Ind-Schemes of Ind-Finite Type and the ! $$!$$ -Tensor Product

Throughout this chapter, 𝕜 $$\Bbbk $$ denotes a fixed ground field. Given two ind-schemes 𝔛 ′ $${\mathfrak X}'$$ and 𝔛 ″ $${\mathfrak X}''$$ (or two schemes X ′ $$X'$$ and X ″ $$X''$$ ) over 𝕜 $$\Bbbk $$ , we denote the fibered product 𝔛 ′ × Spec 𝕜 𝔛 ″ $${\mathfrak X}'\times _{ \operatorname {\mathrm {Spec}}\Bbbk }{\mathfrak X}''$$ (or X ′ × Spec 𝕜 X ″ $$X'\times _{ \operatorname {\mathrm {Spec}}\Bbbk }X''$$ ) simply by 𝔛 ′ × 𝕜 𝔛 ″ $${\mathfrak X}'\times _\Bbbk {\mathfrak X}''$$ (or X ′ × 𝕜 X ″ $$X'\times _\Bbbk X''$$ ) for brevity. (See Sects. 1.1 and 1.2 for a discussion of fibered products of ind-schemes.) Let 𝔛 $$\mathfrak {X}$$ be an ind-separated ind-scheme of ind-finite type over the field 𝕜 $$\Bbb {k}$$ . The aim of this chapter is to describe the cotensor product functor, for a suitable choice of the dualizing complex on 𝔛 $$\mathfrak {X}$$ , as the derived !-restriction to the diagonal of the external tensor product on 𝔛 × k 𝔛 $$\mathfrak {X}\times _k\mathfrak {X}$$ of two given complexes of quasi-coherent sheaves on 𝔛 $$\mathfrak {X}$$ .