Resolutions of co-letterplace ideals and generalizations of Bier spheres

D’Alì, Alessio; Fløystad, Gunnar; Nematbakhsh, Amin

doi:10.1090/tran/7560

Cited by 10 publications

(8 citation statements)

References 24 publications

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“…We also remark that [2,Corollary 4.3] gives a description of the canonical module of the quotient ring of a letterplace ideal, and it also works in the case of Corollary 2.10. However, our description is much simpler in this case.…”

Section: The Construction Of the Dualitymentioning

confidence: 98%

Section: The Construction Of the Dualitymentioning

confidence: 99%

“…For the strongly stable ideal I = (x 3 1 , x 2 1 x 2 , x 1 x 2 2 , x 1 x 2 x 2 3 , x 2 1 x 2 3 ), the generators x 1 x 2 x 2 3 and x 2 1 x 2 3 yield (x 2 1 , x 2 2 , x 2 3 ) and (x 3 1 , x 2 , x 2 3 ), respectively. Next, consider the strongly stable ideal I ′ := I : x ∞ 3 = (x 2 1 , x 1 x 2 ), and it has an irreducible component (x 2 1 , x 2 ) given by x 1 x 2 , which is also an irreducible component of I itself. Finally, I ′ : x ∞ 2 = (x 1 ) itself is an irreducible component of I.…”

Section: Remarks On Irreducible Components Of Strongly Stable Idealsmentioning

confidence: 99%

“…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). The second assertion follows from the fact that any co-letterplace ideal L([n], d; J ) is the b-pol(−) of some strongly stable ideal whose generators all have degree n.…”

Section: The Construction Of the Dualitymentioning

confidence: 99%

“…, and it has an irreducible component (x 2 1 , x 2 ) given by x 1 x 2 , which is also an irreducible component of I itself. Finally, I ′ : x ∞ 2 = (x 1 ) itself is an irreducible component of I.…”

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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: 98%

Section: The Construction Of the Dualitymentioning

confidence: 99%

Section: Remarks On Irreducible Components Of Strongly Stable Idealsmentioning

confidence: 99%

Section: The Construction Of the Dualitymentioning

confidence: 99%

mentioning

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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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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