crystalline-remark-structure-morphism

In Situation \ref{situation-global}. Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$. If we assume that $p$ is locally nilpotent on $S_0$ (which is always the case in practice) then we obtain a situation as in Definition \ref{definition-divided-power-thickening-X} with $S_0$ instead of $X$. Hence we get a site $\text{Cris}(S_0/S)$. If $f : X \to S_0$ is the structure morphism of $X$ over $S$, then we get a commutative diagram of ringed topoi $$ \xymatrix{ (X/S)_{\text{cris}} \ar[r]_{f_{\text{cris}}} \ar[d]_{u_{X/S}} & (S_0/S)_{\text{cris}} \ar[d]^{u_{S_0/S}} \\ \Sh(X_{Zar}) \ar[r]^{f_{small}} & \Sh(S_{0, Zar}) \ar[rd] \\ & & \Sh(S_{Zar}) } $$ see Remark \ref{remark-functoriality-cris}. We think of the composition $(X/S)_{\text{cris}} \to \Sh(S_{Zar})$ as the structure morphism of the crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism $$ \tau_{X/S} : (X/S)_{\text{cris}} \longrightarrow \Sh(S_{Zar}) $$ is defined as we can take the lower route through the diagram above.


crystalline-remark-compatibilities

The morphisms defined above satisfy numerous compatibilities. For example, in the situation of Remark \ref{remark-functoriality-cris} we obtain a commutative diagram of ringed topoi $$ \xymatrix{ (X/S)_{\text{cris}} \ar[d] \ar[r] & (Y/S')_{\text{cris}} \ar[d] \\ \Sh((\Sch/S)_{Zar}) \ar[r] & \Sh((\Sch/S')_{Zar}) } $$ where the vertical arrows are the structure morphisms.


crystalline-remark-crystal

To formulate the general notion of a crystal we use the language of stacks and strongly cartesian morphisms, see Stacks, Definition \ref{stacks-definition-stack} and Categories, Definition \ref{categories-definition-cartesian-over-C}. In Situation \ref{situation-global} let $p : \mathcal{C} \to \text{Cris}(X/S)$ be a stack. A {\it crystal in objects of $\mathcal{C}$ on $X$ relative to $S$} is a {\it cartesian section} $\sigma : \text{Cris}(X/S) \to \mathcal{C}$, i.e., a functor $\sigma$ such that $p \circ \sigma = \text{id}$ and such that $\sigma(f)$ is strongly cartesian for all morphisms $f$ of $\text{Cris}(X/S)$. Similarly for the big crystalline site.


crystalline-remark-first-order-thickening

In Situation \ref{situation-global}. Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$. Write $\Omega_{T/S, \delta} = (\Omega_{X/S})_T$, see Lemma \ref{lemma-module-of-differentials}. We explicitly describe a first order thickening $T'$ of $T$. Namely, set $$ \mathcal{O}_{T'} = \mathcal{O}_T \oplus \Omega_{T/S, \delta} $$ with algebra structure such that $\Omega_{T/S, \delta}$ is an ideal of square zero. Let $\mathcal{J} \subset \mathcal{O}_T$ be the ideal sheaf of the closed immersion $U \to T$. Set $\mathcal{J}' = \mathcal{J} \oplus \Omega_{T/S, \delta}$. Define a divided power structure on $\mathcal{J}'$ by setting $$ \delta_n'(f, \omega) = (\delta_n(f), \delta_{n - 1}(f)\omega), $$ see Lemma \ref{lemma-divided-power-first-order-thickening}. There are two ring maps $$ p_0, p_1 : \mathcal{O}_T \to \mathcal{O}_{T'} $$ The first is given by $f \mapsto (f, 0)$ and the second by $f \mapsto (f, \text{d}_{T/S, \delta}f)$. Note that both are compatible with the divided power structures on $\mathcal{J}$ and $\mathcal{J}'$ and so is the quotient map $\mathcal{O}_{T'} \to \mathcal{O}_T$. Thus we get an object $(U, T', \delta')$ of $\text{Cris}(X/S)$ and a commutative diagram $$ \xymatrix{ & T \ar[ld]_{\text{id}} \ar[d]^i \ar[rd]^{\text{id}} \\ T & T' \ar[l]_{p_0} \ar[r]^{p_1} & T } $$ of $\text{Cris}(X/S)$ such that $i$ is a first order thickening whose ideal sheaf is identified with $\Omega_{T/S, \delta}$ and such that $p_1^* - p_0^* : \mathcal{O}_T \to \mathcal{O}_{T'}$ is identified with the universal derivation $\text{d}_{T/S, \delta}$ composed with the inclusion $\Omega_{T/S, \delta} \to \mathcal{O}_{T'}$.


crystalline-remark-second-order-thickening

In Situation \ref{situation-global}. Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$. Write $\Omega_{T/S, \delta} = (\Omega_{X/S})_T$, see Lemma \ref{lemma-module-of-differentials}. We also write $\Omega^2_{T/S, \delta}$ for its second exterior power. We explicitly describe a second order thickening $T''$ of $T$. Namely, set $$ \mathcal{O}_{T''} = \mathcal{O}_T \oplus \Omega_{T/S, \delta} \oplus \Omega_{T/S, \delta} \oplus \Omega^2_{T/S, \delta} $$ with algebra structure defined in the following way $$ (f, \omega_1, \omega_2, \eta) \cdot (f', \omega_1', \omega_2', \eta') = (ff', f\omega_1' + f'\omega_1, f\omega_2' + f'\omega_2, f\eta' + f'\eta + \omega_1 \wedge \omega_2' + \omega_1' \wedge \omega_2). $$ Let $\mathcal{J} \subset \mathcal{O}_T$ be the ideal sheaf of the closed immersion $U \to T$. Let $\mathcal{J}''$ be the inverse image of $\mathcal{J}$ under the projection $\mathcal{O}_{T''} \to \mathcal{O}_T$. Define a divided power structure on $\mathcal{J}''$ by setting $$ \delta_n''(f, \omega_1, \omega_2, \eta) = (\delta_n(f), \delta_{n - 1}(f)\omega_1, \delta_{n - 1}(f)\omega_2, \delta_{n - 1}(f)\eta + \delta_{n - 2}(f)\omega_1 \wedge \omega_2) $$ see Lemma \ref{lemma-divided-power-second-order-thickening}. There are three ring maps $q_0, q_1, q_2 : \mathcal{O}_T \to \mathcal{O}_{T''}$ given by \begin{align*} q_0(f) & = (f, 0, 0, 0), \\ q_1(f) & = (f, \text{d}f, 0, 0), \\ q_2(f) & = (f, \text{d}f, \text{d}f, 0) \end{align*} where $\text{d} = \text{d}_{T/S, \delta}$. Note that all three are compatible with the divided power structures on $\mathcal{J}$ and $\mathcal{J}''$. There are three ring maps $q_{01}, q_{12}, q_{02} : \mathcal{O}_{T'} \to \mathcal{O}_{T''}$ where $\mathcal{O}_{T'}$ is as in Remark \ref{remark-first-order-thickening}. Namely, set \begin{align*} q_{01}(f, \omega) & = (f, \omega, 0, 0), \\ q_{12}(f, \omega) & = (f, \text{d}f, \omega, \text{d}\omega), \\ q_{02}(f, \omega) & = (f, \omega, \omega, 0) \end{align*} These are also compatible with the given divided power structures. Let's do the verifications for $q_{12}$: Note that $q_{12}$ is a ring homomorphism as \begin{align*} q_{12}(f, \omega)q_{12}(g, \eta) & = (f, \text{d}f, \omega, \text{d}\omega)(g, \text{d}g, \eta, \text{d}\eta) \\ & = (fg, f\text{d}g + g \text{d}f, f\eta + g\omega, f\text{d}\eta + g\text{d}\omega + \text{d}f \wedge \eta + \text{d}g \wedge \omega) \\ & = q_{12}(fg, f\eta + g\omega) = q_{12}((f, \omega)(g, \eta)) \end{align*} Note that $q_{12}$ is compatible with divided powers because \begin{align*} \delta_n''(q_{12}(f, \omega)) & = \delta_n''((f, \text{d}f, \omega, \text{d}\omega)) \\ & = (\delta_n(f), \delta_{n - 1}(f)\text{d}f, \delta_{n - 1}(f)\omega, \delta_{n - 1}(f)\text{d}\omega + \delta_{n - 2}(f)\text{d}(f) \wedge \omega) \\ & = q_{12}((\delta_n(f), \delta_{n - 1}(f)\omega)) = q_{12}(\delta'_n(f, \omega)) \end{align*} The verifications for $q_{01}$ and $q_{02}$ are easier. Note that $q_0 = q_{01} \circ p_0$, $q_1 = q_{01} \circ p_1$, $q_1 = q_{12} \circ p_0$, $q_2 = q_{12} \circ p_1$, $q_0 = q_{02} \circ p_0$, and $q_2 = q_{02} \circ p_1$. Thus $(U, T'', \delta'')$ is an object of $\text{Cris}(X/S)$ and we get morphisms $$ \xymatrix{ T'' \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & T' \ar@<1ex>[r] \ar@<-1ex>[r] & T } $$ of $\text{Cris}(X/S)$ satisfying the relations described above. In applications we will use $q_i : T'' \to T$ and $q_{ij} : T'' \to T'$ to denote the morphisms associated to the ring maps described above.


crystalline-remark-equivalence-more-general

The equivalence of Proposition \ref{proposition-crystals-on-affine} holds if we start with a surjection $P \to C$ where $P/A$ satisfies the strong lifting property of Algebra, Lemma \ref{algebra-lemma-smooth-strong-lift}. To prove this we can argue as in the proof of Lemma \ref{lemma-crystals-on-affine-smooth}. (Details will be added here if we ever need this.) Presumably there is also a direct proof of this result, but the advantage of using polynomial rings is that the rings $D(n)$ are $p$-adic completions of divided power polynomial rings and the algebra is simplified.


crystalline-remark-vanishing

The proof of Proposition \ref{proposition-compare-with-de-Rham} shows that the conclusion $$ Ru_{X/S, *}(\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0 $$ for $i > 0$ is true for any $\mathcal{O}_{X/S}$-module $\mathcal{F}$ which satisfies conditions (1) and (2) of Proposition \ref{proposition-compute-cohomology}. This applies to the following non-crystals: $\Omega^i_{X/S}$ for all $i$, and any sheaf of the form $\underline{\mathcal{F}}$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module. In particular, it applies to the sheaf $\underline{\mathcal{O}_X} = \underline{\mathbf{G}_a}$. But note that we need something like Lemma \ref{lemma-automatic-connection} to produce a de Rham complex which requires $\mathcal{F}$ to be a crystal. Hence (currently) the collection of sheaves of modules for which the full statement of Proposition \ref{proposition-compare-with-de-Rham} holds is exactly the category of crystals in quasi-coherent modules.


crystalline-remark-compute-direct-image

Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let $$ \xymatrix{ X \ar[r]_f \ar[d] & X' \ar[d] \\ S_0 \ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $X'$. Let $\mathcal{F}$ be an $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then $Rf_{\text{cris}, *}\mathcal{F}$ can be computed as follows. \medskip\noindent Given an object $(U', T', \delta')$ of $\text{Cris}(X'/S')$ set $U = X \times_{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote $(T_0, T, \delta)$ the divided power scheme over $S$ such that $$ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } $$ is cartesian in the category of divided power schemes, see Lemma \ref{lemma-fibre-product}. There is an induced morphism $U \to T_0$ and we obtain a morphism $(U/T)_{\text{cris}} \to (X/S)_{\text{cris}}$, see Remark \ref{remark-functoriality-cris}. Let $\mathcal{F}_U$ be the pullback of $\mathcal{F}$. Let $\tau_{U/T} : (U/T)_{\text{cris}} \to T_{Zar}$ be the structure morphism. Then we have \begin{equation} \label{equation-identify-pushforward} \left(Rf_{\text{cris}, *}\mathcal{F}\right)_{T'} = R(T \to T')_*\left(R\tau_{U/T, *} \mathcal{F}_U \right) \end{equation} where the left hand side is the restriction (see Section \ref{section-sheaves}). \medskip\noindent Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense of Sites, Lemma \ref{sites-lemma-localize-topos-site}) of $\text{Cris}(X/S)$ at the sheaf of sets $f_{\text{cris}}^{-1}h_{(U', T', \delta')}$. Next, reduce the statement to the case where $\mathcal{F}$ is an injective module and pushforward of modules using that the pullback of an injective $\mathcal{O}_{X/S}$-module is an injective $\mathcal{O}_{U/T}$-module on $\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward.


crystalline-remark-mayer-vietoris

In the situation of Remark \ref{remark-compute-direct-image} suppose we have an open covering $X = X' \cup X''$. Denote $X''' = X' \cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$ to $X'$, $X''$, and $X'''$. Moreover, let $\mathcal{F}'$, $\mathcal{F}''$, and $\mathcal{F}'''$ be the restriction of $\mathcal{F}$ to the crystalline sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle $$ Rf_{\text{cris}, *}\mathcal{F} \longrightarrow Rf'_{\text{cris}, *}\mathcal{F}' \oplus Rf''_{\text{cris}, *}\mathcal{F}'' \longrightarrow Rf'''_{\text{cris}, *}\mathcal{F}''' \longrightarrow Rf_{\text{cris}, *}\mathcal{F}[1] $$ in $D(\mathcal{O}_{X'/S'})$. \medskip\noindent Hints: This is a formal consequence of the fact that the subcategories $\text{Cris}(X'/S)$, $\text{Cris}(X''/S)$, $\text{Cris}(X'''/S)$ correspond to open subobjects of the final sheaf on $\text{Cris}(X/S)$ and that the last is the intersection of the first two.


crystalline-remark-cech-complex

Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $X$ be a separated\footnote{This assumption is not strictly necessary, as using hypercoverings the construction of the remark can be extended to the general case.} scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. \medskip\noindent Choose an affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$. Write $U_\lambda = \Spec(C_\lambda)$. Choose a polynomial algebra $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$. Having fixed these choices we can construct a {\v C}ech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$. \medskip\noindent Given $n \geq 0$ and $\lambda_0, \ldots, \lambda_n \in \Lambda$ write $U_{\lambda_0 \ldots \lambda_n} = U_{\lambda_0} \cap \ldots \cap U_{\lambda_n}$. This is an affine scheme by assumption. Write $U_{\lambda_0 \ldots \lambda_n} = \Spec(C_{\lambda_0 \ldots \lambda_n})$. Set $$ P_{\lambda_0 \ldots \lambda_n} = P_{\lambda_0} \otimes_A \ldots \otimes_A P_{\lambda_n} $$ which comes with a canonical surjection onto $C_{\lambda_0 \ldots \lambda_n}$. Denote the kernel $J_{\lambda_0 \ldots \lambda_n}$ and set $D_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed divided power envelope of $J_{\lambda_0 \ldots \lambda_n}$ in $P_{\lambda_0 \ldots \lambda_n}$ relative to $\gamma$. Let $M_{\lambda_0 \ldots \lambda_n}$ be the $P_{\lambda_0 \ldots \lambda_n}$-module corresponding to the restriction of $\mathcal{F}$ to $\text{Cris}(U_{\lambda_0 \ldots \lambda_n}/S)$ via Proposition \ref{proposition-crystals-on-affine}. By construction we obtain a cosimplicial divided power ring $D(*)$ having in degree $n$ the ring $$ D(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} D_{\lambda_0 \ldots \lambda_n} $$ (use that divided power envelopes are functorial and the trivial cosimplicial structure on the ring $P(*)$ defined similarly). Since $M_{\lambda_0 \ldots \lambda_n}$ is the ``value'' of $\mathcal{F}$ on the objects $\Spec(D_{\lambda_0 \ldots \lambda_n})$ we see that $M(*)$ defined by the rule $$ M(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} M_{\lambda_0 \ldots \lambda_n} $$ forms a cosimplicial $D(*)$-module. Now we claim that we have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = s(M(*)) $$ Here $s(-)$ denotes the cochain complex associated to a cosimplicial module (see Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}). \medskip\noindent Hints: The proof of this is similar to the proof of Proposition \ref{proposition-compute-cohomology} (in particular the result holds for any module satisfying the assumptions of that proposition).


crystalline-remark-alternating-cech-complex

Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $X$ be a separated quasi-compact scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. \medskip\noindent Choose a finite affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$ and a total ordering on $\Lambda$. Write $U_\lambda = \Spec(C_\lambda)$. Choose a polynomial algebra $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$. Having fixed these choices we can construct an alternating {\v C}ech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$. \medskip\noindent We are going to use the notation introduced in Remark \ref{remark-cech-complex}. Denote $\Omega_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed module of differentials of $D_{\lambda_0 \ldots \lambda_n}$ over $A$ compatible with the divided power structure. Let $\nabla$ be the integrable connection on $M_{\lambda_0 \ldots \lambda_n}$ coming from Proposition \ref{proposition-crystals-on-affine}. Consider the double complex $M^{\bullet, \bullet}$ with terms $$ M^{n, m} = \bigoplus\nolimits_{\lambda_0 < \ldots < \lambda_n} M_{\lambda_0 \ldots \lambda_n} \otimes^\wedge_{D_{\lambda_0 \ldots \lambda_n}} \Omega^m_{D_{\lambda_0 \ldots \lambda_n}}. $$ For the differential $d_1$ (increasing $n$) we use the usual {\v C}ech differential and for the differential $d_2$ we use the connection, i.e., the differential of the de Rham complex. We claim that $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet, \bullet}) $$ Here $\text{Tot}(-)$ denotes the total complex associated to a double complex, see Homology, Definition \ref{homology-definition-associated-simple-complex}. \medskip\noindent Hints: We have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet) $$ by Proposition \ref{proposition-compare-with-de-Rham}. The right hand side of the formula is simply the alternating {\v C}ech complex for the covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ (which induces an open covering of the final sheaf of $\text{Cris}(X/S)$) and the complex $\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet$, see Proposition \ref{proposition-compute-cohomology-crystal}. Now the result follows from a general result in cohomology on sites, namely that the alternating {\v C}ech complex computes the cohomology provided it gives the correct answer on all the pieces (insert future reference here).


crystalline-remark-quasi-coherent

In the situation of Remark \ref{remark-compute-direct-image} assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ the sheaves $R^if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent. \medskip\noindent Hints: We have to show that the restrictions to $T'$ are quasi-coherent $\mathcal{O}_{T'}$-modules, where $(U', T', \delta')$ is any object of $\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine. We use the formula (\ref{equation-identify-pushforward}), the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$ is affine over the base change of $T'$ by $S \to S'$), and Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images} to see that it suffices to show that the sheaves $R^i\tau_{U/T, *}\mathcal{F}_U$ are quasi-coherent. Note that $U \to T_0$ is also quasi-compact and quasi-separated, see Schemes, Lemmas \ref{schemes-lemma-quasi-compact-permanence} and \ref{schemes-lemma-quasi-compact-permanence}. \medskip\noindent This reduces us to proving that $R^i\tau_{X/S, *}\mathcal{F}$ is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here $\tau_{X/S}$ is the structure morphism, see Remark \ref{remark-structure-morphism}. We may work locally on $S$, hence we may assume $S$ affine (see Lemma \ref{lemma-localize}). Induction on the number of affines covering $X$ and Mayer-Vietoris (Remark \ref{remark-mayer-vietoris}) reduces the question to the case where $X$ is also affine (as in the proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}). Say $X = \Spec(C)$ and $S = \Spec(A)$ so that $(A, I, \gamma)$ and $A \to C$ are as in Situation \ref{situation-affine}. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$ as in Section \ref{section-quasi-coherent-crystals}. Let $(M, \nabla)$ be the module corresponding to $\mathcal{F}$, see Proposition \ref{proposition-crystals-on-affine}. Applying  Proposition \ref{proposition-compute-cohomology-crystal} we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is represented by $M \otimes_D \Omega_D^*$. Note that completion isn't necessary as $p$ is nilpotent in $A$! We have to show that this is compatible with taking principal opens in $S = \Spec(A)$. Suppose that $g \in A$. Then we conclude that similarly $R\Gamma(\text{Cris}(X_g/S_g), \mathcal{F})$ is computed by $M_g \otimes_{D_g} \Omega_{D_g}^*$ (again this uses that $p$-adic completion isn't necessary). Hence we conclude because localization is an exact functor on $A$-modules.


crystalline-remark-bounded-cohomology

In the situation of Remark \ref{remark-compute-direct-image} assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is of finite type and quasi-separated. Then there exists an integer $i_0$ such that for any crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ we have $R^if_{\text{cris}, *}\mathcal{F} = 0$ for all $i > i_0$. \medskip\noindent Hints: Arguing as in Remark \ref{remark-quasi-coherent} (using Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}) we reduce to proving that $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i \gg 0$ in the situation of Proposition \ref{proposition-compute-cohomology-crystal} when $C$ is a finite type algebra over $A$. This is clear as we can choose a finite polynomial algebra and we see that $\Omega^i_D = 0$ for $i \gg 0$.


crystalline-remark-bounded-cohomology-over-point

In Situation \ref{situation-global} let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Assume that $S_0$ has a unique point and that $X \to S_0$ is of finite presentation. \begin{enumerate} \item If $\dim X = d$ and $X/S_0$ has embedding dimension $e$, then $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > d + e$. \item If $X$ is separated and can be covered by $q$ affines, and $X/S_0$ has embedding dimension $e$, then $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > q + e$. \end{enumerate} Hints: In case (1) we can use that $$ H^i(\text{Cris}(X/S), \mathcal{F}) = H^i(X_{Zar}, Ru_{X/S, *}\mathcal{F}) $$ and that $Ru_{X/S, *}\mathcal{F}$ is locally calculated by a de Rham complex constructed using an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (see Lemma \ref{lemma-compute-cohomology-crystal-smooth}). These de Rham complexes are zero in all degrees $> e$. Hence (1) follows from Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}. In case (2) we use the alternating {\v C}ech complex (see Remark \ref{remark-alternating-cech-complex}) to reduce to the case $X$ affine. In the affine case we prove the result using the de Rham complex associated to an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (it takes some work to construct such a thing).


crystalline-remark-base-change

In the situation of Remark \ref{remark-compute-direct-image} assume $S = \Spec(A)$ and $S' = \Spec(A')$ are affine. Let $\mathcal{F}'$ be an $\mathcal{O}_{X'/S'}$-module. Let $\mathcal{F}$ be the pullback of $\mathcal{F}'$. Then there is a canonical base change map $$ L(S' \to S)^*R\tau_{X'/S', *}\mathcal{F}' \longrightarrow R\tau_{X/S, *}\mathcal{F} $$ where $\tau_{X/S}$ and $\tau_{X'/S'}$ are the structure morphisms, see Remark \ref{remark-structure-morphism}. On global sections this gives a base change map \begin{equation} \label{equation-base-change-map} R\Gamma(\text{Cris}(X'/S'), \mathcal{F}') \otimes^\mathbf{L}_{A'} A \longrightarrow R\Gamma(\text{Cris}(X/S), \mathcal{F}) \end{equation} in $D(A)$. \medskip\noindent Hint: Compose the very general base change map of Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change} with the canonical map $Lf_{\text{cris}}^*\mathcal{F}' \to f_{\text{cris}}^*\mathcal{F}' = \mathcal{F}$.


crystalline-remark-base-change-isomorphism

The map (\ref{equation-base-change-map}) is an isomorphism provided all of the following conditions are satisfied: \begin{enumerate} \item $p$ is nilpotent in $A'$, \item $\mathcal{F}'$ is a crystal in quasi-coherent $\mathcal{O}_{X'/S'}$-modules, \item $X' \to S'_0$ is a quasi-compact, quasi-separated morphism, \item $X = X' \times_{S'_0} S_0$, \item $\mathcal{F}'$ is a flat $\mathcal{O}_{X'/S'}$-module, \item $X' \to S'_0$ is a local complete intersection morphism (see More on Morphisms, Definition \ref{more-morphisms-definition-lci}; this holds for example if $X' \to S'_0$ is syntomic or smooth), \item $X'$ and $S_0$ are Tor independent over $S'_0$ (see More on Algebra, Definition \ref{more-algebra-definition-tor-independent}; this holds for example if either $S_0 \to S'_0$ or $X' \to S'_0$ is flat). \end{enumerate} Hints: Condition (1) means that in the arguments below $p$-adic completion does nothing and can be ignored. Using condition (3) and Mayer Vietoris (see Remark \ref{remark-mayer-vietoris}) this reduces to the case where $X'$ is affine. In fact by condition (6), after shrinking further, we can assume that $X' = \Spec(C')$ and we are given a presentation $C' = A'/I'[x_1, \ldots, x_n]/(\bar f'_1, \ldots, \bar f'_c)$ where $\bar f'_1, \ldots, \bar f'_c$ is a Koszul-regular sequence in $A'/I'$. (This means that smooth locally $\bar f'_1, \ldots, \bar f'_c$ forms a regular sequence, see More on Algebra, Lemma \ref{more-algebra-lemma-Koszul-regular-flat-locally-regular}.) We choose a lift of $\bar f'_i$ to an element $f'_i \in A'[x_1, \ldots, x_n]$. By (4) we see that $X = \Spec(C)$ with $C = A/I[x_1, \ldots, x_n]/(\bar f_1, \ldots, \bar f_c)$ where $f_i \in A[x_1, \ldots, x_n]$ is the image of $f'_i$. By property (7) we see that $\bar f_1, \ldots, \bar f_c$ is a Koszul-regular sequence in $A/I[x_1, \ldots, x_n]$. The divided power envelope of $I'A'[x_1, \ldots, x_n] + (f'_1, \ldots, f'_c)$ in $A'[x_1, \ldots, x_n]$ relative to $\gamma'$ is $$ D' = A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c \rangle/(\xi_i - f'_i) $$ see Lemma \ref{lemma-describe-divided-power-envelope}. Then you check that $\xi_1 - f'_1, \ldots, \xi_n - f'_n$ is a Koszul-regular sequence in the ring $A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$. Similarly the divided power envelope of $IA[x_1, \ldots, x_n] + (f_1, \ldots, f_c)$ in $A[x_1, \ldots, x_n]$ relative to $\gamma$ is $$ D = A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle/(\xi_i - f_i) $$ and $\xi_1 - f_1, \ldots, \xi_n - f_n$ is a Koszul-regular sequence in the ring $A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$. It follows that $D' \otimes_{A'}^\mathbf{L} A = D$. Condition (2) implies $\mathcal{F}'$ corresponds to a pair $(M', \nabla)$ consisting of a $D'$-module with connection, see Proposition \ref{proposition-crystals-on-affine}. Then $M = M' \otimes_{D'} D$ corresponds to the pullback $\mathcal{F}$. By assumption (5) we see that $M'$ is a flat $D'$-module, hence $$ M = M' \otimes_{D'} D = M' \otimes_{D'} D' \otimes_{A'}^\mathbf{L} A = M' \otimes_{A'}^\mathbf{L} A $$ Since the modules of differentials $\Omega_{D'}$ and $\Omega_D$ (as defined in Section \ref{section-quasi-coherent-crystals}) are free $D'$-modules on the same generators we see that $$ M \otimes_D \Omega^\bullet_D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{D'} D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{A'}^\mathbf{L} A $$ which proves what we want by Proposition \ref{proposition-compute-cohomology-crystal}.


crystalline-remark-rlim

Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ an algebra over $\mathbf{Z}_{(p)}$ with $p$ nilpotent in $A/I$. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $X$ be a scheme over $S_0$ with $p$ locally nilpotent on $X$. Let $\mathcal{F}$ be any $\mathcal{O}_{X/S}$-module. For $e \gg 0$ we have $(p^e) \subset I$ is preserved by $\gamma$, see Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}. Set $S_e = \Spec(A/p^eA)$ for $e \gg 0$. Then $\text{Cris}(X/S_e)$ is a full subcategory of $\text{Cris}(X/S)$ and we denote $\mathcal{F}_e$ the restriction of $\mathcal{F}$ to $\text{Cris}(X/S_e)$. Then $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\lim_e R\Gamma(\text{Cris}(X/S_e), \mathcal{F}_e) $$ \medskip\noindent Hints: Suffices to prove this for $\mathcal{F}$ injective. In this case the sheaves $\mathcal{F}_e$ are injective modules too, the transition maps $\Gamma(\mathcal{F}_{e + 1}) \to \Gamma(\mathcal{F}_e)$ are surjective, and we have $\Gamma(\mathcal{F}) = \lim_e \Gamma(\mathcal{F}_e)$ because any object of $\text{Cris}(X/S)$ is locally an object of one of the categories $\text{Cris}(X/S_e)$ by definition of $\text{Cris}(X/S)$.


crystalline-remark-comparison

Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $Y$ be a smooth scheme over $S$ and set $X = Y \times_S S_0$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then \begin{enumerate} \item $\gamma$ extends to a divided power structure on the ideal of $X$ in $Y$ so that $(X, Y, \gamma)$ is an object of $\text{Cris}(X/S)$, \item the restriction $\mathcal{F}_Y$ (see Section \ref{section-sheaves}) comes endowed with a canonical integrable connection $\nabla : \mathcal{F}_Y \to \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}$, and \item we have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S}) $$ in $D(A)$. \end{enumerate} Hints: See Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends} for (1). See Lemma \ref{lemma-automatic-connection} for (2). For Part (3) note that there is a map, see (\ref{equation-restriction}). This map is an isomorphism when $X$ is affine, see Lemma \ref{lemma-compute-cohomology-crystal-smooth}. This shows that $Ru_{X/S, *}\mathcal{F}$ and $\mathcal{F}_Y \otimes \Omega^\bullet_{Y/S}$ are quasi-isomorphic as complexes on $Y_{Zar} = X_{Zar}$. Since $R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(X_{Zar}, Ru_{X/S, *}\mathcal{F})$ the result follows.


crystalline-remark-perfect

Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules. Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a perfect object of $D(A)$. \medskip\noindent Hints: By Remark \ref{remark-base-change-isomorphism} we have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) \otimes_A^\mathbf{L} A/I \cong R\Gamma(\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)}) $$ By Remark \ref{remark-comparison} we have $$ R\Gamma(\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)}) = R\Gamma(X, \mathcal{F}_X \otimes \Omega^\bullet_{X/S_0}) $$ Using the stupid filtration on the de Rham complex we see that the last displayed complex is perfect in $D(A/I)$ as soon as the complexes $$ R\Gamma(X, \mathcal{F}_X \otimes \Omega^q_{X/S_0}) $$ are perfect complexes in $D(A/I)$, see More on Algebra, Lemma \ref{more-algebra-lemma-two-out-of-three-perfect}. This is true by standard arguments in coherent cohomology using that $\mathcal{F}_X \otimes \Omega^q_{X/S_0}$ is a finite locally free sheaf and $X \to S_0$ is proper and flat (insert future reference here). Applying More on Algebra, Lemma \ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal} we see that $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) \otimes_A^\mathbf{L} A/I^n $$ is a perfect object of $D(A/I^n)$ for all $n$. This isn't quite enough unless $A$ is Noetherian. Namely, even though $I$ is locally nilpotent by our assumption that $p$ is nilpotent, see Divided Power Algebra, Lemma \ref{dpa-lemma-nil}, we cannot conclude that $I^n = 0$ for some $n$. A counter example is $\mathbf{F}_p\langle x \rangle$. To prove it in general when $\mathcal{F} = \mathcal{O}_{X/S}$ the argument of \url{https://math.columbia.edu/~dejong/wordpress/?p=2227} works. When the coefficients $\mathcal{F}$ are non-trivial the argument of \cite{Faltings-very} seems to be as follows. Reduce to the case $pA = 0$ by More on Algebra, Lemma \ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal}. In this case the Frobenius map $A \to A$, $a \mapsto a^p$ factors as $A \to A/I \xrightarrow{\varphi} A$ (as $x^p = 0$ for $x \in I$). Set $X^{(1)} = X \otimes_{A/I, \varphi} A$. The absolute Frobenius morphism of $X$ factors through a morphism $F_X : X \to X^{(1)}$ (a kind of relative Frobenius). Affine locally if $X = \Spec(C)$ then $X^{(1)} = \Spec( C \otimes_{A/I, \varphi} A)$ and $F_X$ corresponds to $C \otimes_{A/I, \varphi} A \to C$, $c \otimes a \mapsto c^pa$. This defines morphisms of ringed topoi $$ (X/S)_{\text{cris}} \xrightarrow{(F_X)_{\text{cris}}} (X^{(1)}/S)_{\text{cris}} \xrightarrow{u_{X^{(1)}/S}} \Sh(X^{(1)}_{Zar}) $$ whose composition is denoted $\text{Frob}_X$. One then shows that $R\text{Frob}_{X, *}\mathcal{F}$ is representable by a perfect complex of $\mathcal{O}_{X^{(1)}}$-modules(!) by a local calculation.


crystalline-remark-complete-perfect

Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules. Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a perfect object of $D(A)$. \medskip\noindent Hints: We know that $K = R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is the derived limit $K = R\lim K_e$ of the cohomologies over $A/p^eA$, see Remark \ref{remark-rlim}. Each $K_e$ is a perfect complex of $D(A/p^eA)$ by Remark \ref{remark-perfect}. Since $A$ is $p$-adically complete the result follows from More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-perfect-gives-complete}.


crystalline-remark-complete-comparison

Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set $X = Y \times_S S_0$. Let $\mathcal{F}$ be a finite type crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then \begin{enumerate} \item there exists a coherent $\mathcal{O}_Y$-module $\mathcal{F}_Y$ endowed with integrable connection $$ \nabla : \mathcal{F}_Y \longrightarrow \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S} $$ such that $\mathcal{F}_Y/p^e\mathcal{F}_Y$ is the module with connection over $A/p^eA$ found in Remark \ref{remark-comparison}, and \item we have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S}) $$ in $D(A)$. \end{enumerate} Hints: The existence of $\mathcal{F}_Y$ is Grothendieck's existence theorem (insert future reference here). The isomorphism of cohomologies follows as both sides are computed as $R\lim$ of the versions modulo $p^e$ (see Remark \ref{remark-rlim} for the left hand side; use the theorem on formal functions, see Cohomology of Schemes, Theorem \ref{coherent-theorem-formal-functions} for the right hand side). Each of the versions modulo $p^e$ are isomorphic by Remark \ref{remark-comparison}.


crystalline-remark-F-crystal-variants

Let $(\mathcal{E}, F)$ be an $F$-crystal as in Definition \ref{definition-F-crystal}. In the literature the nondegeneracy condition is often part of the definition of an $F$-crystal. Moreover, often it is also assumed that $F \circ V = p^n\text{id}$. What is needed for the result below is that there exists an integer $j \geq 0$ such that $\Ker(F)$ and $\Coker(F)$ are killed by $p^j$. If the rank of $\mathcal{E}$ is bounded (for example if $X$ is quasi-compact), then both of these conditions follow from the nondegeneracy condition as formulated in the definition. Namely, suppose $R$ is a ring, $r \geq 1$ is an integer and $K, L \in \text{Mat}(r \times r, R)$ are matrices with $K L = p^i 1_{r \times r}$. Then $\det(K)\det(L) = p^{ri}$.  Let $L'$ be the adjugate matrix of $L$, i.e., $L' L = L L' = \det(L)$. Set $K' = p^{ri} K$ and $j = ri + i$. Then we have $K' L = p^j 1_{r \times r}$ as $K L = p^i$ and $$ L K' = L K \det(L) \det(M) = L K L L' \det(M) = L p^i L' \det(M) = p^j 1_{r \times r} $$ It follows that if $V$ is as in Definition \ref{definition-F-crystal} then setting $V' = p^N V$ where $N > i \cdot \text{rank}(\mathcal{E})$ we get $V' \circ F = p^{N + i}$ and $F \circ V' = p^{N + i}$.


spaces-pushouts-remark-essentially-constant

The meaning of Lemma \ref{lemma-essentially-constant} is the system $X_1 \to X_2 \to X_3 \to \ldots$ is essentially constant with value $X$. See Categories, Definition \ref{categories-definition-essentially-constant-diagram}.


varieties-remark-exact-sequence-induction

Let $k$ be an infinite field. Let $n \geq 1$. Given a finite number of coherent modules $\mathcal{F}_i$ on $\mathbf{P}^n_k$ we can choose a single $s \in \Gamma(\mathbf{P}^n_k, \mathcal{O}(1))$ such that the statement of Lemma \ref{lemma-exact-sequence-induction} works for each of them. To prove this, just apply the lemma to $\bigoplus \mathcal{F}_i$.


varieties-remark-exact-sequence-induction-cohomology

In the situation of Lemmas \ref{lemma-hyperplane} and \ref{lemma-exact-sequence-induction} we have $H \cong \mathbf{P}^{n - 1}_k$ with Serre twists $\mathcal{O}_H(d) = i^*\mathcal{O}_{\mathbf{P}^n_k}(d)$. For every $d \in \mathbf{Z}$ we have a short exact sequence $$ 0 \to \mathcal{F}(d - 1) \to \mathcal{F}(d) \to i_*(\mathcal{G}(d)) \to 0 $$ Namely, tensoring by $\mathcal{O}_{\mathbf{P}^n_k}(d)$ is an exact functor and by the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) we have $i_*(\mathcal{G}(d)) = i_*\mathcal{G} \otimes \mathcal{O}_{\mathbf{P}^n_k}(d)$. We obtain corresponding long exact sequences $$ H^i(\mathbf{P}^n_k, \mathcal{F}(d - 1)) \to H^i(\mathbf{P}^n_k, \mathcal{F}(d)) \to H^i(H, \mathcal{G}(d)) \to H^{i + 1}(\mathbf{P}^n_k, \mathcal{F}(d - 1)) $$ This follows from the above and the fact that we have $H^i(\mathbf{P}^n_k, i_*\mathcal{G}(d)) = H^i(H, \mathcal{G}(d))$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-cohomology} (closed immersions are affine).


varieties-remark-n-fold-relative-frobenius

Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. For $n \geq 1$ $$ X^{(p^n)} = X^{(p^n/S)} = X \times_{S, F_S^n} S $$ viewed as a scheme over $S$. Observe that $X \mapsto X^{(p^n)}$ is a functor. Applying Lemma \ref{lemma-frobenius-endomorphism-identity} we see $F_{X/S, n} = (F_X^n, \text{id}_S) : X \longrightarrow X^{(p^n)}$ is a morphism over $S$ fitting into the commutative diagram $$ \xymatrix{ X \ar[rr]_{F_{X/S, n}} \ar[rrd] \ar@/^1em/[rrrr]^{F_X^n} & & X^{(p^n)} \ar[rr] \ar[d] & & X \ar[d] \\ & & S \ar[rr]^{F_S^n} & & S } $$ where the right square is cartesian. The morphism $F_{X/S, n}$ is sometimes called the {\it $n$-fold relative Frobenius morphism of $X/S$}. This makes sense because we have the formula $$ F_{X/S, n} = F_{X^{(p^{n - 1})}/S} \circ \ldots \circ F_{X^{(p)}/S} \circ F_{X/S} $$ which shows that $F_{X/S, n}$ is the composition of $n$ relative Frobenii. Since we have $$ F_{X^{(p^m)}/S} = F_{X^{(p^{m - 1})}/S}^{(p)} = \ldots = F_{X/S}^{(p^m)} $$ (details omitted) we get also that $$ F_{X/S, n} = F_{X/S}^{(p^{n - 1})} \circ \ldots \circ F_{X/S}^{(p)} \circ F_{X/S} $$


varieties-remark-conductor

Let $A$ be a reduced ring. Let $I, J$ be ideals of $A$ such that $V(I) \cup V(J) = \Spec(A)$. Set $B = A/J$. Then $I \to IB$ is an isomorphism of $A$-modules. Namely, we have $IB = I + J/J = I/(I \cap J)$ and $I \cap J$ is zero because $A$ is reduced and $\Spec(A) = V(I) \cup V(J) = V(I \cap J)$. Thus for any projective $A$-module $P$ we also have $IP = I(P/JP)$.


varieties-remark-useless-generalization

In fact, if $X$ is a scheme whose reduction is a Noetherian separated scheme of dimension $1$, then $X$ has an ample invertible sheaf. The argument to prove this is the same as the proof of Proposition \ref{proposition-dim-1-noetherian-separated-has-ample} except one uses Limits, Lemma \ref{limits-lemma-ample-on-reduction} instead of Cohomology of Schemes, Lemma \ref{coherent-lemma-ample-on-reduction}.


cotangent-remark-variant-cotangent-complex

Let $A \to B$ be a ring map. Let $\mathcal{A}$ be the category of arrows $\psi : C \to B$ of $A$-algebras and let $\mathcal{S}$ be the category of maps $E \to B$ where $E$ is a set. There are adjoint functors $V : \mathcal{A} \to \mathcal{S}$ (the forgetful functor) and $U : \mathcal{S} \to \mathcal{A}$ which sends $E \to B$ to $A[E] \to B$. Let $X_\bullet$ be the simplicial object of $\text{Fun}(\mathcal{A}, \mathcal{A})$ constructed in Simplicial, Section \ref{simplicial-section-standard}. The diagram $$ \xymatrix{ \mathcal{A} \ar[d] \ar[r] & \mathcal{S} \ar@<1ex>[l] \ar[d] \\ \textit{Alg}_A \ar[r] & \textit{Sets} \ar@<1ex>[l] } $$ commutes. It follows that $X_\bullet(\text{id}_B : B \to B)$ is equal to the standard resolution of $B$ over $A$.


cotangent-remark-resolution

Let $A \to B$ be any ring map. Let us call an augmented simplicial $A$-algebra $\epsilon : P_\bullet \to B$ a {\it resolution of $B$ over $A$} if each $P_n$ is a polynomial algebra and $\epsilon$ is a trivial Kan fibration of simplicial sets. If $P_\bullet \to B$ is an augmentation of a simplicial $A$-algebra with each $P_n$ a polynomial algebra surjecting onto $B$, then the following are equivalent \begin{enumerate} \item $\epsilon : P_\bullet \to B$ is a resolution of $B$ over $A$, \item $\epsilon : P_\bullet \to B$ is a quasi-isomorphism on associated complexes, \item $\epsilon : P_\bullet \to B$ induces a homotopy equivalence of simplicial sets. \end{enumerate} To see this use Simplicial, Lemmas \ref{simplicial-lemma-trivial-kan-homotopy}, \ref{simplicial-lemma-homotopy-equivalence}, and \ref{simplicial-lemma-qis-simplicial-abelian-groups}. A resolution $P_\bullet$ of $B$ over $A$ gives a cosimplicial object $U_\bullet$ of $\mathcal{C}_{B/A}$ as in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution} and it follows that $$ L\pi_!\mathcal{F} = \mathcal{F}(P_\bullet) $$ functorially in $\mathcal{F}$, see Lemma \ref{lemma-identify-pi-shriek}. The (formal part of the) proof of Proposition \ref{proposition-polynomial} shows that resolutions exist. We also have seen in the first proof of Lemma \ref{lemma-pi-shriek-standard} that the standard resolution of $B$ over $A$ is a resolution (so that this terminology doesn't lead to a conflict). However, the argument in the proof of Proposition \ref{proposition-polynomial} shows the existence of resolutions without appealing to the simplicial computations in Simplicial, Section \ref{simplicial-section-standard}. Moreover, for {\it any} choice of resolution we have a canonical isomorphism $$ L_{B/A} = \Omega_{P_\bullet/A} \otimes_{P_\bullet, \epsilon} B $$ in $D(B)$ by Lemma \ref{lemma-compute-cotangent-complex}. The freedom to choose an arbitrary resolution can be quite useful.


cotangent-remark-homotopy-triangle

Suppose that we are given a square (\ref{equation-commutative-square}) such that there exists an arrow $\kappa : B \to A'$ making the diagram commute: $$ \xymatrix{ B \ar[r]_\beta \ar[rd]_\kappa & B' \\ A \ar[u] \ar[r]^\alpha & A' \ar[u] } $$ In this case we claim the functoriality map $P_\bullet \to P'_\bullet$ is homotopic to the composition $P_\bullet \to B \to A' \to P'_\bullet$. Namely, using $\kappa$ the functoriality map factors as $$ P_\bullet \to P_{A'/A', \bullet} \to P'_\bullet $$ where $P_{A'/A', \bullet}$ is the standard resolution of $A'$ over $A'$. Since $A'$ is the polynomial algebra on the empty set over $A'$ we see from Simplicial, Lemma \ref{simplicial-lemma-standard-simplicial-homotopy} that the augmentation $\epsilon_{A'/A'} : P_{A'/A', \bullet} \to A'$ is a homotopy equivalence of simplicial rings. Observe that the homotopy inverse map $c : A' \to P_{A'/A', \bullet}$ constructed in the proof of that lemma is just the structure morphism, hence we conclude what we want because the two compositions $$ \xymatrix{ P_\bullet \ar[r] & P_{A'/A', \bullet} \ar@<1ex>[rr]^{\text{id}} \ar@<-1ex>[rr]_{c \circ \epsilon_{A'/A'}} & & P_{A'/A', \bullet} \ar[r] & P'_\bullet } $$ are the two maps discussed above and these are homotopic (Simplicial, Remark \ref{simplicial-remark-homotopy-pre-post-compose}). Since the second map $P_\bullet \to P'_\bullet$ induces the zero map $\Omega_{P_\bullet/A} \to \Omega_{P'_\bullet/A'}$ we conclude that the functoriality map $L_{B/A} \to L_{B'/A'}$ is homotopic to zero in this case.


cotangent-remark-triangle

We sketch an alternative, perhaps simpler, proof of the existence of the fundamental triangle. Let $A \to B \to C$ be ring maps and assume that $B \to C$ is injective. Let $P_\bullet \to B$ be the standard resolution of $B$ over $A$ and let $Q_\bullet \to C$ be the standard resolution of $C$ over $B$. Picture $$ \xymatrix{ P_\bullet : & A[A[A[B]]] \ar[d] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[A[B]] \ar[d] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[B] \ar[d] \ar@<0ex>[l] \ar[r] & B \\ Q_\bullet : & A[A[A[C]]] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[A[C]] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[C] \ar@<0ex>[l] \ar[r] & C } $$ Observe that since $B \to C$ is injective, the ring $Q_n$ is a polynomial algebra over $P_n$ for all $n$. Hence we obtain a cosimplicial object in $\mathcal{C}_{C/B/A}$ (beware reversal arrows). Now set $\overline{Q}_\bullet = Q_\bullet \otimes_{P_\bullet} B$. The key to the proof of Proposition \ref{proposition-triangle} is to show that $\overline{Q}_\bullet$ is a resolution of $C$ over $B$. This follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-O-homology-qis} applied to $\mathcal{C} = \Delta$, $\mathcal{O} = P_\bullet$, $\mathcal{O}' = B$, and $\mathcal{F} = Q_\bullet$ (this uses that $Q_n$ is flat over $P_n$; see Cohomology on Sites, Remark \ref{sites-cohomology-remark-simplicial-modules} to relate simplicial modules to sheaves). The key fact implies that the distinguished triangle of Proposition \ref{proposition-triangle} is the distinguished triangle associated to the short exact sequence of simplicial $C$-modules $$ 0 \to \Omega_{P_\bullet/A} \otimes_{P_\bullet} C \to \Omega_{Q_\bullet/A} \otimes_{Q_\bullet} C \to \Omega_{\overline{Q}_\bullet/B} \otimes_{\overline{Q}_\bullet} C \to 0 $$ which is deduced from the short exact sequences $0 \to \Omega_{P_n/A} \otimes_{P_n} Q_n \to \Omega_{Q_n/A} \to \Omega_{Q_n/P_n} \to 0$ of Algebra, Lemma \ref{algebra-lemma-ses-formally-smooth}. Namely, by Remark \ref{remark-resolution} and the key fact the complex on the right hand side represents $L_{C/B}$ in $D(C)$. \medskip\noindent If $B \to C$ is not injective, then we can use the above to get a fundamental triangle for $A \to B \to B \times C$. Since $L_{B \times C/B} \to L_{B/B} \oplus L_{C/B}$ and $L_{B \times C/A} \to L_{B/A} \oplus L_{C/A}$ are quasi-isomorphism in $D(B \times C)$ (Lemma \ref{lemma-cotangent-complex-product}) this induces the desired distinguished triangle in $D(C)$ by tensoring with the flat ring map $B \times C \to C$.


cotangent-remark-explicit-map

Let $A \to B \to C$ be ring maps with $B \to C$ injective. Recall the notation $P_\bullet$, $Q_\bullet$, $\overline{Q}_\bullet$ of Remark \ref{remark-triangle}. Let $R_\bullet$ be the standard resolution of $C$ over $B$. In this remark we explain how to get the canonical identification of $\Omega_{\overline{Q}_\bullet/B} \otimes_{\overline{Q}_\bullet} C$ with $L_{C/B} = \Omega_{R_\bullet/B} \otimes_{R_\bullet} C$. Let $S_\bullet \to B$ be the standard resolution of $B$ over $B$. Note that the functoriality map $S_\bullet \to R_\bullet$ identifies $R_n$ as a polynomial algebra over $S_n$ because $B \to C$ is injective. For example in degree $0$ we have the map $B[B] \to B[C]$, in degree $1$ the map $B[B[B]] \to B[B[C]]$, and so on. Thus $\overline{R}_\bullet = R_\bullet \otimes_{S_\bullet} B$ is a simplicial polynomial algebra over $B$ as well and it follows (as in Remark \ref{remark-triangle}) from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-O-homology-qis} that $\overline{R}_\bullet \to C$ is a resolution. Since we have a commutative diagram $$ \xymatrix{ Q_\bullet \ar[r] & R_\bullet \\ P_\bullet \ar[u] \ar[r] & S_\bullet \ar[u] \ar[r] & B } $$ we obtain a canonical map $\overline{Q}_\bullet = Q_\bullet \otimes_{P_\bullet} B \to \overline{R}_\bullet$. Thus the maps $$ L_{C/B} = \Omega_{R_\bullet/B} \otimes_{R_\bullet} C \longrightarrow \Omega_{\overline{R}_\bullet/B} \otimes_{\overline{R}_\bullet} C \longleftarrow \Omega_{\overline{Q}_\bullet/B} \otimes_{\overline{Q}_\bullet} C $$ are quasi-isomorphisms (Remark \ref{remark-resolution}) and composing one with the inverse of the other gives the desired identification.


cotangent-remark-make-map

Let $A \to B$ be a ring map.  Working on $\mathcal{C}_{B/A}$ as in Section \ref{section-compute-L-pi-shriek} let $\mathcal{J} \subset \mathcal{O}$ be the kernel of $\mathcal{O} \to \underline{B}$. Note that $L\pi_!(\mathcal{J}) = 0$ by Lemma \ref{lemma-apply-O-B-comparison}. Set $\Omega =  \Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B}$ so that $L_{B/A} = L\pi_!(\Omega)$ by Lemma \ref{lemma-compute-cotangent-complex}. It follows that $L\pi_!(\mathcal{J} \to \Omega) = L\pi_!(\Omega) = L_{B/A}$. Thus, for any object $U = (P \to B)$ of $\mathcal{C}_{B/A}$ we obtain a map \begin{equation} \label{equation-comparison-map-A} (J \to \Omega_{P/A} \otimes_P B) \longrightarrow L_{B/A} \end{equation} where $J = \Ker(P \to B)$ in $D(A)$, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-map-evaluation-to-derived}. Continuing in this manner, note that $L\pi_!(\mathcal{J} \otimes_\mathcal{O}^\mathbf{L} \underline{B}) = L\pi_!(\mathcal{J}) = 0$ by Lemma \ref{lemma-O-homology-B-homology}. Since $\text{Tor}_0^\mathcal{O}(\mathcal{J}, \underline{B}) = \mathcal{J}/\mathcal{J}^2$ the spectral sequence $$ H_p(\mathcal{C}_{B/A}, \text{Tor}_q^\mathcal{O}(\mathcal{J}, \underline{B})) \Rightarrow  H_{p + q}(\mathcal{C}_{B/A}, \mathcal{J} \otimes_\mathcal{O}^\mathbf{L} \underline{B}) = 0 $$ (dual of Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}) implies that $H_0(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) = 0$ and $H_1(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) = 0$. It follows that the complex of $\underline{B}$-modules $\mathcal{J}/\mathcal{J}^2 \to \Omega$ satisfies $\tau_{\geq -1}L\pi_!(\mathcal{J}/\mathcal{J}^2 \to \Omega) = \tau_{\geq -1}L_{B/A}$. Thus, for any object $U = (P \to B)$ of $\mathcal{C}_{B/A}$ we obtain a map \begin{equation} \label{equation-comparison-map} (J/J^2 \to \Omega_{P/A} \otimes_P B) \longrightarrow \tau_{\geq -1}L_{B/A} \end{equation} in $D(B)$, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-map-evaluation-to-derived}.


cotangent-remark-explicit-comparison-map

We can make the comparison map of Lemma \ref{lemma-relation-with-naive-cotangent-complex} explicit in the following way. Let $P_\bullet$ be the standard resolution of $B$ over $A$. Let $I = \Ker(A[B] \to B)$. Recall that $P_0 = A[B]$. The map of the lemma is given by the commutative diagram $$ \xymatrix{ L_{B/A} \ar[d] & \ldots \ar[r] & \Omega_{P_2/A} \otimes_{P_2} B \ar[r] \ar[d] & \Omega_{P_1/A} \otimes_{P_1} B \ar[r] \ar[d] & \Omega_{P_0/A} \otimes_{P_0} B \ar[d] \\ \NL_{B/A} & \ldots \ar[r] & 0 \ar[r] &  I/I^2 \ar[r] & \Omega_{P_0/A} \otimes_{P_0} B } $$ We construct the downward arrow with target $I/I^2$ by sending $\text{d}f \otimes b$ to the class of $(d_0(f) - d_1(f))b$ in $I/I^2$. Here $d_i : P_1 \to P_0$, $i = 0, 1$ are the two face maps of the simplicial structure. This makes sense as $d_0 - d_1$ maps $P_1$ into $I = \Ker(P_0 \to B)$. We omit the verification that this rule is well defined. Our map is compatible with the differential $\Omega_{P_1/A} \otimes_{P_1} B \to \Omega_{P_0/A} \otimes_{P_0} B$ as this differential maps $\text{d}f \otimes b$ to $\text{d}(d_0(f) - d_1(f)) \otimes b$. Moreover, the differential $\Omega_{P_2/A} \otimes_{P_2} B \to \Omega_{P_1/A} \otimes_{P_1} B$ maps $\text{d}f \otimes b$ to $\text{d}(d_0(f) - d_1(f) + d_2(f)) \otimes b$ which are annihilated by our downward arrow. Hence a map of complexes. We omit the verification that this is the same as the map of Lemma \ref{lemma-relation-with-naive-cotangent-complex}.


cotangent-remark-surjection

Adopt notation as in Remark \ref{remark-make-map}. The arguments given there show that the differential $$ H_2(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) \longrightarrow H_0(\mathcal{C}_{B/A}, \text{Tor}_1^\mathcal{O}(\mathcal{J}, \underline{B})) $$ of the spectral sequence is an isomorphism. Let $\mathcal{C}'_{B/A}$ denote the full subcategory of $\mathcal{C}_{B/A}$ consisting of surjective maps $P \to B$. The agreement of the cotangent complex with the naive cotangent complex (Lemma \ref{lemma-relation-with-naive-cotangent-complex}) shows that we have an exact sequence of sheaves $$ 0 \to \underline{H_1(L_{B/A})} \to \mathcal{J}/\mathcal{J}^2 \xrightarrow{\text{d}} \Omega \to \underline{H_2(L_{B/A})} \to 0 $$ on $\mathcal{C}'_{B/A}$. It follows that $\Ker(d)$ and $\Coker(d)$ on the whole category $\mathcal{C}_{B/A}$ have vanishing higher homology groups, since these are computed by the homology groups of constant simplicial abelian groups by Lemma \ref{lemma-identify-pi-shriek}. Hence we conclude that $$ H_n(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) \to H_n(L_{B/A}) $$ is an isomorphism for all $n \geq 2$. Combined with the remark above we obtain the formula $H_2(L_{B/A}) = H_0(\mathcal{C}_{B/A}, \text{Tor}_1^\mathcal{O}(\mathcal{J}, \underline{B}))$.


cotangent-remark-first-homology-symmetric-power

In the situation of Lemma \ref{lemma-vanishing-symmetric-powers} one can show that $H_k(\mathcal{C}, \text{Sym}^k(\mathcal{F})) = \wedge^k_B(H_1(\mathcal{C}, \mathcal{F}))$. Namely, it can be deduced from the proof that $H_k(\mathcal{C}, \text{Sym}^k(\mathcal{F}))$ is the $S_k$-coinvariants of $$ H^{-k}(L\pi_!(\mathcal{F}) \otimes_B^\mathbf{L} L\pi_!(\mathcal{F}) \otimes_B^\mathbf{L} \ldots \otimes_B^\mathbf{L} L\pi_!(\mathcal{F})) = H_1(\mathcal{C}, \mathcal{F})^{\otimes k} $$ Thus our claim is that this action is given by the usual action of $S_k$ on the tensor product multiplied by the sign character. To prove this one has to work through the sign conventions in the definition of the total complex associated to a multi-complex. We omit the verification.


cotangent-remark-elucidate-ss

In the situation of Theorem \ref{theorem-quillen-spectral-sequence} let $I = \Ker(A \to B)$. Then $H^{-1}(L_{B/A}) = H_1(\mathcal{C}_{B/A}, \Omega) = I/I^2$, see Lemma \ref{lemma-surjection}. Hence $H_k(\mathcal{C}_{B/A}, \text{Sym}^k(\Omega)) = \wedge^k_B(I/I^2)$ by Remark \ref{remark-first-homology-symmetric-power}. Thus the $E_1$-page looks like $$ \begin{matrix} B \\ 0 \\ 0 & I/I^2 \\ 0 & H^{-2}(L_{B/A}) \\ 0 & H^{-3}(L_{B/A}) & \wedge^2(I/I^2) \\ 0 & H^{-4}(L_{B/A}) & H_3(\mathcal{C}_{B/A}, \text{Sym}^2(\Omega)) \\ 0 & H^{-5}(L_{B/A}) & H_4(\mathcal{C}_{B/A}, \text{Sym}^2(\Omega)) & \wedge^3(I/I^2) \end{matrix} $$ with horizontal differential. Thus we obtain edge maps $\text{Tor}_i^A(B, B) \to H^{-i}(L_{B/A})$, $i > 0$ and $\wedge^i_B(I/I^2) \to \text{Tor}_i^A(B, B)$. Finally, we have $\text{Tor}_1^A(B, B) = I/I^2$ and there is a five term exact sequence $$ \text{Tor}_3^A(B, B) \to H^{-3}(L_{B/A}) \to \wedge^2_B(I/I^2) \to \text{Tor}_2^A(B, B) \to H^{-2}(L_{B/A}) \to 0 $$ of low degree terms.


cotangent-remark-elucidate-degree-two

Let $A \to B$ be a ring map. Let $P_\bullet$ be a resolution of $B$ over $A$ (Remark \ref{remark-resolution}). Set $J_n = \Ker(P_n \to B)$. Note that $$ \text{Tor}_2^{P_n}(B, B) =  \text{Tor}_1^{P_n}(J_n, B) = \Ker(J_n \otimes_{P_n} J_n \to J_n^2). $$ Hence $H_2(L_{B/A})$ is canonically equal to $$ \Coker(\text{Tor}_2^{P_1}(B, B) \to \text{Tor}_2^{P_0}(B, B)) $$ by Remark \ref{remark-surjection}. To make this more explicit we choose $P_2$, $P_1$, $P_0$ as in Example \ref{example-resolution-length-two}. We claim that $$ \text{Tor}_2^{P_1}(B, B) = \wedge^2(\bigoplus\nolimits_{t \in T} B)\ \oplus \ \bigoplus\nolimits_{t \in T} J_0\ \oplus \ \text{Tor}_2^{P_0}(B, B) $$ Namely, the basis elements $x_t \wedge x_{t'}$ of the first summand corresponds to the element $x_t \otimes x_{t'} - x_{t'} \otimes x_t$ of $J_1 \otimes_{P_1} J_1$. For $f \in J_0$ the element $x_t \otimes f$ of the second summand corresponds to the element $x_t \otimes s_0(f) - s_0(f) \otimes x_t$ of $J_1 \otimes_{P_1} J_1$. Finally, the map $\text{Tor}_2^{P_0}(B, B) \to \text{Tor}_2^{P_1}(B, B)$ is given by $s_0$. The map $d_0 - d_1 : \text{Tor}_2^{P_1}(B, B) \to \text{Tor}_2^{P_0}(B, B)$ is zero on the last summand, maps $x_t \otimes f$ to $f \otimes f_t - f_t \otimes f$, and maps $x_t \wedge x_{t'}$ to $f_t \otimes f_{t'} - f_{t'} \otimes f_t$. All in all we conclude that there is an exact sequence $$ \wedge^2_B(J_0/J_0^2) \to \text{Tor}_2^{P_0}(B, B) \to H^{-2}(L_{B/A}) \to 0 $$ In this way we obtain a direct proof of a consequence of Quillen's spectral sequence discussed in Remark \ref{remark-elucidate-ss}.


cotangent-remark-functoriality-lichtenbaum-schlessinger

Consider a commutative square $$ \xymatrix{ A' \ar[r] & B' \\ A \ar[u] \ar[r] & B \ar[u] } $$ of ring maps. Choose a factorization $$ \xymatrix{ A' \ar[r] & P' \ar[r] & B' \\ A \ar[u] \ar[r] & P \ar[u] \ar[r] & B \ar[u] } $$ with $P$ a polynomial algebra over $A$ and $P'$ a polynomial algebra over $A'$. Choose generators $f_t$, $t \in T$ for $\Ker(P \to B)$. For $t \in T$ denote $f'_t$ the image of $f_t$ in $P'$. Choose $f'_s \in P'$ such that the elements $f'_t$ for $t \in T' = T \amalg S$ generate the kernel of $P' \to B'$. Set $F = \bigoplus_{t \in T} P$ and $F' = \bigoplus_{t' \in T'} P'$. Let $Rel = \Ker(F \to P)$ and $Rel' = \Ker(F' \to P')$ where the maps are given by multiplication by $f_t$, resp.\ $f'_t$ on the coordinates. Finally, set $TrivRel$, resp.\ $TrivRel'$ equal to the submodule of $Rel$, resp.\ $TrivRel$ generated by the elements $(\ldots, f_{t'}, 0, \ldots, 0, -f_t, 0, \ldots)$ for $t, t' \in T$, resp.\ $T'$. Having made these choices we obtain a canonical commutative diagram $$ \xymatrix{ L' : & Rel'/TrivRel' \ar[r] & F' \otimes_{P'} B' \ar[r] & \Omega_{P'/A'} \otimes_{P'} B' \\ L : \ar[u] & Rel/TrivRel \ar[r] \ar[u] & F \otimes_P B \ar[r] \ar[u] & \Omega_{P/A} \otimes_P B \ar[u] } $$ Moreover, tracing through the choices made in the proof of Lemma \ref{lemma-compare-higher} the reader sees that one obtains a commutative diagram $$ \xymatrix{ L_{B'/A'} \ar[r] & L' \\ L_{B/A} \ar[r] \ar[u] & L \ar[u] } $$


cotangent-remark-map-sections-over-U

It is clear from the proof of Lemma \ref{lemma-compute-L-morphism-sheaves-rings} that for any $U \in \Ob(\mathcal{C})$ there is a canonical map $L_{\mathcal{B}(U)/\mathcal{A}(U)} \to L_{\mathcal{B}/\mathcal{A}}(U)$ of complexes of $\mathcal{B}(U)$-modules. Moreover, these maps are compatible with restriction maps and the complex $L_{\mathcal{B}/\mathcal{A}}$ is the sheafification of the rule $U \mapsto L_{\mathcal{B}(U)/\mathcal{A}(U)}$.


cotangent-remark-compute-L-pi-shriek

In the situation above, for every $U \subset X$ open let $P_{\bullet, U}$ be the standard resolution of $\mathcal{O}_X(U)$ over $\Lambda$. Set $\mathbf{A}_{n, U} = \Spec(P_{n, U})$. Then $\mathbf{A}_{\bullet, U}$ is a cosimplicial object of the fibre category $\mathcal{C}_{\mathcal{O}_X(U)/\Lambda}$ of $\mathcal{C}_{X/\Lambda}$ over $U$. Moreover, as discussed in Remark \ref{remark-resolution} we have that $\mathbf{A}_{\bullet, U}$ is a cosimplicial object of $\mathcal{C}_{\mathcal{O}_X(U)/\Lambda}$ as in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}. Since the construction $U \mapsto \mathbf{A}_{\bullet, U}$ is functorial in $U$, given any (abelian) sheaf $\mathcal{F}$ on $\mathcal{C}_{X/\Lambda}$ we obtain a complex of presheaves $$ U \longmapsto \mathcal{F}(\mathbf{A}_{\bullet, U}) $$ whose cohomology groups compute the homology of $\mathcal{F}$ on the fibre category. We conclude by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compute-left-derived-pi-shriek} that the sheafification computes $L_n\pi_!(\mathcal{F})$. In other words, the complex of sheaves whose term in degree $-n$ is the sheafification of $U \mapsto \mathcal{F}(\mathbf{A}_{n, U})$ computes $L\pi_!(\mathcal{F})$.


cotangent-remark-compute-L-pi-shriek-spaces

In the situation above, for every object $U \to X$ of $X_\etale$ let $P_{\bullet, U}$ be the standard resolution of $\mathcal{O}_X(U)$ over $\Lambda$. Set $\mathbf{A}_{n, U} = \Spec(P_{n, U})$. Then $\mathbf{A}_{\bullet, U}$ is a cosimplicial object of the fibre category $\mathcal{C}_{\mathcal{O}_X(U)/\Lambda}$ of $\mathcal{C}_{X/\Lambda}$ over $U$. Moreover, as discussed in Remark \ref{remark-resolution} we have that $\mathbf{A}_{\bullet, U}$ is a cosimplicial object of $\mathcal{C}_{\mathcal{O}_X(U)/\Lambda}$ as in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}. Since the construction $U \mapsto \mathbf{A}_{\bullet, U}$ is functorial in $U$, given any (abelian) sheaf $\mathcal{F}$ on $\mathcal{C}_{X/\Lambda}$ we obtain a complex of presheaves $$ U \longmapsto \mathcal{F}(\mathbf{A}_{\bullet, U}) $$ whose cohomology groups compute the homology of $\mathcal{F}$ on the fibre category. We conclude by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compute-left-derived-pi-shriek} that the sheafification computes $L_n\pi_!(\mathcal{F})$. In other words, the complex of sheaves whose term in degree $-n$ is the sheafification of $U \mapsto \mathcal{F}(\mathbf{A}_{n, U})$ computes $L\pi_!(\mathcal{F})$.


spaces-cohomology-remark-variant

In Lemmas \ref{lemma-check-separated-dvr} and \ref{lemma-check-proper-dvr} it suffices to consider complete discrete valuation rings. To be precise in Lemma \ref{lemma-check-separated-dvr} we can replace condition (3) by the following condition: Given any commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a complete discrete valuation ring with fraction field $K$ there exists at most one dotted arrow making the diagram commute. Namely, given any diagram as in Lemma \ref{lemma-check-separated-dvr} (3) the completion $A^\wedge$ is a discrete valuation ring (More on Algebra, Lemma \ref{more-algebra-lemma-completion-dvr}) and the uniqueness of the arrow $\Spec(A^\wedge) \to X$ implies the uniqueness of the arrow $\Spec(A) \to X$ for example by Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc}. Similarly in Lemma \ref{lemma-check-proper-dvr} we can replace condition (3) by the following condition: Given any commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] & Y } $$ where $A$ is a complete discrete valuation ring with fraction field $K$ there exists an extension $A \subset A'$ of complete discrete valuation rings inducing a fraction field extension $K \subset K'$ such that there exists a unique arrow $\Spec(A') \to X$ making the diagram $$ \xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y } $$ commute. Namely, given any diagram as in Lemma \ref{lemma-check-proper-dvr} part (3) the existence of any commutative diagram $$ \xymatrix{ \Spec(L) \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(B) \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y } $$ for {\it any} extension $A \subset B$ of discrete valuation rings will imply there exists an arrow $\Spec(A) \to X$ fitting into the diagram. This was shown in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-push-down-solution}. In fact, it follows from these considerations that it suffices to look for dotted arrows in diagrams for any class of discrete valuation rings such that, given any discrete valuation ring, there is an extension of it that is in the class. For example, we could take complete discrete valuation rings with algebraically closed residue field.


artin-remark-deformation-category-implies

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be fibred in groupoids over $(\Sch/S)_{fppf}$. Let $k$ be a field of finite type over $S$ and $x_0$ an object of $\mathcal{X}$ over $k$. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be as in (\ref{equation-predeformation-category}). If $\mathcal{F}$ is a deformation category, i.e., if $\mathcal{F}$ satisfies the Rim-Schlessinger condition (RS), then we see that $\mathcal{F}$ satisfies Schlessinger's conditions (S1) and (S2) by Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-implies-S1-S2}. Let $\overline{\mathcal{F}}$ be the functor of isomorphism classes, see Formal Deformation Theory, Remarks \ref{formal-defos-remarks-cofibered-groupoids} (\ref{formal-defos-item-associated-functor-isomorphism-classes}). Then $\overline{\mathcal{F}}$ satisfies (S1) and (S2) as well, see Formal Deformation Theory, Lemma \ref{formal-defos-lemma-S1-S2-associated-functor}. This holds in particular in the situation of Lemma \ref{lemma-deformation-category}.


artin-remark-formal-objects-match

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\xi = (R, \xi_n, f_n)$ be a formal object. Set $k = R/\mathfrak m$ and $x_0 = \xi_1$. The formal object $\xi$ defines a formal object $\xi$ of the predeformation category $\mathcal{F}_{\mathcal{X}, k, x_0}$. This follows immediately from Definition \ref{definition-formal-objects} above, Formal Deformation Theory, Definition \ref{formal-defos-definition-formal-objects}, and our construction of the predeformation category $\mathcal{F}_{\mathcal{X}, k, x_0}$ in Section \ref{section-predeformation-categories}.


artin-remark-strong-effectiveness

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume we have \begin{enumerate} \item an affine open $\Spec(\Lambda) \subset S$, \item an inverse system $(R_n)$ of $\Lambda$-algebras with surjective transition maps whose kernels are locally nilpotent, \item a system $(\xi_n)$ of objects of $\mathcal{X}$ lying over the system $(\Spec(R_n))$. \end{enumerate} In this situation, set $R = \lim R_n$. We say that $(\xi_n)$ is {\it effective} if there exists an object $\xi$ of $\mathcal{X}$ over $\Spec(R)$ whose restriction to $\Spec(R_n)$ gives the system $(\xi_n)$.


artin-remark-trade-openness-versality-diagonal-with-strong-effectiveness

There is a way to deduce openness of versality of the diagonal of an category fibred in groupoids from a strong formal effectiveness axiom. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume \begin{enumerate} \item $\Delta_\Delta : \mathcal{X} \to \mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}$ is representable by algebraic spaces, \item $\mathcal{X}$ has (RS*), \item $\mathcal{X}$ is limit preserving, \item given an inverse system $(R_n)$ of $S$-algebras as in Remark \ref{remark-strong-effectiveness} where $\Ker(R_m \to R_n)$ is an ideal of square zero for all $m \geq n$ the functor $$ \mathcal{X}_{\Spec(\lim R_n)} \longrightarrow \lim_n \mathcal{X}_{\Spec(R_n)} $$ is fully faithful. \end{enumerate} Then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ satisfies openness of versality. This follows by applying Lemma \ref{lemma-SGE-implies-openness-versality} to fibre products of the form $\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, y} (\Sch/V)_{fppf}$ for any affine scheme $V$ locally of finite presentation over $S$ and object $y$ of $\mathcal{X} \times \mathcal{X}$ over $V$. If we ever need this, we will change this remark into a lemma and provide a detailed proof.


artin-remark-functoriality

Assumptions and notation as in Lemma \ref{lemma-properties-lift-RS-star}. Suppose $A \to B$ is a ring map and $y = x|_{\Spec(B)}$. Let $M \in \text{Mod}_A$, $N \in \text{Mod}_B$ and let $M \to N$ an $A$-linear map. Then there are canonical maps $\text{Inf}_x(M) \to \text{Inf}_y(N)$ and $T_x(M) \to T_y(N)$ simply because there is a pullback functor $$ \textit{Lift}(x, A[M]) \to \textit{Lift}(y, B[N]) $$ coming from the ring map $A[M] \to B[N]$. Similarly, given a morphism of deformation situations $(y, B' \to B) \to (x, A' \to A)$ we obtain a pullback functor $\textit{Lift}(x, A') \to \textit{Lift}(y, B')$. Since the construction of the action, the addition, and the scalar multiplication on $\text{Inf}_x$ and $T_x$ use only morphisms in the categories of lifts (see proof of Formal Deformation Theory, Lemma \ref{formal-defos-lemma-linear-functor}) we see that the constructions above are functorial. In other words we obtain $A$-linear maps $$ \text{Inf}_x(M) \to \text{Inf}_y(N) \quad\text{and}\quad T_x(M) \to T_y(N) $$ such that the diagrams $$ \vcenter{ \xymatrix{ \text{Inf}_y(J) \ar[r] & \text{Inf}(y'/y) \\ \text{Inf}_x(I) \ar[r] \ar[u] & \text{Inf}(x'/x) \ar[u] } } \quad\text{and}\quad \vcenter{ \xymatrix{ T_y(J) \times \text{Lift}(y, B') \ar[r] & \text{Lift}(y, B') \\ T_x(I) \times \text{Lift}(x, A') \ar[r] \ar[u] & \text{Lift}(x, A') \ar[u] } } $$ commute. Here $I = \Ker(A' \to A)$, $J = \Ker(B' \to B)$, $x'$ is a lift of $x$ to $A'$ (which may not always exist) and $y' = x'|_{\Spec(B')}$.


artin-remark-automorphisms

Assumptions and notation as in Lemma \ref{lemma-properties-lift-RS-star}. Let $x', x''$ be lifts of $x$ to $A'$. Then we have a composition map $$ \text{Inf}(x'/x) \times \Mor_{\textit{Lift}(x, A')}(x', x'') \times \text{Inf}(x''/x) \longrightarrow \Mor_{\textit{Lift}(x, A')}(x', x''). $$ Since $\textit{Lift}(x, A')$ is a groupoid, if $\Mor_{\textit{Lift}(x, A')}(x', x'')$ is nonempty, then this defines a simply transitive left action of $\text{Inf}(x'/x)$ on $\Mor_{\textit{Lift}(x, A')}(x', x'')$ and a simply transitive right action by $\text{Inf}(x''/x)$. Now the lemma says that $\text{Inf}(x'/x) = \text{Inf}_x(I) = \text{Inf}(x''/x)$. We claim that the two actions described above agree via these identifications. Namely, either $x' \not \cong x''$ in which the claim is clear, or $x' \cong x''$ and in that case we may assume that $x'' = x'$ in which case the result follows from the fact that $\text{Inf}(x'/x)$ is commutative. In particular, we obtain a well defined action $$ \text{Inf}_x(I) \times \Mor_{\textit{Lift}(x, A')}(x', x'') \longrightarrow \Mor_{\textit{Lift}(x, A')}(x', x'') $$ which is simply transitive as soon as $\Mor_{\textit{Lift}(x, A')}(x', x'')$ is nonempty.


artin-remark-short-exact-sequence-thickenings

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $A$ be an $S$-algebra. There is a notion of a {\it short exact sequence} $$ (x, A_1' \to A) \to (x, A_2' \to A) \to (x, A_3' \to A) $$ of deformation situations: we ask the corresponding maps between the kernels $I_i = \Ker(A_i' \to A)$ give a short exact sequence $$ 0 \to I_3 \to I_2 \to I_1 \to 0 $$ of $A$-modules. Note that in this case the map $A_3' \to A_1'$ factors through $A$, hence there is a canonical isomorphism $A_1' = A[I_1]$.


artin-remark-compare-deformation-spaces

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$ has (RS*). Let $k$ be a field of finite type over $S$ and let $x_0$ be an object of $\mathcal{X}$ over $\Spec(k)$. Then we have equalities of $k$-vector spaces $$ T\mathcal{F}_{\mathcal{X}, k, x_0} = T_{x_0}(k) \quad\text{and}\quad \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \text{Inf}_{x_0}(k) $$ where the spaces on the left hand side of the equality signs are given in (\ref{equation-tangent-space}) and (\ref{equation-infinitesimal-automorphisms}) and the spaces on the right hand side are given by Lemma \ref{lemma-properties-lift-RS-star}.


artin-remark-canonical-element

Assumptions and notation as in Lemma \ref{lemma-properties-lift-RS-star}. Choose an affine open $\Spec(\Lambda) \subset S$ such that $\Spec(A) \to S$ corresponds to a ring map $\Lambda \to A$. Consider the ring map $$ A \longrightarrow A[\Omega_{A/\Lambda}], \quad a \longmapsto (a, \text{d}_{A/\Lambda}(a)) $$ Pulling back $x$ along the corresponding morphism $\Spec(A[\Omega_{A/\Lambda}]) \to \Spec(A)$ we obtain a deformation $x_{can}$ of $x$ over $A[\Omega_{A/\Lambda}]$. We call this the {\it canonical element} $$ x_{can} \in T_x(\Omega_{A/\Lambda}) = \text{Lift}(x, A[\Omega_{A/\Lambda}]). $$ Next, assume that $\Lambda$ is Noetherian and $\Lambda \to A$ is of finite type. Let $k = \kappa(\mathfrak p)$ be a residue field at a finite type point $u_0$ of $U = \Spec(A)$. Let $x_0 = x|_{u_0}$. By (RS*) and the fact that $A[k] = A \times_k k[k]$ the space $T_x(k)$ is the tangent space to the deformation functor $\mathcal{F}_{\mathcal{X}, k, x_0}$. Via $$ T\mathcal{F}_{U, k, u_0} = \text{Der}_\Lambda(A, k) = \Hom_A(\Omega_{A/\Lambda}, k) $$ (see Formal Deformation Theory, Example \ref{formal-defos-example-tangent-space-prorepresentable-functor}) and functoriality of $T_x$ the canonical element produces the map on tangent spaces induced by the object $x$ over $U$. Namely, $\theta \in T\mathcal{F}_{U, k, u_0}$ maps to $T_x(\theta)(x_{can})$ in $T_x(k) = T\mathcal{F}_{\mathcal{X}, k, x_0}$.


artin-remark-canonical-isomorphism

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$ satisfies condition (RS*). Let $A$ be an $S$-algebra such that $\Spec(A) \to S$ maps into an affine open and let $x, y$ be objects of $\mathcal{X}$ over $\Spec(A)$. Further, let $A \to B$ be a ring map and let $\alpha : x|_{\Spec(B)} \to y|_{\Spec(B)}$ be a morphism of $\mathcal{X}$ over $\Spec(B)$. Consider the ring map $$ B \longrightarrow B[\Omega_{B/A}], \quad b \longmapsto (b, \text{d}_{B/A}(b)) $$ Pulling back $\alpha$ along the corresponding morphism $\Spec(B[\Omega_{B/A}]) \to \Spec(B)$ we obtain a morphism $\alpha_{can}$ between the pullbacks of $x$ and $y$ over $B[\Omega_{B/A}]$. On the other hand, we can pullback $\alpha$ by the morphism $\Spec(B[\Omega_{B/A}]) \to \Spec(B)$ corresponding to the injection of $B$ into the first summand of $B[\Omega_{B/A}]$. By the discussion of Remark \ref{remark-automorphisms} we can take the difference $$ \varphi(x, y, \alpha) = \alpha_{can} - \alpha|_{\Spec(B[\Omega_{B/A}])} \in \text{Inf}_{x|_{\Spec(B)}}(\Omega_{B/A}). $$ We will call this the {\it canonical automorphism}. It depends on all the ingredients $A$, $x$, $y$, $A \to B$ and $\alpha$.


artin-remark-no-fibre-products

The site $(\textit{Noetherian}/S)_\tau$ does not have fibre products. Hence we have to be careful in working with sheaves. For example, the continuous inclusion functor $(\textit{Noetherian}/S)_\tau \to (\Sch/S)_\tau$ does not define a morphism of sites. See Examples, Section \ref{examples-section-sheaves-locally-Noetherian} for an example in case $\tau = fppf$.


artin-remark-G-rings

In particular, we cannot prove that the desired result is true for every Situation \ref{situation-contractions} because we will need to assume the local rings of $S$ are G-rings. If you can prove the result in general or if you have a counter example, please let us know at \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.


artin-remark-how-to-think-compatibility

In Situation \ref{situation-contractions} let $V$ be a locally Noetherian scheme over $S$. Let $(Z, u', \hat x)$ be a triple satisfying (1), (2), and (3) above. We want to explain a way to think about the compatibility condition (4). It will not be mathematically precise as we are going use a fictitious category $\textit{An}_S$ of analytic spaces over $S$ and a fictitious analytification functor $$ \left\{ \begin{matrix} \text{locally Noetherian formal} \\ \text{algebraic spaces over }S \end{matrix} \right\} \longrightarrow \textit{An}_S, \quad\quad Y \longmapsto Y^{an} $$ For example if $Y = \text{Spf}(k[[t]])$ over $S = \Spec(k)$, then $Y^{an}$ should be thought of as an open unit disc. If $Y = \Spec(k)$, then $Y^{an}$ is a single point. The category $\textit{An}_S$ should have open and closed immersions and we should be able to take the open complement of a closed. Given $Y$ the morphism $Y_{red} \to Y$ should induces a closed immersion $Y_{red}^{an} \to Y^{an}$. We set $Y^{rig} = Y^{an} \setminus Y_{red}^{an}$ equal to its open complement. If $Y$ is an algebraic space and if $Z \subset Y$ is closed, then the morphism $Y_{/Z} \to Y$ should induce an open immersion $Y_{/Z}^{an} \to Y^{an}$ which in turn should induce an open immersion $$ can : (Y_{/Z})^{rig} \longrightarrow (Y \setminus Z)^{an} $$ Also, given a formal modification $g : Y' \to Y$ of locally Noetherian formal algebraic spaces, the induced morphism $g^{rig} : (Y')^{rig} \to Y^{rig}$ should be an isomorphism. Given $\text{An}_S$ and the analytification functor, we can consider the requirement that $$ \xymatrix{ (V_{/Z})^{rig} \ar[rr]_{can} \ar[d]_{(g^{rig})^{-1} \circ \hat x^{an}} & & (V \setminus Z)^{an} \ar[d]^{(u')^{an}} \\ (X'_{/T'})^{rig} \ar[rr]^{can} & & (X' \setminus T')^{an} } $$ commutes. This makes sense as $g^{rig} : (X'_{T'})^{rig} \to W^{rig}$ is an isomorphism and $U' = X' \setminus T'$. Finally, under some assumptions of faithfulness of the analytification functor, this requirement will be equivalent to the compatibility condition formulated above. We hope this will motivate the reader to think of the compatibility of $u'$ and $\hat x$ as the requirement that some maps be equal, rather than asking for the existence of a certain commutative diagram.


artin-remark-diagonal

In Situation \ref{situation-contractions}. Let $V$ be a locally Noetherian scheme over $S$. Let $(Z_i, u'_i, \hat x_i) \in F(V)$ for $i = 1, 2$. Let $V'_i \to V$, $\hat x'_i$ and $x'_i$ witness the compatibility between $u'_i$ and $\hat x_i$ for $i = 1, 2$. \medskip\noindent Set $V' = V'_1 \times_V V'_2$. Let $E' \to V'$ denote the equalizer of the morphisms $$ V' \to V'_1 \xrightarrow{x'_1} X' \quad\text{and}\quad V' \to V'_2 \xrightarrow{x'_2} X' $$ Set $Z = Z_1 \cap Z_2$. Let $E_W \to V_{/Z}$ be the equalizer of the morphisms $$ V_{/Z} \to V_{/Z_1} \xrightarrow{\hat x_1} W \quad\text{and}\quad V_{/Z} \to V_{/Z_2} \xrightarrow{\hat x_2} W $$ Observe that $E' \to V$ is separated and locally of finite type and that $E_W$ is a locally Noetherian formal algebraic space separated over $V$. The compatibilities between the various morphisms involved show that \begin{enumerate} \item $\Im(E' \to V) \cap (Z_1 \cup Z_2)$ is contained in $Z = Z_1 \cap Z_2$,  \item the morphism $E' \times_V (V \setminus Z) \to V \setminus Z$ is a monomorphism and is equal to the equalizer of the restrictions of $u'_1$ and $u'_2$ to $V \setminus (Z_1 \cup Z_2)$, \item the morphism $E'_{/Z} \to V_{/Z}$ factors through $E_W$ and the diagram $$ \xymatrix{ E'_{/Z} \ar[r] \ar[d] & X'_{/T'} \ar[d]^g \\ E_W \ar[r] & W } $$ is cartesian. In particular, the morphism $E'_{/Z} \to E_W$ is a formal modification as the base change of $g$, \item $E'$, $(E' \to V)^{-1}Z$, and $E'_{/Z} \to E_W$ is a triple as in Situation \ref{situation-contractions} with base scheme the locally Noetherian scheme $V$, \item given a morphism $\varphi : A \to V$ of locally Noetherian schemes, the following are equivalent \begin{enumerate} \item $(Z_1, u'_1, \hat x_1)$ and $(Z_2, u'_2, \hat x_2)$ restrict to the same element of $F(A)$, \item $A \setminus \varphi^{-1}(Z) \to V \setminus Z$ factors through $E' \times_V (V \setminus Z)$ and $A_{/\varphi^{-1}(Z)} \to V_{/Z}$ factors through $E_W$. \end{enumerate} \end{enumerate} We conclude, using Lemmas \ref{lemma-solution} and \ref{lemma-functor-is-solution}, that if there is a solution $E \to V$ for the triple in (4), then $E$ represents $F \times_{\Delta, F \times F} V$ on the category of locally Noetherian schemes over $V$.


artin-remark-separated-needed

The proof of Theorem \ref{theorem-contractions} uses that $X'$ and $W$ are separated over $S$ in two places. First, the proof uses this in showing $\Delta : F \to F \times F$ is representable by algebraic spaces. This use of the assumption can be entirely avoided by proving that $\Delta$ is representable by applying the theorem in the separated case to the triples $E'$, $(E' \to V)^{-1}Z$, and $E'_{/Z} \to E_W$ found in Remark \ref{remark-diagonal} (this is the usual bootstrap procedure for the diagonal). Thus the proof of Lemma \ref{lemma-formal-object-effective} is the only place in our proof of Theorem \ref{theorem-contractions} where we really need to use that $X' \to S$ is separated. The reader checks that we use the assumption only to obtain the morphism $x' : V' \to X'$. The existence of $x'$ can be shown, using results in the literature, if $X' \to S$ is quasi-separated, see More on Morphisms of Spaces, Remark \ref{spaces-more-morphisms-remark-weaken-separation-axioms-question}. We conclude the theorem holds as stated with ``separated'' replaced by ``quasi-separated''. If we ever need this we will precisely state and carefully prove this here.


obsolete-remark-composition-of-adjoints-isomorphic-to-identity

The information which used to be contained in this remark is now subsumed in the combination of Categories, Lemmas \ref{categories-lemma-adjoint-fully-faithful} and \ref{categories-lemma-left-adjoint-composed-fully-faithful}.


obsolete-remark-weak-serre-subcategory

The following remarks are obsolete as they are subsumed in Homology, Lemmas \ref{homology-lemma-biregular-ss-converges} and \ref{homology-lemma-first-quadrant-ss}. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} \subset \mathcal{A}$ be a weak Serre subcategory (see Homology, Definition \ref{homology-definition-serre-subcategory}). Suppose that $K^{\bullet, \bullet}$ is a double complex to which Homology, Lemma \ref{homology-lemma-first-quadrant-ss} applies such that for some $r \geq 0$ all the objects ${}'E_r^{p, q}$ belong to $\mathcal{C}$. Then all the cohomology groups $H^n(sK^\bullet)$ belong to $\mathcal{C}$. Namely, the assumptions imply that the kernels and images of ${}'d_r^{p, q}$ are in $\mathcal{C}$. Whereupon we see that each ${}'E_{r + 1}^{p, q}$ is in $\mathcal{C}$. By induction we see that each ${}'E_\infty^{p, q}$ is in $\mathcal{C}$. Hence each $H^n(sK^\bullet)$ has a finite filtration whose subquotients are in $\mathcal{C}$. Using that $\mathcal{C}$ is closed under extensions we conclude that $H^n(sK^\bullet)$ is in $\mathcal{C}$ as claimed. The same result holds for the second spectral sequence associated to $K^{\bullet, \bullet}$. Similarly, if $(K^\bullet, F)$ is a filtered complex to which Homology, Lemma \ref{homology-lemma-biregular-ss-converges} applies and for some $r \geq 0$ all the objects $E_r^{p, q}$ belong to $\mathcal{C}$, then each $H^n(K^\bullet)$ is an object of $\mathcal{C}$.


obsolete-remark-projective-resolution

Let $R$ be a ring. For any set $S$ we let $F(S)$ denote the free $R$-module on $S$. Then any left $R$-module has the following two step resolution $$ F(M \times M) \oplus F(R \times M) \to F(M) \to M \to 0. $$ The first map is given by the rule $$ [m_1, m_2] \oplus [r, m] \mapsto [m_1 + m_2] - [m_1] - [m_2] + [rm] - r[m]. $$


obsolete-remark-section-colimits

This reference/tag used to refer to a Section in the chapter Smoothing Ring Maps, but the material has since been subsumed in Algebra, Section \ref{algebra-section-colimits-flat}.


obsolete-remark-algebra

Let $R$ be a ring. Suppose that we have $F \in R[X, Y]_d$ and $G \in R[X, Y]_e$ such that, setting $S = R[X, Y]/(F)$ we have (1) $S_n$ is finite locally free of rank $d$ for all $n \geq d$, and (2) multiplication by $G$ defines isomorphisms $S_n \to S_{n + e}$ for all $n \geq d$. In this case we may define a finite, locally free $R$-algebra $A$ as follows: \begin{enumerate} \item as an $R$-module $A = S_{ed}$, and \item multiplication $A \times A \to A$ is given by the rule that $H_1 H_2 = H_3$ if and only if $G^d H_3 = H_1 H_2$ in $S_{2ed}$. \end{enumerate} This makes sense because multiplication by $G^d$ induces a bijective map $S_{de} \to S_{2de}$. It is easy to see that this defines a ring structure. Note the confusing fact that the element $G^d$ defines the unit element of the ring $A$.


obsolete-remark-equation-derivatives

This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition \ref{restricted-proposition-approximate} which became unused because of a rearrangement of the material.


obsolete-remark-equation-ci


obsolete-remark-equation-in-ideal


obsolete-remark-equation-derivatives-analogue


obsolete-remark-equation-go-down

This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Lemma \ref{restricted-lemma-lift-approximation} which became unused because of a rearrangement of the material.


obsolete-remark-from-shriek-to-star

Let $U$ be an object of $\mathcal{C}$. For any abelian sheaf $\mathcal{G}$ on $\mathcal{C}/U$ one may wonder whether there is a canonical map $$ c : j_{U!}\mathcal{G} \longrightarrow j_{U*}\mathcal{G} $$ To construct such a thing is the same as constructing a map $j_U^{-1}j_{U!}\mathcal{G} \to \mathcal{G}$. Note that restriction commutes with sheafification. Thus we can use the presheaf of Modules on Sites, Lemma \ref{sites-modules-lemma-extension-by-zero}. Hence it suffices to define for $V/U$ a map $$ \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) \longrightarrow \mathcal{G}(V/U) $$ compatible with restrictions. It looks like we can take the which is zero on all summands except for the one where $\varphi$ is the structure morphism $\varphi_0 : V \to U$ where we take $1$. However, this isn't compatible with restriction mappings: namely, if $\alpha : V' \to V$ is a morphism of $\mathcal{C}$, then denote $V'/U$ the object of $\mathcal{C}/U$ with structure morphism $\varphi'_0 = \varphi_0 \circ \alpha$. We need to check that the diagram $$ \xymatrix{ \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) \ar[d] \ar[r] & \mathcal{G}(V/U) \ar[d] \\ \bigoplus\nolimits_{\varphi' \in \Mor_\mathcal{C}(V', U)} \mathcal{G}(V' \xrightarrow{\varphi'} U) \ar[r] & \mathcal{G}(V'/U) } $$ commutes. The problem here is that there may be a morphism $\varphi : V \to U$ different from $\varphi_0$ such that $\varphi \circ \alpha = \varphi'_0$. Thus the left vertical arrow will send the summand corresponding to $\varphi$ into the summand on which the lower horizontal arrow is equal to $1$ and almost surely the diagram doesn't commute.


obsolete-remark-pullback-K-flat

This remark used to discuss what we know about pullbacks of K-flat complexes being K-flat or not, but is now obsoleted by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-K-flat}.


obsolete-remark-cohomology-topics

This tag used to refer to a section of the chapter on cohomology listing topics to be treated.


obsolete-remark-sites-cohomology-topics


obsolete-remark-V-implies-C

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compare-qc-zar}.


obsolete-remark-V-implies-cohomology

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compare-qc-zar}.


obsolete-remark-induction-step-V-C

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compare-qc-zar}.


obsolete-remark-V-implies-C-etale-fppf

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-fppf-etale}.


obsolete-remark-V-implies-cohomology-etale-fppf

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-fppf-etale}.


obsolete-remark-induction-step-V-C-etale-fppf

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-fppf-etale}.


obsolete-remark-V-implies-C-etale-ph

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-ph-etale}.


obsolete-remark-V-implies-cohomology-etale-ph

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-ph-etale}.


obsolete-remark-V-implies-cohomology-etale-ph-extra

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-cohomology-extra-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-ph-etale}.


obsolete-remark-make-class-zero

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-make-class-zero-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-ph-etale}.


obsolete-remark-induction-step-V-C-etale-ph

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-ph-etale}.


obsolete-remark-V-implies-C-etale-h

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-h-etale}.


obsolete-remark-V-implies-cohomology-etale-h

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-h-etale}.


obsolete-remark-V-implies-cohomology-etale-h-extra

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-cohomology-extra-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-h-etale}.


obsolete-remark-induction-step-V-C-etale-h

This tag used to refer to the special case of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-h-etale}.


obsolete-remark-how-used

This tag used to be in the chapter on \'etale cohomology, but is no longer suitable there because of a reorganization. The content of the tag was the following: \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-when-ctf} can be used to prove that if $f : X \to Y$ is a separated, finite type morphism of schemes and $Y$ is Noetherian, then $Rf_!$ induces a functor $D_{ctf}(X_\etale, \Lambda) \to D_{ctf}(Y_\etale, \Lambda)$. An example of this argument, when $Y$ is the spectrum of a field and $X$ is a curve is given in The Trace Formula, Proposition \ref{trace-proposition-projective-curve-constructible-cohomology}.


obsolete-remark-proof-works-when

This remark used to discuss to what extend the original proof of Lemma \ref{lemma-sheaf-fpqc-quasi-separated} (of December 18, 2009) generalizes.


obsolete-remark-very-reasonable-Zariski-locally-quasi-separated

Very reasonable algebraic spaces form a strictly larger collection than Zariski locally quasi-separated algebraic spaces. Consider an algebraic space of the form $X = [U/G]$ (see Spaces, Definition \ref{spaces-definition-quotient}) where $G$ is a finite group acting without fixed points on a non-quasi-separated scheme $U$. Namely, in this case $U \times_X U = U \times G$ and clearly both projections to $U$ are quasi-compact, hence $X$ is very reasonable. On the other hand, the diagonal $U \times_X U \to U \times U$ is not quasi-compact, hence this algebraic space is not quasi-separated. Now, take $U$ the infinite affine space over a field $k$ of characteristic $\not = 2$ with zero doubled, see Schemes, Example \ref{schemes-example-not-quasi-separated}. Let $0_1, 0_2$ be the two zeros of $U$. Let $G = \{+1, -1\}$, and let $-1$ act by $-1$ on all coordinates, and by switching $0_1$ and $0_2$. Then $[U/G]$ is very reasonable but not Zariski locally quasi-separated (details omitted).


obsolete-remark-different-topologies

We obtain a second topology $\tau_Y$ on $\mathcal{C}_{X/Y}$ by taking the topology inherited from $Y_{Zar}$. There is a third topology $\tau_{X \to Y}$ where a family of morphisms $\{(U_i \to A_i) \to (U \to A)\}$ is a covering if and only if $U = \bigcup U_i$, $V = \bigcup V_i$ and $A_i \cong V_i \times_V A$. This is the topology inherited from the topology on the site $(X/Y)_{Zar}$ whose underlying category is the category of pairs $(U, V)$ as in Lemma \ref{lemma-category-fibred} part (3). The coverings of $(X/Y)_{Zar}$ are families $\{(U_i, V_i) \to (U, V)\}$ such that $U = \bigcup U_i$ and $V = \bigcup V_i$. There are morphisms of topoi $$ \xymatrix{ \Sh(\mathcal{C}_{X/Y}) = \Sh(\mathcal{C}_{X/Y}, \tau_X) & \Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y}) \ar[l] \ar[r] & \Sh(\mathcal{C}_{X/Y}, \tau_Y) } $$ (recall that $\tau_X$ is our ``default'' topology). The pullback functors for these arrows are sheafification and pushforward is the identity on underlying presheaves. The diagram of topoi $$ \xymatrix{ \Sh(X_{Zar}) \ar[d]^f & \Sh(\mathcal{C}_{X/Y}) \ar[l]^\pi & \Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y}) \ar[l] \ar[d] \\ \Sh(Y_{Zar}) & & \Sh(\mathcal{C}_{X/Y}, \tau_Y) \ar[ll] } $$ is {\bf not} commutative. Namely, the pullback of a nonzero abelian sheaf on $Y$ is a nonzero abelian sheaf on $(\mathcal{C}_{X/Y}, \tau_{X \to Y})$, but we can certainly find examples where such a sheaf pulls back to zero on $X$. Note that any presheaf $\mathcal{F}$ on $Y_{Zar}$ gives a sheaf $\underline{\mathcal{F}}$ on $\mathcal{C}_{Y/X}$ by the rule which assigns to $(U \to A/V)$ the set $\mathcal{F}(V)$. Even if $\mathcal{F}$ happens to be a sheaf it isn't true in general that $\underline{\mathcal{F}} = \pi^{-1}f^{-1}\mathcal{F}$. This is related to the noncommutativity of the diagram above, as we can describe $\underline{\mathcal{F}}$ as the pushforward of the pullback of $\mathcal{F}$ to $\Sh(\mathcal{C}_{X/Y}, \tau_{X \to Y})$ via the lower horizontal and right vertical arrows. An example is the sheaf $\underline{\mathcal{O}}_Y$. But what is true is that there is a map $\underline{\mathcal{F}} \to \pi^{-1}f^{-1}\mathcal{F}$ which is transformed (as we shall see later) into an isomorphism after applying $\pi_!$.


obsolete-remark-construction-E

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U$ be another algebraic space over $B$. Denote $q : X \times_B U \to U$ the second projection. Consider the distinguished triangle $$ Lq^*L_{U/B} \to L_{X \times_B U/B} \to E \to Lq^*L_{U/B}[1] $$ of Cotangent, Section \ref{cotangent-section-fibre-product}. For any sheaf $\mathcal{F}$ of $\mathcal{O}_{X \times_B U}$-modules we have the Atiyah class $$ \mathcal{F} \to L_{X \times_B U/B} \otimes_{\mathcal{O}_{X \times_B U}}^\mathbf{L} \mathcal{F}[1] $$ see Cotangent, Section \ref{cotangent-section-atiyah-general}. We can compose this with the map to $E$ and choose a distinguished triangle $$ E(\mathcal{F}) \to \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_{X \times_B U}}^\mathbf{L} E[1] \to E(\mathcal{F})[1] $$ in $D(\mathcal{O}_{X \times_B U})$. By construction the Atiyah class lifts to a map $$ e_\mathcal{F} : E(\mathcal{F}) \longrightarrow Lq^*L_{U/B} \otimes_{\mathcal{O}_{X \times_B U}}^\mathbf{L} \mathcal{F}[1] $$ fitting into a morphism of distinguished triangles $$ \xymatrix{ \mathcal{F} \otimes^\mathbf{L} Lq^*L_{U/B}[1] \ar[r] & \mathcal{F} \otimes^\mathbf{L} L_{X \times_B U/B}[1] \ar[r] & \mathcal{F} \otimes^\mathbf{L} E[1] \\ E(\mathcal{F}) \ar[r] \ar[u]^{e_\mathcal{F}} & \mathcal{F} \ar[r] \ar[u]^{Atiyah} & \mathcal{F} \otimes^\mathbf{L} E[1] \ar[u]^{=} } $$ Given $S, B, X, f, U, \mathcal{F}$ we fix a choice of $E(\mathcal{F})$ and $e_\mathcal{F}$.


obsolete-remark-construction-ob

With notation as in Remark \ref{remark-construction-E} let $i : U \to U'$ be a first order thickening of $U$ over $B$. Let $\mathcal{I} \subset \mathcal{O}_{U'}$ be the quasi-coherent sheaf of ideals cutting out $B$ in $B'$. The fundamental triangle $$ Li^*L_{U'/B} \to L_{U/B} \to L_{U/U'} \to Li^*L_{U'/B}[1] $$ together with the map $L_{U/U'} \to \mathcal{I}[1]$ determine a map $e_{U'} : L_{U/B} \to \mathcal{I}[1]$. Combined with the map $e_\mathcal{F}$ of the previous remark we obtain $$ (\text{id}_\mathcal{F} \otimes Lq^*e_{U'}) \cup e_\mathcal{F} : E(\mathcal{F}) \longrightarrow \mathcal{F} \otimes_{\mathcal{O}_{X \times_B U}} q^*\mathcal{I}[2] $$ (we have also composed with the map from the derived tensor product to the usual tensor product). In other words, we obtain an element $$ \xi_{U'} \in \Ext^2_{\mathcal{O}_{X \times_B U}}( E(\mathcal{F}), \mathcal{F} \otimes_{\mathcal{O}_{X \times_B U}} q^*\mathcal{I}) $$


obsolete-remark-not-true-not-quasi-compact

This remark used to say that it wasn't clear whether the arrows of Chow Homology, Lemma \ref{chow-lemma-cycles-k-group} were isomorphisms in general. However, we've now found a proof of this fact.


obsolete-remark-tangent-spaces

You got here because of a duplicate tag. Please see Formal Deformation Theory, Section \ref{formal-defos-section-tangent-spaces} for the actual content.


obsolete-remark-examples-formal-defos

This tag used to point to a section describing several examples of deformation problems. These now each have their own section. See Deformation Problems, Sections \ref{examples-defos-section-finite-projective-modules}, \ref{examples-defos-section-representations}, \ref{examples-defos-section-continuous-representations}, and \ref{examples-defos-section-graded-algebras}.


stacks-sheaves-remark-ambiguity

We only use this notation when the symbol $\mathcal{X}$ refers to a category fibred in groupoids, and not a scheme, an algebraic space, etc. In this way we will avoid confusion with the small \'etale site of a scheme, or algebraic space which is denoted $X_\etale$ (in which case we use a roman capital instead of a calligraphic one).