The Stacks project

Lemma 99.16.1. 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 $K, E \in D(\mathcal{O}_ X)$. Assume $K$ is pseudo-coherent and $E$ is $Y$-perfect (More on Morphisms of Spaces, Definition 76.52.1). For a field $k$ and a morphism $y : \mathop{\mathrm{Spec}}(k) \to Y$ denote $K_ y$, $E_ y$ the pullback to the fibre $X_ y$.

  1. There is an open $W \subset Y$ characterized by the property

    \[ y \in |W| \Leftrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ y}}(K_ y, E_ y) = 0 \text{ for }i < 0. \]
  2. For any morphism $V \to Y$ factoring through $W$ we have

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

    where $X_ V$ is the base change of $X$ and $K_ V$ and $E_ V$ are the derived pullbacks of $K$ and $E$ to $X_ V$.

  3. The functor $V \mapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ V}}(K_ V, E_ V)$ is a sheaf on $(\textit{Spaces}/W)_{fppf}$ representable by an algebraic space affine and of finite presentation over $W$.

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 15.75.6 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. $\square$

[1] To check the sheaf property for a covering $\{ V_ i \to V\} _{i \in I}$ first consider the Čech fppf hypercovering $a : V_\bullet \to V$ with $V_ n = \coprod _{i_0 \ldots i_ n} V_{i_0} \times _ V \ldots \times _ V V_{i_ n}$ and then set $U_\bullet = V_\bullet \times _{a, V} X_ V$. Then $U_\bullet \to X_ V$ is an fppf hypercovering to which we may apply Simplicial Spaces, Lemma 85.35.1.

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