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Matrix factorizations for domestic triangle singularities

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Kędzierski D., Lenzing H., Meltzer H.

Colloquium Mathematicum

2015

tom 140

nr 2

239-278

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}.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Kędzierski D.

1 Lenzing H.

1 Meltzer H.

1 2015

Wyniki wyszukiwania

Matrix factorizations for domestic triangle singularities