Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes

TL;DR

This paper develops a semi-infinite tensor structure on the semiderived category of quasi-coherent torsion sheaves on ind-schemes, blending cotensor and derived tensor products, advancing semi-infinite algebraic geometry.

Contribution

It introduces a semi-infinite tensor product for semiderived categories on ind-schemes, connecting cotensor and derived tensor products within semi-infinite algebraic geometry.

Findings

Constructed semi-infinite tensor structure on semiderived categories.

Defined the semitensor product as a mixture of cotensor and derived tensor products.

Identified the inverse image of the dualizing complex as the unit object.

Abstract

We construct the semi-infinite tensor structure on the semiderived category of quasi-coherent torsion sheaves on an ind-scheme endowed with a flat affine morphism into an ind-Noetherian ind-scheme with a dualizing complex. The semitensor product is "a mixture of" the cotensor product along the base and the derived tensor product along the fibers. The inverse image of the dualizing complex is the unit object. This construction is a partial realization of the Semi-infinite Algebraic Geometry program, as outlined in the introduction to arXiv:1504.00700.