scholarly journals On Hilbert polynomial of certain determinantal ideals

1991 ◽  
Vol 14 (1) ◽  
pp. 155-162 ◽  
Author(s):  
Shrinivas G. Udpikar
Keyword(s):  
Polynomial Ring ◽  
Invariant Theory ◽  
Fundamental Theorem ◽  
Hilbert Function ◽  
Hilbert Polynomial ◽  
Determinantal Ideals ◽  
The Ideal

LetX=(Xij)be anm(1)bym(2)matrix whose entriesXij,1≤i≤m(1),1≤j≤m(2); are indeterminates over a fieldK. LetK[X]be the polynomial ring in thesem(1)m(2)variables overK. A part of the second fundamental theorem of Invariant Theory says that the idealI[p+1]inK[X], generated by(p+1)by(p+1)minors ofXis prime. More generally in [1], Abhyankar defines an idealI[p+a]inK[X], generated by different size minors ofXand not only proves its primeness but also calculates the Hilbert function as well as the Hilbert polynomial of this ideal. The said Hilbert polynomial is completely determined by certain integer valued functionsFD(m,p,a). In this paper we prove some important properties of these integer valued functions.

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.

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

scholarly journals Stanley’s conjecture for critical ideals

2011 ◽  
Vol 48 (2) ◽  
pp. 220-226
Author(s):  
Azeem Haider ◽  
Sardar Khan
Keyword(s):  
Polynomial Ring ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Stanley Depth

Let S = K[x1,…,xn] be a polynomial ring in n variables over a field K. Stanley’s conjecture holds for the modules I and S/I, when I ⊂ S is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical monomial ideal. For non-critical monomial ideals we show the existence of a Stanley ideal with the same depth and Hilbert function.

scholarly journals The first fundamental theorem of invariant theory for the orthosymplectic super group

2018 ◽  
Vol 327 ◽  
pp. 4-24 ◽  
Author(s):  
P. Deligne ◽  
G.I. Lehrer ◽  
R.B. Zhang
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem

scholarly journals THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

2019 ◽  
pp. 1-25 ◽  
Author(s):  
G. I. LEHRER ◽  
R. B. ZHANG
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem ◽  
Brauer Algebra ◽  
Tensor Space ◽  
General Linear Supergroup ◽  
Special Cases ◽  
Linear Description ◽  
Tensor Representations ◽  
Capelli Identities ◽  
Fundamental Theorems

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$ . This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$ .

scholarly journals Determinantal ideals without minimal free resolutions

1990 ◽  
Vol 118 ◽  
pp. 203-216 ◽  
Author(s):  
Mitsuyasu Hashimoto
Keyword(s):  
Free Resolution ◽  
Unit Element ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Determinantal Ideals ◽  
Arbitrary Base ◽  
Generic Matrix ◽  
Base Ring ◽  
Minimal Free Resolutions ◽  
The Ideal

Let R be a Noetherian commutative ring with, unit element, and Xij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let S = R[xij] be the polynomial ring over R, and It be the ideal in S, generated by the t × t minors of the generic matrix (xij) ∈ Mm, n(S). For many years there has been considerable interest in finding a minimal free resolution of S/It, over arbitrary base ring R. If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗z P. is a resolution of S/It over the base ring R′.

Hilbert's function in a semi-lattice

1959 ◽  
Vol 55 (3) ◽  
pp. 239-243
Author(s):  
A. Learner
Keyword(s):  
Local Ring ◽  
Polynomial Ring ◽  
Hilbert Function ◽  
Homogeneous Ideal ◽  
Samuel 1

Samuel (1) introduced a generalized Hilbert function, written Xq(r, a) and defined for arbitrary ideals a in a local ring Q with maximal ideai m. where q is m-primary.Northcott(2) proved that for a homogeneous ideal ã in a polynomial ring A[X1, …, Xn], where A = Q/q, this is equal to the ordinary Hilbert function χ(r, ã).

scholarly journals R-Sequences and Homological Dimension

1962 ◽  
Vol 20 ◽  
pp. 195-199 ◽  
Author(s):  
Irving Kaplansky
Keyword(s):  
Polynomial Ring ◽  
Prime Ideals ◽  
Homological Dimension ◽  
The Ideal

The motivation for the results in this note comes from a theorem of Macaulay. Let f 1, …, fn be elements of a polynomial ring R over a field, and let I be the ideal they generate. Assume I R and rank (I) = n. Then the theorem of Lasker and Macaulay asserts that I is unmixed (all prime ideals belonging to I have rank n). Macaulay [1, p. 51] proved further that any power of I is unmixed.

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