The Stacks project

Proof. For any morphism $V \to Y$ the complex $K_ V$ is pseudo-coherent (Cohomology on Sites, Lemma 21.45.3) and $E_ V$ is $V$-perfect (More on Morphisms of Spaces, Lemma 76.52.6). Another observation is that given $y : \mathop{\mathrm{Spec}}(k) \to Y$ and a field extension $k'/k$ with $y' : \mathop{\mathrm{Spec}}(k') \to Y$ the induced morphism, we have

by Derived Categories of Schemes, Lemma 36.22.6. Thus the vanishing in (1) is really a property of the induced point $y \in |Y|$. We will use these two observations without further mention in the proof.

Assume first $Y$ is an affine scheme. Then we may apply More on Morphisms of Spaces, Lemma 76.52.11 and find a pseudo-coherent $L \in D(\mathcal{O}_ Y)$ which “universally computes” $Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E)$ in the sense described in that lemma. Unwinding the definitions, we obtain for a point $y \in Y$ the equality

We conclude that

By Derived Categories of Schemes, Lemma 36.31.1 the set $W$ of $y \in Y$ where this happens defines an open of $Y$. This open $W$ then satisfies the requirement in (1) for all morphisms from spectra of fields, by the “universality” of $L$.

Let's go back to $Y$ a general algebraic space. Choose an étale covering $\{ V_ i \to Y\}$ by affine schemes $V_ i$. Then we see that the subset $W \subset |Y|$ pulls back to the corresponding subset $W_ i \subset |V_ i|$ for $X_{V_ i}$, $K_{V_ i}$, $E_{V_ i}$. By the previous paragraph we find that $W_ i$ is open, hence $W$ is open. This proves (1) in general. Moreover, parts (2) and (3) are entirely formulated in terms of the category $\textit{Spaces}/W$ and the restrictions $X_ W$, $K_ W$, $E_ W$. This reduces us to the case $W = Y$.

Assume $W = Y$. We claim that for any algebraic space $V$ over $Y$ we have $Rf_{V, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ V, E_ V)$ has vanishing cohomology sheaves in degrees $< 0$. This will prove (2) because

by Cohomology on Sites, Lemmas 21.35.1 and 21.20.5 and the vanishing of the cohomology sheaves implies the cohomology group $H^ i$ is zero for $i < 0$ by Derived Categories, Lemma 13.16.1.

To prove the claim, we may work étale locally on $V$. In particular, we may assume $Y$ is affine and $W = Y$. Let $L \in D(\mathcal{O}_ Y)$ be as in the second paragraph of the proof. For an algebraic space $V$ over $Y$ denote $L_ V$ the derived pullback of $L$ to $V$. (An important feature we will use is that $L$ “works” for all algebraic spaces $V$ over $Y$ and not just affine $V$.) As $W = Y$ we have $H^ i(L) = 0$ for $i > 0$ (use More on Algebra, Lemma 15.75.6 to go from fibres to stalks). Hence $H^ i(L_ V) = 0$ for $i > 0$. The property defining $L$ is that

Since $L_ V$ sits in degrees $\leq 0$, we conclude that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L_ V, \mathcal{O}_ V)$ sits in degrees $\geq 0$ thereby proving the claim. This finishes the proof of (2).

Assume $W = Y$ but make no assumptions on the algebraic space $Y$. Since we have (2), we see from Simplicial Spaces, Lemma 85.35.1 that the functor $F$ given by $F(V) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ V}}(K_ V, E_ V)$ is a sheaf¹ on $(\textit{Spaces}/Y)_{fppf}$. Thus to prove that $F$ is an algebraic space and that $F \to Y$ is affine and of finite presentation, we may work étale locally on $Y$; see Bootstrap, Lemma 80.11.2 and Morphisms of Spaces, Lemmas 67.20.3 and 67.28.4. We conclude that it suffices to prove $F$ is an affine algebraic space of finite presentation over $Y$ when $Y$ is an affine scheme. In this case we go back to our pseudo-coherent complex $L \in D(\mathcal{O}_ Y)$. Since $H^ i(L) = 0$ for $i > 0$, we can represent $L$ by a complex of the form

with the last term in degree $0$, see More on Algebra, Lemma 15.64.5. Combining the two displayed formulas earlier in the proof we find that

In other words, there is a fibre product diagram

which proves what we want. $\square$