Matrix factorizations for domestic triangle singularities

Dawid Edmund Kędzierski, Helmut Lenzing, and Hagen Meltzer. "Matrix factorizations for domestic triangle singularities." Colloquium Mathematicae 140.2 (2015): 239-278. <http://eudml.org/doc/283591>.

@article{DawidEdmundKędzierski2015,
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 ℤ Δ̃ over k, where Δ̃ is the extended Dynkin diagram corresponding to the Dynkin diagram Δ = [a,b,c]. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the 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,±1.},
author = {Dawid Edmund Kędzierski, Helmut Lenzing, Hagen Meltzer},
journal = {Colloquium Mathematicae},
keywords = {triangle singularity; matrix factorization; weighted projective line; vector bundle; singularity category; Cohen-Macaulay module; projective cover; injective hull},
language = {eng},
number = {2},
pages = {239-278},
title = {Matrix factorizations for domestic triangle singularities},
url = {http://eudml.org/doc/283591},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Dawid Edmund Kędzierski
AU - Helmut Lenzing
AU - Hagen Meltzer
TI - Matrix factorizations for domestic triangle singularities
JO - Colloquium Mathematicae
PY - 2015
VL - 140
IS - 2
SP - 239
EP - 278
AB - 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 ℤ Δ̃ over k, where Δ̃ is the extended Dynkin diagram corresponding to the Dynkin diagram Δ = [a,b,c]. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the 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,±1.
LA - eng
KW - triangle singularity; matrix factorization; weighted projective line; vector bundle; singularity category; Cohen-Macaulay module; projective cover; injective hull
UR - http://eudml.org/doc/283591
ER -