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 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.

scholarly journals RESOLUTIONS AND COHOMOLOGIES OF TORIC SHEAVES: THE AFFINE CASE

2013 ◽  
Vol 24 (09) ◽  
pp. 1350069
Author(s):  
MARKUS PERLING
Keyword(s):  
Polynomial Ring ◽  
Local Cohomology ◽  
Hyperplane Arrangement ◽  
Betti Numbers ◽  
Vector Spaces ◽  
Free Resolutions ◽  
Graded Betti Numbers ◽  
Higher Rank ◽  
Minimal Free Resolutions ◽  
Reflexive Modules

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen–Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the ℤn-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the nonsmooth, simplicial case in dimension d ≥ 3, we construct new examples of indecomposable maximal Cohen–Macaulay modules of rank d – 1.

scholarly journals Generalized Newton complementary duals of monomial ideals

2020 ◽  
pp. 2150021
Author(s):  
Katie Ansaldi ◽  
Kuei-Nuan Lin ◽  
Yi-Huang Shen
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Free Resolutions ◽  
Stable Ideal ◽  
Generalized Newton ◽  
Strongly Stable Ideals ◽  
The Given ◽  
Linear Quotients

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.

scholarly journals Ulrich ideals and modules

2013 ◽  
Vol 156 (1) ◽  
pp. 137-166 ◽  
Author(s):  
SHIRO GOTO ◽  
KAZUHO OZEKI ◽  
RYO TAKAHASHI ◽  
KEI-ICHI WATANABE ◽  
KEN-ICHI YOSHIDA
Keyword(s):  
Numerical Semigroup ◽  
Representation Type ◽  
Local Rings ◽  
Free Resolutions ◽  
Semigroup Rings ◽  
Graded Modules ◽  
Minimal Free Resolutions ◽  
Points Of View ◽  
Numerical Semigroup Rings

AbstractIn this paper we study Ulrich ideals of and Ulrich modules over Cohen--Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.

scholarly journals Regularity and Free Resolution of Ideals Which Are Minimal To $d$-Linearity

2016 ◽  
Vol 118 (2) ◽  
pp. 161 ◽  
Author(s):  
M. Morales ◽  
A. A. Yazdan Pour ◽  
R. Zaare-Nahandi
Keyword(s):  
Polynomial Ring ◽  
Free Resolution ◽  
Free Resolutions ◽  
Edge Ideals ◽  
Homology Manifold ◽  
Linear Resolution ◽  
Vertex Set ◽  
Minimal Free Resolutions ◽  
Positive Integers ◽  
The Ideal

For given positive integers $n\geq d$, a $d$-uniform clutter on a vertex set $[n]=\{1,\dots,n\}$ is a collection of distinct $d$-subsets of $[n]$. Let $\mathscr{C}$ be a $d$-uniform clutter on $[n]$. We may naturally associate an ideal $I(\mathscr{C})$ in the polynomial ring $S=k[x_1,\dots,x_n]$ generated by all square-free monomials \smash{$x_{i_1}\cdots x_{i_d}$} for $\{i_1,\dots,i_d\}\in\mathscr{C}$. We say a clutter $\mathscr{C}$ has a $d$-linear resolution if the ideal \smash{$I(\overline{\mathscr{\mathscr{C}}})$} has a $d$-linear resolution, where \smash{$\overline{\mathscr{C}}$} is the complement of $\mathscr{C}$ (the set of $d$-subsets of $[n]$ which are not in $\mathscr C$). In this paper, we introduce some classes of $d$-uniform clutters which do not have a linear resolution, but every proper subclutter of them has a $d$-linear resolution. It is proved that for any two $d$-uniform clutters $\mathscr{C}_1$, $\mathscr{C}_2$ the regularity of the ideal $I(\overline{\mathscr{C}_1 \cup \mathscr{C}_2})$, under some restrictions on their intersection, is equal to the maximum of the regularities of $I(\overline{\mathscr{C}}_1)$ and $I(\overline{\mathscr{C}}_2)$. As applications, alternative proofs are given for Fröberg's Theorem on linearity of edge ideals of graphs with chordal complement as well as for linearity of generalized chordal hypergraphs defined by Emtander. Finally, we find minimal free resolutions of the ideal of a triangulation of a pseudo-manifold and a homology manifold explicitly.

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.

Association between Acromegaly and Chordomas of the Skull Base. Case Report and Review of the Literature

2014 ◽  
Vol 75 (S 01) ◽  
Author(s):  
Óscar Feo Lee ◽  
Juan Carlos Acevedo G. ◽  
Roberto Díaz O ◽  
Miguel Berbeo C. ◽  
Óscar Zorro G. ◽  
...  
Keyword(s):  
Case Report ◽  
Skull Base ◽  
Base Case ◽  
Review Of The Literature
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