Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals

Fløystad, Gunnar

doi:10.1007/s40306-018-0262-3

Cited by 10 publications

(16 citation statements)

References 14 publications

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“…The next two results are implicitly contained in Fløystad [5]. However they are stated in the context of the preceding papers [6,2], where the words "letterplace ideal" and "coletterplace ideals" are used in the narrow sense (see Remark 2.8 below).…”

Section: The Construction Of the Dualitymentioning

confidence: 99%

“…Remark 2.8. In [5], Fløystad generalized the notions of a (co-)letterplace ideal so that b-pol(I) of any strongly stable ideal I belongs to these classes (one of the crucial points is considering an order ideal J in Hom([n], N), not in Hom([n], [d])). Through this idea, he gave the duality.…”

Section: The Construction Of the Dualitymentioning

confidence: 99%

“…We are grateful to Professor Yuji Yoshino for stimulating discussion and valuable comments. We also thank Professor Gunnar Fløystad for telling us about his paper [5].…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Each approach has each advantage. The paper [5] treats the duality in a much wider context, but if one starts from the generator set G(I), our construction is more direct (Proposition 6.1 and Theorem 5.4 give a simple procedure to compute I * from G(I)). We will give a complete proof of the existence of the duality, since we will re-use ideas of the proof in later sections.…”

Section: Introductionmentioning

confidence: 99%

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Alexander duality for the alternative polarizations of strongly stable ideals

Shibata

Yanagawa

2020

Communications in Algebra

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We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal I ⊂ k[x 1 , . . . , x n ] with deg(m) ≤ d for all m ∈ G(I), its dual I * ⊂ k[y 1 , . . . , y d ] is a strongly stable ideal with deg(m) ≤ n for all m ∈ G(I * ). This duality has been constructed by Fløystad et al. in a different manner, so we emphasis applications here. For example, we will describe the Hilbert series of the local cohomologies H i m (S/I) using the irreducible decomposition of I (through the Betti numbers of I * ).

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Section: The Construction Of the Dualitymentioning

confidence: 99%

Section: The Construction Of the Dualitymentioning

confidence: 99%

“…We are grateful to Professor Yuji Yoshino for stimulating discussion and valuable comments. We also thank Professor Gunnar Fløystad for telling us about his paper [5].…”

Section: Acknowledgmentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Alexander duality for the alternative polarizations of strongly stable ideals

Shibata

Yanagawa

2020

Communications in Algebra

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“…Applications to Stanley-Reisner theory. When P and Q are finite posets we get general constructions, Subsection 3.4, of Alexander dual squarefree monomial ideals, generalizing isotonian ideals and letterplace and co-letterplace ideals, [5], [8], [6], [13], and [12]. In particular, when Q is a chain these constructions have given very large classes of simplicial spheres, [3].…”

Section: Introductionmentioning

confidence: 99%

Profunctors between posets and Alexander duality

Fløystad¹

2021

Preprint

Self Cite

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We advocate profunctors P | −→Q between posets as a more natural and fruitful notion than order preserving maps. We intoduce the graph and ascent of such profunctors. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study the poset of profunctors from N to N. Such profunctors identify as order presering maps f : N → N ∪ {∞}. For our applications when P and Q are infinite, we also introduce a topology on the set of profunctors P | −→Q.

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Profunctors Between Posets and Alexander Duality

Fløystad

2023

Appl Categor Struct

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We consider profunctors "Equation missing" between posets and introduce their graph and ascent. The profunctors $$\text {Pro}(P,Q)$$ Pro ( P , Q ) form themselves a poset, and we consider a partition $$\mathcal {I}\sqcup \mathcal {F}$$ I ⊔ F of this into a down-set $$\mathcal {I}$$ I and up-set $$\mathcal {F}$$ F , called a cut. To elements of $$\mathcal {F}$$ F we associate their graphs, and to elements of $$\mathcal {I}$$ I we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of $$Q \times P$$ Q × P . Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study $$\text {Pro}({\mathbb N}, {\mathbb N})$$ Pro ( N , N ) . Such profunctors identify as order preserving maps $$f: {\mathbb N}\rightarrow {\mathbb N}\cup \{\infty \}$$ f : N → N ∪ { ∞ } . For our applications when P and Q are infinite, we also introduce a topology on $$\text {Pro}(P,Q)$$ Pro ( P , Q ) , in particular on profunctors $$\text {Pro}({\mathbb N},{\mathbb N})$$ Pro ( N , N ) .

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Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals

Cited by 10 publications

References 14 publications

Alexander duality for the alternative polarizations of strongly stable ideals

Alexander duality for the alternative polarizations of strongly stable ideals

Profunctors between posets and Alexander duality

Profunctors Between Posets and Alexander Duality

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