Triangulated categories of matrix factorizations for regular systems of weights with \(\varepsilon = - 1\) (Q1007232): Difference between revisions

One of the results of this article is a categorification of Arnold's strange duality for exceptional unimodular singularities; cf.\ Cor.\ 5.12. Let \(W = (a,b,c;h)\), where \(a,b,c,h\in\mathbb Z_{\geq 1}\) such that \(\gcd(a,b,c)=1\) and \(a+b+c-h= -1\). Suppose that \(T^{-h}(T^h-T^a) (T^h-T^b) (T^h-T^c) (T^a-1)^{-1} (T^b-1)^{-1} (T^c-1)^{-1}\) has no poles except at \(T=0\), and that its Laurent series at \(T = 0\) has coefficient \(0\) for \(T^0\). Consider \(A = \mathbb C[x,y,z]\). Let \(E = h^{-1}(ax{\text{D}}_x + by{\text{D}}_y + cz{\text{D}}_z)\). Let \(A_s\) denote the eigenspace of \(E\) in \(A\) of eigenvalue \(s/2\). We obtain the grading \(A = \bigoplus_{k\in\mathbb Z_{\geq 0}} A_{2k/h}\). Let \(f_W\in A_2\) be such that \(\dim_{\mathbb C} A/({\text{D}}_x f_W,{\text{D}}_y f_W,{\text{D}}_z f_W)\) is finite. If the (combinatorially defined) signature of \(W\) has length 3, then \(f_W\) has an exceptional unimodular singularity at \((x,y,z)=(0,0,0)\). Define the Frobenius category \(\text{MF}^{\text{gr}}_A(f_W)\) (matrix factorisations) to have as objects diagrams \(F_0\rightarrow{f_0} F_1\rightarrow{f_1} F_0\), where \(F_0\) and \(F_1\) are f.g.\ free graded \(A\)-modules, \(f_0\) is a graded \(A\)-linear map of degree \(0\), \(f_1\) of degree \(2\), and where \(f_1\circ f_0 = f_W\cdot{\text{id}}\) and \(f_0\circ f_1 = f_W\cdot{\text{id}}\). Morphisms are compatible pairs of morphisms at \(F_0\) and \(F_1\); admissible short exact sequences are defined pointwise. Write \({\mathcal T}_W\) for the stable category of \(\text{MF}^{\text{gr}}_A(f_W)\). The main result of this article is the construction of a full strongly exceptional sequence for \({\mathcal T}_W\), and, consequently, of an exact equivalence of \({\mathcal T}_W\) with the bounded derived category of an algebra explicitly described via quiver and relations. The authors state that this parallels a result of \textit{H. Lenzing} and \textit{J. de la Pena} [``Extended canonical algebras and Fuchsian singularities'' (arXiv:math/0611532)].