scholarly journals On the set of Hilbert polynomials

2001 ◽  
Vol 64 (2) ◽  
pp. 291-305 ◽  
Author(s):  
Alexander B. Levin
Keyword(s):  
Chromatic Polynomial ◽  
Hilbert Polynomial ◽  
Graded Algebra ◽  
Open Problems ◽  
Graded Algebras ◽  
Hilbert Polynomials

We characterise the set of all Hilbert polynomials of standard graded algebras over a field and give solutions of some open problems on Hilbert polynomials. In particular, we prove that a chromatic polynomial of a graph is a Hilbert polynomial of some standard graded algebra.

Circle Actions

2020 ◽  
pp. 161-166
Author(s):  
Loring W. Tu
Keyword(s):  
Differential Form ◽  
Circle Action ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Circle Actions ◽  
Algebra Isomorphism ◽  
Weil Algebra ◽  
Differential Graded Algebras ◽  
Cartan Model

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

scholarly journals Mora’s holy graal: Algorithms for computing in localizations at prime ideals

2015 ◽  
Vol 25 (07) ◽  
pp. 1125-1143 ◽  
Author(s):  
Magdaleen S. Marais ◽  
Yue Ren
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Computational Approach ◽  
Prime Ideals ◽  
Graded Algebra ◽  
Coordinate Ring ◽  
Localization Formula ◽  
Hilbert Polynomials ◽  
Systems Of Parameters ◽  
Appl Math

This paper discusses a computational treatment of the localization [Formula: see text] of an affine coordinate ring [Formula: see text] at a prime ideal [Formula: see text] and its associated graded algebra [Formula: see text] with the means of computer algebra. Building on Mora’s paper [T. Mora, La queste del Saint [Formula: see text]: A computational approach to local algebra, Discrete Appl. Math. 33 (1991) 161–190], we present shorter proofs on two of the central statements and expand on the applications touched by Mora: resolutions of ideals, systems of parameters and Hilbert polynomials, as well as dimension and regularity of [Formula: see text]. All algorithms are implemented in the library graal.lib for the computer algebra system Singular.

Approximable Algebras as Subalgebras of Section Rings

2020 ◽  
Author(s):  
Catriona Maclean
Keyword(s):  
Line Bundle ◽  
Projective Variety ◽  
Type Theorem ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Graded Ring ◽  
Partial Converse ◽  
Big Line Bundle

Abstract In [2], Huayi Chen introduced approximable graded algebras, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false in [ 8]. Continuing the analysis started in [8], we show that while not every approximable graded algebra is a sub algebra of the section ring of a big line bundle, we can associate to any approximable graded algebra $\textbf{B}$ a projective variety $X(\textbf{B})$ and an infinite divisor $D(\textbf{B}) =\sum _{i=1}^\infty a_i D_i$ with $a_i\rightarrow 0$ such that $\textbf{B}$ is a subalgebra of $$\begin{equation*} R( D(\textbf{B}))=\oplus_n H^0(X(\textbf{B}), n D(\textbf{B})).\end{equation*}$$We also establish a partial converse to these results by showing that if an infinite divisor $D=\sum _i a_iD_i$ converges in the space of numerical classes, then any full-dimensional sub-graded algebra of $\oplus _mH^0(X, \lfloor mD \rfloor ))$ is approximable.

scholarly journals Tight closure of powers of ideals and tight hilbert polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 335-355
Author(s):  
KRITI GOEL ◽  
J. K. VERMA ◽  
VIVEK MUKUNDAN
Keyword(s):  
Local Ring ◽  
Hilbert Function ◽  
Simplicial Complexes ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Prime Characteristic ◽  
Tight Closure ◽  
Powers Of Ideals ◽  
Hilbert Polynomials

AbstractLet (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

Differential Graded Algebras

2020 ◽  
pp. 145-150
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Algebraic Model ◽  
Lie Derivative ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Rham Complex ◽  
De Rham

This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω‎(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω‎ with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.

scholarly journals (q,σ,τ)-Differential Graded Algebras

Universe ◽  
2018 ◽  
Vol 4 (12) ◽  
pp. 138 ◽  
Author(s):  
Viktor Abramov ◽  
Olga Liivapuu ◽  
Abdenacer Makhlouf
Keyword(s):  
Clifford Algebra ◽  
Quantum Plane ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Generalized Clifford Algebra

We propose the notion of ( q , σ , τ ) -differential graded algebra, which generalizes the notions of ( σ , τ ) -differential graded algebra and q-differential graded algebra. We construct two examples of ( q , σ , τ ) -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a ( σ , τ ) -twisted graded q-commutator. In order to construct the second example, we introduce the notion of ( σ , τ ) -pre-cosimplicial algebra.

Division Graded Algebras in the Brauer-Wall Group

1996 ◽  
Vol 39 (1) ◽  
pp. 21-24 ◽  
Author(s):  
Francis Coghlan ◽  
Peter Hoffman
Keyword(s):  
Division Algebra ◽  
Direct Consequence ◽  
Structure Theory ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Central Simple ◽  
Wall Group

AbstractWe show that every element in the Brauer-Wall group of a field with characteristic different from 2 is represented uniquely by a division graded algebra, (i.e. homogeneous elements are invertible) but, of course, not necessarily by a graded (division algebra). This is a fairly direct consequence of Wall's structure theory for central simple Z/2-graded algebras.

scholarly journals Hilbert polynomials of the algebras of $SL_ 2$-invariants

2018 ◽  
Vol 10 (2) ◽  
pp. 303-312
Author(s):  
N.B. Ilash
Keyword(s):  
Quadratic Forms ◽  
Hypergeometric Series ◽  
Hilbert Function ◽  
Dimensional Space ◽  
Hilbert Polynomial ◽  
Binary Forms ◽  
Hilbert Polynomials ◽  
Joint Invariants ◽  
Narayana Numbers ◽  
Classical Invariant

We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group $SL_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in algebraic geometry. It is well known that the Hilbert function of the algebra $SL_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form $\mathcal{C}_{d}= \mathbb{C}[V_d\oplus \mathbb{C}^2]_{SL_2}$ (here $V_d$ is the $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was generalized to the algebra of joint invariants for $n$ binary forms. But the Cayley-Sylvester formula is not expressed in terms of polynomials.In our article we consider the problem of computing the Hilbert polynomials for the algebras of joint invariants and joint covariants of $n$ linear forms and $n$ quadratic forms. We express the Hilbert polynomials $\mathcal{H} \mathcal{I}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i, \mathcal{H}(\mathcal{C}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i,$ $\mathcal{H}(\mathcal{I}^{(n)}_2,i)=\dim(\mathcal{I}^{(n)}_2)_i, \mathcal{H}(\mathcal{C}^{(n)}_2,i)=\dim(\mathcal{C}^{(n)}_2)_i$ of those algebras in terms of quasi-polynomial. We also present them in the form of Narayana numbers and generalized hypergeometric series.

scholarly journals GRADED MORITA EQUIVALENCES FOR GEOMETRIC AS-REGULAR ALGEBRAS

2012 ◽  
Vol 55 (2) ◽  
pp. 241-257 ◽  
Author(s):  
IZURU MORI ◽  
KENTA UEYAMA
Keyword(s):  
Algebraic Geometry ◽  
Three Dimensional ◽  
Koszul Algebras ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Regular Algebra ◽  
Morita Equivalent ◽  
Regular Algebras

AbstractClassification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

scholarly journals Cohomologically Full Rings

2019 ◽  
Author(s):  
Hailong Dao ◽  
Alessandro De Stefani ◽  
Linquan Ma
Keyword(s):  
Local Cohomology ◽  
Projective Dimension ◽  
Base Change ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Local Cohomology Modules ◽  
Flat Base ◽  
Du Bois Singularities ◽  
Cohomology Modules ◽  
Algebra Over A Field

Abstract Inspired by a question raised by Eisenbud–Mustaţă–Stillman regarding the injectivity of maps from ${\operatorname{Ext}}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings, which we call cohomologically full rings. This class of rings includes many well-known singularities: Cohen–Macaulay rings, Stanley–Reisner rings, F-pure rings in positive characteristics, and Du Bois singularities in characteristics $0$. We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. When $R$ is a standard graded algebra over a field of characteristic $0$, we show under certain conditions that being cohomologically full is equivalent to the intermediate local cohomology modules being generated in degree $0$. Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras.

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