The Cotensor Product

The triangulated tensor structure on the coderived category D co ( X – qcoh ) $${\mathsf D}^{\mathsf {co}}(X{{\text{--}\mathsf {qcoh}}})$$ of quasi-coherent sheaves on a Noetherian scheme with a dualizing complex D • $${\mathcal D}^{\scriptstyle \bullet }$$ was introduced in Murfet (The mock homotopy category of projectives and Grothendieck duality. Ph. D. Thesis, Australian National University, September 2007, Propositions 6.2, 8.10, and B.6) and studied in Efimov and Positselski (Algebra Number Theory 9(#5):1159–1292, 2015, Section B.2.5), where it was denoted by □ D • $$\mathbin {\square }_{{\mathcal D}^{\scriptstyle \bullet }}$$ and called the cotensor product of complexes of quasi-coherent sheaves on X over the dualizing complex D • $${\mathcal D}^{\scriptstyle \bullet }$$ . The aim of this chapter is to generalize this construction to complexes of quasi-coherent torsion sheaves on an ind-Noetherian ind-scheme with a dualizing complex and explain the connection with the cotensor product of complexes of comodules over a cocommutative coalgebra.