2021/08/06 by Kalck, Martin · 1 citation
#14B05 #14F08 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2108.03292
We give a complete classification of differential ℤ-graded homotopy categories of matrix factorizations of isolated singularities up to quasi-equivalence. This answers a question of Bernhard Keller and Evgeny Shinder. More generally, we show that a quasi-equivalence between the dg singularity category of a Gorenstein isolated singularity R and the dg singularity category of a complete local Noetherian ℂ-algebra S of different Krull dimension can always be realized by Knörrer's periodicity -- in particular, the existence of such an equivalence implies that R and S are hypersurface singularities. This uses and is complemented by a recent categorical version of the Mather--Yau theorem for hypersurfaces of the same Krull dimension due to Hua & Keller, which completes the classification mentioned above.