scholarly journals COMPUTATION OF THE GROTHENDIECK AND PICARD GROUPS OF A TORIC DM STACK BY USING A HOMOGENEOUS COORDINATE RING FOR

2010 ◽  
Vol 53 (1) ◽  
pp. 97-113 ◽  
Author(s):  
S. PAUL SMITH
Keyword(s):  
Polynomial Ring ◽  
Coordinate Ring ◽  
Graded Modules ◽  
Picard Groups

AbstractWe compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading and suitable irrelevant ideal functions is a homogeneous coordinate ring for the stack.

Locally nilpotent derivations of factorial domains

2019 ◽  
Vol 18 (12) ◽  
pp. 1950222
Author(s):  
M’hammed El Kahoui ◽  
Mustapha Ouali
Keyword(s):  
Recent Result ◽  
Polynomial Ring ◽  
Coordinate Ring ◽  
Locally Nilpotent

Let [Formula: see text] be factorial domains containing [Formula: see text]. In this paper, we give a criterion, in terms of locally nilpotent derivations, for [Formula: see text] to be [Formula: see text]-isomorphic to [Formula: see text], where [Formula: see text] is nonzero and [Formula: see text]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc. 145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent [Formula: see text]-derivations of the polynomial ring in two variables over [Formula: see text].

scholarly journals On some properties of LS algebras

2018 ◽  
Vol 22 (02) ◽  
pp. 1850085
Author(s):  
Rocco Chirivì
Keyword(s):  
Abelian Group ◽  
Polynomial Ring ◽  
Toric Variety ◽  
Ordered Set ◽  
Finite Abelian Group ◽  
Coordinate Ring ◽  
Totally Ordered Set

The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no pseudo-reflection acting on a polynomial ring. This is used to study the Gorenstein property for LS algebras. Further we show that any LS algebra is Koszul.

scholarly journals Foundations of Boij–Söderberg theory for Grassmannians

2018 ◽  
Vol 154 (10) ◽  
pp. 2205-2238 ◽  
Author(s):  
Nicolas Ford ◽  
Jake Levinson
Keyword(s):  
Projective Space ◽  
Polynomial Ring ◽  
Perfect Matchings ◽  
Base Case ◽  
Free Resolutions ◽  
Spectral Sequences ◽  
Graded Modules

Boij–Söderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with Sam, extending the theory to the setting of $\text{GL}_{k}$-equivariant modules and sheaves on Grassmannians. Algebraically, we study modules over a polynomial ring in $kn$ variables, thought of as the entries of a $k\times n$ matrix. We give equivariant analogs of two important features of the ordinary theory: the Herzog–Kühl equations and the pairing between Betti and cohomology tables. As a necessary step, we also extend previous results, concerning the base case of square matrices, to cover complexes other than free resolutions. Our statements specialize to those of ordinary Boij–Söderberg theory when $k=1$. Our proof of the equivariant pairing gives a new proof in the graded setting: it relies on finding perfect matchings on certain graphs associated to Betti tables and to spectral sequences. As an application, we construct three families of extremal rays on the Betti cone for $2\times 3$ matrices.

scholarly journals Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Rosa M. Miró-Roig ◽  
Martí Salat-Moltó
Keyword(s):  
Polynomial Ring ◽  
Toric Variety ◽  
Hilbert Function ◽  
Hilbert Polynomial ◽  
Lattice Polytopes ◽  
Sharp Bounds ◽  
Graded Modules ◽  
Cox Ring ◽  
Regularity Index

Abstract In this paper, we consider Z r \mathbb{Z}^{r} -graded modules on the Cl ⁡ ( X ) \operatorname{Cl}(X) -graded Cox ring C ⁢ [ x 1 , … , x r ] \mathbb{C}[x_{1},\dotsc,x_{r}] of a smooth complete toric variety 𝑋. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive Z s + r + 2 \mathbb{Z}^{s+r+2} -graded modules over any non-standard bigraded polynomial ring C ⁢ [ x 0 , … , x s , y 0 , … , y r ] \mathbb{C}[x_{0},\dotsc,x_{s},\allowbreak y_{0},\dotsc,y_{r}] . In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

scholarly journals Graded Bourbaki ideals of graded modules

2021 ◽  
Author(s):  
Jürgen Herzog ◽  
Shinya Kumashiro ◽  
Dumitru I. Stamate
Keyword(s):  
Graded Modules

On Stanley Depths of Certain Monomial Factor Algebras

2015 ◽  
Vol 58 (2) ◽  
pp. 393-401
Author(s):  
Zhongming Tang
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Primary Decomposition ◽  
Factor Algebra ◽  
Stanley Depth ◽  
Stanley Conjecture ◽  
Stanley Decomposition

AbstractLet S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

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