Locally class-presentable and class-accessible categories

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory

A note on the Bloch-Beilinson conjecture for K3 surfaces and spherical objects

For a projective K3 surface X we introduce the dense triangulated subcategory S^* of the bounded derived category D^b(Coh(X)) of coherent sheaves on X that is generated by spherical objects. For a K3 surface X over \bar Q it is shown that S^* admits a bounded t-structure if and only if the Bloch-Beilinson conjecture holds for X, i.e. CH^2(X)=Z.Comment: 8 page

Testing the Titius-Bode law predictions for Kepler multi-planet systems

We use three and half years of Kepler Long Cadence data to search for the 97 predicted planets of Bovaird & Lineweaver (2013) in 56 of the multi-planet systems, based on a general Titius-Bode relation. Our search yields null results in the majority of systems. We detect five planetary candidates around their predicted periods. We also find an additional transit signal beyond those predicted in these systems. We discuss the possibility that the remaining predicted planets are not detected in the Kepler data due to their non-coplanarity or small sizes. We find that the detection rate is beyond the lower boundary of the expected number of detections, which indicates that the prediction power of the TB relation in general extra solar planetary systems is questionable. Our analysis of the distribution of the adjacent period ratios of the systems suggests that the general Titius-Bode relation may over-predict the presence of planet pairs near the 3:2 resonance.Comment: Accepted by MNRA

Morita theory and singularity categories

We propose an analogue of the bounded derived category for an augmented ring spectrum, defined in terms of a notion of Noether normalization. In many cases we show this category is independent of the chosen normalization. Based on this, we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are Koszul dual in the style of the BGG correspondence. Examples of interest include Koszul algebras and Ginzburg DG-algebras, $C^*(BG)$ for finite groups (or for compact Lie groups with orientable adjoint representation), cochains in rational homotopy theory and various examples from chromatic homotopy theory.Comment: Final version, accepted for publication in Advances in Mathematics, 49 page