Matrix factorizations for domestic triangle singularities

DOI10.4064/CM140-2-6zbMATH Open1348.14047arXiv1507.07832OpenAlexW3098017516WikidataQ114022032 ScholiaQ114022032MaRDI QIDQ5261970

Hagen Meltzer, Dawid Edmund Kędzierski, Helmut Lenzing

Publication date: 8 July 2015

Published in: Colloquium Mathematicum (Search for Journal in Brave)

Abstract: Working over an algebraically closed field

k

of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities

f = x^{a} + y^{b} + z^{c}

of domestic type, that is, we assume that

(a, b, c)

are integers at least two, satisfying

1 / a + 1 / b + 1 / c > 1

. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type

(a, b, c)

. Equivalently, in a representation-theoretic context, we can work in the mesh category of

m a t h b b Z i l d e D e l t a

over

k

, where

i l d e D e l t a

is the extended Dynkin diagram, corresponding to the Dynkin diagram

D e l t a = [a, b, c]

. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the

m a t h b b Z

-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from

0, p m 1

.

Full work available at URL: https://arxiv.org/abs/1507.07832

zbMATH Keywords

Cohen-Macaulay module vector bundle injective hull matrix factorization singularity category weighted projective line projective cover triangle singularity

Mathematics Subject Classification ID

Representation type (finite, tame, wild, etc.) of associative algebras (16G60) Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers (16G70) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) (13H10) Singularities in algebraic geometry (14B05) Singularities of surfaces or higher-dimensional varieties (14J17)

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