scholarly journals Non-commutative algebraic geometry of semi-graded rings

2017 ◽  
Vol 27 (04) ◽  
pp. 361-389 ◽  
Author(s):  
Oswaldo Lezama ◽  
Edward Latorre
Keyword(s):  
Algebraic Geometry ◽  
Hilbert Series ◽  
Commutative Rings ◽  
Hilbert Polynomial ◽  
Graded Rings ◽  
Projective Scheme ◽  
New Type ◽  
Kirillov Dimension

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

scholarly journals A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

2009 ◽  
Vol 52 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Jakob Cimprič
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Representation Theorem ◽  
Commutative Rings ◽  
Real Algebraic Geometry ◽  
New Approach ◽  
Noncommutative Version ◽  
Quadratic Modules ◽  
Rings With Involution ◽  
Associative Rings

AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

scholarly journals COMBINATORICS AND N-KOSZUL ALGEBRAS

2008 ◽  
Vol 05 (08) ◽  
pp. 1205-1214
Author(s):  
ROLAND BERGER
Keyword(s):  
Hilbert Series ◽  
Natural Generalization ◽  
Koszul Algebras ◽  
Polynomial Algebras ◽  
New Type ◽  
Master Theorem

The numerical Hilbert series combinatorics for quadratic Koszul algebras was extended to N-Koszul algebras by Dubois-Violette and Popov [9]. In this paper, we give a striking application of this extension when the relations of the algebra are all the antisymmetric tensors of degree N over given variables. Furthermore, we present a new type of Hilbert series combinatorics, called comodule Hilbert series combinatorics, and due to Hai, Kriegk and Lorenz [15]. When the relations are all the antisymmetric tensors, a natural generalization of the MacMahon Master Theorem (MMT) is obtained from the comodule level, the original MMT corresponding to N = 2 and to polynomial algebras.

Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz

1975 ◽  
Vol 77 (1) ◽  
pp. 7-19 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Algebraic Geometry ◽  
Matrix Ring ◽  
Commutative Rings ◽  
Commutative Case ◽  
Matrix Rings ◽  
Finite Dimensional ◽  
Existentially Closed ◽  
Closed Fields ◽  
Skew Fields ◽  
Algebraically Closed Fields

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.

On the length formula of Hoskin and Deligne and associated graded rings of two-dimensional regular local rings

1992 ◽  
Vol 111 (3) ◽  
pp. 423-432 ◽  
Author(s):  
Bernard L. Johnston ◽  
Jugal Verma
Keyword(s):  
Hilbert Series ◽  
Primary Ideal ◽  
Local Rings ◽  
Classical Theorem ◽  
Two Dimensional ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Associated Graded Rings ◽  
Image Position

Let (R, m) be a 2-dimensional regular local ring and I an m-primary ideal. The aim of this paper is to find conditions on I so that the associated graded ring of I,and the Rees ring of I,where t is an indeterminate, are Cohen–Macaulay (resp. Gorenstein). To this end, we use the results and techniques from Zariski's theory of complete ideals ([14], appendix 5) and its later generalizations and refinements due to Huneke [7] and Lipman[8]. The main result is an application of three deep theorems: (i) a generalization of Macaulay's classical theorem on Hilbert series of Gorenstein graded rings [13], (ii) a generalization of the Briançon–Skoda theorem due to Lipman and Sathaye [9], and (iii) a formula for the length of R/I, where I is a complete m-primary ideal, due to Hoskin[4] and Deligne[1].

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

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 881 ◽  
Author(s):  
Oswaldo Lezama ◽  
Jaime Gomez
Keyword(s):  
Mirror Symmetry ◽  
Hilbert Series ◽  
Poincaré Series ◽  
Graded Rings ◽  
Yoneda Algebra ◽  
Graded Algebras ◽  
Poincare Series ◽  
Rings And Algebras ◽  
Lattice Of Ideals ◽  
Point Modules

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.

The Gelfand-Kirillov dimension and the Hilbert polynomial

2003 ◽  
pp. 239-261
Author(s):  
José Bueso ◽  
José Gómez-Torrecillas ◽  
Alain Verschoren
Keyword(s):  
Hilbert Polynomial ◽  
Kirillov Dimension

scholarly journals Hilbert series and applications to graded rings

2003 ◽  
Vol 2003 (7) ◽  
pp. 397-403 ◽  
Author(s):  
Selma Altinok
Keyword(s):  
Hilbert Series ◽  
Graded Rings ◽  
Singular Surfaces ◽  
Explicit Formulas ◽  
Small Denominator ◽  
Small Codimension

This paper contains a number of practical remarks on Hilbert series that we expect to be useful in various contexts. We use the fractional Riemann-Roch formula of Fletcher and Reid to write out explicit formulas for the Hilbert seriesP(t)in a number of cases of interest for singular surfaces (see Lemma 2.1) and3-folds. IfXis aℚ-Fano3-fold andS∈ |−KX|aK3surface in its anticanonical system (or the general elephant ofX), polarised withD=𝒪S (−KX), we determine the relation betweenPX(t)andPS,D(t). We discuss the denominator∏(1−tai)ofP(t)and, in particular, the question of how to choose a reasonably small denominator. This idea has applications to findingK3surfaces and Fano3-folds whose corresponding graded rings have small codimension. Most of the information about the anticanonical ring of a Fano3-fold orK3surface is contained in its Hilbert series. We believe that, by using information on Hilbert series, the classification ofℚ-Fano3-folds is too close. FindingK3surfaces are important because they occur as the general elephant of aℚ-Fano 3-fold.

scholarly journals A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type

2020 ◽  
Vol 16 (31) ◽  
pp. 27-52
Author(s):  
Armando Reyes ◽  
Jason Hernández-Mogollón
Keyword(s):  
Algebraic Geometry ◽  
Commutative Rings ◽  
Ring Theory ◽  
Noncommutative Ring ◽  
Polynomial Type ◽  
Noncommutative Ring Theory

In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.

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