scholarly journals VECTOR SPACE BASES FOR THE HOMOGENEOUS PARTS IN HOMOGENEOUS IDEALS AND GRADED MODULES OVER A POLYNOMIAL RING

2014 ◽  
Vol 93 (6) ◽  
Author(s):  
N. Duck ◽  
K.-H. Zimmermann
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Graded Modules ◽  
Homogeneous Ideals

scholarly journals Multivariate Alexander colorings

2018 ◽  
Vol 27 (14) ◽  
pp. 1850076 ◽  
Author(s):  
Lorenzo Traldi
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Knot Theory ◽  
Laurent Polynomial ◽  
Module Structure ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Linear Maps ◽  
Laurent Polynomial Ring ◽  
Special Case

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

scholarly journals On representation of polynomial ring on a vector space via a linear transformation

2017 ◽  
Vol 893 ◽  
pp. 012011
Author(s):  
S Wahyuni ◽  
I E Wijayanti ◽  
N Hijriati
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Linear Transformation

scholarly journals Modular representations of abelian groups with regular rings of invariants

1982 ◽  
Vol 86 ◽  
pp. 229-248 ◽  
Author(s):  
Haruhisa Nakajima
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Finite Subgroup ◽  
Dimensional Vector ◽  
Modular Representations ◽  
Invariant Polynomials ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Regular Rings

Let k be a field of characteristic p and G a finite subgroup of GL(V) where V is a finite dimensional vector space over k. Then G acts naturally on the symmetric algebra k[V] of V. We denote by k[V]G the subring of k[V] consisting of all invariant polynomials under this action of G. The following theorem is well known.Theorem 1.1 (Chevalley-Serre, cf. [1, 2, 3]). Assume that p = 0 or (|G|, p) = 1. Then k[V]G is a polynomial ring if and only if G is generated by pseudo-reflections in GL(V).

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 NILPOTENT MODULES OVER POLYNOMIAL RINGS

2020 ◽  
pp. 20-25
Author(s):  
А. Petravchuk ◽  
Ie. Chapovskyi ◽  
I. Klimenko ◽  
M. Sidorov
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Polynomial Rings ◽  
Closed Field ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Groups Of Automorphisms

Let K be an algebra ically closed field of characteristic zero, K[X ] the polynomial ring in n variables. The vector space Tn = K[X] is a K[X ] -module with the action i = xi 'x  v v for vTn . Every finite dimensional submodule V of Tn is nilpotent, i.e. every f  K[X ] acts nilpotently (by multiplication) on V . We prove that every nilpotent K[X ] -module V of finite dimension over K with one-dimensional socle can be isomorphically embedded in the module Tn . The groups of automorphisms of the module Tn and its finite dimensional monomial submodules are found. Similar results are obtained for (non-nilpotent) finite dimensional K[X ] -modules with one dimensional socle.

scholarly journals Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application to Singer's cohomological transfer

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Open Problem ◽  
General Linear Group ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Ext Groups ◽  
Hit Problem ◽  
Prime 2

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This problem is the content of "hit problem" of Frank Peterson. We study the $q$-th Singer algebraic transfer $Tr_q^{A}$, which is a homomorphism from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the Adams $E_2$-term, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The Singer transfer is one of the useful tools for describing mysterious Ext groups. In the present study, by using techniques of the hit problem of four variables, we explicitly determine the structure of the spaces $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and the representation of the fourth transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. These new results confirmed Singer's conjecture for the monomorphism of the rank $4$ transfer. Our approach is different from that of Singer.

The structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application, I

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Open Problem ◽  
General Linear Group ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Ext Groups ◽  
Hit Problem ◽  
Prime 2

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, can one write down a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This problem is the content of "hit problem" of Frank Peterson. We study the $q$-th Singer algebraic transfer $Tr_q^{A}$, which is a homomorphism from the space of $GL_q(k)$-coinvariants $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the Adams $E_2$-terms, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The Singer transfer is one of the useful tools for describing mysterious Ext groups. In the present study, by using techniques of the hit problem of four variables, we explicitly determine the structure of the spaces $k\otimes _{GL_4(k)} P((P_4)_{n_j}^{*})$ in some generic degrees. Applying these results and the representation of the transfer $Tr_4^{A}$ over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. Our approach is different from that of Singer.

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 Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Open Problem ◽  
Cohomology Group ◽  
General Linear Group ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Hit Problem ◽  
Prime 2

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This is the content of the classical "hit problem" of Frank Peterson. Based on this problem, we are interested in the $q$-th algebraic transfer $Tr_q^{A}$ of W. Singer \cite{W.S1}, which is one of the useful tools for describing mod-2 cohomology of the algebra $A.$ This transfer is a linear map from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The present paper is to investigate this algebraic transfer for the cohomological degree $q = 4.$ More specifically, basing the techniques of the hit problem of four variables, we explicitly determine the structure of $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and a representation of the rank 4 transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. Also, we give some conjectures on the dimensions of $k\otimes_{GL_q(k)} ((P_4)_n^{*})$ for the remaining degrees $n.$ As a consequence, Singer's conjecture for the algebraic transfer is true in the rank 4 case. This study and our previous results \cite{D.P11, D.P12} have been provided a panorama of the behavior of $Tr_4^{A}.$

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