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 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 Linearity defect and regularity over a Koszul algebra

2009 ◽  
Vol 104 (2) ◽  
pp. 205 ◽  
Author(s):  
Kohji Yanagawa
Keyword(s):  
Complete Intersection ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Exterior Algebra ◽  
Finitely Generated ◽  
Koszul Algebra ◽  
Algebra Over A Field

Let $A = \bigoplus_{i\in \mathsf{N}}A_i$ be a Koszul algebra over a field $K = A_0$, and $*\operatorname{mod} A$ the category of finitely generated graded left $A$-modules. The linearity defect $\mathrm{ld}_A(M)$ of $M \in *\operatorname{mod} A$ is an invariant defined by Herzog and Iyengar. An exterior algebra $E$ is a Koszul algebra which is the Koszul dual of a polynomial ring. Eisenbud et al. showed that $\mathrm{ld}_E(M) < \infty$ for all $M \in *\operatorname{mod} E$. Improving this, we show that the Koszul dual $A^!$ of a Koszul commutative algebra $A$ satisfies the following. Let $M \in *\operatorname{mod} A^!$. If $\{\dim_K M_i \mid i \in {\mathsf Z}\}$ is bounded, then $\mathrm{ld}_{A^!}(M) < \infty$. If $A$ is complete intersection, then $\mathrm{reg}_{A^!}(M) < \infty$ and $\mathrm{ld}_{A^!}(M) < \infty$ for all $M \in *\operatorname{mod} A^!$. If $E=\bigwedge \langle y_1, \ldots, y_n\rangle$ is an exterior algebra, then $\mathrm{ld}_E(M)\leq c^{n!} 2^{(n-1)!}$ for $M \in *\operatorname{mod} E$ with $c := \max \{\dim_K M_i \mid i \in{\mathsf Z}\}$.

scholarly journals Nonnoetherian geometry

2016 ◽  
Vol 15 (09) ◽  
pp. 1650176 ◽  
Author(s):  
Charlie Beil
Keyword(s):  
Commutative Algebra ◽  
Projective Dimension ◽  
Krull Dimension ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
One Dimensional ◽  
Simple Modules ◽  
Commutative Algebras

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of [Formula: see text] are all isomorphic if and only if [Formula: see text] is noetherian, if and only if the center [Formula: see text] of [Formula: see text] is noetherian, if and only if [Formula: see text] is a finitely generated [Formula: see text]-module. Furthermore, we show that [Formula: see text] is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

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

Embedding Theorems for L-Algebras and Anti-commutative Algebras

2019 ◽  
Vol 26 (01) ◽  
pp. 51-64
Author(s):  
Qiuhui Mo
Keyword(s):  
Associative Algebra ◽  
Commutative Algebra ◽  
Embedding Theorems ◽  
Commutative Algebras

Bokut, Chen and Huang proved that every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. In this paper, we prove that every countably generated L-algebra can be embedded into a simple two-generated L-algebra. We also prove that every anti-commutative algebra can be embedded into a simple anti-commutative algebra, and that every countably generated anti-commutative algebra can be embedded into a simple two-generated anti-commutative algebra. Finally, we prove that every anti-commutative algebra can be embedded into its universal enveloping non-associative algebra.

scholarly journals Prime ideals in skew Laurent polynomial rings

1993 ◽  
Vol 36 (2) ◽  
pp. 299-317 ◽  
Author(s):  
K. W. Mackenzie
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Ideal Structure ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Laurent Polynomial Ring

Let R be a commutative ring and {σ1,…,σn} a set of commuting automorphisms of R. Let T = be the skew Laurent polynomial ring in n indeterminates over R and let be the Laurent polynomial ring in n central indeterminates over R. There is an isomorphism φ of right R-modules between T and S given by φ(θj) = xj. We will show that the map φ induces a bijection between the prime ideals of T and the Γ-prime ideals of S, where Γ is a certain set of endomorphisms of the ℤ-module S. We can study the structure of the lattice of Γ-prime ideals of the ring S by using commutative algebra, and this allows us to deduce results about the prime ideal structure of the ring T. As an example, if R is a Cohen-Macaulay ℂ-algebra and the action of the σj on R is locally finite-dimensional, we will show that the ring T is catenary.

Born geometry on ρ-commutative algebra

2020 ◽  
Vol 17 (14) ◽  
pp. 2050210
Author(s):  
Zahra Bagheri ◽  
Esmaeil Peyghan
Keyword(s):  
Commutative Algebra ◽  
Riemannian Geometry ◽  
Fundamental Theorem ◽  
Existence And Uniqueness ◽  
Commutative Algebras

The aim of this paper is to establish a generalization of the Born geometry to [Formula: see text]-commutative algebras. We introduce the notion of Born [Formula: see text]-commutative algebras and study the existence and uniqueness of a torsion connection which preserves the Born structure. Also, an analogue of the fundamental theorem of Riemannian geometry will be proved for these algebras.

ANTI-COMMUTATIVE ALGEBRAS WITH SKEW-SYMMETRIC IDENTITIES

2009 ◽  
Vol 08 (02) ◽  
pp. 157-180 ◽  
Author(s):  
A. S. DZHUMADIL'DAEV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Algebra ◽  
Symmetric Polynomial ◽  
Commutative Algebras ◽  
Associative Commutative Algebra ◽  
Symmetric Identities ◽  
Generalized Lie Algebras

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If sn is a standard skew-symmetric polynomial of degree n, then any s4-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D2) are s4-Lie. An algebra (U, id ∧ D3 - 2D ∧ D2) is s'5-Lie, where s'5 is a non-standard skew-symmetric polynomial of degree 5.

XXV.—Properties of I-extensions of Anti-commutative Algebras

1961 ◽  
Vol 65 (4) ◽  
pp. 345-357
Author(s):  
E. W. Wallace
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Algebra ◽  
Characteristic Zero ◽  
Cartesian Product ◽  
Paper Properties ◽  
Commutative Algebras ◽  
The Given

SYNOPSISS. T. Tsou and A. G. Walker have defined the I-extension of a given Lie algebra as a certain Lie algebra on the Cartesian product of the given algebra and one of its ideals (Tsou 1955). I-extensions have been studied also in connection with metrisable Lie groups and metrisable Lie algebras. The definition can be applied immediately to any anti-commutative algebra, and in this paper properties of such I-extensions are established. A list of all proper I-extensions of dimension not greater than four over a field of characteristic zero is also given together with a set of characters.

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