 follows from Corollary 3.10.(1,2) and the fact that d rC = f C for all C ∈ C n .2. Statement 2 follows from Corollary 3.10.(1,3) and the fact that ( d ′ C ) r = f C for all C ∈ C n . (cid:3) Proof of Theorem 3.2 . 1. By Theorem 3.9.(1), the restriction map J ( I n ) → J ( D n ) σ,τ is anisomorphism and its inverse map is the extension map. By Corollary 3.11.(1), the restrictionmap J ( D n ) σ,τ → J ( F n ) σ,τ is an isomorphism and its inverse map is the extension map. Now,statement 1 is obvious.2. Similarly, by Theorem 3.9.(2), the restriction map J ( A n ) → J ( D n ) σ,τ is an isomorphismand its inverse map is the extension map. By Corollary 3.11.(2), the restriction map J ( D n ) σ,τ →J ( F n, ) σ,τ is an isomorphism and its inverse map is the extension map. Now, statement 2 isobvious. (cid:3) Theorem 3.12
Let
Id( I n ) be the set of all the idempotent ideals of the algebra I n . Then . the restriction map I ( I n ) → Id( I n ) , a a e := a ∩I n is a bijection such that ( a ∗ a ) r = a r ∗ a r for ∗ ∈ { + , · , ∩} , and its inverse is the extension map b b e := I n b I n .2. The restriction map Id( I n ) → J ( F n , ) σ,τ , b b r := b ∩ F n, , is a bijection such that ( b ∗ b ) r = b r ∗ b r for ∗ ∈ { + , · , ∩} , and its inverse is the extension map c c e := I n c I n .Proof . 1. Statement 1 follows from Theorem 3.2.(1) and statement 2.2. Statement 2 follows at once from a classification of the idempotent ideals of the algebra S n ≃ I n (Theorem 7.2, [13]). (cid:3) I n The aim of this section is to show that the factor algebra I n / a n of the algebra I n at its maximalideal a n = p + · · · + p n is the only Noetherian factor algebra of the algebra I n (Proposition 4.1). The factor algebra I n / a n . Recall that the Weyl algebra A n is the generalized Weyl algebra P n (( σ , ..., σ n ) , ( H , . . . , H n )). Denote by S n the multiplicative submonoid of P n generated bythe elements H i + j where i = 1 , . . . , n and j ∈ Z . It follows from the above presentation ofthe Weyl algebra A n as a GWA that S n is an Ore set in A n , and, using the Z n -grading of A n ,that the (two-sided) localization A n := S − n A n of the Weyl algebra A n at S n is the skew Laurentpolynomial ring A n = S − n P n [ x ± , . . . , x ± n ; σ , ..., σ n ] (15)with coefficients in the algebra L n := S − n P n = K [ H ± , ( H ± − , ( H ± − , . . . , H ± n , ( H n ± − , ( H n ± − , . . . ] , which is the localization of P n at S n . We identify the Weyl algebra A n with its image in thealgebra A n via the monomorphism, A n → A n , x i x i , ∂ i H i x − i , i = 1 , . . . , n. Let k n be the n ’th Weyl skew field , that is the full ring of quotients of the n ’th Weyl algebra A n (it exists by Goldie’s Theorem since A n is a Noetherian domain). Then the algebra A n is a K -subalgebra of k n generated by the elements x i , x − i , H i and H − i , i = 1 , . . . , n since, for all j ∈ N , ( H i ∓ j ) − = x ± ji H − i x ∓ ji , i = 1 , . . . , n. (16)Clearly, A n ≃ A ⊗ · · · ⊗ A ( n times).Recall that the algebra I n is a subalgebra of A n and the extension a en of the maximal ideal a n of the algebra I n is the maximal ideal of the algebra A n . By (22) of [11], there is the algebraisomorphism (where a := a + a en ): A n / a en → A n , x i x i , ∂ i H i x − i , H ± i H ± i , i = 1 , . . . , n. Since a ern = a n (Theorem 3.1), the algebra B n := I n / a n is a subalgebra of the algebra A n / a en , andso there is the algebra monomorphism (where a := a + a en ): B n → A n , x i x i , ∂ i H i x − i , Z i x i H − i , H i H i , i = 1 , . . . , n. It follows that there is the algebra isomorphism: B n → n O i =1 K [ H i ][ ∂ i , ∂ − i ; τ i ] = P n [ ∂ ± , . . . , ∂ ± n ; τ , . . . , τ n ] , P n = K [ H , . . . , H n ] where τ i ( H j ) = H j + δ ij . It is a standard fact that B n = ( A n ) ∂ ,...,∂ n (17)where ( A n ) ∂ ,...,∂ n is the localization of the Weyl algebra A n at the Ore subset of A n which is thesubmonoid of A n generated by the elements ∂ , . . . , ∂ n . Note that ( A n ) ∂ ,...,∂ n ≃ ( A n ) x ,...,x n . Itis well-known that the algebra B n is a simple, Noetherian, finitely generated algebra of Gelfand-Kirillov dimension 2 n and l . gldim( B n ) = r . gldim( B n ) = n . Proposition 4.1
Let a be an ideal of the algebra I n such that a = I n . The following statementsare equivalent.1. The factor algebra I n / a is a left Noetherian algebra.2. The factor algebra I n / a is a right Noetherian algebra.3. The factor algebra I n / a is a Noetherian algebra.4. a = a n .Proof . Note that the algebra B n = I n / a n is a Noetherian algebra as a two-sided localization ofthe Noetherian algebra A n . Suppose that a = a n . Fix p ∈ Min( a ). Then p = p I := P i ∈ I p i for anon-empty subset I of the set { , . . . , n } with m := | I | < n (Corollary 3.3.(10) and Corollary 3.4).The factor algebra I n / p ≃ B m ⊗ I n − m is neither left nor right Noetherian since the algebra I n − m is so. The algebra I n / p is a factor algebra of the algebra I n / a . Then the algebra I n / a is neitherleft nor right Noetherian. Now, the proposition is obvious. (cid:3) Lemma 4.2
Let a be an ideal of the algebra I n distinct from I n . Then GK ( I n / a ) = 2 n .Proof . It is well-known that GK ( B n ) = 2 n . Now, 2 n = GK ( I n ) ≥ GK ( I n / a ) ≥ GK ( I n / a n ) =GK ( B n ) = 2 n . Therefore, GK ( I n / a ) = 2 n . (cid:3) I n and its centre In this section, the group I ∗ n of units of the algebra I n is described (Theorem 5.6.(1)) and itscentre is found (Theorem 5.6.(2)). It is proved that the algebra I n is central (Lemma 5.4.(2)) andself-dual. The involution ∗ on the algebra I n . Using the defining relations in Proposition 2.2.(1), wesee that the algebra I n admits the involution: ∗ : I n → I n , ∂ i Z i , Z i ∂ i , H i H i , i = 1 , . . . , n, (18)i.e. it is a K -algebra anti-isomorphism (( ab ) ∗ = b ∗ a ∗ ) such that ∗ ◦ ∗ = id I n . Therefore, thealgebra I n is self-dual , i.e. is isomorphic to its opposite algebra I opn . As a result, the left and theright properties of the algebra I n are the same. For all elements α, β ∈ N n , e ∗ αβ = e βα . (19)An element a ∈ I n is called hermitian if a ∗ = a . Lemma 5.1 a ∗ = a for all ideals a of the algebra I n .2. ( I n,α ) ∗ = I n, − α for all α ∈ Z n . . The set Fix I n ( ∗ ) = { a ∈ I n | a ∗ = a } of all the hermitian elements of the algebra I n is thecommutative subalgebra D n of the algebra I n .Proof . 1. By (19), p ∗ i = p i for all i = 1 , . . . , n (see Corollary 3.3.(2)). By Corollary 3.3.(4,9), a ∗ = a .2. Note that D ∗ n = D n and v ∗ α = v − α . By Proposition 2.2.(2), ( I n,α ) ∗ = ( v α D n ) ∗ = D n v − α = I n, − α .3. By statement 2, Fix I n ( ∗ ) ⊆ D n . The opposite inclusion is obvious. Therefore, Fix I n ( ∗ ) = D n . (cid:3) The involution ∗ of the algebra I n respects the maximal ideal a n ( a ∗ n = a n ). Therefore, thefactor algebra B n = I n / a n inherits the involution ∗ : ∂ ∗ i = ∂ − i , x ∗ i = x i + ∂ − i , H ∗ i = H i for i = 1 , . . . , n (since ∂ ∗ i = R i = ∂ − i and x ∗ i = ( ∂ i H i ) ∗ = H i ∂ − i = ∂ i x i ∂ − i = x i + ∂ − i ).The involution ∗ of the algebra I n can be extended to an involution of the algebra A n by setting x ∗ i = H i ∂ i , ∂ ∗ i = Z i , ( H ± i ) ∗ = H ± i , i = 1 , . . . , n. This can be checked using the defining relations coming from the presentation of the algebra A n as a GWA. Note that y ∗ i = ( H − i ∂ i ) ∗ = R i H − i = x i H − i , A ∗ n A n , S ∗ n S n , but I ∗ n = I n where I n is the algebra of integro-differential operators with constant coefficients.For a subset S of a ring R , the sets l . ann R ( S ) := { r ∈ R | rS = 0 } and r . ann R ( S ) := { r ∈ R | Sr = 0 } are called the left and the right annihilators of the set S in R . Using the fact that thealgebra I n is a GWA and its Z n -grading, we see thatl . ann I n ( Z i ) = M k ∈ N Ke k ( i ) O O i = j I ( j ) , r . ann I n ( Z i ) = 0 . (20)r . ann I n ( ∂ i ) = M k ∈ N Ke k ( i ) O O i = j I ( j ) , l . ann I n ( ∂ i ) = 0 . (21)Recall that a submodule of a module that intersects non-trivially each nonzero submodule of themodule is called an essential submodule. Lemma 5.2
1. For all nonzero ideals a of the algebra I n , l . ann I n ( a ) = r . ann I n ( a ) = 0 .2. Each nonzero ideal of the algebra I n is an essential left and right submodule of I n .Proof . The algebra I n is self-dual, so it suffices to prove only, say, the left versions of thestatements.1. Suppose that b := l . ann I n ( a ) = 0, we seek a contradiction. By Corollary 3.3.(8), the nonzeroideals a and b contain the ideal F n . Then 0 = ba ⊇ F n = F n = 0, a contradiction. Therefore, b = 0.2. Let I be a nonzero left ideal of the algebra I n . By statement 1, 0 = F n I ⊆ F n ∩ I . Therefore, F n is an essential left submodule of the algebra I n . Then so are all the nonzero ideals of the algebra I n since F n is the least nonzero ideal of the algebra I n . (cid:3) Corollary 5.3
Let A be a K -algebra. Then the algebra I n ⊗ A is a prime algebra iff the algebra A is so.Proof . It is obvious that if the algebra A is not prime ( ab = 0 for some nonzero ideals a and b of A ) then the algebra I n ⊗ A is neither (since I n ⊗ a · I n ⊗ b = 0).It suffices to show that if the algebra A is prime then so is the algebra I n ⊗ A . Let c be anonzero ideal of the algebra I n ⊗ A . Then F n c = 0, by Lemma 5.2.(1). Note that F n c ⊆ c . Let u = E αβ ⊗ a + · · · + E σρ ⊗ a ′ be a nonzero element of F n c where E αβ , . . . , E σρ are distinct matrixunits; a, . . . , a ′ ∈ A , and a = 0. Then 0 = E αβ ⊗ a = E αα uE ββ ∈ a , and so F n ⊗ AaA ⊆ c . Let d
15e a nonzero ideal of the algebra I n ⊗ A . Then F n ⊗ AbA ⊆ d for some nonzero element b ∈ A .Then cd ⊇ F n ⊗ AaA · F n ⊗ AbA = F n ⊗ ( AaA · AbA ) = 0since F n = F n and AaA · AbA = 0 ( A is a prime algebra). Therefore, I n ⊗ A is a prime algebra. (cid:3) The centre of the algebra I n . For an algebra A and its subset S , the subalgebra of A ,Cen A ( S ) := { a ∈ A | as = sa for all s ∈ S } , is called the centralizer of S in A . The next lemmashows that the algebra I n is a central algebra, i.e. its centre Z ( I n ) is K . Lemma 5.4 Cen I n ( F n, ) = Cen I n ( D n ) = D n .2. The centre of the algebra I n is K .3. Cen I n ( I n ) = K .Proof . 1. Since F n, ⊂ D n and D n is a commutative algebra, we have the inclusions D n ⊆ Cen I n ( D n ) ⊆ Cen I n ( F n, ). It remains to show that the inclusion C := Cen I n ( F n, ) ⊆ D n holds.Recall that the algebra I n = L α ∈ Z n I n,α is a Z n -graded algebra with F n, ⊂ D n = I n, . Therefore, C is a homogeneous subalgebra of I n , i.e. C = L α ∈ Z n C α where C α := C ∩ I n,α . We have to showthat C α = 0 for all α = 0. Let c ∈ C α for some α = 0. Then c = v α, + dv α, − for some element d ∈ D n (the elements v α, + and v α, − are defined in Theorem 3.9.(1)). For all elements E ββ ∈ F n, where β ∈ N n , cE ββ = v α, + dτ α, − ( E ββ ) v α, − = v α, + dE β − α − ,β − α − v α, − ,E ββ c = v α, + τ α, + ( E ββ ) dv α, − = v α, + E β − α + ,β − α + dv α, − , where τ α, − := Q α i < τ i , τ α, + := Q α i > τ i , α − := − P α i < α i e i and α + := P α i > α i e i ( E st = 0if either s N n or t N n ). Since cE ββ = E ββ c and the map a v α, + av α, − is injective (its leftinverse is the map a u α, − au α, + , see (14)), we have the equality E β − α + ,β − α + d = E β − α − ,β − α − d for each β ∈ N n . Since L γ ∈ N n KE γγ is the direct sum of ideals of the algebra D n , it follows that E γγ d = 0 for all elements γ ∈ N n . Then it is not difficult to show that d = 0 (using the fact thateach polynomial of K [ H , . . . , H n ] is uniquely determined by its values on the set N n ).2. By statement 1, the centre Z of the algebra I n is a subalgebra of D n . Let d ∈ Z . For allelements i = 1 , . . . , n , 0 = dx i − x i d = x i ( τ i ( d ) − d ). Since I n ⊆ A n , we see that 0 = y i x i ( τ i ( d ) − d ) = τ i ( d ) − d , and so d ∈ T ni =1 ker D n ( τ i −
1) = K . Therefore, Z = K .3. By (12), F n, ⊆ I n . This implies that C := Cen I n ( I n ) ⊆ Cen I n ( F n, ) = D n , by statement1. Let d ∈ C . Then0 = ∂ i · ∂ i ( d Z i − Z i d ) = ∂ i Z i ( τ i ( d ) − d ) = τ i ( d ) − d for all i = 1 , . . . , n, where τ i ( a ) = ∂ i a R i . Hence d ∈ T ni =1 ker D n ( τ i −
1) = K , and so C = K . (cid:3) Lemma 5.5
Let C = P n , K [ ∂ , . . . , ∂ n ] , K [ R , . . . , R n ] or D n . Then Cen I n ( C ) = C and C is amaximal commutative subalgebra of the algebra I n .Proof . The first statement, Cen I n ( C ) = C , follows from the fact that the algebra I n is Z n -graded and the canonical generators of the algebra C are homogeneous elements of the algebra I n (we leave this as an exercise for the reader). Then C is a maximal commutative subalgebra of thealgebra I n since Cen I n ( C ) = C and C is a commutative algebra. (cid:3) The group I ∗ n of units of the algebra I n and its centre . The group A ∗ of units of thealgebra A contains the following infinite discrete subgroup Theorem 4.2, [11]: H := { Y i ≥ ( H + i ) n i · Y i ≥ ( H − i ) n − i | ( n i ) i ∈ Z ∈ Z ( Z ) } ≃ Z ( Z ) . (22)16or each tensor multiple A ( i ) of the algebra A n = N ni =1 A ( i ), let H ( i ) be the correspondinggroup H . Their (direct) product H n := H (1) · · · H ( n ) = n Y i =1 H ( i ) (23)is a (discrete) subgroup of the group A ∗ n of units of the algebra A n , and H n ≃ H n ≃ ( Z n ) ( Z ) .Note that A ∗ n = K ∗ × ( H n ⋉ (1 + a en ) ∗ ) and Z ( A ∗ n ) = K ∗ (Theorem 4.4, [11]). A similar resultholds for the group I ∗ n of the algebra I n (Theorem 5.6). Since a n is an ideal of the algebra I n , theintersection (1 + a n ) ∗ := I ∗ n ∩ (1 + a n ) is a subgroup of the group I ∗ n of units of the algebra I n . Theorem 5.6
1. Let F n := L ni =1 ( K + F ( i )) . Then I ∗ n = K ∗ × (1 + a n ) ∗ and I ∗ n ⊇ (1 + F n ∩ a n ) ∗ ≃ GL ∞ ( K ) ⋉ · · · ⋉ GL ∞ ( K ) | {z } n − .
2. The centre of the group I ∗ n is K ∗ .Proof . 1. The commutative diagram of algebra homomorphisms I n (cid:15) (cid:15) / / A n (cid:15) (cid:15) B n / / A n yields the commutative diagram of group homomorphisms I ∗ n (cid:15) (cid:15) / / A ∗ n (cid:15) (cid:15) B ∗ n / / A ∗ n . Since B ∗ n = S α ∈ Z n K ∗ ∂ α and A ∗ n = K ∗ × ( H n ⋉ (1 + a en ) ∗ ), we see that K ∗ × (1 + a n ) ∗ ⊆ I ∗ n ⊆ I n ∩ A ∗ n = K ∗ × ( I n ∩ (1 + a en ) ∗ ) = K ∗ × (1 + a ern ) ∗ = K ∗ × (1 + a n ) ∗ since a ern = a n (Theorem 3.1). Therefore, I ∗ n = K ∗ × (1 + a n ) ∗ .Since F n ⊂ I n ⊂ A n , it is obvious that I ∗ n ⊇ (1 + F n ∩ a n ) ∗ = (1 + F n ∩ a en ) ∗ ≃ GL ∞ ( K ) ⋉ · · · ⋉ GL ∞ ( K ) | {z } n − . The isomorphism is established in Corollary 7.3, [14].2. Let S be the set of elements of the type 1 + Q i ∈ I e s i s i ( i ) where ∅ 6 = I ⊆ { , . . . , n } .Then S ⊆ I ∗ n and Cen I n ( S ) = Cen I n ( F n, ) = D n , by Lemma 5.4.(1). Therefore, Cen I ∗ n ( S ) =Cen I n ( S ) ∩ I ∗ n = D n ∩ I ∗ n = D ∗ n = F ∗ n, . We see that Cen F n, ( S ) = K . Therefore, the centre of thegroup I ∗ n is K ∗ . (cid:3) The group of units (1 + F ) ∗ and I ∗ . Recall that the algebra (without 1) F = L i,j ∈ N Ke ij isthe union M ∞ ( K ) := S d ≥ M d ( K ) = lim −→ M d ( K ) of the matrix algebras M d ( K ) := L ≤ i,j ≤ d − Ke ij ,i.e. F = M ∞ ( K ). For each d ≥
1, consider the (usual) determinant det d = det : 1 + M d ( K ) → K , u det( u ). These determinants determine the (global) determinant ,det : 1 + M ∞ ( K ) = 1 + F → K, u det( u ) , (24)where det( u ) is the common value of all the determinants det d ( u ), d ≫
1. The (global) determinanthas usual properties of the determinant. In particular, for all u, v ∈ M ∞ ( K ), det( uv ) =17et( u ) · det( v ). It follows from this equality and the Cramer’s formula for the inverse of a matrixthat the group GL ∞ ( K ) := (1 + M ∞ ( K )) ∗ of units of the monoid 1 + M ∞ ( K ) is equal toGL ∞ ( K ) = { u ∈ M ∞ ( K ) | det( u ) = 0 } . (25)Therefore, (1 + F ) ∗ = { u ∈ F | det( u ) = 0 } = GL ∞ ( K ) . (26) Corollary 5.7 I ∗ = K ∗ × (1 + F ) ∗ = K ∗ × GL ∞ ( K ) , i.e. I ∗ = { λ (1 + f ) | det(1 + f ) = 0 , λ ∈ K ∗ , f ∈ F } . The elements λ ∈ K ∗ , µe ij where µ ∈ K and i = j , and γe where γ ∈ K \{− } are generators for the group I ∗ n . I n In this section, we prove that the weak dimension of the algebra I n and of all its prime factoralgebras is n (Theorem 6.2). An analogue of Hilbert’s Syzygy Theorem is established for thealgebra I n and for all its prime factor algebras (Theorem 6.5). The weak dimension of the algebra I n . Let S be a non-empty multiplicatively closedsubset of a ring R , and let ass( S ) := { r ∈ R | sr = 0 for some s ∈ S } . Then a left quotient ring of R with respect to S is a ring Q together with a homomorphism ϕ : R → Q such that(i) for all s ∈ S , ϕ ( s ) is a unit in Q ;(ii) for all q ∈ Q , q = ϕ ( s ) − ϕ ( r ) for some r ∈ R and s ∈ S , and(iii) ker( ϕ ) = ass( S ).If there exists a left quotient ring Q of R with respect to S then it is unique up to isomorphism,and it is denoted S − R . It is also said that the ring Q is the left localization of the ring R at S . Example 1 . Let S := S ∂ := { ∂ i , i ≥ } and R = I . Then ass( S ) = F , I / ass( S ) = B and theconditions (i)-(iii) hold where Q = B . This means that the ring B = I /F is the left quotientring of I at S , i.e. B ≃ S − ∂ I . Example 2 . Let S := S ∂ ,...,∂ n := { ∂ α , α ∈ N n } and R = I n . Then ass( S ∂ ,...,∂ n ) = a n , I n / a n = B n , and S − ∂ ,...,∂ n I n ≃ B n , (27)i.e. B n is the left quotient ring of I n at S ∂ ,...,∂ n . Note that the right localization I n S − ∂ ,...,∂ n of I n at S ∂ ,...,∂ n does not exist. Otherwise, we would have S − ∂ ,...,∂ n I n ≃ I n S − ∂ ,...,∂ n but all the elements ∂ α are left regular, and we would have a monomorphism I n → S − ∂ ,...,∂ n I n ≃ B n , which would beimpossible since the elements ∂ i of the algebra I n are not regular. By applying the involution ∗ to (27), we see that I n S − R ,..., R n ≃ B n , (28)i.e. the algebra B n is the right localization of I n at the multiplicatively closed set S R ,..., R n := { R α | α ∈ N n } .Given a ring R and modules R M and N R , we denote by pd( R M ) and pd( N R ) their projectivedimensions. Let us recall a result which will be used repeatedly in proofs later.It is obvious that P n ≃ A n / P ni =1 A n ∂ i . A similar result is true for the I n -module P n (Propo-sition 6.1.(2)). Note that pd A n ( P n ) = n but pd I n ( P n ) = 0 (Proposition 6.1.(3)). Proposition 6.1 I = I ∂ L I e and I = R I L e I .2. I n P n ≃ I n / P ni =1 I n ∂ i .3. The I n -module P n is projective.4. F n = F ⊗ n is a left and right projective I n -module.5. The projective dimension of the left and right I n -module I n /F n is . . For each element α ∈ N n , the I n -module I n / I n ∂ α is projective, moreover, I n / I n ∂ α ≃ L ni =1 ( K [ x i ] N N j = i I ( i )) α j .Proof . 1. Using the equality R ∂ = 1 − e , we see that I = I ∂ + I e . Since ∂e = 0 and e = e , we have I ∂ ∩ I e = ( I ∂ ∩ I e ) e ⊆ I ∂e = 0. Therefore, I = I ∂ L I e . Thenapplying the involution ∗ to this equality we obtain the equality I = R I L e I .2. Since I P ≃ I e = L i ∈ N Ke i , 1 e , we have I P ≃ I / I ∂ , by statement 1.Therefore, I n P n ≃ N ni =1 P ( i ) ≃ N ni =1 I ( i ) / I ( i ) ∂ i ≃ I n / P ni =1 I n ∂ i .3. By statement 1, I n = n O i =1 I ( i ) = n O i =1 ( I ( i ) ∂ i M I ( i ) e ( i )) = I n n Y i =1 e ( i ) M ( n X i =1 I n ∂ i ) ≃ P n M ( n X i =1 I n ∂ i ) . Therefore, P n is a projective I n -module.4. Note that the left I -module F = L i ≥ I E ii ≃ L i ≥ P is projective by statement 2.Therefore, F n = F ⊗ n is a projective left I n -module. Since the ideal F n is stable under theinvolution ∗ , F ∗ n = F n , the right I n -module F n is projective.5. The short exact sequence of left and right I n -modules 0 → F n → I n → I n /F n → F n is an essential left and right submodule of I n (Lemma 5.2.(2)). By statement 4, theprojective dimension of the left and right I n -module I n /F n is 1.6. Let Z n = L ni =1 Z e i where e , . . . , e n is the canonical free Z -basis for Z n . Let m = | α | . Fix achain of elements of Z n , β = 0 , β , . . . , β m = α such that, for all i , β i +1 = β i + e j for some index j = j ( i ). Then all the factors of the chain of left ideals I n ∂ α = I n ∂ β m ⊂ I n ∂ β m − ⊂ · · · ⊂ I n ∂ β ⊂ I n are projective I n -modules since I n ∂ β i / I n ∂ β i +1 ≃ I n / I n ∂ j ≃ K [ x j ] ⊗ I n − is the projective I n -module (statement 3). The first isomorphism is due to the fact that the element ∂ i is left regular,i.e. a∂ i = b∂ i implies a = b (by multiplying the equation on the right by R i ). Therefore, the I n -module I n / I n ∂ α is projective. Moreover, I n / I n ∂ α ≃ L ni =1 ( K [ x i ] N N j = i I ( i )) α j . (cid:3) Theorem 6.2
Let I n,m := B n − m ⊗ I m where m = 0 , , . . . , n and I = B := K . Then wdim( I n,m ) = n for all m = 0 , , . . . , n . In particular, wdim( I n ) = n .Proof . The algebra B n is Noetherian, hence n = l . gldim( S − ∂ ,...,∂ n I n,m ) = wdim( B n ) ≤ wdim( I n,m ) (Corollary 7.4.3, [49]). To finish the proof of the theorem it suffices to show thatthe inequality wdim( I n,m ) ≤ n holds for all numbers n and m . We use induction on n . Thecase n = 0 is trivial. So, let n ≥ n ′ < n and all m = 0 , , . . . , n ′ . For n , we use the second induction on m = 0 , , . . . , n . When m = 0, theinequality holds since I n, = B n and wdim( B n ) = n .Suppose that m > I n,m ′ ) ≤ n for all m ′ < m . We have to show that wdim( I n,m ) ≤ n or, equivalently, fd I n,m ( M ) ≤ n for all I n,m -modules M (fd denotes the flat dimension). Changingthe order of the tensor multiples we can write I n,m = I ⊗ I n − ,m − . Then wdim( I n − ,m − ) ≤ n − B = S − ∂ I = I /F and every ∂ -torsion I -module V isa direct sum of several (maybe an infinite number of) copies of the projective simple I -module K [ x ] (Proposition 6.1.(6)), hence V is projective, hence V is flat. Note that S − ∂ I n,m ≃ I n,m − andwdim( I n,m − ) ≤ n , by the inductive hypothesis. The I n,m -module tor ∂ ( M ) := { m ∈ M | ∂ i m = 0for some i } is the ∂ - torsion submodule of the I n,m -module M . There are two short exact sequencesof I n,m -modules, 0 → tor ∂ ( M ) → M → M → , (29)0 → M → S − ∂ M → M ′ → , (30)where the I n,m -modules tor ∂ ( M ) and M ′ are ∂ -torsion, and the I n -module S − ∂ M is ∂ -torsionfree. To prove that fd I n,m ( M ) ≤ n it suffices to show that the flat dimensions of the I n,m -modules tor ∂ ( M ), S − ∂ M and M ′ are less or equal to n . Indeed, then by (30), fd I n,m ( M ) ≤ { fd I n,m ( S − ∂ M ) , fd I n,m ( M ′ ) } ≤ n ; and by (29), fd I n,m ( M ) ≤ max { fd I n,m (tor ∂ ( M )) , fd I n,m ( M ) }≤ n .The I -module tor ∂ ( M ) (where I n,m = I ⊗ I n − ,m − ) is a direct sum of copies (may be infinitelymany) of the projective simple I -module K [ x ]. Note that End I ( K [ x ]) ≃ ker K [ x ] ( ∂ ) = K since I K [ x ] ≃ I / I ∂ . Using this fact and Proposition 6.1.(6), for each finitely generated submodule T of the I n,m -module tor ∂ ( M ) there exists a family { T i } i ∈ I of its submodules T i where ( I, ≤ ) is a well-ordered set such that if i, j ∈ I and i ≤ j then T i ⊆ T j , T = S i ∈ I T i and T i / S j
1. The module tor ∂ ( M ) = S θ ∈ Θ T θ is the union of its finitely generated submodules T θ , hencefd I n,m (tor ∂ ( M )) = fd I n,m ( [ θ ∈ Θ T θ ) ≤ sup { fd I n,m ( T θ ) } θ ∈ Θ = n − . Similarly, fd I n,m ( M ) ≤ n − I n,m -module M is ∂ -torsion.It remains to show that fd I n,m ( S − ∂ ( M )) ≤ n . By (27), the left I -module B is flat, hence theleft I n,m -module B ⊗ I n − ,m − is flat. Then, by Proposition 7.2.2.(ii), [49],fd I n,m ( S − ∂ M ) ≤ fd B ⊗ I n − ,m − ( S − ∂ M ) + fd I n,m ( B ⊗ I n − ,m − ) ≤ wdim( I n,m − ) ≤ n. The proof of the theorem is complete. (cid:3)
Corollary 6.3
Let M be a ∂ -torsion I n,m -module, i.e. S − ∂ M = 0 , where S − ∂ : I n,m = I ⊗ I n − ,m − → B ⊗ I n − ,m − = I n,m − is the localization map and n, m ≥ . Then there exists afamily { T i } i ∈ I of I n,m -submodules of M such that M = S i ∈ I T i , ( I, ≤ ) is a well-ordered set suchthat if i, j ∈ I and i ≤ j then T i ⊆ T j , and T i / S j
Let A be a prime factor algebra of the algebra I n . Then wdim( A ) = n .Proof . By Corollary 3.3.(10), the algebra A is isomorphic to the algebra I n,m for some m . Now,the corollary follows from Theorem 6.2. (cid:3) The next theorem is an analogue of Hilbert’s Syzygy Theorem for the algebra I n and its primefactor algebras. The flat dimension of an A -module M is denoted by fd A ( M ). Theorem 6.5
Let K be an algebraically closed uncountable field of characteristic zero. Let A be aprime factor algebra of I n (for example, A = I n ) and B be a Noetherian finitely generated algebraover K . Then wdim( A ⊗ B ) = wdim( A ) + wdim( B ) = n + wdim( B ) .Proof . Recall that A ≃ I n,m for some m ∈ { , , . . . , n } and wdim( I n,m ) = n (Theorem 6.2).Since n + wdim( B ) = wdim( I n,m ) + wdim( B ) ≤ wdim( I n,m ⊗ B ) , it suffices to show that wdim( I n,m ⊗ B ) ≤ n + wdim( B ) for all numbers n and m . We use inductionon n . The case n = 0 is trivial since A = K . So, let n ≥
1, and we assume that the inequalityholds for all n ′ < n and all m ′ = 0 , , . . . , n ′ . For the number n ≥
1, we use the second inductionon m = 0 , , . . . , n . The case m = 0, i.e. I n, = B n , is known, Corollary 6.3, [8] (this can also bededuced from Proposition 9.1.12, [49]; see also [9]).20o, let m > m ′ < m . Let M bean I n,m ⊗ B -module. We have to show that fd I n,m ⊗ B ( M ) ≤ n + wdim( B ). We can treat M asan I n,m -module. Then we have the short exact sequences (29) and (30) which are, in fact, shortexact sequence of I n,m ⊗ B -modules. To prove that fd I n,m ⊗ B ( M ) ≤ n + wdim( B ) it suffices toshow that the flat dimensions of the I n,m ⊗ B -modules tor ∂ ( M ), S − ∂ M and M ′ are less or equalto n + wdim( B ), by the same reason as in the proof of Theorem 6.2. Repeating the same argumentas at the end of the proof of Theorem 6.2, for each finitely generated submodule T of the I n,m ⊗ B -module tor ∂ ( M ) (where I n,m = I ⊗ I n − ,m − ) there exists a family { T i } i ∈ I of its submodules T i where ( I, ≤ ) is a well-ordered set such that if i, j ∈ I and i ≤ j then T i ⊆ T j , T = S i ∈ I T i and T i / S j
Let M be a module over an algebra A , I a non-empty well-ordered set, { M i } i ∈ I be a family of submodules of M such that if i, j ∈ I and i ≤ j then M i ⊆ M j . If M = S i ∈ I M i and pd A ( M i /M I n,m ′ ) ≤ n + m ′ for all m ′ < m . We have to show thatgldim( I n,m ) ≤ n + m , or equivalently pd I n,m ( M ) ≤ n + m for all I n,m -modules M . Changing21he order of the tensor multiples we can write I n,m = I ⊗ I n − ,m − . For the I n,m -module M we have the short exact sequence of I n,m -modules (29). By Corollary 6.3, for the I n,m -moduletor ∂ ( M ) there exists a family { T i } i ∈ I of its submodules T i where ( I, ≤ ) is a well-ordered set suchthat if i, j ∈ I and i ≤ j then T i ⊆ T j , tor ∂ ( M ) = S i ∈ I T i and T i / S j
2. Note that pd I n,m ( I n,m − ) ≤ → F → I → B → I -module B ) and, by Proposition 7.2.2.(ii), [49],pd I n,m ( M ) ≤ pd I n,m − ( M ) + pd I n,m ( I n,m − ) ≤ n + m − n + m. By (29), pd I n,m ( M ) ≤ max { pd I n,m (tor ∂ ( M )) , pd I n,m ( M ) } ≤ n + m, as required. The proof of the theorem is complete. (cid:3) Conjecture . gldim( I n ) = n . A n In this section, we prove that the weak dimension of the Jacobian algebra A n and of all its primefactor algebras is n (Theorem 7.2, Corollary 7.3). An analogue of Hilbert’s Syzygy Theorem isestablished for the Jacobian algebras A n and for all its prime factor algebras (Theorem 7.4).A K -algebra R has the endomorphism property over K if, for each simple R -module M ,End R ( M ) is algebraic over K . Theorem 7.1 [10]
Let K be a field of characteristic zero.1. The algebra A n is a simple, affine, Noetherian domain.2. The Gelfand-Kirillov dimension GK ( A n ) = 3 n ( = 2 n = GK ( A n )) .3. The (left and right) global dimension gl . dim( A n ) = n .4. The (left and right) Krull dimension K . dim( A n ) = n .5. Let d = gl . dim or d = K . dim . Let R be a Noetherian K -algebra with d( R ) < ∞ such that R [ t ] , the polynomial ring in a central indeterminate, has the endomorphism property over K . Then d( A ⊗ R ) = d( R ) + 1 . If, in addition, the field K is algebraically closed anduncountable, and the algebra R is affine, then d( A n ⊗ R ) = d( R ) + n . GK ( A ) = 3 is due to A. Joseph [41], p. 336; see also [47], Example 4.11, p. 45. The Jacobian algebra A n is a localization of the algebra I n . Using the presentations ofthe algebras I n and A n as GWAs, it is obvious that the algebra I n is the two-sided localization, A n = S − I n = I n S − , (31)of the algebra I n at the multiplicatively closed subset S := { Q ni =1 ( H i + α i ) n i ∗ | ( α i ) ∈ Z n , ( n i ) ∈ N n } of I n where ( H i + α i ) ∗ := ( H i + α i if α i ≥ , ( H i + α i ) if α i < , since, for all elements β ∈ Z n , v β n Y i =1 ( H i + α i ) n i ∗ = n Y i =1 ( H i + α i − β i ) n i ∗ v β . (32)22he left (resp. right) localization of the Jacobian algebra A n = K h y , . . . , y n , H ± , . . . , H ± n , x , . . . , x n i , (where y i := H − i x i )at the multiplicatively closed set S y ,...,y n := { y α | α ∈ N n } (resp. S x ,...,x n := { x α | α ∈ N n } ) isthe algebra A n ≃ S − y ,...,y n A n ≃ A n S − x ,...,x n . (33)The algebra A n has the involution ∗ . The algebra A n ≃ A n / a e inherits the involution ∗ since( a en ) ∗ = a en , and so do the algebras A n,m := A n − m ⊗ A m where m = 0 , , . . . , n and A = A := K .Therefore, the algebras A n,m are self-dual, and so l . gldim( A n,m ) = r . gldim( A n,m ) := gldim( A n,m ). Theorem 7.2
Let A n,m := A n − m ⊗ A m where m = 0 , , . . . , n and A = A := K . Then wdim( A n,m ) = n for all m = 0 , , . . . , n . In particular, wdim( A n ) = n .Proof . By Theorem 7.1.(1,3) and (33), n = gldim( A n ) = wdim( A n ) = l . gldim( S − y ,...,y n A n ) ≤ wdim( S − y ,...,y n − m A n ) = wdim( A n,m ) ≤ wdim( I n,m ) = n (by (31) and Theorem 6.2) . Therefore, wdim( A n,m ) = n for all n and m . (cid:3) Corollary 7.3
Let A be a prime factor algebra of the algebra A n . Then wdim( A ) = n .Proof . By Corollary 3.5, [11], the algebra A is isomorphic to the algebra A n,m for some m .Now, the corollary follows from Theorem 7.2. (cid:3) The next theorem is an analogue of Hilbert’s Syzygy Theorem for the Jacobian algebras andtheir prime factor algebras.
Theorem 7.4
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