The Stacks project

Lemma 99.16.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper, flat, and of finite presentation. Let $E \in D(\mathcal{O}_ X)$. Assume

1. $E$ is $S$-perfect (More on Morphisms of Spaces, Definition 76.52.1), and

2. for every point $s \in S$ we have

   \[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ s}}(E_ s, E_ s) = 0 \quad \text{for}\quad i < 0 \]

   where $E_ s$ is the pullback to the fibre $X_ s$.

Then

1. (1) and (2) are preserved by arbitrary base change $V \to Y$,

2. $\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ V}}(E_ V, E_ V) = 0$ for $i < 0$ and all $V$ over $Y$,

3. $V \mapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ V}}(E_ V, E_ V)$ is representable by an algebraic space affine and of finite presentation over $Y$.

Here $X_ V$ is the base change of $X$ and $E_ V$ is the derived pullback of $E$ to $X_ V$.