Invariance Under Postcomposition with a Smooth Morphism

Let 𝔛 $${\mathfrak X}$$ be an ind-semi-separated ind-Noetherian ind-scheme, and let τ : 𝔛 ′ → 𝔛 $$\tau \colon {\mathfrak X}'\longrightarrow {\mathfrak X}$$ be a smooth affine morphism of finite type. Let π ′ : 𝔜 → 𝔛 ′ $$\pi '\colon {\boldsymbol {\mathfrak Y}}\longrightarrow {\mathfrak X}'$$ be a flat affine morphism, and let π : 𝔜 → 𝔛 $$\pi \colon {\boldsymbol {\mathfrak Y}}\longrightarrow {\mathfrak X}$$ denote the composition π = τ π ′ $$\pi =\tau \pi '$$ . Let D • $${\mathcal D}^{\scriptstyle \bullet }$$ be a dualizing complex on 𝔛 $${\mathfrak X}$$ ; then D ′ • = τ ∗ D • $${\mathcal D}'{ }^{\scriptstyle \bullet }= \tau ^*{\mathcal D}^{\scriptstyle \bullet }$$ is a dualizing complex on 𝔛 ′ $${\mathfrak X}'$$ . The aim of this chapter is to show that the constructions of Chaps. 7–8, including the semiderived category of quasi-coherent torsion sheaves on 𝔜 $${\boldsymbol {\mathfrak Y}}$$ and the semitensor product operation on it, are preserved by the passage from the flat affine moprhism π ′ $$\pi '$$ to the flat affine morphism π $$\pi $$ .