Some homological properties of skew PBW extensions arising in non-commutative algebraic geometry

In this short paper we study for the skew PBW (Poincar\'e-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the parametrization of its point modules.


Introduction
In the study of non-commutative algebraic geometry many important classes of non-commutative rings and algebras have been investigated intensively in the last years. Actually, the non-commutative algebraic geometry consists in generalizing some classical results of commutative algebraic geometry to some special non-commutative rings and algebras. Probably the most important types of such rings are the Artin-Schelter regular algebras, Auslander regular algebras, the Auslander Gorenstein rings, the Cohen-Macaulay rings, and the strongly Noetherian algebras. In non-commutative algebraic geometry, Artin-Schelter regular algebras are the analog of the commutative polynomials in commutative algebraic geometry, and in addition, all examples studied of Artin-Schelter regular algebras are Auslander regular, Auslander-Gorenstein, Cohen-Macaulay, and strongly Noetherian. In the present paper we are interested in the Auslander-Gorenstein, Cohen-Macaulay, and strongly Noetherian conditions for skew P BW extensions (the Auslander regular condition was investigated in [20]). Skew P BW extensions are rings of polynomial type and generalize the classical P BW extensions (see [14], [15]). Many ring-theoretical and homological properties of skew P BW extensions have been studied in the last years (Hilbert basis theorem, global dimension, Krull dimension, Goldie dimension, prime ideals, etc., see [1], [2], [15], [20]); however, the three homological conditions mentioned before arising in non-commutative algebraic geometry have not been considered for the skew P BW extensions.
In the first and second sections we recall the definition of skew P BW extensions and some important examples of this type of non-commutative rings. In the third section we discuss the Auslander-Gorenstein condition. The next two sections are dedicated to investigate the Cohen-Macaulay and the strongly Noetherian properties. In the last section we present an application of the strongly noetherianity to parametrize the point modules of some key skew P BW extensions. Definition 1.1 ( [14]). Let R and A be rings. We say that A is a skew P BW extension of R (also called a σ − P BW extension of R) if the following conditions hold: (ii) There exist finitely many elements x 1 , . . . , x n ∈ A such A is a left R-free module with basis The set M on(A) is called the set of standard monomials of A.
Associated to a skew P BW extension A = σ(R) x 1 , . . . , x n , there are n injective endomorphisms σ 1 , . . . , σ n of R and σ i -derivations, as the following proposition shows. Proposition 1.2. Let A be a skew P BW extension of R. Then, for every 1 ≤ i ≤ n, there exist an injective ring endomorphism σ i : R → R and a σ i -derivation δ i : R → R such that Proof. See [14], Proposition 3.
A particular case of skew P BW extension is when all derivations δ i are zero. Another interesting case is when all σ i are bijective and the constants c ij are invertible. We recall the following definition (cf. [14]).
(iv) Let f be as in (iii), then deg(f ) := max{deg(X i )} t i=1 . The next theorems establish some results for skew P BW extensions that we will use later, for their proofs see [15]. Theorem 1.5. Let A be an arbitrary skew P BW extension of the ring R. Then, A is a filtered ring with filtration given by and the corresponding graded ring Gr(A) is a quasi-commutative skew P BW extension of R. Moreover, if A is bijective, then Gr(A) is quasi-commutative bijective skew P BW extension of R. Theorem 1.6. Let A be a quasi-commutative skew P BW extension of a ring R. Then,

1.
A is isomorphic to an iterated skew polynomial ring of endomorphism type, i.e., 2. If A is bijective, then each endomorphism θ i is bijective, 1 ≤ i ≤ n. Theorem 1.7 (Hilbert Basis Theorem). Let A be a bijective skew P BW extension of R. If R is a left (right) Noetherian ring then A is also a left (right) Noetherian ring.

Examples
In order to understand the importance of results of the next three sections, next we recall some examples of skew P BW extensions; for more details see [15] and [20]. K denotes a field. Example 1.8 (P BW extensions, [7]). Any P BW extension is a bijective skew P BW extension since in this case σ i = i R for each 1 ≤ i ≤ n, and c i,j = 1 for every 1 ≤ i, j ≤ n. Thus, for P BW extensions we have A = i(R) x 1 , . . . , x n . Some examples of P BW extensions are the following: (b) Any skew polynomial ring of derivation type A = R[x; σ, δ], i.e., with σ = i R . In general, any Ore extension of derivation type R[x 1 ; σ 1 , δ 1 ] · · · [x n ; σ n , δ n ], i.e., such that σ i = i R , for any 1 ≤ i ≤ n. (d) The universal enveloping algebra of a finite dimensional Lie algebra U(G). In this case, x j ] ∈ G = K + Kx 1 + · · · + Kx n , for any r ∈ K and 1 ≤ i, j ≤ n.
(e) The crossed product R * U(G), in particular, the tensor product A : Example 1.9 (Ore extensions of bijective type). Any skew polynomial ring R[x; σ, δ] of bijective type, i.e., with σ bijective, is a bijective skew P BW extension. In this case we have be an iterated skew polynomial ring of bijective type, i.e., the following conditions hold: • for 1 ≤ i ≤ n, σ i is bijective; • for every r ∈ R and 1 ≤ i ≤ n, σ i (r) ∈ R, δ i (r) ∈ R; • for i < j, σ j (x i ) = cx i + d, with c, d ∈ R and c has a left inverse; Some concrete examples of Ore algebras of bijective type are the following: The algebra for multidimensional discrete linear systems.
Example 1.10 (Operator algebras). Some important and well-known operator algebras like: (a) Algebra of linear partial differential operators.
(b) Algebra of linear partial shift operators.
(c) Algebra of linear partial difference operators.
(d) Algebra of linear partial q−dilations operators.
(h) The algebra U .
(i) The coordinate algebra of the quantum matrix space M q (2).
(m) Quantum Weyl algebra of Maltsiniotis A q,λ n . (n) The algebra of differential operators D q (S q ) on a quantum space S q .
These algebras are examples of bijective skew P BW extensions.
Example 1.14 (n−multiparametric skew quantum space over R). Let R be a ring with a fixed matrix of parameters q := [q ij ] ∈ M n (R), n ≥ 2, such that q ii = 1 = q ij q ji for every 1 ≤ i, j ≤ n, and suppose also that it is given a system σ 1 , . . . , σ n of automorphisms of R. The quasi-commutative bijective skew P BW extension R q,σ [x 1 , . . . , s n ] defined by is called the n−multiparametric skew quantum space over R. When all automorphisms of the extension are trivial, i.e., x n ] is called n−multiparametric skew quantum space, and the case particular case n = 2 is called skew quantum plane, for trivial automorphisms we have the n−multiparametric quantum space and the quantum plane.
Example 1.15 (Skew quantum polynomials). Let R be a ring with a fixed matrix of parameters q := [q ij ] ∈ M n (R), n ≥ 2, such that q ii = 1 = q ij q ji for every 1 ≤ i, j ≤ n, and suppose also that it is given a system σ 1 , . . . , σ n of automorphisms of R. The ring of skew quantum polynomials over R, denoted by Q r,n q,σ := R q,σ [x ±1 1 , . . . , x ±1 r , x r+1 , . . . , x n ], is defined as follows: r , x r+1 , . . . , x n ] is a free left R−module with basis {x α1 1 · · · x αn n |α i ∈ Z for 1 ≤ i ≤ r and α i ∈ N for r + 1 ≤ i ≤ n}.

Auslander regularity conditions
In this section we will study the Auslander regularity and the Auslander-Gorenstein conditions for skew P BW extensions. The first condition was studied by Björk in [8], [9] and by Ekström in [10] for filtered Zariski rings and Ore extensions; in [20] was studied the Auslander condition for skew P BW extensions. We will consider the second condition, but for completeness, we will integrate the first one in the statements of the results below. (iii) A is Auslander-Gorenstein (AG) if A is Noetherian (i.e., two-sided Noetherian), which satisfies the Auslander condition, id( A A) < ∞, and id(A A ) < ∞.

Definition 2.3. Let
A be a filtered ring with filtration {F n (A)} n∈Z .

The Rees ring associated to A is a graded ring defined by
A := n∈Z F n (A).
2. The filtration {F n (A)} n∈Z is left (right) Zariskian, and A is called a left (right) Zariski ring, if F −1 (A) ⊆ Rad(F 0 (A)) and the associated Rees ring A is left (right) Noetherian.
Proposition 2.6. If A is AG, respectively AR, then the skew polynomial ring A[x; σ, δ] with σ bijective is also AG, respectively AR.
The following proposition states that the AG (AR) conditions are preserved under arbitrary localizations.
Proposition 2.7. Let A be an AG ring, respectively AR, and S a multiplicative Ore set of regular elements of A. Then so too is S −1 A (and also AS −1 ).

Lemma 2.8. If
A is a bijective skew P BW extension of a Noetherian ring R, then A is a left and right Zariski ring.
Proof. Since A is N−filtered, 0 = F −1 (A) ⊆ Rad(F 0 (A)) = Rad(R). By Theorem 1.6, Gr(A) is isomorphic to an iterated skew polynomial ring R[z 1 ; θ 1 ] · · · [z n ; θ n ], with θ i is bijective, 1 ≤ i ≤ n. Whence Gr(A) is Noetherian. Proposition 2.4 says that A is a left and right Zariski ring. Theorem 2.9. Let A be a bijective skew P BW extension of a ring R such that R is AG, respectively AR, then so too is A.
Proof. Acoording to Theorem 1.5, Gr(A) is a quasi-commutative skew P BW extension, and by the hypothesis, Gr(A) is also bijective. By Theorem 1.6, Gr(A) is isomorphic to an iterated skew polynomial ring R[z 1 ; θ 1 ] . . . [z n ; θ n ] such that each θ i is bijective, 1 i n. Proposition 2.6 says that Gr(A) is AG, respectively AR. From Lemma 2.8, A is a left and right Zariski ring, so by Proposition 2.5, A is AG, respectively AR. Corollary 2.10. If R is AG, respectively AR, then the ring of skew quantum polynomials is AG, respectively AR.
Proof. Let R be AG; according to Example 1.15, Q r,n q,σ (R) is a localization of a bijective skew P BW extension A of the ring R by a multiplicative Ore set of regular elements of A. From Theorem 2.9 and Proposition 2.7 we get that Q r,n q,σ (R) is AG. If R is AR, then R is AG and gld(R) < ∞, then gld(Q r,n q,σ (R)) < ∞ and Q r,n q,σ (R) is AG, so Q r,n q,σ (R) is AR.
Example 2.11. All examples in the section 1.1 are AG rings (respectively AR) assuming that the ring of coefficients (R or K) is AG (AR). If the ring the coefficients is a field, then of course, it is an AG ring (respectively AR).

Cohen-Macaulayness
In this section we study the Cohen-Macaulay property for skew P BW extensions.
Definition 3.1. Let A be an algebra over a field K, we say that A is Cohen-Macaulay (CM) with respect to the classical Gelfand-Kirillov dimension if: for every non-zero Noetherian A−module M .
Recall that if A is a K-algebra, then the classical Gelfand-Kirillov dimension of A is defined by where V vanishes over all frames of A and V n := K v 1 · · · v n |v i ∈ V (a frame of A is a finite dimensional K-subspace of A such that 1 ∈ V ; since A is a K-algebra, then K ֒→ A, and hence, K is a frame of A of dimension 1). For a K−algebra A, an automorphism σ of A is said to be locally algebraic if for any a ∈ A the set {σ m (a)|m ∈ N} is cointained in a finite dimensional subspace of A.
The classical Gelfand-Kirillov dimension for skew P BW extensions was studied in [19] (in [16] has been studied the Gelfand-Kirillov dimension of skew P BW extensions assuming that the ring R of coefficients is a left Noetherian domain).
Proposition 3.2. Let R be a K−algebra with a finite dimensional generating subspace V and let A = σ(R) x 1 , . . . , x n be a bijective skew P BW extension of R. If σ n (V ) ⊆ V or σ n is locally algebraic, then The following proposition in the classical case is also known. Proof. See [11], Proposition 6.6.  (Gr(M )) (see [9], Proof of Theorem 3.9). Now, it is possible to prove the following proposition. Therefore A is CM .
Proposition 3.6. Suppose that R is AR (AG) and CM ring. Let R[x; σ, δ] be an Ore extension with σ bijective.
It is clear that every central element is local. The next proposition says that CM property is preserved under certain localizations.
Proposition 3.8. Let A be an AG ring, and S a multiplicatively closed set of local normal elements in Proof. cf. [3], Theorem 2.4. Theorem 3.9. Let A be a bijective skew P BW extension of a ring R such that R is AG, CM , and R = i≥0 ⊕R i is a connected graded K−algebra such that σ j (R i ) ⊆ R i for each i ≥ 0 and 1 ≤ j ≤ n, then A is CM .
Proof. From Theorem 1.5 it is clear that A is a K−algebra with a finite filtration and Gr(A) is a quasicommutative skew P BW extensions, and by the hypothesis, Gr(A) is also bijective. By Theorem 1.6, Gr(A) is isomorphic to an iterated skew polynomial ring R[z 1 ; θ 1 ] . . . [z n ; θ n ] such that each θ i is bijective, 1 i n (according to the proof of Theorem 1.6 in [15], θ j (r) = σ j (r) for every r ∈ R). Proposition 2.6 says that Gr(A) is AG. From Lemma 2.8, A is a left and right Zariski ring, and by Proposition 3.6 Gr(A) is CM , so by Proposition 3.5 A is CM . Corollary 3.10. Let R = i≥0 ⊕R i be a connected graded K−algebra such that σ j (R i ) ⊆ R i for every i ≥ 0 and all σ j in definition of skew quantum polynomials Q r,n q,σ . If R is AG and CM , then Q r,n q,σ is CM . Proof. By Example 1.15, Q r,n q,σ is a localization of a bijective skew P BW extension A of R by a multiplicative Ore set of regular elements of A, and the multiplicative set generated by x 1 , . . . , x r consists of monomials, which are local normal elements. From Theorem 3.9 and Proposition 3.8 we get that Q r,n q,σ is CM .

Strongly noetherian algebras
Now we will consider the strongly noetherian property for skew P BW extensions, this condition was studied by Artin, Small and Zhang in [4], and appears naturally in the study of point modules in noncommutative algebraic geometry (see [5], [6] and [21]).
Definition 4.1. Let K be a commutative ring and let A be a left Noetherian K-algebra. We say that A is left strongly Noetherian if for any commutative Noetherian K-algebra C, C ⊗ K A is left Noetherian.
Some examples of strongly Noetherian algebras include Weyl algebras, Sklyanin algebras over a field K and universal enveloping algebras of finite dimensional Lie algebras (see [4], Corollaries 4.11 and 4.12). Moreover, all known examples of Artin-Shelter regular algebras are strongly Noetherian, is an open question if every Artin-Shelter regular algebra is strongly Noetherian.
Proposition 4.2. Let K be a commutative ring and let A be a K−algebra.
(i) If A is left strongly Noetherian, then A[x; σ, δ] is left strongly Noetherian when σ is bijective.
(ii) If A is N−filtered and Gr(A) is left strongly Noetherian, then A is left strongly Noetherian.
(iii) If S is a multiplicative Ore set of regular elements of A, then S −1 A is left strongly Noetherian.
With the previous result we get the main result of the present section.   5 Point modules of some skew P BW extensions As application of strongly Noetherian algebras, we present next some examples of finitely graded algebras that are bijective skew P BW extensions and such that they have nice spaces parametrizing its point modules. It is important to remark that Artin, Tate and Van den Bergh studied point modules in order to complete the classification of Artin-Shelter regular algebras of dimension 3 (see [5]).
Recall that a K-algebra A is finitely graded if it is connected N−graded and finitely generated as a K−algebra.
Definition 5.1. Let A be a finitely graded K−algebra that is generated in degree 1. A point module for A is a graded left module M such that M is cyclic, generated in degree 0, and dim K (M n ) = 1 for all n ≥ 0.
If A is commutative, then its point modules naturally correspond to the (closed) points of the scheme proj(A). Similarly, many non-commutative graded rings also have nice parameter spaces of point modules. For example, the point modules of the quantum plane or Jordan plane are parametrized by P 1 , i.e., there exists a bijective correspondence between the projective space P 1 and the collection of isomorphism classes of point modules of the quantum plane or Jordan plane (see [21], Example 3.2. ). The following proposition states that for finitely graded strongly Noetherian algebras is possible to find a projective scheme that parametrizes the set of its point modules.
Proposition 5.2 ([6], Corollary [4.12). Let A be a finitely graded K−algebra which is generated in degree 1. If A is strongly Noetherian, then the point modules of A are naturally parametrized by a commutative projective scheme over K.
(iv) Multiplicative analogue of the Weyl algebra.
According to Theorem 4.3 they are left strongly noetherian and by Proposition 5.2 there exists a commutative projective scheme that parametrizes the set of point modules of A .
Corollary 5.3. Let A be a bijective skew P BW extension as above, then there exists a commutative projective scheme over K that parametrizes the set of point modules of A.