## 99.16 Moduli of complexes on a proper morphism

The title and the material of this section are taken from [lieblich-complexes]. Let $S$ be a scheme and let $f : X \to B$ be a proper, flat, finitely presented morphism of algebraic spaces. We will prove that there is an algebraic stack

\[ \mathcal{C}\! \mathit{omplexes}_{X/B} \]

parametrizing “families” of objects of $D^ b_{\textit{Coh}}$ of the fibres with vanishing negative self-exts. More precisely a family is given by a relatively perfect object of the derived category of the total space; this somewhat technical notion is studied in More on Morphisms of Spaces, Section 76.52.

Already if $X$ is a proper algebraic space over a field $k$ we obtain a very interesting algebraic stack. Namely, there is an embedding

\[ \mathcal{C}\! \mathit{oh}_{X/k} \longrightarrow \mathcal{C}\! \mathit{omplexes}_{X/k} \]

since for any $\mathcal{O}$-module $\mathcal{F}$ (on any ringed topos) we have $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {O}(\mathcal{F}, \mathcal{F}) = 0$ for $i < 0$. Although this certainly shows our stack is nonempty, the true motivation for the study of $\mathcal{C}\! \mathit{omplexes}_{X/k}$ is that there are often objects of the derived category $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ with vanishing negative self-exts and nonvanishing cohomology sheaves in more than one degree. For example, $X$ could be derived equivalent to another proper algebraic space $Y$ over $k$, i.e., we have a $k$-linear equivalence

\[ F : D^ b_{\textit{Coh}}(\mathcal{O}_ Y) \longrightarrow D^ b_{\textit{Coh}}(\mathcal{O}_ X) \]

There are cases where this happens and $F$ is not given by an isomorphism between $X$ and $Y$; for example in the case of an abelian variety and its dual. In this situation $F$ induces an isomorphism of algebraic stacks

\[ \mathcal{C}\! \mathit{omplexes}_{Y/k} \longrightarrow \mathcal{C}\! \mathit{omplexes}_{X/k} \]

(insert future reference here) and in particular the stack of coherent sheaves on $Y$ maps into the stack of complexes on $X$. Turning this around, if we can understand well enough the geometry of $\mathcal{C}\! \mathit{omplexes}_{X/k}$, then we can try to use this to study all possible derived equivalent $Y$.

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 $X_ y$.

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

\[ y \in |W| \Leftrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{X_ y}}(K_ y, E_ y) = 0 \text{ for }i < 0. \]

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 $X_ V$.

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 sheaf^{1} 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.
$\square$

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

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

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

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

$V \mapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ V}}(E_ V, E_ V)$ 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 $X_ V$.

**Proof.**
Immediate consequence of Lemma 99.16.1.
$\square$

Situation 99.16.3. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper, flat, and of finite presentation. We denote $\mathcal{C}\! \mathit{omplexes}_{X/B}$ the category whose objects are triples $(T, g, E)$ where

$T$ is a scheme over $S$,

$g : T \to B$ is a morphism over $S$, and setting $X_ T = T \times _{g, B} X$

$E$ is an object of $D(\mathcal{O}_{X_ T})$ satisfying conditions (1) and (2) of Lemma 99.16.2.

A morphism $(T, g, E) \to (T', g', E')$ is given by a pair $(h, \varphi )$ where

$h : T \to T'$ is a morphism of schemes over $B$ (i.e., $g' \circ h = g$), and

$\varphi : L(h')^*E' \to E$ is an isomorphism of $D(\mathcal{O}_{X_ T})$ where $h' : X_ T \to X_{T'}$ is the base change of $h$.

Thus $\mathcal{C}\! \mathit{omplexes}_{X/B}$ is a category and the rule

\[ p : \mathcal{C}\! \mathit{omplexes}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}, \quad (T, g, E) \longmapsto T \]

is a functor. For a scheme $T$ over $S$ we denote $\mathcal{C}\! \mathit{omplexes}_{X/B, T}$ the fibre category of $p$ over $T$. These fibre categories are groupoids.

Lemma 99.16.4. In Situation 99.16.3 the functor $p : \mathcal{C}\! \mathit{omplexes}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}$ is fibred in groupoids.

**Proof.**
We show that $p$ is fibred in groupoids by checking conditions (1) and (2) of Categories, Definition 4.35.1. Given an object $(T', g', E')$ of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ and a morphism $h : T \to T'$ of schemes over $S$ we can set $g = h \circ g'$ and $E = L(h')^*E'$ where $h' : X_ T \to X_{T'}$ is the base change of $h$. Then it is clear that we obtain a morphism $(T, g, E) \to (T', g', E')$ of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ lying over $h$. This proves (1). For (2) suppose we are given morphisms

\[ (h_1, \varphi _1) : (T_1, g_1, E_1) \to (T, g, E) \quad \text{and}\quad (h_2, \varphi _2) : (T_2, g_2, E_2) \to (T, g, E) \]

of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ and a morphism $h : T_1 \to T_2$ such that $h_2 \circ h = h_1$. Then we can let $\varphi $ be the composition

\[ L(h')^*E_2 \xrightarrow {L(h')^*\varphi _2^{-1}} L(h')^*L(h_2)^*E = L(h_1)^*E \xrightarrow {\varphi _1} E_1 \]

to obtain the morphism $(h, \varphi ) : (T_1, g_1, E_1) \to (T_2, g_2, E_2)$ that witnesses the truth of condition (2).
$\square$

Lemma 99.16.5. In Situation 99.16.3. Denote $\mathcal{X} = \mathcal{C}\! \mathit{omplexes}_{X/B}$. Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces.

**Proof.**
Consider two objects $x = (T, g, E)$ and $y = (T, g', E')$ of $\mathcal{X}$ over a scheme $T$. We have to show that $\mathit{Isom}_\mathcal {X}(x, y)$ is an algebraic space over $T$, see Algebraic Stacks, Lemma 94.10.11. If for $h : T' \to T$ the restrictions $x|_{T'}$ and $y|_{T'}$ are isomorphic in the fibre category $\mathcal{X}_{T'}$, then $g \circ h = g' \circ h$. Hence there is a transformation of presheaves

\[ \mathit{Isom}_\mathcal {X}(x, y) \longrightarrow \text{Equalizer}(g, g') \]

Since the diagonal of $B$ is representable (by schemes) this equalizer is a scheme. Thus we may replace $T$ by this equalizer and $E$ and $E'$ by their pullbacks. Thus we may assume $g = g'$.

Assume $g = g'$. After replacing $B$ by $T$ and $X$ by $X_ T$ we arrive at the following problem. Given $E, E' \in D(\mathcal{O}_ X)$ satisfying conditions (1), (2) of Lemma 99.16.2 we have to show that $\mathit{Isom}(E, E')$ is an algebraic space. Here $\mathit{Isom}(E, E')$ is the functor

\[ (\mathit{Sch}/B)^{opp} \to \textit{Sets},\quad T \mapsto \{ \varphi : E_ T \to E'_ T \text{ isomorphism in }D(\mathcal{O}_{X_ T})\} \]

where $E_ T$ and $E'_ T$ are the derived pullbacks of $E$ and $E'$ to $X_ T$. Now, let $W \subset B$, resp. $W' \subset B$ be the open subspace of $B$ associated to $E, E'$, resp. to $E', E$ by Lemma 99.16.1. Clearly, if there exists an isomorphism $E_ T \to E'_ T$ as in the definition of $\mathit{Isom}(E, E')$, then we see that $T \to B$ factors into both $W$ and $W'$ (because we have condition (1) for $E$ and $E'$ and we'll obviously have $E_ t \cong E'_ t$ so no nonzero maps $E_ t[i] \to E_ t$ or $E'_ t[i] \to E_ t$ over the fibre $X_ t$ for $i > 0$. Thus we may replace $B$ by the open $W \cap W'$. In this case the functor $H = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E')$

\[ (\mathit{Sch}/B)^{opp} \to \textit{Sets},\quad T \mapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ T}}(E_ T, E'_ T) \]

is an algebraic space affine and of finite presentation over $B$ by Lemma 99.16.1. The same is true for $H' = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E', E)$, $I = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E)$, and $I' = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E', E')$. Therefore we can repeat the argument of the proof of Proposition 99.4.3 to see that

\[ \mathit{Isom}(E, E') = (H' \times _ B H) \times _{c, I \times _ B I', \sigma } B \]

for some morphisms $c$ and $\sigma $. Thus $\mathit{Isom}(E, E')$ is an algebraic space.
$\square$

Lemma 99.16.6. In Situation 99.16.3 the functor $p : \mathcal{C}\! \mathit{omplexes}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}$ is a stack in groupoids.

**Proof.**
To prove that $\mathcal{C}\! \mathit{omplexes}_{X/B}$ is a stack in groupoids, we have to show that the presheaves $\mathit{Isom}$ are sheaves and that descent data are effective. The statement on $\mathit{Isom}$ follows from Lemma 99.16.5, see Algebraic Stacks, Lemma 94.10.11. Let us prove the statement on descent data.

Suppose that $\{ a_ i : T_ i \to T\} $ is an fppf covering of schemes over $S$. Let $(\xi _ i, \varphi _{ij})$ be a descent datum for $\{ T_ i \to T\} $ with values in $\mathcal{C}\! \mathit{omplexes}_{X/B}$. For each $i$ we can write $\xi _ i = (T_ i, g_ i, E_ i)$. Denote $\text{pr}_0 : T_ i \times _ T T_ j \to T_ i$ and $\text{pr}_1 : T_ i \times _ T T_ j \to T_ j$ the projections. The condition that $\xi _ i|_{T_ i \times _ T T_ j} \cong \xi _ j|_{T_ i \times _ T T_ j}$ implies in particular that $g_ i \circ \text{pr}_0 = g_ j \circ \text{pr}_1$. Thus there exists a unique morphism $g : T \to B$ such that $g_ i = g \circ a_ i$, see Descent on Spaces, Lemma 74.7.2. Denote $X_ T = T \times _{g, B} X$. Set $X_ i = X_{T_ i} = T_ i \times _{g_ i, B} X = T_ i \times _{a_ i, T} X_ T$ and

\[ X_{ij} = X_{T_ i} \times _{X_ T} X_{T_ j} = X_ i \times _{X_ T} X_ j \]

with projections $\text{pr}_ i$ and $\text{pr}_ j$ to $X_ i$ and $X_ j$. Observe that the pullback of $(T_ i, g_ i, E_ i)$ by $\text{pr}_0 : T_ i \times _ T T_ j \to T_ i$ is given by $(T_ i \times _ T T_ j, g_ i \circ \text{pr}_0, L\text{pr}_ i^*E_ i)$. Hence a descent datum for $\{ T_ i \to T\} $ in $\mathcal{C}\! \mathit{omplexes}_{X/B}$ is given by the objects $(T_ i, g \circ a_ i, E_ i)$ and for each pair $i, j$ an isomorphism in $D\mathcal{O}_{X_{ij}})$

\[ \varphi _{ij} : L\text{pr}_ i^*E_ i \longrightarrow L\text{pr}_ j^*E_ j \]

satisfying the cocycle condition over the pullback of $X$ to $T_ i \times _ T T_ j \times _ T T_ k$. Using the vanishing of negative Exts provided by (b) of Lemma 99.16.2, we may apply Simplicial Spaces, Lemma 85.35.2 to obtain descent^{2} for these complexes. In other words, we find there exists an object $E$ in $D_\mathit{QCoh}(\mathcal{O}_{X_ T})$ restricting to $E_ i$ on $X_{T_ i}$ compatible with $\varphi _{ij}$. Recall that being $T$-perfect signifies being pseudo-coherent and having locally finite tor dimension over $f^{-1}\mathcal{O}_ T$. Thus $E$ is $T$-perfect by an application of More on Morphisms of Spaces, Lemmas 76.54.1 and 76.54.2. Finally, we have to check condition (2) from Lemma 99.16.2 for $E$. This immediately follows from the description of the open $W$ in Lemma 99.16.1 and the fact that (2) holds for $E_ i$ on $X_{T_ i}/T_ i$.
$\square$

Lemma 99.16.8. In Situation 99.16.3 assume that $B \to S$ is locally of finite presentation. Then $p : \mathcal{C}\! \mathit{omplexes}_{X/B} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition 98.11.1).

**Proof.**
Write $B(T)$ for the discrete category whose objects are the $S$-morphisms $T \to B$. Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a filtered limit of affine schemes over $S$. Assigning to an object $(T, h, E)$ of $\mathcal{C}\! \mathit{omplexes}_{X/B, T}$ the object $h$ of $B(T)$ gives us a commutative diagram of fibre categories

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{C}\! \mathit{omplexes}_{X/B, T_ i} \ar[r] \ar[d] & \mathcal{C}\! \mathit{omplexes}_{X/B, T} \ar[d] \\ \mathop{\mathrm{colim}}\nolimits B(T_ i) \ar[r] & B(T) } \]

We have to show the top horizontal arrow is an equivalence. Since we have assume that $B$ is locally of finite presentation over $S$ we see from Limits of Spaces, Remark 70.3.11 that the bottom horizontal arrow is an equivalence. This means that we may assume $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a filtered limit of affine schemes over $B$. Denote $g_ i : T_ i \to B$ and $g : T \to B$ the corresponding morphisms. Set $X_ i = T_ i \times _{g_ i, B} X$ and $X_ T = T \times _{g, B} X$. Observe that $X_ T = \mathop{\mathrm{colim}}\nolimits X_ i$. By More on Morphisms of Spaces, Lemma 76.52.9 the category of $T$-perfect objects of $D(\mathcal{O}_{X_ T})$ is the colimit of the categories of $T_ i$-perfect objects of $D(\mathcal{O}_{X_{T_ i}})$. Thus all we have to prove is that given an $T_ i$-perfect object $E_ i$ of $D(\mathcal{O}_{X_{T_ i}})$ such that the derived pullback $E$ of $E_ i$ to $X_ T$ satisfies condition (2) of Lemma 99.16.2, then after increasing $i$ we have that $E_ i$ satisfies condition (2) of Lemma 99.16.2. Let $W \subset |T_ i|$ be the open constructed in Lemma 99.16.1 for $E_ i$ and $E_ i$. By assumption on $E$ we find that $T \to T_ i$ factors through $T$. Hence there is an $i' \geq i$ such that $T_{i'} \to T_ i$ factors through $W$, see Limits, Lemma 32.4.10 Then $i'$ works by construction of $W$.
$\square$

Lemma 99.16.9. In Situation 99.16.3. Let

\[ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' } \]

be a pushout in the category of schemes over $S$ where $Z \to Z'$ is a finite order thickening and $Z \to Y$ is affine, see More on Morphisms, Lemma 37.14.3. Then the functor on fibre categories

\[ \mathcal{C}\! \mathit{omplexes}_{X/B, Y'} \longrightarrow \mathcal{C}\! \mathit{omplexes}_{X/B, Y} \times _{\mathcal{C}\! \mathit{omplexes}_{X/B, Z}} \mathcal{C}\! \mathit{omplexes}_{X/B, Z'} \]

is an equivalence.

**Proof.**
Observe that the corresponding map

\[ B(Y') \longrightarrow B(Y) \times _{B(Z)} B(Z') \]

is a bijection, see Pushouts of Spaces, Lemma 81.6.1. Thus using the commutative diagram

\[ \xymatrix{ \mathcal{C}\! \mathit{omplexes}_{X/B, Y'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{omplexes}_{X/B, Y} \times _{\mathcal{C}\! \mathit{omplexes}_{X/B, Z}} \mathcal{C}\! \mathit{omplexes}_{X/B, Z'} \ar[d] \\ B(Y') \ar[r] & B(Y) \times _{B(Z)} B(Z') } \]

we see that we may assume that $Y'$ is a scheme over $B'$. By Remark 99.16.7 we may replace $B$ by $Y'$ and $X$ by $X \times _ B Y'$. Thus we may assume $B = Y'$.

Assume $B = Y'$. We first prove fully faithfulness of our functor. To do this, let $\xi _1, \xi _2$ be two objects of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ over $Y'$. Then we have to show that

\[ \mathit{Isom}(\xi _1, \xi _2)(Y') \longrightarrow \mathit{Isom}(\xi _1, \xi _2)(Y) \times _{\mathit{Isom}(\xi _1, \xi _2)(Z)} \mathit{Isom}(\xi _1, \xi _2)(Z') \]

is bijective. However, we already know that $\mathit{Isom}(\xi _1, \xi _2)$ is an algebraic space over $B = Y'$. Thus this bijectivity follows from Artin's Axioms, Lemma 98.4.1 (or the aforementioned Pushouts of Spaces, Lemma 81.6.1).

Essential surjectivity. Let $(E_ Y, E_{Z'}, \alpha )$ be a triple, where $E_ Y \in D(\mathcal{O}_ Y)$ and $E_{Z'} \in D(\mathcal{O}_{X_{Z'}})$ are objects such that $(Y, Y \to B, E_ Y)$ is an object of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ over $Y$, such that $(Z', Z' \to B, E_{Z'})$ is an object of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ over $Z'$, and $\alpha : L(X_ Z \to X_ Y)^*E_ Y \to L(X_ Z \to X_{Z'})^*E_{Z'}$ is an isomorphism in $D(\mathcal{O}_{Z'})$. That is to say

\[ ((Y, Y \to B, E_ Y), (Z', Z' \to B, E_{Z'}), \alpha ) \]

is an object of the target of the arrow of our lemma. Observe that the diagram

\[ \xymatrix{ X_ Z \ar[r] \ar[d] & X_{Z'} \ar[d] \\ X_ Y \ar[r] & X_{Y'} } \]

is a pushout with $X_ Z \to X_ Y$ affine and $X_ Z \to X_{Z'}$ a thickening (see Pushouts of Spaces, Lemma 81.6.7). Hence by Pushouts of Spaces, Lemma 81.8.1 we find an object $E_{Y'} \in D(\mathcal{O}_{X_{Y'}})$ together with isomorphisms $L(X_ Y \to X_{Y'})^*E_{Y'} \to E_ Y$ and $L(X_{Z'} \to X_{Y'})^*E_{Y'} \to E_ Z$ compatible with $\alpha $. Clearly, if we show that $E_{Y'}$ is $Y'$-perfect, then we are done, because property (2) of Lemma 99.16.2 is a property on points (and $Y$ and $Y'$ have the same points). This follows from More on Morphisms of Spaces, Lemma 76.54.4.
$\square$

Lemma 99.16.10. In Situation 99.16.3 assume that $S$ is a locally Noetherian scheme and $B \to S$ is locally of finite presentation. Let $k$ be a finite type field over $S$ and let $x_0 = (\mathop{\mathrm{Spec}}(k), g_0, E_0)$ be an object of $\mathcal{X} = \mathcal{C}\! \mathit{omplexes}_{X/B}$ over $k$. Then the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ (Artin's Axioms, Section 98.8) are finite dimensional.

**Proof.**
Observe that by Lemma 99.16.9 our stack in groupoids $\mathcal{X}$ satisfies property (RS*) defined in Artin's Axioms, Section 98.18. In particular $\mathcal{X}$ satisfies (RS). Hence all associated predeformation categories are deformation categories (Artin's Axioms, Lemma 98.6.1) and the statement makes sense.

In this paragraph we show that we can reduce to the case $B = \mathop{\mathrm{Spec}}(k)$. Set $X_0 = \mathop{\mathrm{Spec}}(k) \times _{g_0, B} X$ and denote $\mathcal{X}_0 = \mathcal{C}\! \mathit{omplexes}_{X_0/k}$. In Remark 99.16.7 we have seen that $\mathcal{X}_0$ is the $2$-fibre product of $\mathcal{X}$ with $\mathop{\mathrm{Spec}}(k)$ over $B$ as categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Thus by Artin's Axioms, Lemma 98.8.2 we reduce to proving that $B$, $\mathop{\mathrm{Spec}}(k)$, and $\mathcal{X}_0$ have finite dimensional tangent spaces and infinitesimal automorphism spaces. The tangent space of $B$ and $\mathop{\mathrm{Spec}}(k)$ are finite dimensional by Artin's Axioms, Lemma 98.8.1 and of course these have vanishing $\text{Inf}$. Thus it suffices to deal with $\mathcal{X}_0$.

Let $k[\epsilon ]$ be the dual numbers over $k$. Let $\mathop{\mathrm{Spec}}(k[\epsilon ]) \to B$ be the composition of $g_0 : \mathop{\mathrm{Spec}}(k) \to B$ and the morphism $\mathop{\mathrm{Spec}}(k[\epsilon ]) \to \mathop{\mathrm{Spec}}(k)$ coming from the inclusion $k \to k[\epsilon ]$. Set $X_0 = \mathop{\mathrm{Spec}}(k) \times _ B X$ and $X_\epsilon = \mathop{\mathrm{Spec}}(k[\epsilon ]) \times _ B X$. Observe that $X_\epsilon $ is a first order thickening of $X_0$ flat over the first order thickening $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k[\epsilon ])$. Observe that $X_0$ and $X_\epsilon $ give rise to canonically equivalent small étale topoi, see More on Morphisms of Spaces, Section 76.9. By More on Morphisms of Spaces, Lemma 76.54.4 we see that $T\mathcal{F}_{\mathcal{X}_0, k, x_0}$ is the set of isomorphism classes of lifts of $E_0$ to $X_\epsilon $ in the sense of Deformation Theory, Lemma 91.16.7. We conclude that

\[ T\mathcal{F}_{\mathcal{X}_0, k, x_0} = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_{X_0}}(E_0, E_0) \]

Here we have used the identification $\epsilon k[\epsilon ] \cong k$ of $k[\epsilon ]$-modules. Using Deformation Theory, Lemma 91.16.7 once more we see that there is a surjection

\[ \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) \leftarrow \mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_{X_0}}(E_0, E_0) \]

of $k$-vector spaces. As $E_0$ is pseudo-coherent it lies in $D^-_{\textit{Coh}}(\mathcal{O}_{X_0})$ by Derived Categories of Spaces, Lemma 75.13.7. Since $E_0$ locally has finite tor dimension and $X_0$ is quasi-compact we see $E_0 \in D^ b_{\textit{Coh}}(\mathcal{O}_{X_0})$. Thus the $\mathop{\mathrm{Ext}}\nolimits $s above are finite dimensional $k$-vector spaces by Derived Categories of Spaces, Lemma 75.8.4.
$\square$

Lemma 99.16.11. In Situation 99.16.3 assume $B = S$ is locally Noetherian. Then strong formal effectiveness in the sense of Artin's Axioms, Remark 98.20.2 holds for $p : \mathcal{C}\! \mathit{omplexes}_{X/S} \to (\mathit{Sch}/S)_{fppf}$.

**Proof.**
Let $(R_ n)$ be an inverse system of $S$-algebras with surjective transition maps whose kernels are locally nilpotent. Set $R = \mathop{\mathrm{lim}}\nolimits R_ n$. Let $(\xi _ n)$ be a system of objects of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ lying over $(\mathop{\mathrm{Spec}}(R_ n))$. We have to show $(\xi _ n)$ is effective, i.e., there exists an object $\xi $ of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ lying over $\mathop{\mathrm{Spec}}(R)$.

Write $X_ R = \mathop{\mathrm{Spec}}(R) \times _ S X$ and $X_ n = \mathop{\mathrm{Spec}}(R_ n) \times _ S X$. Of course $X_ n$ is the base change of $X_ R$ by $R \to R_ n$. Since $S = B$, we see that $\xi _ n$ corresponds simply to an $R_ n$-perfect object $E_ n \in D(\mathcal{O}_{X_ n})$ satisfying condition (2) of Lemma 99.16.2. In particular $E_ n$ is pseudo-coherent. The isomorphisms $\xi _{n + 1}|_{\mathop{\mathrm{Spec}}(R_ n)} \cong \xi _ n$ correspond to isomorphisms $L(X_ n \to X_{n + 1})^*E_{n + 1} \to E_ n$. Therefore by Flatness on Spaces, Theorem 77.13.6 we find a pseudo-coherent object $E$ of $D(\mathcal{O}_{X_ R})$ with $E_ n$ equal to the derived pullback of $E$ for all $n$ compatible with the transition isomorphisms.

Observe that $(R, \mathop{\mathrm{Ker}}(R \to R_1))$ is a henselian pair, see More on Algebra, Lemma 15.11.3. In particular, $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical of $R$. Then we may apply More on Morphisms of Spaces, Lemma 76.54.5 to see that $E$ is $R$-perfect.

Finally, we have to check condition (2) of Lemma 99.16.2. By Lemma 99.16.1 the set of points $t$ of $\mathop{\mathrm{Spec}}(R)$ where the negative self-exts of $E_ t$ vanish is an open. Since this condition is true in $V(\mathop{\mathrm{Ker}}(R \to R_1))$ and since $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical of $R$ we conclude it holds for all points.
$\square$

reference
Theorem 99.16.12 (Algebraicity of moduli of complexes on a proper morphism). Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is proper, flat, and of finite presentation. Then $\mathcal{C}\! \mathit{omplexes}_{X/B}$ is an algebraic stack over $S$.

**Proof.**
Set $\mathcal{X} = \mathcal{C}\! \mathit{omplexes}_{X/B}$. We have seen that $\mathcal{X}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ with diagonal representable by algebraic spaces (Lemmas 99.16.6 and 99.16.5). Hence it suffices to find a scheme $W$ and a surjective and smooth morphism $W \to \mathcal{X}$.

Let $B'$ be a scheme and let $B' \to B$ be a surjective étale morphism. Set $X' = B' \times _ B X$ and denote $f' : X' \to B'$ the projection. Then $\mathcal{X}' = \mathcal{C}\! \mathit{omplexes}_{X'/B'}$ is equal to the $2$-fibre product of $\mathcal{X}$ with the category fibred in sets associated to $B'$ over the category fibred in sets associated to $B$ (Remark 99.16.7). By the material in Algebraic Stacks, Section 94.10 the morphism $\mathcal{X}' \to \mathcal{X}$ is surjective and étale. Hence it suffices to prove the result for $\mathcal{X}'$. In other words, we may assume $B$ is a scheme.

Assume $B$ is a scheme. In this case we may replace $S$ by $B$, see Algebraic Stacks, Section 94.19. Thus we may assume $S = B$.

Assume $S = B$. Choose an affine open covering $S = \bigcup U_ i$. Denote $\mathcal{X}_ i$ the restriction of $\mathcal{X}$ to $(\mathit{Sch}/U_ i)_{fppf}$. If we can find schemes $W_ i$ over $U_ i$ and surjective smooth morphisms $W_ i \to \mathcal{X}_ i$, then we set $W = \coprod W_ i$ and we obtain a surjective smooth morphism $W \to \mathcal{X}$. Thus we may assume $S = B$ is affine.

Assume $S = B$ is affine, say $S = \mathop{\mathrm{Spec}}(\Lambda )$. Write $\Lambda = \mathop{\mathrm{colim}}\nolimits \Lambda _ i$ as a filtered colimit with each $\Lambda _ i$ of finite type over $\mathbf{Z}$. For some $i$ we can find a morphism of algebraic spaces $X_ i \to \mathop{\mathrm{Spec}}(\Lambda _ i)$ which is proper, flat, of finite presentation and whose base change to $\Lambda $ is $X$. See Limits of Spaces, Lemmas 70.7.1, 70.6.12, and 70.6.13. If we show that $\mathcal{C}\! \mathit{omplexes}_{X_ i/\mathop{\mathrm{Spec}}(\Lambda _ i)}$ is an algebraic stack, then it follows by base change (Remark 99.16.7 and Algebraic Stacks, Section 94.19) that $\mathcal{X}$ is an algebraic stack. Thus we may assume that $\Lambda $ is a finite type $\mathbf{Z}$-algebra.

Assume $S = B = \mathop{\mathrm{Spec}}(\Lambda )$ is affine of finite type over $\mathbf{Z}$. In this case we will verify conditions (1), (2), (3), (4), and (5) of Artin's Axioms, Lemma 98.17.1 to conclude that $\mathcal{X}$ is an algebraic stack. Note that $\Lambda $ is a G-ring, see More on Algebra, Proposition 15.50.12. Hence all local rings of $S$ are G-rings. Thus (5) holds. To check (2) we have to verify axioms [-1], [0], [1], [2], and [3] of Artin's Axioms, Section 98.14. We omit the verification of [-1] and axioms [0], [1], [2], [3] correspond respectively to Lemmas 99.16.6, 99.16.8, 99.16.9, 99.16.10. Condition (3) follows from Lemma 99.16.11. Condition (1) is Lemma 99.16.5.

It remains to show condition (4) which is openness of versality. To see this we will use Artin's Axioms, Lemma 98.20.3. We have already seen that $\mathcal{X}$ has diagonal representable by algebraic spaces, has (RS*), and is limit preserving (see lemmas used above). Hence we only need to see that $\mathcal{X}$ satisfies the strong formal effectiveness formulated in Artin's Axioms, Lemma 98.20.3. This follows from Lemma 99.16.11 and the proof is complete.
$\square$

## Comments (1)

Comment #8771 by Thiago Solovera e Nery on