Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the-suitably graded-triangle singularities 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm140-2-6,
author = {Dawid Edmund K\k edzierski and Helmut Lenzing and Hagen Meltzer},
title = {Matrix factorizations for domestic triangle singularities},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {239-278},
zbl = {06456785},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm140-2-6}
}
Dawid Edmund Kędzierski; Helmut Lenzing; Hagen Meltzer. Matrix factorizations for domestic triangle singularities. Colloquium Mathematicae, Tome 139 (2015) pp. 239-278. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm140-2-6/