spaces-perfect-remark-DQCoh-is-Ddga-with-support

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. The analogue of Theorem \ref{theorem-DQCoh-is-Ddga} holds for $D_{\QCoh, T}(\mathcal{O}_X)$. This follows from the exact same argument as in the proof of the theorem, using Lemmas \ref{lemma-generator-with-support} and \ref{lemma-compact-is-perfect-with-support} and a variant of Lemma \ref{lemma-tensor-with-QCoh-complex} with supports. If we ever need this, we will precisely state the result here and give a detailed proof.


spaces-perfect-remark-independence-choice

Let $X$ be a quasi-compact and quasi-separated algebraic space over a ring $R$. By the construction of the proof of Theorem \ref{theorem-DQCoh-is-Ddga} there exists a differential graded algebra $(A, \text{d})$ over $R$ such that $D_\QCoh(X)$ is $R$-linearly equivalent to $D(A, \text{d})$ as a triangulated category. One may ask: how unique is $(A, \text{d})$? The answer is (only) slightly better than just saying that $(A, \text{d})$ is well defined up to derived equivalence. Namely, suppose that $(B, \text{d})$ is a second such pair. Then we have $$ (A, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet) $$ and $$ (B, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, L^\bullet) $$ for some K-injective complexes $K^\bullet$ and $L^\bullet$ of $\mathcal{O}_X$-modules corresponding to perfect generators of $D_\QCoh(\mathcal{O}_X)$. Set $$ \Omega = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, L^\bullet) \quad \Omega' = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, K^\bullet) $$ Then $\Omega$ is a differential graded $B^{opp} \otimes_R A$-module and $\Omega'$ is a differential graded $A^{opp} \otimes_R B$-module. Moreover, the equivalence $$ D(A, \text{d}) \to D_\QCoh(\mathcal{O}_X) \to D(B, \text{d}) $$ is given by the functor $- \otimes_A^\mathbf{L} \Omega'$ and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark \ref{dga-remark-hochschild-cohomology}. If we ever need this remark we will provide a precise statement with a detailed proof here.


spaces-perfect-remark-explain-consequence

Let $S$ be a scheme. Let $(U \subset X, f : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $X$, $U$, $V$ are quasi-compact and quasi-separated. By Lemma \ref{lemma-better-coherator} the functors $DQ_X$, $DQ_U$, $DQ_V$, $DQ_{U \times_X V}$ exist. Moreover, there is a canonical distinguished triangle $$ DQ_X(K) \to Rj_{U, *}DQ_U(K|_U) \oplus Rj_{V, *}DQ_V(K|_V) \to Rj_{U \times_X V, *}DQ_{U \times_X V}(K|_{U \times_X V}) \to $$ for any $K \in D(\mathcal{O}_X)$. This follows by applying the exact functor $DQ_X$ to the distinguished triangle of Lemma \ref{lemma-exact-sequence-j-star} and using Lemma \ref{lemma-pushforward-better-coherator} three times.


spaces-perfect-remark-multiplication-map

With notation as in Lemma \ref{lemma-affine-morphism-and-hom-out-of-perfect}. The diagram $$ \xymatrix{ R\Hom_X(M, Rg'_*L) \otimes_R^\mathbf{L} R' \ar[r] \ar[d]_\mu & R\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \ar[d]^a \\ R\Hom_X(M, R(g')_*L) \ar@{=}[r] & R\Hom_{X'}(L(g')^*M, L) } $$ is commutative where the top horizontal arrow is the map from the lemma, $\mu$ is the multiplication map, and $a$ comes from the adjunction map $L(g')^*Rg'_*L \to L$. The multiplication map is the adjunction map $K' \otimes_R^\mathbf{L} R' \to K'$ for any $K' \in D(R')$.


spaces-perfect-remark-base-change-of-L

The pseudo-coherent complex $L$ of part (B) of Lemma \ref{lemma-compute-ext} is canonically associated to the situation. For example, formation of $L$ as in (B) is compatible with base change. In other words, given a cartesian diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of schemes we have canonical functorial isomorphisms $$ \Ext^i_{\mathcal{O}_{Y'}}(Lg^*L, \mathcal{F}') \longrightarrow \Ext^i_{\mathcal{O}_X}(L(g')^*E, (g')^*\mathcal{G}^\bullet \otimes_{\mathcal{O}_{X'}} (f')^*\mathcal{F}') $$ for $\mathcal{F}'$ quasi-coherent on $Y'$. Obsere that we do {\bf not} use derived pullback on $\mathcal{G}^\bullet$ on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof.


spaces-perfect-remark-explain-perfect-direct-image

Let $R$ be a ring. Let $X$ be an algebraic space of finite presentation over $R$. Let $\mathcal{G}$ be a finitely presented $\mathcal{O}_X$-module flat over $R$ with support proper over $R$. By Lemma \ref{lemma-base-change-tensor-perfect} there exists a finite complex of finite projective $R$-modules $M^\bullet$ such that we have $$ R\Gamma(X_{R'}, \mathcal{G}_{R'}) = M^\bullet \otimes_R R' $$ functorially in the $R$-algebra $R'$.


dualizing-remark-matlis

Let $(R, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa$ over $R$. Here is an addendum to Matlis duality: If $N$ is an $\mathfrak m$-power torsion module and $M = \Hom_R(N, E)$ is a finite module over the completion of $R$, then $N$ satisfies the descending chain condition. Namely, for any submodules $N'' \subset N' \subset N$ with $N'' \not = N'$, we can find an embedding $\kappa \subset N''/N'$ and hence a nonzero map $N' \to E$ annihilating $N''$ which we can extend to a map $N \to E$ annihilating $N''$. Thus $N \supset N' \mapsto M' = \Hom_R(N/N', E) \subset M$ is an inclusion preserving map from submodules of $N$ to submodules of $M$, whence the conclusion.


dualizing-remark-exact-support

Let $A$ be a ring and let $I \subset A$ be an ideal. Set $B = A/I$. In this case the functor $\Hom_A(B, -)$ is equal to the functor $$ \text{Mod}_A \longrightarrow \text{Mod}_B,\quad M \longmapsto M[I] $$ which sends $M$ to the submodule of $I$-torsion.


dualizing-remark-vanishing-for-arbitrary-modules

Let $(A, \mathfrak m)$ and $\omega_A^\bullet$ be as in Lemma \ref{lemma-sitting-in-degrees}. By More on Algebra, Lemma \ref{more-algebra-lemma-injective-amplitude} we see that $\omega_A^\bullet$ has injective-amplitude in $[-d, 0]$ because part (3) of that lemma applies. In particular, for any $A$-module $M$ (not necessarily finite) we have $\Ext^i_A(M, \omega_A^\bullet) = 0$ for $i \not \in \{-d, \ldots, 0\}$.


dualizing-remark-specific-injective-hull

Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with a normalized dualizing complex $\omega_A^\bullet$. By Lemma \ref{lemma-local-cohomology-of-dualizing} above we see that $R\Gamma_Z(\omega_A^\bullet)$ is an injective hull of the residue field placed in degree $0$. In fact, this gives a ``construction'' or ``realization'' of the injective hull which is slightly more canonical than just picking any old injective hull. Namely, a normalized dualizing complex is unique up to isomorphism, with group of automorphisms the group of units of $A$, whereas an injective hull of $\kappa$ is unique up to isomorphism, with group of automorphisms the group of units of the completion $A^\wedge$ of $A$ with respect to $\mathfrak m$.


properties-remark-normal-connected-irreducible

Let $X$ be a normal scheme. If $X$ is locally Noetherian then we see that $X$ is integral if and only if $X$ is connected, see Lemma \ref{lemma-normal-locally-Noetherian}. But there exists a connected affine scheme $X$ such that $\mathcal{O}_{X, x}$ is a domain for all $x \in X$, but $X$ is not irreducible, see Examples, Section \ref{examples-section-connected-locally-integral-not-integral}. This example is even a normal scheme (proof omitted), so beware!


properties-remark-non-integral-Japanese

In \cite{Hoobler-finite} a (locally Noetherian) scheme $X$ is called Japanese if for every $x \in X$ and every associated prime $\mathfrak p$ of $\mathcal{O}_{X, x}$ the ring $\mathcal{O}_{X, x}/\mathfrak p$ is Japanese. We do not use this definition since there exists a one dimensional Noetherian domain with excellent (in particular Japanese) local rings whose normalization is not finite. See \cite[Example 1]{Hochster-loci} or \cite{Heinzer-Levy} or \cite[Expos\'e XIX]{Traveaux}. On the other hand, we could circumvent this problem by calling a scheme $X$ Japanese if for every affine open $\Spec(A) \subset X$ the ring $A/\mathfrak p$ is Japanese for every associated prime $\mathfrak p$ of $A$.


properties-remark-neurotic

With assumptions and notation of Lemma \ref{lemma-ample-quasi-coherent}. Denote the displayed map of the lemma by $\theta_\mathcal{F}$. Note that the isomorphism $f^*\mathcal{O}_Y(n) \to \mathcal{L}^{\otimes n}$ of Lemma \ref{lemma-ample-gcd-is-one} is just $\theta_{\mathcal{L}^{\otimes n}}$. Consider the multiplication maps $$ \widetilde{M} \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(n) \longrightarrow \widetilde{M(n)} $$ see Constructions, Equation (\ref{constructions-equation-multiply-more-generally}). Pull this back to $X$ and consider $$ \xymatrix{ f^*\widetilde{M} \otimes_{\mathcal{O}_X} f^*\mathcal{O}_Y(n) \ar[r] \ar[d]_{\theta_\mathcal{F} \otimes \theta_{\mathcal{L}^{\otimes n}}} & f^*\widetilde{M(n)} \ar[d]^{\theta_{\mathcal{F} \otimes \mathcal{L}^{\otimes n}}} \\ \mathcal{F} \otimes \mathcal{L}^{\otimes n} \ar[r]^{\text{id}} & \mathcal{F} \otimes \mathcal{L}^{\otimes n} } $$ Here we have used the obvious identification $M(n) = \Gamma_*(X, \mathcal{L}, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$. This diagram commutes. Proof omitted.


properties-remark-maximal-points-affine

Lemma \ref{lemma-maximal-points-affine} above is false if $X$ is not quasi-separated. Here is an example. Take $R = \mathbf{Q}[x, y_1, y_2, \ldots]/((x-i)y_i)$. Consider the minimal prime ideal $\mathfrak p = (y_1, y_2, \ldots)$ of $R$. Glue two copies of $\Spec(R)$ along the (not quasi-compact) open $\Spec(R) \setminus V(\mathfrak p)$ to get a scheme $X$ (glueing as in Schemes, Example \ref{schemes-example-affine-space-zero-doubled}). Then the two maximal points of $X$ corresponding to $\mathfrak p$ are not contained in a common affine open. The reason is that any open of $\Spec(R)$ containing $\mathfrak p$ contains infinitely many of the ``lines'' $x = i$, $y_j = 0$, $j \not = i$ with parameter $y_i$. Details omitted.


quot-remark-hom-base-change

In Situation \ref{situation-hom} let $B' \to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \times_B B'$ and denote $\mathcal{F}'$, $\mathcal{G}'$ the pullback of $\mathcal{F}$, $\mathcal{G}$ to $X'$. Then we obtain a functor $\mathit{Hom}(\mathcal{F}', \mathcal{G}') : (\Sch/B')^{opp} \to \textit{Sets}$ associated to the base change $f' : X' \to B'$. For a scheme $T$ over $B'$ it is clear that we have $$ \mathit{Hom}(\mathcal{F}', \mathcal{G}')(T) = \mathit{Hom}(\mathcal{F}, \mathcal{G})(T) $$ where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \to B' \to B$. This trivial remark will occasionally be useful to change the base algebraic space.


quot-remark-coherent-base-change

In Situation \ref{situation-coherent} the rule $(T, g, \mathcal{F}) \mapsto (T, g)$ defines a $1$-morphism $$ \Cohstack_{X/B} \longrightarrow \mathcal{S}_B $$ of stacks in groupoids (see Lemma \ref{lemma-coherent-stack}, Algebraic Stacks, Section \ref{algebraic-section-split}, and Examples of Stacks, Section \ref{examples-stacks-section-stack-associated-to-sheaf}). Let $B' \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{S}_{B'} \to \mathcal{S}_B$ be the associated $1$-morphism of stacks fibred in sets. Set $X' = X \times_B B'$. We obtain a stack in groupoids $\Cohstack_{X'/B'} \to (\Sch/S)_{fppf}$ associated to the base change $f' : X' \to B'$. In this situation the diagram $$ \vcenter{ \xymatrix{ \Cohstack_{X'/B'} \ar[r] \ar[d] & \Cohstack_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_B } } \quad \begin{matrix} \text{or in} \\ \text{another} \\ \text{notation} \end{matrix} \quad \vcenter{ \xymatrix{ \Cohstack_{X'/B'} \ar[r] \ar[d] & \Cohstack_{X/B} \ar[d] \\ \Sch/B' \ar[r] & \Sch/B } } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space.


quot-remark-q-base-change

In Situation \ref{situation-q} let $B' \to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \times_B B'$ and denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X'$. Thus we have the functor $Q_{\mathcal{F}'/X'/B'}$ on the category of schemes over $B'$. For a scheme $T$ over $B'$ it is clear that we have $$ Q_{\mathcal{F}'/X'/B'}(T) = Q_{\mathcal{F}/X/B}(T) $$ where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \to B' \to B$. Similar remarks apply to $\text{Q}^{fp}_{\mathcal{F}/X/B}$. These trivial remarks will occasionally be useful to change the base algebraic space.


quot-remark-q-sheaf

Let $S$ be a scheme, $X$ an algebraic space over $S$, and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module. Suppose that $\{f_i : X_i \to X\}_{i \in I}$ is an fpqc covering and for each $i, j \in I$ we are given an fpqc covering $\{X_{ijk} \to X_i \times_X X_j\}$. In this situation we have a bijection $$ \left\{ \begin{matrix} \text{quotients }\mathcal{F} \to \mathcal{Q}\text{ where } \\ \mathcal{Q}\text{ is a quasi-coherent }\\ \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{families of quotients }f_i^*\mathcal{F} \to \mathcal{Q}_i \text{ where } \\ \mathcal{Q}_i\text{ is quasi-coherent and } \mathcal{Q}_i\text{ and }\mathcal{Q}_j\\ \text{ restrict to the same quotient on }X_{ijk} \end{matrix} \right\} $$ Namely, let $(f_i^*\mathcal{F} \to \mathcal{Q}_i)_{i \in I}$ be an element of the right hand side. Then since $\{X_{ijk} \to X_i \times_X X_j\}$ is an fpqc covering we see that the pullbacks of $\mathcal{Q}_i$ and $\mathcal{Q}_j$ restrict to the same quotient of the pullback of $\mathcal{F}$ to $X_i \times_X X_j$ (by fully faithfulness in Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}). Hence we obtain a descent datum for quasi-coherent modules with respect to $\{X_i \to X\}_{i \in I}$. By Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we find a map of quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F} \to \mathcal{Q}$ whose restriction to $X_i$ recovers the given maps $f_i^*\mathcal{F} \to \mathcal{Q}_i$. Since the family of morphisms $\{X_i \to X\}$ is jointly surjective and flat, for every point $x \in |X|$ there exists an $i$ and a point $x_i \in |X_i|$ mapping to $x$. Note that the induced map on local rings $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_i, \overline{x_i}}$ is faithfully flat, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-flat}. Thus we see that $\mathcal{F} \to \mathcal{Q}$ is surjective.


quot-remark-q-obs

In Situation \ref{situation-q} {\bf assume} that $\mathcal{F}$ is flat over $B$. Let $T \subset T'$ be an first order thickening of schemes over $B$ with ideal sheaf $\mathcal{J}$. Then $X_T \subset X_{T'}$ is a first order thickening of algebraic spaces whose ideal sheaf $\mathcal{I}$ is a quotient of $f_T^*\mathcal{J}$. We will think of sheaves on $X_{T'}$, resp.\ $T'$ as sheaves on $X_T$, resp.\ $T$ using the fundamental equivalence described in More on Morphisms of Spaces, Section \ref{spaces-more-morphisms-section-thickenings}. Let $$ 0 \to \mathcal{K} \to \mathcal{F}_T \to \mathcal{Q} \to 0 $$ define an element $x$ of $Q_{\mathcal{F}/X/B}(T)$. Since $\mathcal{F}_{T'}$ is flat over $T'$ we have a short exact sequence $$ 0 \to f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T \xrightarrow{i} \mathcal{F}_{T'} \xrightarrow{\pi} \mathcal{F}_T \to 0 $$ and we have $f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T = \mathcal{I} \otimes_{\mathcal{O}_{X_T}} \mathcal{F}_T$, see Deformation Theory, Lemma \ref{defos-lemma-deform-module-ringed-topoi}. Let us use the abbreviation $ f_T^*\mathcal{J} \otimes_{\mathcal{O}_{X_T}} \mathcal{G} = \mathcal{G} \otimes_{\mathcal{O}_T} \mathcal{J} $ for an $\mathcal{O}_{X_T}$-module $\mathcal{G}$. Since $\mathcal{Q}$ is flat over $T$, we obtain a short exact sequence $$ 0 \to \mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{F}_T \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J} \to \to 0 $$ Combining the above we obtain an canonical extension $$ 0 \to \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J} \to \pi^{-1}(\mathcal{K})/i(\mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J}) \to \mathcal{K} \to 0 $$ of $\mathcal{O}_{X_T}$-modules. This defines a canonical class $$ o_x(T') \in \Ext^1_{\mathcal{O}_{X_T}}(\mathcal{K}, \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J}) $$ If $o_x(T')$ is zero, then we obtain a splitting of the short exact sequence defining it, in other words, we obtain a $\mathcal{O}_{X_{T'}}$-submodule $\mathcal{K}' \subset \pi^{-1}(\mathcal{K})$ sitting in a short exact sequence $0 \to \mathcal{K} \otimes_{\mathcal{O}_T} \mathcal{J} \to \mathcal{K}' \to \mathcal{K} \to 0$. Then it follows from the lemma reference above that $\mathcal{Q}' = \mathcal{F}_{T'}/\mathcal{K}'$ is a lift of $x$ to an element of $Q_{\mathcal{F}/X/B}(T')$. Conversely, the reader sees that the existence of a lift implies that $o_x(T')$ is zero. Moreover, if $x \in Q_{\mathcal{F}/X/B}^{fp}(T)$, then automatically $x' \in Q_{\mathcal{F}/X/B}^{fp}(T')$ by Deformation Theory, Lemma \ref{defos-lemma-deform-fp-module-ringed-topoi}. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi).


quot-remark-q-defos

In Situation \ref{situation-q} {\bf assume} that $\mathcal{F}$ is flat over $B$. We continue the discussion of Remark \ref{remark-q-obs}. Assume $o_x(T') = 0$. Then we claim that the set of lifts $x' \in Q_{\mathcal{F}/X/B}(T')$ is a principal homogeneous space under the group $$ \Hom_{\mathcal{O}_{X_T}}(\mathcal{K}, \mathcal{Q} \otimes_{\mathcal{O}_T} \mathcal{J}) $$ Namely, given any $\mathcal{F}_{T'} \to \mathcal{Q}'$ flat over $T'$ lifting the quotient $\mathcal{Q}$ we obtain a commutative diagram with exact rows and columns $$ \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \mathcal{K} \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{F}_T \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{Q} \otimes \mathcal{J} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] \ar[d] & \mathcal{F}_{T'} \ar[r] \ar[d] & \mathcal{Q}' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[d] \ar[r] & \mathcal{F}_T \ar[d] \ar[r] & \mathcal{Q} \ar[d] \ar[r] & 0 \\ & 0 & 0 & 0 } $$ (to see this use the observations made in the previous remark). Given a map $\varphi : \mathcal{K} \to \mathcal{Q} \otimes \mathcal{J}$ we can consider the subsheaf $\mathcal{K}'_\varphi \subset \mathcal{F}_{T'}$ consisting of those local sections $s$ whose image in $\mathcal{F}_T$ is a local section $k$ of $\mathcal{K}$ and whose image in $\mathcal{Q}'$ is the local section $\varphi(k)$ of $\mathcal{Q} \otimes \mathcal{J}$. Then set $\mathcal{Q}'_\varphi = \mathcal{F}_{T'}/\mathcal{K}'_\varphi$. Conversely, any second lift of $x$ corresponds to one of the qotients constructed in this manner. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi).


quot-remark-quot-via-artins-axioms

Let $S$ be a Noetherian scheme all of whose local rings are G-rings. Let $X$ be an algebraic space over $S$ whose structure morphism $f : X \to S$ is of finite presentation and separated. Let $\mathcal{F}$ be a finitely presented quasi-coherent sheaf on $X$ flat over $S$. In this remark we sketch how one can use Artin's axioms to prove that $\Quotfunctor_{\mathcal{F}/X/S}$ is an algebraic space locally of finite presentation over $S$ and avoid using the algebraicity of the stack of coherent sheaves as was done in the proof of Proposition \ref{proposition-quot}. \medskip\noindent We check the conditions listed in Artin's Axioms, Proposition \ref{artin-proposition-spaces-diagonal-representable}. Representability of the diagonal of $\Quotfunctor_{\mathcal{F}/X/S}$ can be seen as follows: suppose we have two quotients $\mathcal{F}_T \to \mathcal{Q}_i$, $i = 1, 2$. Denote $\mathcal{K}_1$ the kernel of the first one. Then we have to show that the locus of $T$ over which $u : \mathcal{K}_1 \to \mathcal{Q}_2$ becomes zero is representable. This follows for example from Flatness on Spaces, Lemma \ref{spaces-flat-lemma-F-zero-closed-proper} or from a discussion of the $\mathit{Hom}$ sheaf earlier in this chapter. Axioms [0] (sheaf), [1] (limits), [2] (Rim-Schlessinger) follow from Lemmas \ref{lemma-quot-sheaf}, \ref{lemma-q-limit-preserving}, and \ref{lemma-q-RS-star} (plus some extra work to deal with the properness condition). Axiom [3] (finite dimensionality of tangent spaces) follows from the description of the infinitesimal deformations in Remark \ref{remark-q-defos} and finiteness of cohomology of coherent sheaves on proper algebraic spaces over fields (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent}). Axiom [4] (effectiveness of formal objects) follows from Grothendieck's existence theorem (More on Morphisms of Spaces, Theorem \ref{spaces-more-morphisms-theorem-grothendieck-existence}). As usual, the trickiest to verify is axiom [5] (openness of versality). One can for example use the obstruction theory described in Remark \ref{remark-q-obs} and the description of deformations in Remark \ref{remark-q-defos} to do this using the criterion in Artin's Axioms, Lemma \ref{artin-lemma-get-openness-obstruction-theory}. Please compare with the second proof of Lemma \ref{lemma-coherent-defo-thy}.


quot-remark-spaces-base-change

Let $B$ be an algebraic space over $\Spec(\mathbf{Z})$. Let $B\textit{-Spaces}'_{ft}$ be the category consisting of pairs $(X \to S, h : S \to B)$ where $X \to S$ is an object of $\Spacesstack'_{ft}$ and $h : S \to B$ is a morphism. A morphism $(X' \to S', h') \to (X \to S, h)$ in $B\textit{-Spaces}'_{ft}$ is a morphism $(f, g)$ in $\Spacesstack'_{ft}$ such that $h \circ g = h'$. In this situation the diagram $$ \xymatrix{ B\textit{-Spaces}'_{ft} \ar[r] \ar[d] & \Spacesstack'_{ft} \ar[d] \\ (\Sch/B)_{fppf} \ar[r] & \Sch_{fppf} } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\Spacesstack'_{ft}$ to the case of families over a given base algebraic space. Of course, a similar construction works for $B\textit{-Spaces}'_{fp, flat, proper}$


quot-remark-spaces-defo-thy

Lemma \ref{lemma-spaces-defo-thy} can also be shown using either Artin's Axioms, Lemma \ref{artin-lemma-dual-openness} (as in the first proof of Lemma \ref{lemma-coherent-defo-thy}), or using an obstruction theory as in Artin's Axioms, Lemma \ref{artin-lemma-get-openness-obstruction-theory} (as in the second proof of Lemma \ref{lemma-coherent-defo-thy}). In both cases one uses the deformation and obstruction theory developed in Cotangent, Section \ref{cotangent-section-deformations-ringed-topoi} to translate the needed properties of deformations and obstructions into $\Ext$-groups to which Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-compute-ext} can be applied. The second method (using an obstruction theory and therefore using the full cotangent complex) is perhaps the ``standard'' method used in most references.


quot-remark-polarized-base-change

Let $B$ be an algebraic space over $\Spec(\mathbf{Z})$. Let $B\textit{-Polarized}$ be the category consisting of triples $(X \to S, \mathcal{L}, h : S \to B)$ where $(X \to S, \mathcal{L})$ is an object of $\Polarizedstack$ and $h : S \to B$ is a morphism. A morphism $(X' \to S', \mathcal{L}', h') \to (X \to S, \mathcal{L}, h)$ in $B\textit{-Polarized}$ is a morphism $(f, g, \varphi)$ in $\Polarizedstack$ such that $h \circ g = h'$. In this situation the diagram $$ \xymatrix{ B\textit{-Polarized} \ar[r] \ar[d] & \Polarizedstack \ar[d] \\ (\Sch/B)_{fppf} \ar[r] & \Sch_{fppf} } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\Polarizedstack$ to the case of families over a given base algebraic space.


quot-remark-polarized-defo-thy

Lemma \ref{lemma-polarized-defo-thy} can also be shown using an obstruction theory as in Artin's Axioms, Lemma \ref{artin-lemma-get-openness-obstruction-theory} (as in the second proof of Lemma \ref{lemma-coherent-defo-thy}). To do this one has to generalize the deformation and obstruction theory developed in Cotangent, Section \ref{cotangent-section-deformations-ringed-topoi} to the case of pairs of algebraic spaces and quasi-coherent modules. Another possibility is to use that the $1$-morphism $\Polarizedstack \to \Spacesstack'_{fp, flat, proper}$ is algebraic (Lemma \ref{lemma-polarized-to-spaces-algebraic}) and the fact that we know openness of versality for the target (Lemma \ref{lemma-spaces-defo-thy} and Remark \ref{remark-spaces-defo-thy}).


quot-remark-curves-base-change

Let $B$ be an algebraic space over $\Spec(\mathbf{Z})$. Let $B\text{-}\Curvesstack$ be the category consisting of pairs $(X \to S, h : S \to B)$ where $X \to S$ is an object of $\Curvesstack$ and $h : S \to B$ is a morphism. A morphism $(X' \to S', h') \to (X \to S, h)$ in $B\text{-}\Curvesstack$ is a morphism $(f, g)$ in $\Curvesstack$ such that $h \circ g = h'$. In this situation the diagram $$ \xymatrix{ B\text{-}\Curvesstack \ar[r] \ar[d] & \Curvesstack \ar[d] \\ (\Sch/B)_{fppf} \ar[r] & \Sch_{fppf} } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\Curvesstack$ to the case of families of curves over a given base algebraic space.


quot-remark-alternative-approach-curves

Consider the $2$-fibre product $$ \xymatrix{ \Curvesstack \times_{\Spacesstack'_{fp, flat, proper}} \Polarizedstack \ar[r] \ar[d] & \Polarizedstack \ar[d] \\ \Curvesstack \ar[r] & \Spacesstack'_{fp, flat, proper} } $$ This fibre product parametrized polarized curves, i.e., families of curves endowed with a relatively ample invertible sheaf. It turns out that the left vertical arrow $$ \textit{PolarizedCurves} \longrightarrow \Curvesstack $$ is algebraic, smooth, and surjective. Namely, this $1$-morphism is algebraic (as base change of the arrow in Lemma \ref{lemma-polarized-to-spaces-algebraic}), every point is in the image, and there are no obstructions to deforming invertible sheaves on curves (see proof of Lemma \ref{lemma-curves-existence}). This gives another approach to the algebraicity of $\Curvesstack$. Namely, by Lemma \ref{lemma-curves-open-and-closed-in-spaces} we see that $\textit{PolarizedCurves}$ is an open and closed substack of the algebraic stack $\Polarizedstack$ and any stack in groupoids which is the target of a smooth algebraic morphism from an algebraic stack is an algebraic stack.


quot-remark-complexes-base-change

In Situation \ref{situation-complexes} the rule $(T, g, E) \mapsto (T, g)$ defines a $1$-morphism $$ \Complexesstack_{X/B} \longrightarrow \mathcal{S}_B $$ of stacks in groupoids (see Lemma \ref{lemma-complexes-stack}, Algebraic Stacks, Section \ref{algebraic-section-split}, and Examples of Stacks, Section \ref{examples-stacks-section-stack-associated-to-sheaf}). Let $B' \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{S}_{B'} \to \mathcal{S}_B$ be the associated $1$-morphism of stacks fibred in sets. Set $X' = X \times_B B'$. We obtain a stack in groupoids $\Complexesstack_{X'/B'} \to (\Sch/S)_{fppf}$ associated to the base change $f' : X' \to B'$. In this situation the diagram $$ \vcenter{ \xymatrix{ \Complexesstack_{X'/B'} \ar[r] \ar[d] & \Complexesstack_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_B } } \quad \begin{matrix} \text{or in} \\ \text{another} \\ \text{notation} \end{matrix} \quad \vcenter{ \xymatrix{ \Complexesstack_{X'/B'} \ar[r] \ar[d] & \Complexesstack_{X/B} \ar[d] \\ \Sch/B' \ar[r] & \Sch/B } } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space.


coherent-remark-chow-Noetherian

In the situation of Chow's Lemma \ref{lemma-chow-Noetherian}: \begin{enumerate} \item The morphism $\pi$ is actually H-projective (hence projective, see Morphisms, Lemma \ref{morphisms-lemma-H-projective}) since the morphism $X' \to \mathbf{P}^n_S \times_S X = \mathbf{P}^n_X$ is a closed immersion (use the fact that $\pi$ is proper, see Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). \item We may assume that $\pi^{-1}(U)$ is scheme theoretically dense in $X'$. Namely, we can simply replace $X'$ by the scheme theoretic closure of $\pi^{-1}(U)$. In this case we can think of $U$ as a scheme theoretically dense open subscheme of $X'$. See Morphisms, Section \ref{morphisms-section-scheme-theoretic-image}. \item If $X$ is reduced then we may choose $X'$ reduced. This is clear from (2). \end{enumerate}


coherent-remark-explain-perfect-direct-image

A consequence of Lemma \ref{lemma-perfect-direct-image} is that there exists a finite complex of finite projective $A$-modules $M^\bullet$ such that we have $$ H^i(X_{A'}, \mathcal{F}_{A'}) = H^i(M^\bullet \otimes_A A') $$ functorially in $A'$. The condition that $\mathcal{F}$ is flat over $A$ is essential, see \cite{Hartshorne}.


coherent-remark-inverse-systems-kernel-cokernel-annihilated-by

Let $X$ be a Noetherian scheme and let $\mathcal{I}, \mathcal{K} \subset \mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $\alpha : (\mathcal{F}_n) \to (\mathcal{G}_n)$ be a morphism of $\textit{Coh}(X, \mathcal{I})$. Given an affine open $\Spec(A) = U \subset X$ with $\mathcal{I}|_U, \mathcal{K}|_U$ corresponding to ideals $I, K \subset A$ denote $\alpha_U : M \to N$ of finite $A^\wedge$-modules which corresponds to $\alpha|_U$ via Lemma \ref{lemma-inverse-systems-affine}. We claim the following are equivalent \begin{enumerate} \item there exists an integer $t \geq 1$ such that $\Ker(\alpha_n)$ and $\Coker(\alpha_n)$ are annihilated by $\mathcal{K}^t$ for all $n \geq 1$, \item for any affine open $\Spec(A) = U \subset X$ as above the modules $\Ker(\alpha_U)$ and $\Coker(\alpha_U)$ are annihilated by $K^t$ for some integer $t \geq 1$, and \item there exists a finite affine open covering $X = \bigcup U_i$ such that the conclusion of (2) holds for $\alpha_{U_i}$. \end{enumerate} If these equivalent conditions hold we will say that $\alpha$ is a {\it map whose kernel and cokernel are annihilated by a power of $\mathcal{K}$}. To see the equivalence we use the following commutative algebra fact: suppose given an exact sequence $$ 0 \to T \to M \to N \to Q \to 0 $$ of $A$-modules with $T$ and $Q$ annihilated by $K^t$ for some ideal $K \subset A$. Then for every $f, g \in K^t$ there exists a canonical map $"fg": N \to M$ such that $M \to N \to M$ is equal to multiplication by $fg$. Namely, for $y \in N$ we can pick $x \in M$ mapping to $fy$ in $N$ and then we can set $"fg"(y) = gx$. Thus it is clear that $\Ker(M/JM \to N/JN)$ and $\Coker(M/JM \to N/JN)$ are annihilated by $K^{2t}$ for any ideal $J \subset A$. \medskip\noindent Applying the commutative algebra fact to $\alpha_{U_i}$ and $J = I^n$ we see that (3) implies (1). Conversely, suppose (1) holds and $M \to N$ is equal to $\alpha_U$. Then there is a $t \geq 1$ such that $\Ker(M/I^nM \to N/I^nN)$ and $\Coker(M/I^nM \to N/I^nN)$ are annihilated by $K^t$ for all $n$. We obtain maps $"fg" : N/I^nN \to M/I^nM$ which in the limit induce a map $N \to M$ as $N$ and $M$ are $I$-adically complete. Since the composition with $N \to M \to N$ is multiplication by $fg$ we conclude that $fg$ annihilates $T$ and $Q$. In other words $T$ and $Q$ are annihilated by $K^{2t}$ as desired.


coherent-remark-reformulate-existence-theorem

Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \Spec(A)$ and $S_n = \Spec(A/I^n)$. Let $X \to S$ be a separated morphism of finite type. For $n \geq 1$ we set $X_n = X \times_S S_n$. Picture: $$ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots & X \ar[d] \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots & S } $$ In this situation we consider systems $(\mathcal{F}_n, \varphi_n)$ where \begin{enumerate} \item $\mathcal{F}_n$ is a coherent $\mathcal{O}_{X_n}$-module, \item $\varphi_n : i_n^*\mathcal{F}_{n + 1} \to \mathcal{F}_n$ is an isomorphism, and \item $\text{Supp}(\mathcal{F}_1)$ is proper over $S_1$. \end{enumerate} Theorem \ref{theorem-grothendieck-existence} says that the completion functor $$ \begin{matrix} \text{coherent }\mathcal{O}_X\text{-modules }\mathcal{F} \\ \text{with support proper over }A \end{matrix} \quad \longrightarrow \quad \begin{matrix} \text{systems }(\mathcal{F}_n) \\ \text{as above} \end{matrix} $$ is an equivalence of categories. In the special case that $X$ is proper over $A$ we can omit the conditions on the supports.


formal-defos-remark-predeformation-functor

We say that a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$ is a {\it predeformation functor} if the associated cofibered set is a predeformation category, i.e.\ if $F(k)$ is a one element set.  Thus if $\mathcal{F}$ is a predeformation category, then $\overline{\mathcal{F}}$ is a predeformation functor.


formal-defos-remark-localize-cofibered-groupoid

Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids, and let $x \in \Ob(\mathcal{F}(k))$.  We denote by $\mathcal{F}_x$ the category of objects over $x$. An object of $\mathcal{F}_x$ is an arrow $y \to x$. A morphism $(y \to x) \to (z \to x)$ in $\mathcal{F}_x$ is a commutative diagram $$ \xymatrix{ y \ar[rr] \ar[dr] & & z \ar[dl] \\ & x & } $$ There is a forgetful functor $\mathcal{F}_x \to \mathcal{F}$. We define the functor $p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda$ as the composition $\mathcal{F}_x \to \mathcal{F} \xrightarrow{p} \mathcal{C}_\Lambda$. Then $p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda$ is a predeformation category (proof omitted). In this way we can pass from an arbitrary category cofibered in groupoids over $\mathcal{C}_\Lambda$ to a predeformation category at any $x \in \Ob(\mathcal{F}(k))$.


formal-defos-remark-different-sequence-ideals

Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. Suppose that for each $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$ we are given a filtration $\mathcal{I}_R$ of $R$ by ideals. If $\mathcal{I}_R$ induces the $\mathfrak m_R$-adic topology on $R$ for all $R$, then one can define a category $\widehat{\mathcal{F}}_\mathcal{I}$ by mimicking the definition of $\widehat{\mathcal{F}}$. This category comes equipped with a morphism $\widehat{p}_\mathcal{I} : \widehat{\mathcal{F}}_\mathcal{I} \to \widehat{\mathcal{C}}_\Lambda$ making it into a category cofibered in groupoids such that $\widehat{\mathcal{F}}_\mathcal{I}(R)$ is isomorphic to $\widehat{\mathcal{F}}_{\mathcal{I}_R}(R)$ as defined above. The categories cofibered in groupoids $\widehat{\mathcal{F}}_\mathcal{I}$ and $\widehat{\mathcal{F}}$ are equivalent, by using over an object $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$ the equivalence of Lemma \ref{lemma-formal-objects-different-filtration}.


formal-defos-remark-completion-functor

Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Identifying functors with cofibered sets, the completion of $F$ is the functor $\widehat{F} : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ given by $\widehat{F}(S) = \lim F(S/\mathfrak{m}_S^{n})$.  This agrees with the definition in Schlessinger's paper \cite{Sch}.


formal-defos-remark-restrict-completion

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. We claim that there is a canonical equivalence $$ can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \longrightarrow \mathcal{F}. $$ Namely, let $A \in \Ob(\mathcal{C}_\Lambda)$ and let $(A, \xi_n, f_n)$ be an object of $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}(A)$. Since $A$ is Artinian there is a minimal $m \in \mathbf{N}$ such that $\mathfrak m_A^m = 0$. Then $can$ sends $(A, \xi_n, f_n)$ to $\xi_m$. This functor is an equivalence of categories cofibered in groupoids by Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories} because it is an equivalence on all fibre categories by Lemma \ref{lemma-formal-objects-different-filtration} and the fact that the $\mathfrak m_A$-adic topology on a local Artinian ring $A$ comes from the zero ideal. We will frequently identify $\mathcal{F}$ with a full subcategory of $\widehat{\mathcal{F}}$ via a quasi-inverse to the functor $can$.


formal-defos-remark-completion-morphism

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Then there is an induced morphism $\widehat{\varphi}: \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ of categories cofibered in groupoids over $\widehat{\mathcal{C}}_\Lambda$. It sends an object $\xi = (R, \xi_n, f_n)$ of $\widehat{\mathcal{F}}$ to $(R, \varphi(\xi_n), \varphi(f_n))$, and it sends a morphism $(a_0 : R \to S, a_n : \xi_n \to \eta_n)$ between objects $\xi$ and $\eta$ of $\widehat{\mathcal{F}}$ to $(a_0 : R \to S, \varphi(a_n) : \varphi(\xi_n) \to \varphi(\eta_n))$. Finally, if $t : \varphi \to \varphi'$ is a $2$-morphism between $1$-morphisms $\varphi, \varphi': \mathcal{F} \to \mathcal{G}$ of categories cofibred in groupoids, then we obtain a $2$-morphism $\widehat{t} : \widehat{\varphi} \to \widehat{\varphi}'$. Namely, for $\xi = (R, \xi_n, f_n)$ as above we set $\widehat{t}_\xi = (t_{\varphi(\xi_n)})$. Hence completion defines a functor between $2$-categories $$ \widehat{~} : \text{Cof}(\mathcal{C}_\Lambda) \longrightarrow \text{Cof}(\widehat{\mathcal{C}}_\Lambda) $$ from the $2$-category of categories cofibred in groupoids over $\mathcal{C}_\Lambda$ to the $2$-category of categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda$.


formal-defos-remark-completion-restriction-adjoint

We claim the completion functor of Remark \ref{remark-completion-morphism} and the restriction functor $|_{\mathcal{C}_\Lambda} : \text{Cof}(\widehat{\mathcal{C}}_\Lambda) \to \text{Cof}(\mathcal{C}_\Lambda)$ of Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-restricting-base-category}) are ``2-adjoint'' in the following precise sense. Let $\mathcal{F} \in \Ob(\text{Cof}(\mathcal{C}_\Lambda))$ and let $\mathcal{G} \in \Ob(\text{Cof}(\widehat{\mathcal{C}}_\Lambda))$. Then there is an equivalence of categories $$ \Phi : \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}}) $$ To describe this equivalence, we define canonical morphisms $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$ and $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ as follows \begin{enumerate} \item Let $R \in \Ob(\widehat{\mathcal{C}}_\Lambda))$ and let $\xi$ be an object of the fiber category $\mathcal{G}(R)$. Choose a pushforward $\xi \to \xi_n$ of $\xi$ to $R/\mathfrak m_R^n$ for each $n \in \mathbf{N}$, and let $f_n : \xi_{n + 1} \to \xi_n$ be the induced morphism. Then $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}$ sends $\xi$ to $(R, \xi_n, f_n)$. \item This is the equivalence $can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ of Remark \ref{remark-restrict-completion}. \end{enumerate} Having said this, the equivalence $\Phi : \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})  \to \Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})$ sends a morphism $\varphi : \mathcal{G}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ to $$ \mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}} \xrightarrow{\widehat{\varphi}} \widehat{\mathcal{F}} $$ There is a quasi-inverse $\Psi : \Mor_{\widehat{\mathcal{C}}_\Lambda}( \mathcal{G}, \widehat{\mathcal{F}}) \to \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})$ to $\Phi$ which sends $\psi : \mathcal{G} \to \widehat{\mathcal{F}}$ to $$ \mathcal{G}|_{\mathcal{C}_\Lambda} \xrightarrow{\psi|_{\mathcal{C}_\Lambda}} \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}. $$ We omit the verification that $\Phi$ and $\Psi$ are quasi-inverse. We also do not address functoriality of $\Phi$ (because it would lead into 3-category territory which we want to avoid at all cost).


formal-defos-remark-completion-restriction-cofset-adjoint

For a category $\mathcal{C}$ we denote by $\text{CofSet}(\mathcal{C})$ the category of cofibered sets over $\mathcal{C}$. It is a $1$-category isomorphic the category of functors $\mathcal{C} \to \textit{Sets}$. See Remarks \ref{remarks-cofibered-groupoids} (\ref{item-convention-cofibered-sets}). The completion and restriction functors restrict to functors $\widehat{~} : \text{CofSet}(\mathcal{C}_\Lambda) \to \text{CofSet}(\widehat{\mathcal{C}}_\Lambda)$ and $|_{\mathcal{C}_\Lambda} : \text{CofSet}(\widehat{\mathcal{C}}_\Lambda) \to \text{CofSet}(\mathcal{C}_\Lambda)$ which we denote by the same symbols. As functors on the categories of cofibered sets, completion and restriction are adjoints in the usual 1-categorical sense: the same construction as in Remark \ref{remark-completion-restriction-adjoint} defines a functorial bijection $$ \Mor_{\mathcal{C}_\Lambda}(G|_{\mathcal{C}_\Lambda}, F) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}(G, \widehat{F}) $$ for $F \in \Ob(\text{CofSet}(\mathcal{C}_\Lambda))$ and $G \in \Ob(\text{CofSet}(\widehat{\mathcal{C}}_\Lambda))$. Again the map $\widehat{F}|_{\mathcal{C}_\Lambda} \to F$ is an isomorphism.


formal-defos-remark-restrict-complete-continuous-functor

Let $G : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ be a functor that commutes with limits. Then the map $G \to \widehat{G|_{\mathcal{C}_\Lambda}}$ described in Remark \ref{remark-completion-restriction-adjoint} is an isomorphism. Indeed, if $S$ is  an object of $\widehat{\mathcal{C}}_\Lambda$, then we have canonical bijections $$ \widehat{G|_{\mathcal{C}_\Lambda}}(S) = \lim_n G(S/\mathfrak{m}_S^n) = G(\lim_n S/\mathfrak{m}_S^n) = G(S). $$ In particular, if $R$ is an object of $\widehat{\mathcal{C}}_\Lambda$ then $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$ because the representable functor $\underline{R}$ commutes with limits by definition of limits.


formal-defos-remark-formal-objects-yoneda

Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda$.  It defines a functor $\underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ as described in Remarks \ref{remarks-cofibered-groupoids} (\ref{item-definition-yoneda}). As usual we identify this functor with the associated cofibered set.  If $\mathcal{F}$ is a cofibered category over $\mathcal{C}_\Lambda$, then there is an equivalence of categories \begin{equation} \label{equation-formal-objects-maps} \Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \longrightarrow \widehat{\mathcal{F}}(R). \end{equation} It is given by the composition $$ \Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \xrightarrow{\Phi} \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow{\sim} \widehat{\mathcal{F}}(R) $$ where $\Phi$ is as in Remark \ref{remark-completion-restriction-adjoint} and the second equivalence comes from the 2-Yoneda lemma (the cofibered analogue of Categories, Lemma \ref{categories-lemma-yoneda-2category}). Explicitly, the equivalence sends a morphism $\varphi : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ to the formal object $(R, \varphi(R \to R/\mathfrak{m}_R^n), \varphi(f_n))$ in $\widehat{\mathcal{F}}(R)$, where $f_n : R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$ is the projection. \medskip\noindent Assume a choice of pushforwards for $\mathcal{F}$ has been made. Given any $\xi \in \Ob(\widehat{\mathcal{F}}(R))$ we construct an explicit $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ which maps to $\xi$ under (\ref{equation-formal-objects-maps}). Namely, say $\xi = (R, \xi_n, f_n)$. An object $\alpha$ in $\underline{R}|_{\mathcal{C}_\Lambda}$ is the same thing as a morphism $\alpha : R \to A$ of $\widehat{\mathcal{C}}_\Lambda$ with $A$ Artinian. Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$. Then $\alpha$ factors through a unique $\alpha_m : R/\mathfrak m_R^m \to A$ and we can set $\underline{\xi}(\alpha) = \alpha_{m, *}\xi_m$. We omit the description of $\underline{\xi}$ on morphisms and we omit the proof that $\underline{\xi}$ maps to $\xi$ via (\ref{equation-formal-objects-maps}). \medskip\noindent Assume a choice of pushforwards for $\widehat{\mathcal{F}}$ has been made. In this case the proof of Categories, Lemma \ref{categories-lemma-yoneda-2category} gives an explicit quasi-inverse $$ \iota : \widehat{\mathcal{F}}(R) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}}) $$ to the 2-Yoneda equivalence which takes $\xi$ to the morphism $\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}$ sending $f \in \underline{R}(S) = \Mor_{\mathcal{C}_\Lambda}(R, S)$ to $f_*\xi$. A quasi-inverse to (\ref{equation-formal-objects-maps}) is then $$ \widehat{\mathcal{F}}(R) \xrightarrow{\iota} \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow{\Psi} \Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F}) $$ where $\Psi$ is as in Remark \ref{remark-completion-restriction-adjoint}. Given $\xi \in \Ob(\widehat{\mathcal{F}}(R))$ we have $\Psi(\iota(\xi)) \cong \underline{\xi}$ where $\underline{\xi}$ is as in the previous paragraph, because both are mapped to $\xi$ under the equivalence of categories (\ref{equation-formal-objects-maps}). Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}$ (see Remark \ref{remark-restrict-complete-continuous-functor}) and unwinding the definitions of $\Phi$ and $\Psi$ we conclude that $\iota(\xi)$ is isomorphic to the completion of $\underline{\xi}$.


formal-defos-remark-formal-objects-yoneda-map

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. Let $\xi = (R, \xi_n, f_n)$ and $\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$. Let $a = (a_n) : \xi \to \eta$ be a morphism of formal objects, i.e., a morphism of $\widehat{\mathcal{F}}$. Let $f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in $\widehat{\mathcal{C}}_\Lambda$. Then we obtain a $2$-commutative diagram $$ \xymatrix{ \underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & & \underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\ & \mathcal{F} } $$ where $\underline{\xi}$ and $\underline{\eta}$ are the morphisms constructed in Remark \ref{remark-formal-objects-yoneda}. To see this let $\alpha : S \to A$ be an object of $\underline{S}|_{\mathcal{C}_\Lambda}$ (see loc.\ cit.). Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_A^m = 0$. We get a commutative diagram $$ \xymatrix{ R \ar[d]^f \ar[r] & R/\mathfrak m_R^m \ar[d]_{f_m} \ar[rd]^{\beta_m} \\ S \ar[r] & S/\mathfrak m_S^m \ar[r]^{\alpha_m} & A } $$ such that the bottom arrows compose to give $\alpha$. Then $\underline{\eta}(\alpha) = \alpha_{m, *}\eta_m$ and $\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m$. The morphism $a_m : \xi_m \to \eta_m$ lies over $f_m$ hence we obtain a canonical morphism $$ \underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m \longrightarrow \underline{\eta}(\alpha) = \alpha_{m, *}\eta_m $$ lying over $\text{id}_A$ such that $$ \xymatrix{ \xi_m \ar[r] \ar[d]^{a_m} &  \beta_{m, *}\xi_m \ar[d] \\ \eta_m \ar[r] &  \alpha_{m, *}\eta_m } $$ commutes by the axioms of a category cofibred in groupoids. This defines a transformation of functors $\underline{\xi} \circ f \to \underline{\eta}$ which witnesses the 2-commutativity of the first diagram of this remark.


formal-defos-remark-spell-out-formal-object

According to Remark \ref{remark-formal-objects-yoneda}, giving a formal object $\xi$ of $\mathcal{F}$ is equivalent to giving a prorepresentable functor $U : \mathcal{C}_\Lambda \to \textit{Sets}$ and a morphism $U \to \mathcal{F}$.


formal-defos-remark-smoothness-2-categorical

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.  Let $B \to A$ be a ring map in $\mathcal{C}_\Lambda$.  Choices of pushforwards along $B \to A$ for objects in the fiber categories $\mathcal{F}(B)$ and $\mathcal{G}(B)$ determine functors $\mathcal{F}(B) \to \mathcal{F}(A)$ and $\mathcal{G}(B) \to \mathcal{G}(A)$ fitting into a $2$-commutative diagram $$ \xymatrix{ \mathcal{F}(B) \ar[r]^{\varphi} \ar[d] & \mathcal{G}(B) \ar[d] \\ \mathcal{F}(A) \ar[r]^{\varphi}        & \mathcal{G}(A) . } $$ Hence there is an induced functor $\mathcal{F}(B) \to \mathcal{F}(A) \times_{\mathcal{G}(A)} \mathcal{G}(B)$.  Unwinding the definitions shows that $\varphi : \mathcal{F} \to \mathcal{G}$ is smooth if and only if this induced functor is essentially surjective whenever $B \to A$ is surjective (or equivalently, by Lemma \ref{lemma-smoothness-small-extensions}, whenever $B \to A$ is a small extension).


formal-defos-remark-compare-smooth-schlessinger

The characterization of smooth morphisms in Remark \ref{remark-smoothness-2-categorical} is analogous to Schlessinger's notion of a smooth morphism of functors, cf.\ \cite[Definition 2.2.]{Sch}. In fact, when $\mathcal{F}$ and $\mathcal{G}$ are cofibered in sets then our notion is equivalent to Schlessinger's. Namely, in this case let $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$ be the corresponding functors, see Remarks \ref{remarks-cofibered-groupoids} (\ref{item-convention-cofibered-sets}). Then $F \to G$ is smooth if and only if for every surjection of rings $B \to A$ in $\mathcal{C}_\Lambda$ the map $F(B) \to F(A) \times_{G(A)} G(B)$ is surjective.


formal-defos-remark-smooth-to-iso-classes

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Then the morphism $\mathcal{F} \to \overline{\mathcal{F}}$ is smooth. Namely, suppose that $f : B \to A$ is a ring map in $\mathcal{C}_\Lambda$. Let $x \in \Ob(\mathcal{F}(A))$ and let $\overline{y} \in \overline{\mathcal{F}}(B)$ be the isomorphism class of $y \in \Ob(\mathcal{F}(B))$ such that $\overline{f_*y} = \overline{x}$. Then we simply take $x' = y$, the implied morphism $x' = y \to x$ over $B \to A$, and the equality $\overline{x'} = \overline{y}$ as the solution to the problem posed in Definition \ref{definition-smooth-morphism}.


formal-defos-remark-versal-object

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$, and let $\xi$ be a formal object of $\mathcal{F}$.  It follows from the definition of smoothness that versality of $\xi$ is equivalent to the following condition: If $$ \xymatrix{ & y \ar[d] \\ \xi \ar[r] & x } $$ is a diagram in $\widehat{\mathcal{F}}$ such that $y \to x$ lies over a surjective map $B \to A$ of Artinian rings (we may assume it is a small extension),  then there exists a morphism $\xi \to y$ such that $$ \xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar[ur] & x } $$ commutes. In particular, the condition that $\xi$ be versal does not depend on the choices of pushforwards made in the construction of $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ in Remark \ref{remark-formal-objects-yoneda}.


formal-defos-remark-smooth-on-top

Suppose $\mathcal{F}$ is a predeformation category admitting a smooth morphism $\varphi : \mathcal U \to \mathcal{F}$ from a predeformation category $\mathcal{U}$.  Then by Lemma \ref{lemma-smooth-morphism-essentially-surjective} $\varphi$ is essentially surjective, so by Lemma \ref{lemma-smooth-properties} $p: \mathcal{F} \to \mathcal{C}_\Lambda$ is smooth if and only if the composition $\mathcal U \xrightarrow{\varphi} \mathcal{F} \xrightarrow{p} \mathcal{C}_\Lambda$ is smooth, i.e.\ $\mathcal{F}$ is smooth if and only if $\mathcal{U}$ is smooth.


formal-defos-remark-compare-S1-S2-schlessinger

When $\mathcal{F}$ is cofibered in sets, conditions (S1) and (S2) are exactly conditions (H1) and (H2) from Schlessinger's paper \cite{Sch}. Namely, for a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$, conditions (S1) and (S2) state: \begin{enumerate} \item [(S1)] If $A_1 \to A$ and $A_2 \to A$ are maps in $\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective, then the induced map $F(A_1 \times_A A_2) \to F(A_1) \times_{F(A)} F(A_2)$ is surjective. \item [(S2)]  If $A \to k$ is a map in $\mathcal{C}_\Lambda$, then the induced map $F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])$ is bijective. \end{enumerate} The injectivity of the map $F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])$ comes from the second part of condition (S2) and the fact that morphisms are identities.


formal-defos-remark-linear-enriched-over-modules

One can define the notion of an $R$-linearity for any functor between categories enriched over $\text{Mod}_R$. We made the definition specifically for functors $L: \text{Mod}^{fg}_R \to \text{Mod}_R$ and $L: \text{Mod}_R \to \text{Mod}_R$ because these are the cases that we have needed so far.


formal-defos-remark-linear-functor

If $L: \text{Mod}^{fg}_R \to \text{Mod}_R$ is an $R$-linear functor, then $L$ preserves finite products and sends the zero module to the zero module, see Homology, Lemma \ref{homology-lemma-additive-additive}. On the other hand, if a functor $\text{Mod}^{fg}_R \to \textit{Sets}$ preserves finite products and sends the zero module to a one element set, then it has a unique lift to a $R$-linear functor, see Lemma \ref{lemma-linear-functor}.


formal-defos-remark-tangent-space-cofibered-groupoid

We can globalize the notions of tangent space and differential to arbitrary categories cofibered in groupoids as follows. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$, and let $x \in \Ob(\mathcal{F}(k))$. As in Remark \ref{remark-localize-cofibered-groupoid}, we get a predeformation category $\mathcal{F}_x$. We define $$ T_x\mathcal{F} = T\mathcal{F}_x $$ to be the {\it tangent space of $\mathcal{F}$ at $x$}. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$ and $x \in \Ob(\mathcal{F}(k))$, then there is an induced morphism $\varphi_x: \mathcal{F}_x \to \mathcal{G}_{\varphi(x)}$. We define the {\it differential $d_x \varphi : T_x \mathcal{F} \to T_{\varphi(x)} \mathcal{G}$ of $\varphi$ at $x$} to be the map $d \varphi_x: T \mathcal{F}_x \to T \mathcal{G}_{\varphi(x)}$. If both $\mathcal{F}$ and $\mathcal{G}$ satisfy (S2) then all of these tangent spaces have a natural $k$-vector space structure and all the differentials $d_x \varphi : T_x \mathcal{F} \to T_{\varphi(x)} \mathcal{G}$ are $k$-linear (use Lemmas \ref{lemma-S1-S2-localize} and \ref{lemma-k-linear-differential}).


formal-defos-remark-compare-schlessinger-H3-pre

Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a predeformation functor satisfying (S1) and (S2). The condition $\dim_k TF < \infty$ is precisely condition (H3) from Schlessinger's paper. Recall that (S1) and (S2) correspond to conditions (H1) and (H2), see Remark \ref{remark-compare-S1-S2-schlessinger}. Thus Lemma \ref{lemma-versal-object-existence} tells us $$ (H1) + (H2) + (H3) \Rightarrow \text{ there exists a versal formal object} $$ for predeformation functors. We will make the link with hulls in Remark \ref{remark-compare-schlessinger-H3}.


formal-defos-remark-compare-schlessinger-H3

Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a predeformation functor satisfying (S1) and (S2) and $\dim_k TF < \infty$. Recall that these conditions correspond to the conditions (H1), (H2), and (H3) from Schlessinger's paper, see Remark \ref{remark-compare-schlessinger-H3-pre}. Now, in the classical case (or if $k' \subset k$ is separable) following Schlessinger we introduce the notion of a hull: a {\it hull} is a versal formal object $\xi \in \widehat{F}(R)$ such that $d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF$ is an isomorphism, i.e., (\ref{equation-bijective}) holds. Thus Theorem \ref{theorem-miniversal-object-existence} tells us $$ (H1) + (H2) + (H3) \Rightarrow \text{ there exists a hull} $$ in the classical case. In other words, our theorem recovers Schlessinger's theorem on the existence of hulls.


formal-defos-remark-compose-minimal-into-iso-classes

Let $\mathcal{F}$ be a predeformation category. Recall that $\mathcal{F} \to \overline{\mathcal{F}}$ is smooth, see Remark \ref{remark-smooth-to-iso-classes}. Hence if $\xi \in \widehat{\mathcal{F}}(R)$ is a versal formal object, then the composition $$ \underline{R}|_{\mathcal{C}_\Lambda} \longrightarrow \mathcal{F} \longrightarrow \overline{\mathcal{F}} $$ is smooth (Lemma \ref{lemma-smooth-properties}) and we conclude that the image $\overline{\xi}$ of $\xi$ in $\overline{\mathcal{F}}$ is a versal formal object. If (\ref{equation-bijective}) holds, then $\overline{\xi}$ induces an isomorphism $T\underline{R}|_{\mathcal{C}_\Lambda} \to T\overline{\mathcal{F}}$ because $\mathcal{F} \to \overline{\mathcal{F}}$ identifies tangent spaces. Hence in this case $\overline{\xi}$ is a hull for $\overline{\mathcal{F}}$, see Remark \ref{remark-compare-schlessinger-H3}. By Theorem \ref{theorem-miniversal-object-existence} we can always find such a $\xi$ if $k' \subset k$ is separable and $\mathcal{F}$ is a predeformation category satisfying (S1), (S2), and $\dim_k T\mathcal{F} < \infty$.


formal-defos-remark-compare-schlessinger-H4

When $\mathcal{F}$ is cofibered in sets, condition (RS) is exactly condition (H4) from Schlessinger's paper \cite[Theorem 2.11]{Sch}.  Namely, for a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$, condition (RS) states: If $A_1 \to A$ and $A_2 \to A$ are maps in $\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective, then the induced map $F(A_1 \times_A A_2) \to F(A_1) \times_{F(A)} F(A_2)$ is bijective.


formal-defos-remark-deformation-functor

We say that a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$ is a {\it deformation functor} if the associated cofibered set is a deformation category, i.e.\ if $F(k)$ is a one element set and $F$ satisfies (RS). If $\mathcal{F}$ is a deformation category, then $\overline{\mathcal{F}}$ is a predeformation functor but not necessarily a deformation functor, as Lemma \ref{lemma-RS-associated-functor} shows.


formal-defos-remark-omit-arrow

When the map $f: A' \to A$ is clear from the context, we may write $\textit{Lift}(x, A')$ and $\text{Lift}(x, A')$ in place of $\textit{Lift}(x, f)$ and $\text{Lift}(x, f)$.


formal-defos-remark-tangent-space-lifting

Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$. Let $V$ be a finite dimensional vector space. Then $\text{Lift}(x_0, k[V])$ is the set of isomorphism classes of $\mathcal{F}_{x_0}(k[V])$ where $\mathcal{F}_{x_0}$ is the predeformation category of objects in $\mathcal{F}$ lying over $x_0$, see Remark \ref{remark-localize-cofibered-groupoid}. Hence if $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$ (see Lemma \ref{lemma-S1-S2-localize}) and by Lemma \ref{lemma-tangent-space-vector-space} we see that $$ \text{Lift}(x_0, k[V]) = T\mathcal{F}_{x_0} \otimes_k V $$ as $k$-vector spaces.


formal-defos-remark-lift-bijections

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$ satisfying (RS).  Let $$ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A } $$ be a fibre square in $\mathcal{C}_\Lambda$ such that either $A_1 \to A$ or $A_2 \to A$ is surjective. Let $x \in \Ob(\mathcal{F}(A))$. Given lifts $x_1 \to x$ and $x_2 \to x$ of $x$ to $A_1$ and $A_2$, we get by (RS) a lift $x_1 \times_x x_2 \to x$ of $x$ to $A_1 \times_A A_2$. Conversely, by Lemma \ref{lemma-RS-fiber-square} any lift of $x$ to $A_1 \times_A A_2$ is of this form. Hence a bijection $$ \text{Lift}(x, A_1) \times \text{Lift}(x, A_2) \longrightarrow \text{Lift}(x, A_1 \times_A A_2). $$ Similarly, if $x_1 \to x$ is a fixed lifting of $x$ to $A_1$, then there is a bijection $$ \text{Lift}(x_1, A_1 \times_A A_2) \longrightarrow \text{Lift}(x, A_2). $$ Now let $$ \xymatrix{ A_1' \times_A A_2 \ar[r] \ar[d] & A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1' \ar[r] & A_1 \ar[r] & A } $$ be a composition of fibre squares in $\mathcal{C}_\Lambda$ with both $A'_1 \to A_1$ and $A_1 \to A$ surjective. Let $x_1 \to x$ be a morphism lying over $A_1 \to A$. Then by the above we have bijections \begin{align*} \text{Lift}(x_1, A_1' \times_A A_2) & = \text{Lift}(x_1, A_1') \times \text{Lift}(x_1, A_1 \times_A A_2) \\ & = \text{Lift}(x_1, A_1') \times \text{Lift}(x, A_2). \end{align*}


formal-defos-remark-free-transitive-action-functorial

The action of Lemma \ref{lemma-free-transitive-action} is functorial. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of deformation categories. Let $A' \to A$ be a surjective ring map whose kernel $I$ is annihilated by $\mathfrak m_{A'}$. Let $x \in \Ob(\mathcal{F}(A))$. In this situation $\varphi$ induces the vertical arrows in the following commutative diagram $$ \xymatrix{ \text{Lift}(x, A') \times (T\mathcal{F} \otimes_k I) \ar[d]_{(\varphi, d\varphi \otimes \text{id}_I)} \ar[r] & \text{Lift}(x, A') \ar[d]^\varphi \\ \text{Lift}(\varphi(x), A') \times (T\mathcal{G} \otimes_k I) \ar[r] & \text{Lift}(\varphi(x), A') } $$ The commutativity follows as each of the maps (\ref{equation-two}), (\ref{equation-one}), and (\ref{equation-three}) of the proof of Lemma \ref{lemma-free-transitive-action} gives rise to a similar commutative diagram.


formal-defos-remark-choice-pushforward-immaterial-infinitesimal-aut

Up to canonical isomorphism $\text{Inf}_{x_0}(\mathcal{F})$ does not depend on the choice of pushforward $x_0 \to x_0'$ because any two pushforwards are canonically isomorphic. Moreover, if $y_0 \in \mathcal{F}(k)$ and $x_0 \cong y_0$ in $\mathcal{F}(k)$, then $\text{Inf}_{x_0}(\mathcal{F}) \cong \text{Inf}_{y_0}(\mathcal{F})$ where the isomorphism depends (only) on the choice of an isomorphism $x_0 \to y_0$. In particular, $\text{Aut}_k(x_0)$ acts on $\text{Inf}_{x_0}(\mathcal{F})$.


formal-defos-remark-trivial-aut-point

Assume $\mathcal{F}$ is a predeformation category. Then \begin{enumerate} \item for $x_0 \in \Ob(\mathcal{F}(k))$ the automorphism group $\text{Aut}_k(x_0)$ is trivial and hence $\text{Inf}_{x_0}(\mathcal{F}) = \text{Aut}_{k[\epsilon]}(x'_0)$, and \item for $x_0, y_0 \in \Ob(\mathcal{F}(k))$ there is a unique isomorphism $x_0 \to y_0$ and hence a canonical identification $\text{Inf}_{x_0}(\mathcal{F}) = \text{Inf}_{y_0}(\mathcal{F})$. \end{enumerate} Since $\mathcal{F}(k)$ is nonempty, choosing $x_0 \in \Ob(\mathcal{F}(k))$ and setting $$ \text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F}) $$ we get a well defined {\it group of infinitesimal automorphisms of $\mathcal{F}$}. With this notation we have $\text{Inf}(\mathcal{F}_{x_0}) = \text{Inf}_{x_0}(\mathcal{F})$. Please compare with the equality $T\mathcal{F}_{x_0} = T_{x_0}\mathcal{F}$ in Remark \ref{remark-tangent-space-cofibered-groupoid}.


formal-defos-remark-infaut-lifting-equalities

We point out some basic relationships between infinitesimal automorphism groups, liftings, and tangent spaces to automorphism functors. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $x' \to x$ be a morphism lying over a ring map $A' \to A$. Then from the definitions we have an equality $$ \text{Inf}(x'/x) = \text{Lift}(\text{id}_x, A') $$ where the liftings are of $\text{id}_x$ as an object of $\mathit{Aut}(x')$.  If $x_0 \in \Ob(\mathcal{F}(k))$ and $x'_0$ is the pushforward to $\mathcal{F}(k[\epsilon])$, then applying this to $x'_0 \to x_0$ we get $$ \text{Inf}_{x_0}(\mathcal{F}) = \text{Lift}(\text{id}_{x_0}, k[\epsilon]) = T_{\text{id}_{x_0}} \mathit{Aut}(x_0), $$ the last equality following directly from the definitions.


formal-defos-remark-confusion-groupoids-in-functors

A groupoid in functors on $\mathcal{C}$ amounts to the data of a functor $\mathcal{C} \to \textit{Groupoids}$, and a morphism of groupoids in functors on $\mathcal{C}$ amounts to a morphism of the corresponding functors $\mathcal{C} \to \textit{Groupoids}$ (where $\textit{Groupoids}$ is regarded as a 1-category).  However, for our purposes it is more convenient to use the terminology of groupoids in functors. In fact, thinking of a groupoid in functors as the corresponding functor $\mathcal{C} \to \textit{Groupoids}$, or equivalently as the category cofibered in groupoids associated to that functor, can lead to confusion (Remark \ref{remark-smooth-groupoid-in-functors-warning}).


formal-defos-remark-identity-inverse

Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$. There are unique morphisms $e : U \to R$ and $i : R \to R$ such that for every object $T$ of $\mathcal{C}$, $e: U(T) \to R(T)$ sends $x \in U(T)$ to the identity morphism on $x$ and $i: R(T) \to R(T)$ sends $a \in U(T)$ to the inverse of $a$ in the groupoid category $(U(T), R(T), s, t, c)$. We will sometimes refer to $s$, $t$, $c$, $e$, and $i$ as ``source'', ``target'', ``composition'', ``identity'', and ``inverse''.


formal-defos-remark-reason-existence-coproduct

Hence a representable groupoid in functors on $\mathcal{C}$ is given by objects $U$ and $R$ of $\mathcal{C}$ and morphisms $s, t : U \to R$ and $c : R \to R \amalg_{s, U, t} R$ such that $(\underline{U}, \underline{R}, s, t, c)$ satisfies the condition of Definition \ref{definition-groupoid-in-functors}. The reason for requiring the existence of the pushout $R \amalg_{s, U, t} R$ is so that the composition morphism $c$ is defined at the level of morphisms in $\mathcal{C}$. This requirement will always be satisfied below when we consider representable groupoids in functors on $\widehat{\mathcal{C}}_\Lambda$, since by Lemma \ref{lemma-CLambdahat-pushouts} the category $\widehat{\mathcal{C}}_\Lambda$ admits pushouts.


formal-defos-remark-simplify-terminology

We will say ``{\it let $(\underline{U}, \underline{R}, s, t, c)$ be a groupoid in functors on $\mathcal{C}$}'' to mean that we have a representable groupoid in functors. Thus this means that $U$ and $R$ are objects of $\mathcal{C}$, there are morphisms $s, t : U \to R$, the pushout $R \amalg_{s, U, t} R$ exists, there is a morphism $c : R \to R \amalg_{s, U, t} R$, and $(\underline{U}, \underline{R}, s, t, c)$ is a groupoid in functors on $\mathcal{C}$.


formal-defos-remark-notation-restriction

In the situation of Definition \ref{definition-restricting-groupoids-in-functors}, we often denote $s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'}$ simply by $s, t, c$.


formal-defos-remark-groupoid-in-functors-complete-restrict

Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$. Then there is a canonical isomorphism $(U, R, s, t, c)^{\wedge}|_{\mathcal{C}_\Lambda} \cong (U, R, s, t, c)$, see Remark \ref{remark-restrict-completion}. On the other hand, let $(U, R, s, t, c)$ be a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda$ such that $U, R : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ both commute with limits, e.g.\ if $U, R$ are representable. Then there is a canonical isomorphism $((U, R, s, t, c)|_{\mathcal{C}_\Lambda})^{\wedge} \cong (U, R, s, t, c)$. This follows from Remark \ref{remark-restrict-complete-continuous-functor}.


formal-defos-remark-smooth-groupoid-in-functors-warning

We note that this terminology is potentially confusing: if $(U, R, s, t, c)$ is a smooth groupoid in functors, then the quotient $[U/R]$ need not be a smooth category cofibred in groupoids as defined in Definition \ref{definition-cofibered-groupoid-projection-smooth}. However smoothness of $(U, R, s, t, c)$ does imply (in fact is equivalent to) smoothness of the quotient morphism $U \to [U/R]$ as we shall see in Lemma \ref{lemma-smooth-quotient-morphism}.


formal-defos-remark-smooth-power-series-prorepresentable-smooth-groupoid-in-functors

Let $(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}$ be a prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$. Then $(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}$ is smooth if and only if $R_1$ is a power series over $R_0$ via both $s$ and $t$. This follows from Lemma \ref{lemma-smooth-morphism-power-series}.


formal-defos-remarks-cofibered-groupoids

Everything about categories fibered in groupoids translates directly to the cofibered setting. The following remarks are meant to fix notation. Let $\mathcal{C}$ be a category. \begin{enumerate} \item We often omit the functor $p: \mathcal{F} \to \mathcal{C}$ from the notation. \item The fiber category over an object $U$ in $\mathcal{C}$ is denoted by $\mathcal{F}(U)$. Its objects are those of $\mathcal{F}$ lying over $U$ and its morphisms are those of $\mathcal{F}$ lying over $\text{id}_U$. If $x, y$  are objects of $\mathcal{F}(U)$, we sometimes write $\Mor_U(x, y)$ for $\Mor_{\mathcal{F}(U)}(x, y)$. \item The fibre categories $\mathcal{F}(U)$ are groupoids, see Categories, Lemma \ref{categories-lemma-fibred-groupoids}. Hence the morphisms in $\mathcal{F}(U)$ are all isomorphisms. We sometimes write $\text{Aut}_U(x)$ for $\Mor_{\mathcal{F}(U)}(x, x)$. \item \label{item-pushforward} Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and let $x \in \Ob(\mathcal{F}(U))$. A {\it pushforward} of $x$ along $f$ is a morphism $x \to y$ of $\mathcal{F}$ lying over $f$. A pushforward is unique up to unique isomorphism (see the discussion following Categories, Definition \ref{categories-definition-cartesian-over-C}). We sometimes write $x \to f_*x$ for ``the'' pushforward of $x$ along $f$. \item A {\it choice of pushforwards for $\mathcal{F}$} is the choice of a pushforward of $x$ along $f$ for every pair $(x, f)$ as above. We can make such a choice of pushforwards for $\mathcal{F}$ by the axiom of choice. \item Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}$. Given a choice of pushforwards for $\mathcal{F}$, there is an associated pseudo-functor $\mathcal{C} \to \textit{Groupoids}$. We will never use this construction so we give no details. \item \label{item-cofibered-morphism} A morphism of categories cofibered in groupoids over $\mathcal{C}$ is a functor commuting with the projections to $\mathcal{C}$. If $\mathcal{F}$ and $\mathcal{F}'$ are categories cofibered in groupoids over $\mathcal{C}$, we denote the morphisms from $\mathcal{F}$ to $\mathcal{F}'$ by $\Mor_\mathcal{C}(\mathcal{F}, \mathcal{F}')$. \item \label{item-definition-cofibered-groupoids-2-category} Categories cofibered in groupoids form a $(2, 1)$-category $\text{Cof}(\mathcal{C})$. Its 1-morphisms are the morphisms described in (\ref{item-cofibered-morphism}). If $p : \mathcal{F} \to C$ and $p': \mathcal{F}' \to \mathcal{C}$ are categories cofibered in groupoids and $\varphi, \psi : \mathcal{F} \to \mathcal{F}'$ are $1$-morphisms, then a 2-morphism $t : \varphi \to \psi$ is a morphism of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{F})$. \item \label{item-construction-associated-cofibered-groupoid} Let $F : \mathcal{C} \to \textit{Groupoids}$ be a functor. There is a category cofibered in groupoids $\mathcal{F} \to \mathcal{C}$ associated to $F$ as follows. An object of $\mathcal{F}$ is a pair $(U, x)$ where $U \in \Ob(\mathcal{C})$ and $x \in \Ob(F(U))$. A morphism $(U, x) \to (V, y)$ is a pair $(f, a)$ where $f \in \Mor_\mathcal{C}(U, V)$ and $a \in \Mor_{F(V)}(F(f)(x), y)$. The functor $\mathcal{F} \to \mathcal{C}$ sends $(U, x)$ to $U$. See Categories, Section \ref{categories-section-presheaves-groupoids}. \item \label{item-associated-functor-isomorphism-classes} Let $\mathcal{F}$ be cofibered in groupoids over $\mathcal{C}$. For $U \in \Ob(\mathcal{C})$ set $\overline{\mathcal{F}}(U)$ equal to the set of isomorphisms classes of the category $\mathcal{F}(U)$. If $f : U \to V$ is a morphism of $\mathcal{C}$, then we obtain a map of sets $\overline{\mathcal{F}}(U) \to \overline{\mathcal{F}}(V)$ by mapping the isomorphism class of $x$ to the isomorphism class of a pushforward $f_*x$ of $x$ see (\ref{item-pushforward}). Then $\overline{\mathcal{F}} : \mathcal{C} \to \textit{Sets}$ is a functor. Similarly, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of cofibered categories, we denote by $\overline{\varphi}: \overline{\mathcal{F}} \to  \overline{\mathcal{G}}$ the associated morphism of functors. \item \label{item-convention-cofibered-sets} Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. We can think of a set as a discrete category, i.e., as a groupoid with only identity morphisms. Then the construction (\ref{item-construction-associated-cofibered-groupoid}) associates to $F$ a category cofibered in sets. This defines a fully faithful embedding of the category of functors $\mathcal{C} \to \textit{Sets}$ to the category of categories cofibered in groupoids over $\mathcal{C}$. We identify the category of functors with its image under this embedding. Hence if $F : \mathcal{C} \to \textit{Sets}$ is a functor, we denote the associated category cofibered in sets also by $F$; and if $\varphi : F \to G$ is a morphism of functors, we denote still by $\varphi$ the corresponding morphism of categories cofibered in sets, and vice-versa. See Categories, Section \ref{categories-section-fibred-in-sets}. \item \label{item-definition-yoneda} Let $U$ be an object of $\mathcal{C}$.  We write $\underline{U}$ for the functor $\Mor_\mathcal{C}(U, -): \mathcal{C} \to \textit{Sets}$.  This defines a fully faithful embedding of $\mathcal C^{opp}$ into the category of functors $\mathcal{C} \to \textit{Sets}$. Hence, if $f : U \to V$ is a morphism, we are justified in denoting still by $f$ the induced morphism $\underline{V} \to \underline{U}$, and vice-versa. \item \label{item-fibre-product} Fiber products of categories cofibered in groupoids: If $\mathcal{F} \to \mathcal{H}$ and $\mathcal{G} \to \mathcal{H}$ are morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$, then a construction of their 2-fiber product is given by the construction for their 2-fiber product as categories over $\mathcal{C}_\Lambda$, as described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. \item \label{item-product} Products of categories cofibered in groupoids: If $\mathcal{F}$ and $\mathcal{G}$ are categories cofibered in groupoids over $\mathcal{C}_\Lambda$ then their product is defined to be the $2$-fiber product $\mathcal{F} \times_{\mathcal{C}_\Lambda} \mathcal{G}$ as described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. \item \label{item-definition-restricting-base-category} Restricting the base category: Let $p : \mathcal{F} \to \mathcal{C}$ be a category cofibered in groupoids, and let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$. The restriction $\mathcal{F}|_{\mathcal{C}'}$ is the full subcategory of $\mathcal{F}$ whose objects lie over objects of $\mathcal{C}'$. It is a category cofibered in groupoids via the functor $p|_{\mathcal{C}'}: \mathcal{F}|_{\mathcal{C}'} \to \mathcal{C}'$. \end{enumerate}


adequate-remark-settheoretic

Consider the category $\textit{Alg}_{fp, A}$ whose objects are $A$-algebras $B$ of the form $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$ and whose morphisms are $A$-algebra maps. Every $A$-algebra $B$ is a filtered colimit of finitely presented $A$-algebra, i.e., a filtered colimit of objects of $\textit{Alg}_{fp, A}$. By Lemma \ref{lemma-adequate-finite-presentation} we conclude every adequate functor $F$ is determined by its restriction to $\textit{Alg}_{fp, A}$. For some questions we can therefore restrict to functors on $\textit{Alg}_{fp, A}$. For example, the category of adequate functors does not depend on the choice of the big $\tau$-site chosen in Section \ref{section-conventions}.


adequate-remark-linearly-adequate

Let $A$ be a ring. The proof of Lemma \ref{lemma-extension-adequate-key} shows that any extension $0 \to \underline{M} \to E \to L \to 0$ of module-valued functors on $\textit{Alg}_A$ with $L$ linearly adequate splits. It uses only the following properties of the module-valued functor $F = \underline{M}$: \begin{enumerate} \item $F(B) \otimes_B B' \to F(B')$ is an isomorphism for a flat ring map $B \to B'$, and \item $F(C)^{(1)} = F(p_1)(F(B)^{(1)}) \oplus F(p_2)(F(B)^{(1)})$ where $B = A[x_1, \ldots, x_n]/(\sum a_{ij}x_j)$ and $C = A[x_1, \ldots, x_n, y_1, \ldots, y_n]/ (\sum a_{ij}x_j, \sum a_{ij}y_j)$. \end{enumerate} These two properties hold for any adequate functor $F$; details omitted. Hence we see that $L$ is a projective object of the abelian category of adequate functors.


adequate-remark-compare

Let $S$ be a scheme. We have functors $u : \QCoh(\mathcal{O}_S) \to \textit{Adeq}(\mathcal{O})$ and $v : \textit{Adeq}(\mathcal{O}) \to \QCoh(\mathcal{O}_S)$. Namely, the functor $u : \mathcal{F} \mapsto \mathcal{F}^a$ comes from taking the associated $\mathcal{O}$-module which is adequate by Lemma \ref{lemma-adequate-characterize}. Conversely, the functor $v$ comes from restriction $v : \mathcal{G} \mapsto \mathcal{G}|_{S_{Zar}}$, see Lemma \ref{lemma-same-cohomology-adequate}. Since $\mathcal{F}^a$ can be described as the pullback of $\mathcal{F}$ under a morphism of ringed topoi $((\Sch/S)_\tau, \mathcal{O}) \to (S_{Zar}, \mathcal{O}_S)$, see Descent, Remark \ref{descent-remark-change-topologies-ringed-sites} and since restriction is the pushforward we see that $u$ and $v$ are adjoint as follows $$ \SheafHom_{\mathcal{O}_S}(\mathcal{F}, v\mathcal{G}) = \SheafHom_\mathcal{O}(u\mathcal{F}, \mathcal{G}) $$ where $\mathcal{O}$ denotes the structure sheaf on the big site. It is immediate from the description that the adjunction mapping $\mathcal{F} \to vu\mathcal{F}$ is an isomorphism for all quasi-coherent sheaves.


adequate-remark-D-adeq-independence-topology

Let $S$ be a scheme. Let $\tau, \tau' \in \{Zar, \etale, smooth, syntomic, fppf\}$. Denote $\mathcal{O}_\tau$, resp.\ $\mathcal{O}_{\tau'}$ the structure sheaf $\mathcal{O}$ viewed as a sheaf on $(\Sch/S)_\tau$, resp.\ $(\Sch/S)_{\tau'}$. Then $D_{\textit{Adeq}}(\mathcal{O}_\tau)$ and $D_{\textit{Adeq}}(\mathcal{O}_{\tau'})$ are canonically isomorphic. This follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compare-topologies-derived-adequate-modules}. Namely, assume $\tau$ is stronger than the topology $\tau'$, let $\mathcal{C} = (\Sch/S)_{fppf}$, and let $\mathcal{B}$ the collection of affine schemes over $S$. Assumptions (1) and (2) we've seen above. Assumption (3) is clear and assumption (4) follows from Lemma \ref{lemma-same-cohomology-adequate}.


adequate-remark-D-adeq-and-D-QCoh

Let $S$ be a scheme. The morphism $f$ see (\ref{equation-compare-big-small}) induces adjoint functors $Rf_* : D_{\textit{Adeq}}(\mathcal{O}) \to D_\QCoh(S)$ and $Lf^* : D_\QCoh(S) \to D_{\textit{Adeq}}(\mathcal{O})$. Moreover $Rf_* Lf^* \cong \text{id}_{D_\QCoh(S)}$. \medskip\noindent We sketch the proof. By Remark \ref{remark-D-adeq-independence-topology} we may assume the topology $\tau$ is the Zariski topology. We will use the existence of the unbounded total derived functors $Lf^*$ and $Rf_*$ on $\mathcal{O}$-modules and their adjointness, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}. In this case $f_*$ is just the restriction to the subcategory $S_{Zar}$ of $(\Sch/S)_{Zar}$. Hence it is clear that $Rf_* = f_*$ induces $Rf_* : D_{\textit{Adeq}}(\mathcal{O}) \to D_\QCoh(S)$. Suppose that $\mathcal{G}^\bullet$ is an object of $D_\QCoh(S)$. We may choose a system $\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \ldots$ of bounded above complexes of flat $\mathcal{O}_S$-modules whose transition maps are termwise split injectives and a diagram $$ \xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}\mathcal{G}^\bullet \ar[r] & \tau_{\leq 2}\mathcal{G}^\bullet \ar[r] & \ldots } $$ with the properties (1), (2), (3) listed in Derived Categories, Lemma \ref{derived-lemma-special-direct-system} where $\mathcal{P}$ is the collection of flat $\mathcal{O}_S$-modules. Then $Lf^*\mathcal{G}^\bullet$ is computed by $\colim f^*\mathcal{K}_n^\bullet$, see Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-pullback-K-flat} and \ref{sites-cohomology-lemma-derived-base-change} (note that our sites have enough points by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-points-fppf}). We have to see that $H^i(Lf^*\mathcal{G}^\bullet) = \colim H^i(f^*\mathcal{K}_n^\bullet)$ is adequate for each $i$. By Lemma \ref{lemma-abelian-adequate} we conclude that it suffices to show that each $H^i(f^*\mathcal{K}_n^\bullet)$ is adequate. \medskip\noindent The adequacy of $H^i(f^*\mathcal{K}_n^\bullet)$ is local on $S$, hence we may assume that $S = \Spec(A)$ is affine. Because $S$ is affine $D_\QCoh(S) = D(\QCoh(\mathcal{O}_S))$, see the discussion in Derived Categories of Schemes, Section \ref{perfect-section-derived-quasi-coherent}. Hence there exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{K}_n^\bullet$ where $\mathcal{F}^\bullet$ is a bounded above complex of flat quasi-coherent modules. Then $f^*\mathcal{F}^\bullet \to f^*\mathcal{K}_n^\bullet$ is a quasi-isomorphism, and the cohomology sheaves of $f^*\mathcal{F}^\bullet$ are adequate. \medskip\noindent The final assertion $Rf_* Lf^* \cong \text{id}_{D_\QCoh(S)}$ follows from the explicit description of the functors above. (In plain English: if $\mathcal{F}$ is quasi-coherent and $p > 0$, then $L_pf^*\mathcal{F}$ is a parasitic adequate module.)


adequate-remark-conclusion

Remark \ref{remark-D-adeq-and-D-QCoh} above implies we have an equivalence of derived categories $$ D_{\textit{Adeq}}(\mathcal{O})/D_\mathcal{C}(\mathcal{O}) \longrightarrow D_\QCoh(S) $$ where $\mathcal{C}$ is the category of parasitic adequate modules. Namely, it is clear that $D_\mathcal{C}(\mathcal{O})$ is the kernel of $Rf_*$, hence a functor as indicated. For any object $X$ of $D_{\textit{Adeq}}(\mathcal{O})$ the map $Lf^*Rf_*X \to X$ maps to a quasi-isomorphism in $D_\QCoh(S)$, hence $Lf^*Rf_*X \to X$ is an isomorphism in $D_{\textit{Adeq}}(\mathcal{O})/D_\mathcal{C}(\mathcal{O})$. Finally, for $X, Y$ objects of $D_{\textit{Adeq}}(\mathcal{O})$ the map $$ Rf_* : \Hom_{D_{\textit{Adeq}}(\mathcal{O})/D_\mathcal{C}(\mathcal{O})}(X, Y) \to \Hom_{D_\QCoh(S)}(Rf_*X, Rf_*Y) $$ is bijective as $Lf^*$ gives an inverse (by the remarks above).


spaces-topologies-remark-change-topologies-ringed

The sites $(\textit{Spaces}/X)_\etale$ and $X_{spaces, \etale}$ come with structure sheaves. For the small \'etale site we have seen this in Properties of Spaces, Section \ref{spaces-properties-section-structure-sheaf}. The structure sheaf $\mathcal{O}$ on the big \'etale site $(\textit{Spaces}/X)_\etale$ is defined by assigning to an object $U$ the global sections of the structure sheaf of $U$. This makes sense because after all $U$ is an algebraic space itself hence has a structure sheaf. Since $\mathcal{O}_U$ is a sheaf on the \'etale site of $U$, the presheaf $\mathcal{O}$ so defined satisfies the sheaf condition for coverings of $U$, i.e., $\mathcal{O}$ is a sheaf. We can upgrade the morphisms $i_f$, $\pi_X$, $i_X$, $f_{small}$, and $f_{big}$ defined above to morphisms of ringed sites, respectively topoi. Let us deal with these one by one. \begin{enumerate} \item In Lemma \ref{lemma-put-in-T-etale} denote $\mathcal{O}$ the structure sheaf on $(\textit{Spaces}/X)_\etale$. We have $(i_f^{-1}\mathcal{O})(U/Y) = \mathcal{O}_U(U) = \mathcal{O}_Y(U)$ by construction. Hence an isomorphism $i_f^\sharp : i_f^{-1}\mathcal{O} \to \mathcal{O}_Y$. \item In Lemma \ref{lemma-at-the-bottom-etale} it was noted that $i_X$ is a special case of $i_f$ with $f = \text{id}_X$ hence we are back in case (1). \item In Lemma \ref{lemma-at-the-bottom-etale} the morphism $\pi_X$ satisfies $(\pi_{X, *}\mathcal{O})(U) = \mathcal{O}(U) = \mathcal{O}_X(U)$. Hence we can use this to define $\pi_X^\sharp : \mathcal{O}_X \to \pi_{X, *}\mathcal{O}$. \item In Lemma \ref{lemma-morphism-big-small-etale} the extension of $f_{small}$ to a morphism of ringed topoi was discussed in Properties of Spaces, Lemma \ref{spaces-properties-lemma-morphism-ringed-topoi}. \item In Lemma \ref{lemma-morphism-big-small-etale} the functor $f_{big}^{-1}$ is simply the restriction via the inclusion functor $(\textit{Spaces}/Y)_\etale \to (\textit{Spaces}/X)_\etale$. Let $\mathcal{O}_1$ be the structure sheaf on $(\textit{Spaces}/X)_\etale$ and let $\mathcal{O}_2$ be the structure sheaf on $(\textit{Spaces}/Y)_\etale$. We obtain a canonical isomorphism $f_{big}^\sharp : f_{big}^{-1}\mathcal{O}_1 \to \mathcal{O}_2$. \end{enumerate} Moreover, with these definitions compositions work out correctly too. We omit giving a detailed statement and proof.


proetale-remark-size-w

Let $A$ be a ring. Let $\kappa$ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of $A_w$ (Lemma \ref{lemma-make-w-local}) is at most $\kappa$. Namely, each $A_E$ has cardinality at most $\kappa$ and the set of finite subsets of $A$ has cardinality at most $\kappa$ as well. Thus the result follows as $\kappa \otimes \kappa = \kappa$, see Sets, Section \ref{sets-section-cardinals}.


proetale-remark-slightly-stronger

In each of Lemmas \ref{lemma-construct}, \ref{lemma-construct-profinite}, Proposition \ref{proposition-maps-wich-identify-local-rings}, and Lemma \ref{lemma-find-Zariski-w-contractible} we find an ind-Zariski ring map with some properties. In the paper \cite{BS} the authors use the notion of an ind-(Zariski localization) which is a filtered colimit of finite products of principal localizations. It is possible to replace ind-Zariski by ind-(Zariski localization) in each of the results listed above. However, we do not need this and the notion of an ind-Zariski homomorphism of rings as defined here has slightly better formal properties. Moreover, the notion of an ind-Zariski ring map is the natural analogue of the notion of an ind-\'etale ring map defined in the next section.


proetale-remark-size-T

Let $A$ be a ring. Let $\kappa$ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of $T(A)$ is at most $\kappa$. Namely, each $B_E$ has cardinality at most $\kappa$ and the index set $I(A)$ has cardinality at most $\kappa$ as well. Thus the result follows as $\kappa \otimes \kappa = \kappa$, see Sets, Section \ref{sets-section-cardinals}. It follows that the ring constructed in the proof of Lemma \ref{lemma-first-construction} has cardinality at most $\kappa$ as well.


proetale-remark-first-construction-functorial

The construction $A \mapsto T(A)$ is functorial in the following sense: If $A \to A'$ is a ring map, then we can construct a commutative diagram $$ \xymatrix{ A \ar[r] \ar[d] & T(A) \ar[d] \\ A' \ar[r] & T(A') } $$ Namely, given $(A \to A[x_1, \ldots, x_n]/(f_1, \ldots, f_n))$ in $S(A)$ we can use the ring map $\varphi : A \to A'$ to obtain a corresponding element $(A' \to A'[x_1, \ldots, x_n]/(f^\varphi_1, \ldots, f^\varphi_n))$ of $S(A')$ where $f^\varphi$ means the polynomial obtained by applying $\varphi$ to the coefficients of the polynomial $f$. Moreover, there is a commutative diagram $$ \xymatrix{ A \ar[r] \ar[d] & A[x_1, \ldots, x_n]/(f_1, \ldots, f_n) \ar[d] \\ A' \ar[r] & A'[x_1, \ldots, x_n]/(f^\varphi_1, \ldots, f^\varphi_n) } $$ which is a in the category of rings. For $E \subset S(A)$ finite, set $E' = \varphi(E)$ and define $B_E \to B_{E'}$ in the obvious manner. Taking the colimit gives the desired map $T(A) \to T(A')$, see Categories, Lemma \ref{categories-lemma-functorial-colimit}.


proetale-remark-h-limit-preserving

Let $S$ be a scheme contained in a big site $\Sch_h$. Let $F$ be a sheaf of sets on $(\Sch/S)_h$ such that $F(T) = \colim F(T_i)$ whenever $T = \lim T_i$ is a directed limit of affine schemes in $(\Sch/S)_h$. In this situation $F$ extends uniquely to a contravariant functor $F'$ on the category of all schemes over $S$ such that (a) $F'$ satisfies the sheaf property for the h topology and (b) $F'$ is limit preserving. See More on Flatness, Lemma \ref{flat-lemma-extend-sheaf-h}. In this situation Lemma \ref{lemma-h-limit-preserving} tells us that $F'$ satisfies the sheaf property for the V topology.


proetale-remark-size-w-contractible

Let $A$ be a ring. Let $\kappa$ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of the ring $D$ constructed in Proposition \ref{proposition-find-w-contractible} is at most $$ \kappa^{2^{2^{2^\kappa}}}. $$ Namely, the ring map $A \to D$ is constructed as a composition $$ A \to A_w = A' \to C' \to C \to D. $$ Here the first three steps of the construction are carried out in the first paragraph of the proof of Lemma \ref{lemma-get-w-local-algebraic-residue-field-extensions}. For the first step we have $|A_w| \leq \kappa$ by Remark \ref{remark-size-w}. We have $|C'| \leq \kappa$ by Remark \ref{remark-size-T}. Then $|C| \leq \kappa$ because $C$ is a localization of $(C')_w$ (it is constructed from $C'$ by an application of Lemma \ref{lemma-localize-along-closed-profinite} in the proof of Lemma \ref{lemma-w-local-algebraic-residue-field-extensions}). Thus $C$ has at most $2^\kappa$ maximal ideals. Finally, the ring map $C \to D$ identifies local rings and the cardinality of the set of maximal ideals of $D$ is at most $2^{2^{2^\kappa}}$ by Topology, Remark \ref{topology-remark-size-projective-cover}. Since $D \subset \prod_{\mathfrak m \subset D} D_\mathfrak m$ we see that $D$ has at most the size displayed above.


proetale-remark-extend-to-all

Let $X$ be a scheme. Because $X_\proetale$ has enough weakly contractible objects for all $K$ in $D(X_\proetale)$ we have $K = R\lim \tau_{\geq -n}K$ by Cohomology on Sites, Proposition \ref{sites-cohomology-proposition-enough-weakly-contractibles}. Since $R\Gamma$ commutes with $R\lim$ by Injectives, Lemma \ref{injectives-lemma-RF-commutes-with-Rlim} we see that $$ R\Gamma(X, K) = R\lim R\Gamma(X, \tau_{\geq -n}K) $$ in $D(\textit{Ab})$. This will sometimes allow us to extend results from bounded below complexes to all complexes.


formal-spaces-remark-mcquillan

There is a variant of the construction of formal schemes due to McQuillan, see \cite{McQuillan}. He suggests a slight weakening of the condition of admissibility. Namely, recall that an admissible topological ring is a complete (and separated by our conventions) topological ring $A$ which is linearly topologized such that there exists an ideal of definition: an open ideal $I$ such that any neighbourhood of $0$ contains $I^n$ for some $n \geq 1$. McQuillan works with what we will call {\it weakly admissible} topological rings. A weakly admissible topological ring $A$ is a complete (and separated by our conventions) topological ring which is linearly topologized such that there exists an {\it weak ideal of definition}: an open ideal $I$ such that for all $f \in I$ we have $f^n \to 0$ for $n \to \infty$. Similarly to the admissible case, if $I$ is a weak ideal of definition and $J \subset A$ is an open ideal, then $I \cap J$ is a weak ideal of definition. Thus the weak ideals of definition form a fundamental system of open neighbourhoods of $0$ and one can proceed along much the same route as above to define a larger category of formal schemes based on this notion. The analogues of Lemmas \ref{lemma-fully-faithful} and \ref{lemma-formal-scheme-sheaf-fppf} still hold in this setting (with the same proof).


formal-spaces-remark-sheafification-of-presheaves-in-top

\begin{reference} \cite{Gray} \end{reference} In this remark we briefly discuss sheafification of presheaves of topological spaces. The exact same arguments work for presheaves of topological abelian groups, topological rings, and topological modules (over a given topological ring). In order to do this in the correct generality let us work over a site $\mathcal{C}$. The reader who is interested in the case of (pre)sheaves over a topological space $X$ should think of objects of $\mathcal{C}$ as the opens of $X$, of morphisms of $\mathcal{C}$ as inclusions of opens, and of coverings in $\mathcal{C}$ as coverings in $X$, see Sites, Example \ref{sites-example-site-topological}. Denote $\Sh(\mathcal{C}, \textit{Top})$ the category of sheaves of topological spaces on $\mathcal{C}$ and denote $\textit{PSh}(\mathcal{C}, \textit{Top})$ the category of presheaves of topological spaces on $\mathcal{C}$. Let $\mathcal{F}$ be a presheaf of topological spaces on $\mathcal{C}$. The sheafification $\mathcal{F}^\#$ should satisfy the formula $$ \Mor_{\textit{PSh}(\mathcal{C}, \textit{Top})}(\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(\mathcal{C}, \textit{Top})}(\mathcal{F}^\#, \mathcal{G}) $$ functorially in $\mathcal{G}$ from $\Sh(\mathcal{C}, \textit{Top})$. In other words, we are trying to construct the left adjoint to the inclusion functor $\Sh(\mathcal{C}, \textit{Top}) \to \textit{PSh}(\mathcal{C}, \textit{Top})$. We first claim that $\Sh(\mathcal{C}, \textit{Top})$ has limits and that the inclusion functor commutes with them. Namely, given a category $\mathcal{I}$ and a functor $i \mapsto \mathcal{G}_i$ into $\Sh(\mathcal{C}, \textit{Top})$ we simply define $$ (\lim \mathcal{G}_i)(U) = \lim \mathcal{G}_i(U) $$ where we take the limit in the category of topological spaces (Topology, Lemma \ref{topology-lemma-limits}). This defines a sheaf because limits commute with limits (Categories, Lemma \ref{categories-lemma-colimits-commute}) and in particular products and equalizers (which are the operations used in the sheaf axiom). Finally, a morphism of presheaves from $\mathcal{F} \to \lim \mathcal{G}_i$ is clearly the same thing as a compatible system of morphisms $\mathcal{F} \to \mathcal{G}_i$. In other words, the object $\lim \mathcal{G}_i$ is the limit in the category of presheaves of topological spaces and a fortiori in the category of sheaves of topological spaces. Our second claim is that any morphism of presheaves $\mathcal{F} \to \mathcal{G}$ with $\mathcal{G}$ an object of $\Sh(\mathcal{C}, \textit{Top})$ factors through a subsheaf $\mathcal{G}' \subset \mathcal{G}$ whose size is bounded. Here we define the {\it size} $|\mathcal{H}|$ of a sheaf of topological spaces $\mathcal{H}$ to be the cardinal $\sup_{U \in \Ob(\mathcal{C})} |\mathcal{H}(U)|$. To prove our claim we let $$ \mathcal{G}'(U) = \left\{ \quad s \in \mathcal{G}(U) \quad \middle| \quad \begin{matrix} \text{there exists a covering }\{U_i \to U\}_{i \in I} \\ \text{such that } s|_{U_i} \in \Im(\mathcal{F}(U_i) \to \mathcal{G}(U_i)) \end{matrix} \quad \right\} $$ We endow $\mathcal{G}'(U)$ with the induced topology. Then $\mathcal{G}'$ is a sheaf of topological spaces (details omitted) and $\mathcal{G}' \to \mathcal{G}$ is a morphism through which the given map $\mathcal{F} \to \mathcal{G}$ factors. Moreover, the size of $\mathcal{G}'$ is bounded by some cardinal $\kappa$ depending only on $\mathcal{C}$ and the presheaf $\mathcal{F}$ (hint: use that coverings in $\mathcal{C}$ form a set by our conventions). Putting everything together we see that the assumptions of Categories, Theorem \ref{categories-theorem-adjoint-functor} are satisfied and we obtain sheafification as the left adjoint of the inclusion functor from sheaves to presheaves. Finally, let $p$ be a point of the site $\mathcal{C}$ given by a functor $u : \mathcal{C} \to \textit{Sets}$, see Sites, Definition \ref{sites-definition-point}. For a topological space $M$ the presheaf defined by the rule $$ U \mapsto \text{Map}(u(U), M) = \prod\nolimits_{x \in u(U)} M $$ endowed with the product topology is a sheaf of topological spaces. Hence the exact same argument as given in the proof of Sites, Lemma \ref{sites-lemma-point-pushforward-sheaf} shows that $\mathcal{F}_p = \mathcal{F}^\#_p$, in other words, sheafification commutes with taking stalks at a point.


formal-spaces-remark-compare-with-affine-formal-schemes

The classical affine formal algebraic spaces correspond to the affine formal schemes considered in EGA (\cite{EGA}). To explain this we assume our base scheme is $\Spec(\mathbf{Z})$. Let $\mathfrak X = \text{Spf}(A)$ be an affine formal scheme. Let $h_\mathfrak X$ be its functor of points as in Lemma \ref{lemma-fully-faithful}. Then $h_\mathfrak X = \colim h_{\Spec(A/I)}$ where the colimit is over the collection of ideals of definition of the admissible topological ring $A$. This follows from (\ref{equation-morphisms-affine-formal-schemes}) when evaluating on affine schemes and it suffices to check on affine schemes as both sides are fppf sheaves, see Lemma \ref{lemma-formal-scheme-sheaf-fppf}. Thus $h_\mathfrak X$ is an affine formal algebraic space. In fact, it is a classical affine formal algebraic space by Definition \ref{definition-types-affine-formal-algebraic-space}. Thus Lemma \ref{lemma-fully-faithful} tells us the category of affine formal schemes is equivalent to the category of classical affine formal algebraic spaces.


formal-spaces-remark-compare-with-formal-schemes

Modulo set theoretic issues the category of formal schemes \`a la EGA (see Section \ref{section-formal-schemes-EGA}) is equivalent to a full subcategory of the category of formal algebraic spaces. To explain this we assume our base scheme is $\Spec(\mathbf{Z})$. By Lemma \ref{lemma-formal-scheme-sheaf-fppf} the functor of points $h_\mathfrak X$ associated to a formal scheme $\mathfrak X$ is a sheaf in the fppf topology. By Lemma \ref{lemma-fully-faithful} the assignment $\mathfrak X \mapsto h_\mathfrak X$ is a fully faithful embedding of the category of formal schemes into the category of fppf sheaves. Given a formal scheme $\mathfrak X$ we choose an open covering $\mathfrak X = \bigcup \mathfrak X_i$ with $\mathfrak X_i$ affine formal schemes. Then $h_{\mathfrak X_i}$ is an affine formal algebraic space by Remark \ref{remark-compare-with-affine-formal-schemes}. The morphisms $h_{\mathfrak X_i} \to h_\mathfrak X$ are representable and open immersions. Thus $\{h_{\mathfrak X_i} \to h_\mathfrak X\}$ is a family as in Definition \ref{definition-formal-algebraic-space} and we see that $h_\mathfrak X$ is a formal algebraic space.