Matrix factorizations for domestic triangle singularities

Dawid Edmund Kędzierski¹ ; Helmut Lenzing² ; Hagen Meltzer³

¹ Instytut Matematyki Uniwersytet Szczeciński 70-451 Szczecin, Poland
² Institut für Mathematik Universität Paderborn 33095 Paderborn, Germany
³ Instytut Matematyki Uniwersytet Szczeciński 70451 Szczecin, Poland

Colloquium Mathematicum, Tome 140 (2015) no. 2, pp. 239-278

Résumé

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 $\mathbb {Z}\tilde\varDelta $ over $k$, where $\tilde\varDelta $ is the extended Dynkin diagram corresponding to the Dynkin diagram $\varDelta =[a,b,c]$. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the $\mathbb {Z}$-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 $\{0,\pm 1\}$.

Détail

DOI : 10.4064/cm140-2-6

Keywords: working algebraically closed field characteristic determine matrix factorizations suitably graded triangle singularities y domestic type assume integers least satisfying using work kussin lenzing meltzer achieved determining projective covers frobenius category vector bundles weighted projective line weight type equivalently representation theoretic context work mesh category mathbb tilde vardelta where tilde vardelta extended dynkin diagram corresponding dynkin diagram vardelta work related methods results different determination matrix factorizations mathbb graded simple singularities kajiura saito takahashi particular obtain symmetric matrix factorizations whose entries scalar multiples monomials scalars taken