The Stacks project

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

of $\mathcal{C}\! \mathit{oh}_{X/B}$ 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

to obtain the morphism $(h, \varphi ) : (T_1, g_1, \mathcal{F}_1) \to (T_2, g_2, \mathcal{F}_2)$ that witnesses the truth of condition (2). $\square$

Proof. Consider two objects $x = (T, g, \mathcal{F})$ and $y = (T, h, \mathcal{G})$ 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 $a : T' \to T$ the restrictions $x|_{T'}$ and $y|_{T'}$ are isomorphic in the fibre category $\mathcal{X}_{T'}$, then $g \circ a = h \circ a$. Hence there is a transformation of presheaves

Since the diagonal of $B$ is representable (by schemes) this equalizer is a scheme. Thus we may replace $T$ by this equalizer and the sheaves $\mathcal{F}$ and $\mathcal{G}$ by their pullbacks. Thus we may assume $g = h$. In this case we have $\mathit{Isom}_\mathcal {X}(x, y) = \mathit{Isom}(\mathcal{F}, \mathcal{G})$ and the result follows from Proposition 99.4.3. $\square$

Proof. To prove that $\mathcal{C}\! \mathit{oh}_{X/B}$ 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 99.5.3, 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 $\mathcal{C}\! \mathit{oh}_{X/B}$. For each $i$ we can write $\xi _ i = (T_ i, g_ i, \mathcal{F}_ i)$. 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 $\xi _ i|_{T_ i \times _ T T_ j} = \xi _ j|_{T_ i \times _ T T_ j}$ 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

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

satisfying the cocycle condition over (the pullback of $X$ to) $T_ i \times _ T T_ j \times _ T T_ k$. Ok, and now we simply use that $\{ X_ i \to X_ T\} $ is an fppf covering so that we can view $(\mathcal{F}_ i, \varphi _{ij})$ as a descent datum for this covering. By Descent on Spaces, Proposition 74.4.1 this descent datum is effective and we obtain a quasi-coherent sheaf $\mathcal{F}$ over $X_ T$ restricting to $\mathcal{F}_ i$ on $X_ i$. By Morphisms of Spaces, Lemma 67.31.5 we see that $\mathcal{F}$ is flat over $T$ and Descent on Spaces, Lemma 74.6.2 guarantees that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module. Finally, by Descent on Spaces, Lemma 74.11.19 we see that the scheme theoretic support of $\mathcal{F}$ is proper over $T$ as we've assumed the scheme theoretic support of $\mathcal{F}_ i$ is proper over $T_ i$ (note that taking scheme theoretic support commutes with flat base change by Morphisms of Spaces, Lemma 67.30.10). In this way we obtain our desired object over $T$. $\square$

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 $(T, h, \mathcal{F})$ of $\mathcal{C}\! \mathit{oh}_{X/B, T}$ the object $h$ of $B(T)$ gives us a commutative diagram of fibre categories

We have to show the top horizontal arrow is an equivalence. Since we have assumed 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 $X_ T = \mathop{\mathrm{colim}}\nolimits X_ i$ and that the algebraic spaces $X_ i$ and $X_ T$ are quasi-separated and quasi-compact (as they are of finite presentation over the affines $T_ i$ and $T$). By Limits of Spaces, Lemma 70.7.2 we see that

where $\textit{FP}(W)$ is short hand for the category of finitely presented $\mathcal{O}_ W$-modules. The results of Limits of Spaces, Lemmas 70.6.12 and 70.12.3 tell us the same thing is true if we replace $\textit{FP}(X_ i)$ and $\textit{FP}(X_ T)$ by the full subcategory of objects flat over $T_ i$ and $T$ with scheme theoretic support proper over $T_ i$ and $T$. This proves the lemma. $\square$

Proof. Observe that the corresponding map

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

we see that we may assume that $Y'$ is a scheme over $B'$. By Remark 99.5.5 we may replace $B$ by $Y'$ and $X$ by $X \times _ B Y'$. Thus we may assume $B = Y'$. In this case the statement follows from Pushouts of Spaces, Lemma 81.6.6. $\square$

Proof. Recall that a finitely presented module is quasi-coherent hence the equivalence of (1) and (2) and (3) and (4). The equivalence of (2) and (4) is a special case of Deformation Theory, Lemma 91.11.3. $\square$

Proof. Observe that by Lemma 99.5.7 our stack in groupoids $\mathcal{X}$ satisfies property (RS*) defined in Artin's Axioms, Section 98.21. 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 $\mathcal{X}_0 = \mathcal{C}\! \mathit{oh}_{X_0/k}$. In Remark 99.5.5 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 $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k[\epsilon ])$. Unwinding the definitions and using Lemma 99.5.8 we see that $T\mathcal{F}_{\mathcal{X}_0, k, x_0}$ is the set of lifts of $\mathcal{G}_0$ to a flat module on $X_\epsilon $. By Deformation Theory, Lemma 91.12.1 we conclude that

Here we have used the identification $\epsilon k[\epsilon ] \cong k$ of $k[\epsilon ]$-modules. Using Deformation Theory, Lemma 91.12.1 once more we see that

These spaces are finite dimensional over $k$ as $\mathcal{G}_0$ has support proper over $\mathop{\mathrm{Spec}}(k)$. Namely, $X_0$ is of finite presentation over $\mathop{\mathrm{Spec}}(k)$, hence Noetherian. Since $\mathcal{G}_0$ is of finite presentation it is a coherent $\mathcal{O}_{X_0}$-module. Thus we may apply Derived Categories of Spaces, Lemma 75.8.4 to conclude the desired finiteness. $\square$

Proof. Let $A$ be an $S$-algebra which is a complete local Noetherian ring with maximal ideal $\mathfrak m$ whose residue field $k$ is of finite type over $S$. We have to show that the category of objects over $A$ is equivalent to the category of formal objects over $A$. Since we know this holds for the category $\mathcal{S}_ B$ fibred in sets associated to $B$ by Artin's Axioms, Lemma 98.9.5, it suffices to prove this for those objects lying over a given morphism $\mathop{\mathrm{Spec}}(A) \to B$.

Set $X_ A = \mathop{\mathrm{Spec}}(A) \times _ B X$ and $X_ n = \mathop{\mathrm{Spec}}(A/\mathfrak m^ n) \times _ B X$. By Grothendieck's existence theorem (More on Morphisms of Spaces, Theorem 76.42.11) we see that the category of coherent modules $\mathcal{F}$ on $X_ A$ with support proper over $\mathop{\mathrm{Spec}}(A)$ is equivalent to the category of systems $(\mathcal{F}_ n)$ of coherent modules $\mathcal{F}_ n$ on $X_ n$ with support proper over $\mathop{\mathrm{Spec}}(A/\mathfrak m^ n)$. The equivalence sends $\mathcal{F}$ to the system $(\mathcal{F} \otimes _ A A/\mathfrak m^ n)$. See discussion in More on Morphisms of Spaces, Remark 76.42.12. To finish the proof of the lemma, it suffices to show that $\mathcal{F}$ is flat over $A$ if and only if all $\mathcal{F} \otimes _ A A/\mathfrak m^ n$ are flat over $A/\mathfrak m^ n$. This follows from More on Morphisms of Spaces, Lemma 76.24.3. $\square$

First proof. This proof is based on the criterion of Artin's Axioms, Lemma 98.24.4. Let $U \to S$ be of finite type morphism of schemes, $x$ an object of $\mathcal{X}$ over $U$ and $u_0 \in U$ a finite type point such that $x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$ is a closed point (Morphisms, Lemma 29.16.1) and $U = \mathop{\mathrm{Spec}}(A)$ with $U \to S$ mapping into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$ of $S$. Let $\mathcal{F}$ be the coherent module on $X_ A = \mathop{\mathrm{Spec}}(A) \times _ S X$ flat over $A$ corresponding to the given object $x$.

According to Deformation Theory, Lemma 91.12.1 we have an isomorphism of functors

and given any surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have an obstruction class

This uses that for any $A' \to A$ as above the base change $X_{A'} = \mathop{\mathrm{Spec}}(A') \times _ B X$ is flat over $A'$. Moreover, the construction of the obstruction class is functorial in the surjection $A' \to A$ (for fixed $A$) by Deformation Theory, Lemma 91.12.3. Apply Derived Categories of Spaces, Lemma 75.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms

for $i \leq m$ compatible with boundary maps. This object $K$, together with the displayed identifications above gives us a datum as in Artin's Axioms, Situation 98.24.2. Finally, condition (iv) of Artin's Axioms, Lemma 98.24.3 holds by Deformation Theory, Lemma 91.12.5. Thus Artin's Axioms, Lemma 98.24.4 does indeed apply and the lemma is proved. $\square$

Second proof. This proof is based on Artin's Axioms, Lemma 98.22.2. Conditions (1), (2), and (3) of that lemma correspond to Lemmas 99.5.3, 99.5.7, and 99.5.6.

We have constructed an obstruction theory in the chapter on deformation theory. Namely, given an $S$-algebra $A$ and an object $x$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $\mathop{\mathrm{Spec}}(A)$ given by $\mathcal{F}$ on $X_ A$ we set $\mathcal{O}_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ and if $A' \to A$ is a surjection with kernel $I$, then as obstruction element we take the element

of Deformation Theory, Lemma 91.12.1. All properties of an obstruction theory as defined in Artin's Axioms, Definition 98.22.1 follow from this lemma except for functoriality of obstruction classes as formulated in condition (ii) of the definition. But as stated in the footnote to assumption (4) of Artin's Axioms, Lemma 98.22.2 it suffices to check functoriality of obstruction classes for a fixed $A$ which follows from Deformation Theory, Lemma 91.12.3. Deformation Theory, Lemma 91.12.1 also tells us that $T_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^1_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for any $A$-module $M$.

To finish the proof it suffices to show that $T_ x(\prod M_ n) = \prod T_ x(M_ n)$ and $\mathcal{O}_ x(\prod M_ n) = \prod \mathcal{O}_ x(M)$. Apply Derived Categories of Spaces, Lemma 75.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(\mathcal{F}, \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We find a perfect object $K \in D(A)$ and functorial isomorphisms

for $i = 1, 2$. A straightforward argument shows that

whenever $K$ is a pseudo-coherent object of $D(A)$. In fact, this property (for all $i$) characterizes pseudo-coherent complexes, see More on Algebra, Lemma 15.65.5. $\square$

Proof. Set $\mathcal{X} = \mathcal{C}\! \mathit{oh}_{X/B}$. We have seen that $\mathcal{X}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ with diagonal representable by algebraic spaces (Lemmas 99.5.4 and 99.5.3). 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 $\mathcal{X}' = \mathcal{C}\! \mathit{oh}_{X'/B'}$ 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 99.5.5). 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 of finite presentation, separated, and flat and whose base change to $\Lambda $ is $X$. See Limits of Spaces, Lemmas 70.7.1, 70.6.9, and 70.6.12. If we show that $\mathcal{C}\! \mathit{oh}_{X_ i/\mathop{\mathrm{Spec}}(\Lambda _ i)}$ is an algebraic stack, then it follows by base change (Remark 99.5.5 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. By Lemma 99.5.11 we have that $\mathcal{X}$ satisfies openness of versality, hence (4) 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 99.5.4, 99.5.6, 99.5.7, 99.5.9. Condition (3) follows from Lemma 99.5.10. Finally, condition (1) is Lemma 99.5.3. This finishes the proof of the theorem. $\square$