scholarly journals LCM-Lattice, Taylor Bases and Minimal Free Resolutions of a Monomial Ideal

2021 ◽  
Author(s):  
Ri-Xiang Chen
Keyword(s):  
Monomial Ideal ◽  
Free Resolutions ◽  
Minimal Free Resolutions ◽  
Lcm Lattice

scholarly journals Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 605
Author(s):  
Lukas Katthän
Keyword(s):  
Simplicial Complex ◽  
Linear Part ◽  
Monomial Ideal ◽  
Short Note ◽  
Free Resolution ◽  
Free Resolutions ◽  
Restriction Maps ◽  
Minimal Free Resolution ◽  
Linear Maps ◽  
Minimal Free Resolutions

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex Δ . Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of Δ . Along the way, we also show that if a monomial ideal has at least one generator of degree 2, then the linear strand of its minimal free resolution can be written using only ± 1 coefficients.

scholarly journals Minimal free resolutions of 2 × n domino tilings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950118
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Domino Tilings ◽  
Minimal Free Resolutions ◽  
The Ideal

We introduce a squarefree monomial ideal associated to the set of domino tilings of a [Formula: see text] rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo–Mumford regularity of the ideal.

scholarly journals Bundles on with vanishing lower cohomologies

2020 ◽  
pp. 1-20
Author(s):  
Mengyuan Zhang
Keyword(s):  
Hilbert Function ◽  
Betti Numbers ◽  
Projective Spaces ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Rational Varieties ◽  
Minimal Free Resolutions ◽  
Graded Lattice

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

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′.

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