Betti tables for indecomposable matrix factorizations of $XY(X-Y)(X-λY)$

Gélinas, Vincent

doi:10.48550/arxiv.1712.07043

2017

DOI: 10.48550/arxiv.1712.07043

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Betti tables for indecomposable matrix factorizations of $XY(X-Y)(X-λY)$

Vincent Gélinas¹

Abstract: We classify the Betti tables of indecomposable graded matrix factorizations over the simple elliptic singularity f λ = XY (X − Y )(X − λY ) by making use of an associated weighted projective line of genus one.

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“…There are a number of Z-graded Gorenstein rings R such that the stable categories CM Z 0 R admit tilting objects, see e.g. [AIR,DL1,DL2,FU,G1,G2,HIMO,IO,IT,JKS,KST1,KST2,Ki1,Ki2,KLM,LP,LZ,MU,U1,U2,Ya] and a survey article [I]. Therefore the following problem is of our central interest.…”

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confidence: 99%

Tilting theory for Gorenstein rings in dimension one

Buchweitz,

Iyama,

Yamaura

2018

Preprint

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For a Z-graded Gorenstein ring R = i≥0 R i such that R 0 is a field, we study the stable category CM Z R of Z-graded maximal Cohen-Macaulay R-modules, which is canonically triangle equivalent to the singularity category of Buchweitz and Orlov. Its thick subcategory CM Z 0 R is central in representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split triangles. It is important to understand when the triangulated category CM Z 0 R admits a tilting (respectively, silting) object. In this paper, we give a complete answer to this question in the case dim R = 1: We prove that CM Z 0 R always admits a silting object, and that CM Z 0 R admits a tilting object if and only if either R is regular or the a-invariant of R is nonnegative. We also show that, if R is reduced and non-regular, then its a-invariant is non-negative and the above tilting object gives a full strong exceptional collection in CM Z 0 R = CM Z R.

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confidence: 99%

Tilting theory for Gorenstein rings in dimension one

Buchweitz,

Iyama,

Yamaura

2018

Preprint

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Betti tables for indecomposable matrix factorizations of $XY(X-Y)(X-λY)$

Abstract: We classify the Betti tables of indecomposable graded matrix factorizations over the simple elliptic singularity f λ = XY (X − Y )(X − λY ) by making use of an associated weighted projective line of genus one.

Cited by 1 publication

References 8 publications

Tilting theory for Gorenstein rings in dimension one

Tilting theory for Gorenstein rings in dimension one

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