Linear strands of multigraded free resolutions

Brown, Michael K.; Erman, Daniel

doi:10.1007/s00208-024-02803-1

Cited by 2 publications

(1 citation statement)

References 26 publications

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“…The proof of the proposition follows from an adaptation to the multi-graded context [ 5 ] of the classical BGG correspondence, as described in [ 7 ].…”

Section: Hilbert Series For Koszul Modules Of Simplicial Complexesmentioning

confidence: 99%

Higher resonance schemes and Koszul modules of simplicial complexes

Aprodu,

Farkas,

Raicu

et al. 2024

J Algebr Comb

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Each connected graded, graded-commutative algebra A of finite type over a field $$\Bbbk $$ k of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When $$A=\Bbbk \langle \Delta \rangle $$ A = k ⟨ Δ ⟩ is the exterior Stanley–Reisner algebra associated to a finite simplicial complex $$\Delta $$ Δ , we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

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“…The proof of the proposition follows from an adaptation to the multi-graded context [ 5 ] of the classical BGG correspondence, as described in [ 7 ].…”

Section: Hilbert Series For Koszul Modules Of Simplicial Complexesmentioning

confidence: 99%

Higher resonance schemes and Koszul modules of simplicial complexes

Aprodu,

Farkas,

Raicu

et al. 2024

J Algebr Comb

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A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

Brown,

Sayrafi

2024

Alg. Number Th.

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Given a smooth projective toric variety X of Picard rank 2, we resolve the diagonal sheaf on X × X by a linear complex of length dim X consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Erman and Smith that predicts a version of Hilbert's syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories.

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Linear strands of multigraded free resolutions

Cited by 2 publications

References 26 publications

Higher resonance schemes and Koszul modules of simplicial complexes

Higher resonance schemes and Koszul modules of simplicial complexes

A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

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