Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras

Abstract

In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series.

Computation of point modules of finitely semi-graded rings

“…The second and the third sections contain the novelty of the paper. The main results are Theorem 2.7, which completes the parametrization started in [10], and Theorem 3.1 where we apply the results of the previous sections to compute the point modules of some skew P BW extensions, and hence, the point modules of some examples of non N-graded quantum algebras. Concrete examples are presented in Examples 3.2 and 3.3.…”

“…In this section we complete the parametrization of point modules of F SG rings that was initiated in [10]. The main result of the present section is Theorem 2.7 below.…”

“…The main result of the present section is Theorem 2.7 below. For completeness, we include the basic facts of parametrization studied in [10], omitting the proofs. Definition 2.1.…”

“…Thus, the semi-graded rings are nice objects for the investigation of geometric properties of non-commutative algebras that are not N-graded, and include as a particular case the class of finitely graded algebras. In [10] was initiated the study the point modules of finitely semi-graded rings, in the present paper we complete this investigation and we will compute the point modules of finitely semi-graded rings generated in degree one. In particular, from the parametrization of the point modules for the quantum affine n-space, the set of point modules for some important examples of non N-graded quantum algebras is computed The paper is organized as follows: In the first section we recall some basic facts and examples on point modules for classical N-graded algebras, we include a complete proof (usually not available in the literature) of the parametrization of the point modules for the quantum affine n-space (Example 1.7); we review the definition and elementary properties of semi-graded rings, in particular, the subclass of finitely semi-graded rings and modules.…”

Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions

“…Last, but not least, we consider as a possible future work to investigate minimal prime ideals and the 2-primal property in a more general context of noncommutative rings than skew PBW extensions such as for example the semi-graded rings introduced by Lezama [33] (see also [29] and [32]), or maybe considering a weak notion of compatibility following the ideas presented recently by the second author [57], and of course, in the setting of modules over these extensions, see [36], [42], and [48].…”