scholarly journals Regularity bounds for twisted commutative algebras

2020 ◽  
Vol 52 (2) ◽  
pp. 299-315
Author(s):  
Steven V Sam ◽  
Andrew Snowden
Keyword(s):  
Commutative Algebras ◽  
Twisted Commutative Algebras

scholarly journals GL-EQUIVARIANT MODULES OVER POLYNOMIAL RINGS IN INFINITELY MANY VARIABLES. II

2019 ◽  
Vol 7 ◽  
Author(s):  
STEVEN V SAM ◽  
ANDREW SNOWDEN
Keyword(s):  
Commutative Algebra ◽  
Polynomial Rings ◽  
Commutative Algebras ◽  
Representation Stability ◽  
Veronese Embeddings ◽  
Twisted Commutative Algebras

Twisted commutative algebras (tca’s) have played an important role in the nascent field of representation stability. Let $A_{d}$ be the tca freely generated by $d$ indeterminates of degree 1. In a previous paper, we determined the structure of the category of $A_{1}$-modules (which is equivalent to the category of $\mathbf{FI}$-modules). In this paper, we establish analogous results for the category of $A_{d}$-modules, for any $d$. Modules over $A_{d}$ are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra.

scholarly journals Noetherianity of some degree two twisted commutative algebras

2015 ◽  
Vol 22 (2) ◽  
pp. 913-937 ◽  
Author(s):  
Rohit Nagpal ◽  
Steven V Sam ◽  
Andrew Snowden
Keyword(s):  
Commutative Algebras ◽  
Twisted Commutative Algebras

scholarly journals Generalizations of Stillman’s Conjecture via Twisted Commutative Algebra

2019 ◽  
Author(s):  
Daniel Erman ◽  
Steven V Sam ◽  
Andrew Snowden
Keyword(s):  
Polynomial Ring ◽  
Commutative Algebra ◽  
Commutative Algebras ◽  
Twisted Commutative Algebras ◽  
Broad Generalization

Abstract Combining recent results on Noetherianity of twisted commutative algebras by Draisma and the resolution of Stillman’s conjecture by Ananyan–Hochster, we prove a broad generalization of Stillman’s conjecture. Our theorem yields an array of boundedness results in commutative algebra that only depend on the degrees of the generators of an ideal and not the number of variables in the ambient polynomial ring.

scholarly journals Hilbert series for twisted commutative algebras

2018 ◽  
Vol 1 (1) ◽  
pp. 147-172 ◽  
Author(s):  
Steven V Sam ◽  
Andrew Snowden
Keyword(s):  
Hilbert Series ◽  
Commutative Algebras ◽  
Twisted Commutative Algebras

scholarly journals Serre–Swan theorem for non-commutative -algebras

2003 ◽  
Vol 48 (2-3) ◽  
pp. 275-296 ◽  
Author(s):  
Katsunori Kawamura
Keyword(s):  
Commutative Algebras

scholarly journals The Homological Kähler-de Rham Differential Mechanism: II. Sheaf-Theoretic Localization of Quantum Dynamics

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Anastasios Mallios ◽  
Elias Zafiris
Keyword(s):  
Quantum Dynamics ◽  
Dynamical Behavior ◽  
Quantum Regime ◽  
Commutative Algebras ◽  
Differential Mechanism ◽  
De Rham ◽  
Physical Fields

The homological Kähler-de Rham differential mechanism models the dynamical behavior of physical fields by purely algebraic means and independently of any background manifold substratum. This is of particular importance for the formulation of dynamics in the quantum regime, where the adherence to such a fixed substratum is problematic. In this context, we show that the functorial formulation of the Kähler-de Rham differential mechanism in categories of sheaves of commutative algebras, instantiating generalized localization environments of physical observables, induces a consistent functorial framework of dynamics in the quantum regime.

scholarly journals Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Laurent Poinsot
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Semidirect Product ◽  
Hopf Algebras ◽  
Formal Series ◽  
Product Structure ◽  
Locally Finite ◽  
Commutative Algebras ◽  
Affine Groups ◽  
Formal Power

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.

THE DYNAMICS OF NQ-SYSTEMS IN THE PLANE

2009 ◽  
Vol 19 (01) ◽  
pp. 117-133 ◽  
Author(s):  
MATEJ MENCINGER ◽  
MILAN KUTNJAK
Keyword(s):  
Algebraic Approach ◽  
Dynamical Analysis ◽  
Algebraic Classification ◽  
Commutative Algebras ◽  
Planar Maps ◽  
Points Of View ◽  
One To One

The dynamics of discrete homogeneous quadratic planar maps is considered via the algebraic approach. There is a one-to-one correspondence between these systems and 2D commutative algebras (c.f. [Markus, 1960]). In particular, we consider the systems corresponding to algebras which contain some nilpotents of rank two (i.e. NQ-systems). Markus algebraic classification is used to obtain the class representatives. The case-by-case dynamical analysis is presented. It is proven that there is no chaos in NQ-systems. Yet, some cases are really interesting from the dynamical and bifurcational points of view.

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