Abstract
We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.
Highlights
If m = n R is the hypersurface ring R = S/(det φ) and R is Gorenstein
Using Grothendieck-Serre duality for the projective morphism π we show that it suffices to establish the claims for c
Replacing Rj∗OZ by its locally free OY -resolution described in 5.2 above, we find that Rj∗(j∗p∗Mba(−c)) is represented in the derived category of Y by a complex C with terms
Summary
Meaning a commutative base ring, most often a field projective K-modules of finite ranks m n indicated exterior power of F determinant of F , rank F F symmetric power HomK (G, F ) S(H∨), a polynomial algebra over K dual over K or S, as context implies abelian category with enough projectives and the homotopy category of its projectives. R, R Lα free S-modules induced from G and F the generic S-linear map generic (m × n)-matrix of local coordinates on Spec S the quotient of S determined by the maximal minors of X. OP-module of degree-a differential forms the tautological subbundle of rank m − 1 in π∗F the tautological quotient bundle of rank m − 1 of π∗F ∨. Beılinson quiver on F path algebra of Q doubled Beılinson quiver on F and G quiverized Clifford algebra, path algebra of Q infinite doubled Beılinson quiver and its path algebra
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