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
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_{y'}}}(K_{y'}, E_{y'}) = \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ y}}(K_ y, E_ y) \otimes _ k k' \]
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
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\kappa (y)}(L \otimes _{\mathcal{O}_ Y}^\mathbf {L} \kappa (y), \kappa (y)) = \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ y}}(K_ y, E_ y) \]
We conclude that
\[ H^ i(L \otimes _{\mathcal{O}_ Y}^\mathbf {L} \kappa (y)) = 0 \text{ for } i > 0 \Leftrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ y}}(K_ y, E_ y) = 0 \text{ for }i < 0. \]
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
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ V}}(K_ V, E_ V) = H^ i(X_ V, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ V, E_ V)) = H^ i(V, Rf_{V, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ V, E_ V)) \]
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 15.75.5 to go from fibres to stalks). Hence $H^ i(L_ V) = 0$ for $i > 0$. The property defining $L$ is that
\[ Rf_{V, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ V, E_ V) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L_ V, \mathcal{O}_ V) \]
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
\[ \ldots \to \mathcal{O}_ Y^{\oplus m_1} \to \mathcal{O}_ Y^{\oplus m_0} \to 0 \to \ldots \]
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
\[ F(V) = \mathop{\mathrm{Ker}}( \mathop{\mathrm{Hom}}\nolimits _ V(\mathcal{O}_ V^{\oplus m_0}, \mathcal{O}_ V) \to \mathop{\mathrm{Hom}}\nolimits _ V(\mathcal{O}_ V^{\oplus m_1}, \mathcal{O}_ V) ) \]
In other words, there is a fibre product diagram
\[ \xymatrix{ F \ar[d] \ar[r] & Y \ar[d]^0 \\ \mathbf{A}_ Y^{m_0} \ar[r] & \mathbf{A}_ Y^{m_1} } \]
which proves what we want.
$\square$
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