Algebraický seminář

Michal Hrbek (IM CAS): The standard t-structure on a scheme is usually not singly generated

Abstract: The standard t-structure on the derived module category of any ring is generated by the ring itself. Bondal and Van den Bergh showed that the (quasi-coherent) derived category of any quasi-compact and quasi-separated scheme admits a single perfect generator. This could lead one to suspect that standard t-structure is always generated by a single perfect complex for non-affine schemes. However, we find that this is typically not the case. Specifically, we show that this fails for any non-affine Noetherian scheme which is proper over a field, thus including all non-finite projective schemes. More generally, we show that any proper but non-finite map of Noetherian schemes gives rise to a counter-example. Our results hold true also in the generality of algebraic spaces and certain stacks. This is a joint work with A. Bhaduri, T. De Deyn, P. Lank, and K. Manali-Rahul (arXiv:2511.01622).