stacks-remark-stack-make-small

(Cutting down a ``big'' stack to get a stack.) Let $\mathcal{C}$ be a site. Suppose that $p : \mathcal{S} \to \mathcal{C}$ is functor from a ``big'' category to $\mathcal{C}$, i.e., suppose that the collection of objects of $\mathcal{S}$ forms a proper class. Finally, suppose that $p : \mathcal{S} \to \mathcal{C}$ satisfies conditions (1), (2), (3) of Definition \ref{definition-stack}. In general there is no way to replace $p : \mathcal{S} \to \mathcal{C}$ by a equivalent category such that we obtain a stack. The reason is that it can happen that a fibre categories $\mathcal{S}_U$ may have a proper class of isomorphism classes of objects. On the other hand, suppose that \begin{enumerate} \item[(4)] for every $U \in \Ob(\mathcal{C})$ there exists a set $S_U \subset \Ob(\mathcal{S}_U)$ such that every object of $\mathcal{S}_U$ is isomorphic in $\mathcal{S}_U$ to an element of $S_U$. \end{enumerate} In this case we can find a full subcategory $\mathcal{S}_{small}$ of $\mathcal{S}$ such that, setting $p_{small} = p|_{\mathcal{S}_{small}}$, we have \begin{enumerate} \item[(a)] the functor $p_{small} : \mathcal{S}_{small} \to \mathcal{C}$ defines a stack, and \item[(b)] the inclusion $\mathcal{S}_{small} \to \mathcal{S}$ is fully faithful and essentially surjective. \end{enumerate} (Hint: For every $U \in \Ob(\mathcal{C})$ let $\alpha(U)$ denote the smallest ordinal such that $\Ob(\mathcal{S}_U) \cap V_{\alpha(U)}$ surjects onto the set of isomorphism classes of $\mathcal{S}_U$, and set $\alpha = \sup_{U \in \Ob(\mathcal{C})} \alpha(U)$. Then take $\Ob(\mathcal{S}_{small}) = \Ob(\mathcal{S}) \cap V_\alpha$. For notation used see Sets, Section \ref{sets-section-sets-hierarchy}.)


stacks-remarks-definition-descent-datum

Two remarks on Definition \ref{definition-descent-data} are in order. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $\{f_i : U_i \to U\}_{i \in I}$, and $(X_i, \varphi_{ij})$ be as in Definition \ref{definition-descent-data}. \begin{enumerate} \item There is a diagonal morphism $\Delta : U_i \to U_i \times_U U_i$. We can pull back $\varphi_{ii}$ via this morphism to get an automorphism $\Delta^\ast \varphi_{ii} \in \text{Aut}_{U_i}(x_i)$. On pulling back the cocycle condition for the triple $(i, i, i)$ by $\Delta_{123} : U_i \to U_i \times_U U_i \times_U U_i$ we deduce that $\Delta^\ast \varphi_{ii} \circ \Delta^\ast \varphi_{ii} = \Delta^\ast \varphi_{ii}$; thus $\Delta^\ast \varphi_{ii} = \text{id}_{x_i}$. \item There is a morphism $\Delta_{13}: U_i \times_U U_j \to U_i \times_U U_j \times_U U_i$ and we can pull back the cocycle condition for the triple $(i, j, i)$ to get the identity $(\sigma^\ast \varphi_{ji}) \circ \varphi_{ij} = \text{id}_{\text{pr}_0^\ast x_i}$, where $\sigma : U_i \times_U U_j \to U_j \times_U U_i$ is the switching morphism. \end{enumerate}


spaces-simplicial-remark-augmentation-site

In Situation \ref{situation-simplicial-site} an {\it augmentation $a_0$ towards a site $\mathcal{D}$} will mean \begin{enumerate} \item[(A)] $a_0 : \mathcal{C}_0 \to \mathcal{D}$ is a morphism of sites given by a continuous functor $u_0 : \mathcal{D} \to \mathcal{C}_0$ such that for all $\varphi, \psi : [0] \to [n]$ we have $u_\varphi \circ u_0 = u_\psi \circ u_0$. \item[(B)] $a_0 : \Sh(\mathcal{C}_0) \to \Sh(\mathcal{D})$ is a morphism of topoi given by a cocontinuous functor $u_0 : \mathcal{C}_0 \to \mathcal{D}$ such that for all $\varphi, \psi : [0] \to [n]$ we have $u_0 \circ u_\varphi = u_0 \circ u_\psi$. \end{enumerate}


spaces-simplicial-remark-morphism-simplicial-sites

Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and $\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in Situation \ref{situation-simplicial-site}. A {\it morphism $h$ between simplicial sites} will mean \begin{enumerate} \item[(A)] Morphisms of sites $h_n : \mathcal{C}_n \to \mathcal{C}'_n$ such that $f'_\varphi \circ h_n = h_m \circ f_\varphi$ as morphisms of sites for all $\varphi : [m] \to [n]$. \item[(B)] Cocontinuous functors $v_n : \mathcal{C}_n \to \mathcal{C}'_n$ inducing morphisms of topoi $h_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}'_n)$ such that $u'_\varphi \circ v_n = v_m \circ u_\varphi$ as functors for all $\varphi : [m] \to [n]$. \end{enumerate} In both cases we have $f'_\varphi \circ h_n = h_m \circ f_\varphi$ as morphisms of topoi, see Sites, Lemma \ref{sites-lemma-composition-cocontinuous} for case B and Sites, Definition \ref{sites-definition-composition-morphisms-sites} for case A.


spaces-simplicial-remark-morphism-augmentation-simplicial-sites

Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and $\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in Situation \ref{situation-simplicial-site}. Let $a_0$, resp.\ $a'_0$ be an augmentation towards a site $\mathcal{D}$, resp.\ $\mathcal{D}'$ as in Remark \ref{remark-augmentation-site}. Let $h$ be a morphism between simplicial sites as in Remark \ref{remark-morphism-simplicial-sites}. We say a morphism of topoi $h_{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')$ is {\it compatible with $h$, $a_0$, $a'_0$} if \begin{enumerate} \item[(A)] $h_{-1}$ comes from a morphism of sites $h_{-1} : \mathcal{D} \to \mathcal{D}'$ such that $a'_0 \circ h_0 = h_{-1} \circ a_0$ as morphisms of sites. \item[(B)] $h_{-1}$ comes from a cocontinuous functor $v_{-1} : \mathcal{D} \to \mathcal{D}'$ such that $u'_0 \circ v_0 = v_{-1} \circ u_0$ as functors. \end{enumerate} In both cases we have $a'_0 \circ h_0 = h_{-1} \circ a_0$ as morphisms of topoi, see Sites, Lemma \ref{sites-lemma-composition-cocontinuous} for case B and Sites, Definition \ref{sites-definition-composition-morphisms-sites} for case A.


spaces-simplicial-remark-morphism-simplicial-sites-modules

Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and $\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$ and $\mathcal{O}'$ be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$. We will say that $(h, h^\sharp)$ is a {\it morphism between ringed simplicial sites} if $h$ is a morphism between simplicial sites as in Remark \ref{remark-morphism-simplicial-sites} and $h^\sharp : h_{total}^{-1}\mathcal{O}' \to \mathcal{O}$ or equivalently $h^\sharp : \mathcal{O}' \to h_{total, *}\mathcal{O}$ is a homomorphism of sheaves of rings.


spaces-simplicial-remark-warning-cartesian-modules

Lemma \ref{lemma-Serre-subcat-cartesian-modules} notwithstanding, it can happen that the category of cartesian $\mathcal{O}$-modules is abelian without being a Serre subcategory of $\textit{Mod}(\mathcal{O})$. Namely, suppose that we only know that $f_{\delta_1^1}$ and $f_{\delta_0^1}$ are flat. Then it follows easily from Lemma \ref{lemma-characterize-cartesian-modules} that the category of cartesian $\mathcal{O}$-modules is abelian. But if $f_{\delta_0^2}$ is not flat (for example), there is no reason for the inclusion functor from the category of cartesian $\mathcal{O}$-modules to all $\mathcal{O}$-modules to be exact.


spaces-simplicial-remark-semi-representable-over-object

Let $\mathcal{C}$ be a site. Let $X \in \Ob(\mathcal{C})$. The category $\text{SR}(\mathcal{C}, X)$ of {\it semi-representable objects over $X$} is defined by the formula $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$. See Hypercoverings, Definition \ref{hypercovering-definition-SR}. Thus we may apply the above discussion to the site $\mathcal{C}/X$. Briefly, the constructions above give \begin{enumerate} \item a site $\mathcal{C}/K$ for $K$ in $\text{SR}(\mathcal{C}, X)$, \item a decomposition $\Sh(\mathcal{C}/K) = \prod \Sh(\mathcal{C}/U_i)$ if $K = \{U_i/X\}$, \item a localization functor $j : \mathcal{C}/K \to \mathcal{C}/X$, \item a morphism $f : \Sh(\mathcal{C}/K) \to \Sh(\mathcal{C}/L)$ for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$. \end{enumerate} All results of this section hold in this situation by replacing $\mathcal{C}$ everywhere by $\mathcal{C}/X$.


spaces-simplicial-remark-semi-representable-ringed

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$. In this case, for any semi-representable object $K$ of $\mathcal{C}$ the site $\mathcal{C}/K$ is a ringed site with sheaf of rings $\mathcal{O}_K = j^{-1}\mathcal{O}_\mathcal{C}$. The constructions above give \begin{enumerate} \item a ringed site $(\mathcal{C}/K, \mathcal{O}_K)$ for $K$ in $\text{SR}(\mathcal{C})$, \item a decomposition $\textit{Mod}(\mathcal{O}_K) = \prod \textit{Mod}(\mathcal{O}_{U_i})$ if $K = \{U_i\}$, \item a localization morphism $j : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})$ of ringed topoi, \item a morphism $f : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}/L), \mathcal{O}_L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C})$. \end{enumerate} Many of the results above hold in this setting. For example, the functor $j^*$ has an exact left adjoint $$ j_! : \textit{Mod}(\mathcal{O}_K) \to \textit{Mod}(\mathcal{O}_\mathcal{C}), $$ which in terms of the product decomposition given in (2) sends $(\mathcal{F}_i)_{i \in I}$ to $\bigoplus j_{i, !}\mathcal{F}_i$. Similarly, given $f : K \to L$ as above, the functor $f^*$ has an exact left adjoint $f_! : \textit{Mod}(\mathcal{O}_K) \to \textit{Mod}(\mathcal{O}_L)$. Thus the functors $j^*$ and $f^*$ are exact, i.e., $j$ and $f$ are flat morphisms of ringed topoi (also follows from the equalities $\mathcal{O}_K = j^{-1}\mathcal{O}_\mathcal{C}$ and $\mathcal{O}_K = f^{-1}\mathcal{O}_L$).


spaces-simplicial-remark-semi-representable-ringed-over-object

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings on $\mathcal{C}$. Let $X \in \Ob(\mathcal{C})$ and denote $\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/U}$. Then we can combine the constructions given in Remarks \ref{remark-semi-representable-over-object} and \ref{remark-semi-representable-ringed} to get \begin{enumerate} \item a ringed site $(\mathcal{C}/K, \mathcal{O}_K)$ for $K$ in $\text{SR}(\mathcal{C}, X)$, \item a decomposition $\textit{Mod}(\mathcal{O}_K) = \prod \textit{Mod}(\mathcal{O}_{U_i})$ if $K = \{U_i\}$, \item a localization morphism $j : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}/X), \mathcal{O}_X)$ of ringed topoi, \item a morphism $f : (\Sh(\mathcal{C}/K), \mathcal{O}_K) \to (\Sh(\mathcal{C}/L), \mathcal{O}_L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$. \end{enumerate} Of course all of the results mentioned in Remark \ref{remark-semi-representable-ringed} hold in this setting as well.


spaces-simplicial-remark-augmentation-over-object

Let $\mathcal{C}$ be a site. Let $X \in \Ob(\mathcal{C})$. Recall that we have a category $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$ of semi-representable objects over $X$, see Remark \ref{remark-semi-representable-over-object}. We may apply the above discussion to the site $\mathcal{C}/X$. Briefly, the constructions above give \begin{enumerate} \item a site $(\mathcal{C}/K)_{total}$ for a simplicial $K$ object of $\text{SR}(\mathcal{C}, X)$, \item a localization functor $j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}/X$, \item localization functors $j_n : \mathcal{C}/K_n \to \mathcal{C}/X$, \item a morphism of topoi $a : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C}/X)$, \item morphisms of topoi $a_n : \Sh(\mathcal{C}/K_n) \to \Sh(\mathcal{C}/X)$, \item a functor $a^{Sh}_! : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C}/X)$ left adjoint to $a^{-1}$, and \item a functor $a_! : \textit{Ab}((\mathcal{C}/K)_{total}) \to \textit{Ab}(\mathcal{C}/X)$ left adjoint to $a^{-1}$. \end{enumerate} All of the results of this section hold in this setting. To prove this one replaces the site $\mathcal{C}$ everywhere by $\mathcal{C}/X$.


spaces-simplicial-remark-augmentation-ringed

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Given a simplicial semi-representable object $K$ of $\mathcal{C}$ we set $\mathcal{O} = a^{-1}\mathcal{O}_\mathcal{C}$, where $a$ is as in Lemmas \ref{lemma-augmentation-simplicial-semi-representable} and \ref{lemma-comparison}. The constructions above, keeping track of the sheaves of rings as in Remark \ref{remark-semi-representable-ringed}, give \begin{enumerate} \item a ringed site $((\mathcal{C}/K)_{total}, \mathcal{O})$ for a simplicial $K$ object of $\text{SR}(\mathcal{C})$, \item a morphism of ringed topoi $a : (\Sh((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})$, \item morphisms of ringed topoi $a_n : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})$, \item a functor $a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_\mathcal{C})$ left adjoint to $a^*$. \end{enumerate} The functor $a_!$ exists (but in general is not exact) because $a^{-1}\mathcal{O}_\mathcal{C} = \mathcal{O}$ and we can replace the use of Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint} in the proof of Lemma \ref{lemma-comparison} by Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}. As discussed in Remark \ref{remark-semi-representable-ringed} there are exact functors $a_{n!} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O}_\mathcal{C})$ left adjoint to $a_n^*$. Consequently, the morphisms $a$ and $a_n$ are flat. Remark \ref{remark-semi-representable-ringed} implies the morphism of ringed topoi $f_\varphi : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}/K_m), \mathcal{O}_m)$ for $\varphi : [m] \to [n]$ is flat and there exists an exact functor $f_{\varphi !} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O}_m)$ left adjoint to $f_\varphi^*$. This in turn implies that for the flat morphism of ringed topoi $g_n : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh((\mathcal{C}/K)_{total}), \mathcal{O})$ the functor $g_{n!} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O})$ left adjoint to $g_n^*$ is exact, see Lemma \ref{lemma-exactness-g-shriek-modules}.


spaces-simplicial-remark-augmentation-ringed-over-object

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $X \in \Ob(\mathcal{C})$ and denote $\mathcal{O}_X = \mathcal{O}_\mathcal{C}|_{\mathcal{C}/X}$. Then we can combine the constructions given in Remarks \ref{remark-augmentation-over-object} and \ref{remark-augmentation-ringed} to get \begin{enumerate} \item a ringed site $((\mathcal{C}/K)_{total}, \mathcal{O})$ for a simplicial $K$ object of $\text{SR}(\mathcal{C}, X)$, \item a morphism of ringed topoi $a : (\Sh((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\Sh(\mathcal{C}/X), \mathcal{O}_X)$, \item morphisms of ringed topoi $a_n : (\Sh(\mathcal{C}/K_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}/X), \mathcal{O}_X)$, \item a functor $a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_X)$ left adjoint to $a^*$. \end{enumerate} Of course, all the results mentioned in Remark \ref{remark-augmentation-ringed} hold in this setting as well.


spaces-simplicial-remark-compare-cohomology-hypercovering-presheaf

Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets on $\mathcal{C}$. If $\mathcal{C}$ has equalizers and fibre products, then we've defined the notion of a hypercovering of $\mathcal{G}$ in Hypercoverings, Definition \ref{hypercovering-definition-hypercovering-variant}. We claim that all the results in this section have a valid counterpart in this setting. To see this, define the localization $\mathcal{C}/\mathcal{G}$ of $\mathcal{C}$ at $\mathcal{G}$ exactly as in Sites, Lemma \ref{sites-lemma-localize-topos-site} (which is stated only for sheaves; the topos $\Sh(\mathcal{C}/\mathcal{G})$ is equal to the localization of the topos $\Sh(\mathcal{C})$ at the sheaf $\mathcal{G}^\#$). Then the reader easily shows that the site $\mathcal{C}/\mathcal{G}$ has fibre products and equalizers and that a hypercovering of $\mathcal{G}$ in $\mathcal{C}$ is the same thing as a hypercovering for the site $\mathcal{C}/\mathcal{G}$. Hence replacing the site $\mathcal{C}$ by $\mathcal{C}/\mathcal{G}$ in the lemmas on hypercoverings above we obtain proofs of the corresponding results for hypercoverings of $\mathcal{G}$. Example: for a hypercovering $K$ of $\mathcal{G}$ we have $$ R\Gamma(\mathcal{C}/\mathcal{G}, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}E) $$ for $E \in D^+(\mathcal{C}/\mathcal{G})$ where $a : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C}/\mathcal{G})$ is the canonical augmentation. This is Lemma \ref{lemma-compare-cohomology-hypercovering}. Let $R\Gamma(\mathcal{G}, -) : D(\mathcal{C}) \to D(\textit{Ab})$ be defined as the derived functor of the functor $H^0(\mathcal{G}, -) = H^0(\mathcal{G}^\#, -)$ discussed in Hypercoverings, Section \ref{hypercovering-section-hypercoverings-verdier} and Cohomology on Sites, Section \ref{sites-cohomology-section-limp}. We have $$ R\Gamma(\mathcal{G}, E) = R\Gamma(\mathcal{C}/\mathcal{G}, j^{-1}E) $$ by the analogue of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open} for the localization fuctor $j : \mathcal{C}/\mathcal{G} \to \mathcal{C}$. Putting everything together we obtain $$ R\Gamma(\mathcal{G}, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}j^{-1}E) = R\Gamma((\mathcal{C}/K)_{total}, g^{-1}E) $$ for $E \in D^+(\mathcal{C})$ where $g : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C})$ is the composition of $a$ and $j$.


examples-stacks-remark-higher-rank

Note that the whole discussion in this section works if we want to consider those quasi-coherent sheaves which are locally generated by at most $\kappa$ sections, for some infinite cardinal $\kappa$, e.g., $\kappa = \aleph_0$.


examples-stacks-remark-stack-spaces

Ignoring set theoretical difficulties\footnote{The difficulty is not that $\Spacesstack$ is a proper class, since by our definition of an algebraic space over $S$ there is only a set worth of isomorphism classes of algebraic spaces over $S$. It is rather that arbitrary disjoint unions of algebraic spaces may end up being too large, hence lie outside of our chosen ``partial universe'' of sets.} $\Spacesstack$ also satisfies descent for objects and hence is a stack. Namely, we have to show that given \begin{enumerate} \item an fppf covering $\{U_i \to U\}_{i \in I}$, \item for each $i \in I$ an algebraic space $X_i/U_i$, and \item for each $i, j \in I$ an isomorphism $\varphi_{ij} : X_i \times_U U_j \to U_i \times_U X_j$ of algebraic spaces over $U_i \times_U U_j$ satisfying the cocycle condition over $U_i \times_U U_j \times_U U_k$, \end{enumerate} there exists an algebraic space $X/U$ and isomorphisms $X_{U_i} \cong X_i$ over $U_i$ recovering the isomorphisms $\varphi_{ij}$. First, note that by Sites, Lemma \ref{sites-lemma-glue-sheaves} there exists a sheaf $X$ on $(\Sch/U)_{fppf}$ recovering the $X_i$ and the $\varphi_{ij}$. Then by Bootstrap, Lemma \ref{bootstrap-lemma-locally-algebraic-space} we see that $X$ is an algebraic space (if we ignore the set theoretic condition of that lemma). We will use this argument in the next section to show that if we consider only algebraic spaces of finite type, then we obtain a stack.


examples-stacks-remark-higher-cardinality-spaces

Note that the whole discussion in this section works if we want to consider those algebraic spaces $X/U$ which are locally of finite type such that the inverse image in $X$ of an affine open of $U$ can be covered by countably many affines. If needed we can also introduce the notion of a morphism of $\kappa$-type (meaning some bound on the number of generators of ring extensions and some bound on the cardinality of the affines over a given affine in the base) where $\kappa$ is a cardinal, and then we can produce a stack $$ \Spacesstack_\kappa \longrightarrow (\Sch/S)_{fppf} $$ in exactly the same manner as above (provided we make sure that $\Sch$ is large enough depending on $\kappa$).


examples-stacks-remark-principal-stack-in-groupoids

We conjecture that up to a replacement as in Stacks, Remark \ref{stacks-remark-stack-make-small} the functor $$ p : G\textit{-Principal} \longrightarrow (\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\Sch/S)_{fppf}$. This would follow if one could show that given \begin{enumerate} \item a covering $\{U_i \to U\}_{i \in I}$ of $(\Sch/S)_{fppf}$, \item an group algebraic space $H$ over $U$, \item for every $i$ a principal homogeneous $H_{U_i}$-space $X_i$ over $U_i$, and \item $H$-equivariant isomorphisms $\varphi_{ij} : X_{i, U_i \times_U U_j} \to X_{j, U_i \times_U U_j}$ satisfying the cocycle condition, \end{enumerate} there exists a principal homogeneous $H$-space $X$ over $U$ which recovers $(X_i, \varphi_{ij})$. The technique of the proof of Bootstrap, Lemma \ref{bootstrap-lemma-descent-torsor} reduces this to a set theoretical question, so the reader who ignores set theoretical questions will ``know'' that the result is true. In \url{https://math.columbia.edu/~dejong/wordpress/?p=591} there is a suggestion as to how to approach this problem.


examples-stacks-remark-X-mod-G-group

Let $S$ be a scheme. Let $G$ be an abstract group. Let $X$ be an algebraic space over $S$. Let $G \to \text{Aut}_S(X)$ be a group homomorphism. In this setting we can define $[[X/G]]$ similarly to the above as follows: \begin{enumerate} \item An object of $[[X/G]]$ consists of a triple $(U, P, \varphi : P \to X)$ where \begin{enumerate} \item $U$ is an object of $(\Sch/S)_{fppf}$, \item $P$ is a sheaf on $(\Sch/U)_{fppf}$ which comes with an action of $G$ that turns it into a torsor under the constant sheaf with value $G$, and \item $\varphi : P \to X$ is a $G$-equivariant map of sheaves. \end{enumerate} \item A morphism $(f, g) : (U, P, \varphi) \to (U', P', \varphi')$ is given by a morphism of schemes $f : T \to T'$ and a $G$-equivariant isomorphism $g : P \to f^{-1}P'$ such that $\varphi = \varphi' \circ g$. \end{enumerate} In exactly the same manner as above we obtain a functor $$ [[X/G]] \longrightarrow (\Sch/S)_{fppf} $$ which turns $[[X/G]]$ into a stack in groupoids over $(\Sch/S)_{fppf}$. The constant sheaf $\underline{G}$ is (provided the cardinality of $G$ is not too large) representable by $G_S$ on $(\Sch/S)_{fppf}$ and this version of $[[X/G]]$ is equivalent to the stack $[[X/G_S]]$ introduced above.


models-remark-genus-equality

Let $n, m_i, a_{ij}, w_i, g_i$ be a minimal numerical type with $n > 1$. Equality $g = g_{top}$ can hold in Lemma \ref{lemma-genus-nonnegative}. For example, if $m_i = w_i = 1$ and $g_i = 0$ for all $i$ and $a_{ij} \in \{0, 1\}$ for $i < j$.


models-remark-numerical-type-not-from-model

Not every numerical type comes from a model for the silly reason that there exist numerical types whose genus is negative. There exist a minimal numerical types of positive genus which are not the numerical type associated to a model (over some dvr) of a smooth projective geometrically irreducible curve (over the fraction field of the dvr). A simple example is $n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 6$, $g_1 = 1$. Namely, in this case the special fibre $X_k$ would not be geometrically connected because it would live over an extension $\kappa$ of $k$ of degree $6$. This is a contradiction with the fact that the generic fibre is geometrically connected (see More on Morphisms, Lemma \ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}). Similarly, $n = 2$, $m_1 = m_2 = 1$, $-a_{11} = -a_{22} = a_{12} = a_{21} = 6$, $w_1 = w_2 = 6$, $g_1 = g_2 = 1$ would be an example for the same reason (details omitted). But if the gcd of the $w_i$ is $1$ we do not have an example.


models-remark-genus-change

In the situation of Lemma \ref{lemma-blowdown-regular-model} we can also say exactly how the genus $g_i$ of $C_i$ and the genus $g'_i$ of $C'_i$ are related. The formula is $$ g'_i = \frac{w_i}{w'_i}(g_i - 1) + 1 + \frac{(C_i \cdot C_n)^2 - w_n(C_i \cdot C_n)}{2w'_iw_n} $$ where $w_i = [\kappa_i : k]$, $w_n = [\kappa_n : k]$, and $w'_i = [\kappa'_i : k]$. To prove this we consider the short exact sequence $$ 0 \to \mathcal{O}_{X'}(-C'_i) \to \mathcal{O}_{X'} \to \mathcal{O}_{C'_i} \to 0 $$ and its pullback to $X$ which reads $$ 0 \to \mathcal{O}_X(-C'_i - e_iC_n) \to \mathcal{O}_X \to \mathcal{O}_{C_i + e_i C_n} \to 0 $$ with $e_i$ as in the proof of Lemma \ref{lemma-blowdown-regular-model}. Since $Rf_*f^*\mathcal{L} = \mathcal{L}$ for any invertible module $\mathcal{L}$ on $X'$ (details omitted), we conclude that $$ Rf_*\mathcal{O}_{C_i + e_i C_n} = \mathcal{O}_{C'_i} $$ as complexes of coherent sheaves on $X'_k$. Hence both sides have the same Euler characteristic and this agrees with the Euler characteristic of $\mathcal{O}_{C_i + e_i C_n}$ on $X_k$. Using the exact sequence $$ 0 \to \mathcal{O}_{C_i + e_i C_n} \to \mathcal{O}_{C_i} \oplus \mathcal{O}_{e_iC_n} \to \mathcal{O}_{C_i \cap e_iC_n} \to 0 $$ and further filtering $\mathcal{O}_{e_iC_n}$ (details omitted) we find $$ \chi(\mathcal{O}_{C'_i}) = \chi(\mathcal{O}_{C_i}) - {e_i + 1 \choose 2}(C_n \cdot C_n) - e_i(C_i \cdot C_n) $$ Since $e_i = -(C_i \cdot C_n)/(C_n \cdot C_n)$ and $(C_n \cdot C_n) = -w_n$ this leads to the formula stated at the start of this remark. If we ever need this we will formulate this as a lemma and provide a detailed proof.


models-remark-compare-contractions

Let $f : X \to X'$ be as in Lemma \ref{lemma-blowdown-regular-model}. Let $n, m_i, a_{ij}, w_i, g_i$ be the numerical type associated to $X$ and let $n', m'_i, a'_{ij}, w'_i, g'_i$ be the numerical type associated to $X'$. It is clear from Lemma \ref{lemma-blowdown-regular-model} and Remark \ref{remark-genus-change} that this agrees with the contraction of numerical types in Lemma \ref{lemma-contract} except for the value of $w'_i$. In the geometric situation $w'_i$ is some positive integer dividing both $w_i$ and $w_n$. In the numerical case we chose $w'_i$ to be the largest possible integer dividing $w_i$ such that $g'_i$ (as given by the formula) is an integer. This works well in the numerical setting in that it helps compare the Picard groups of the numerical types, see Lemma \ref{lemma-contract-picard-group} (although only injectivity is every used in the following and this injectivity works as well for smaller $w'_i$).


models-remark-improving-bound

Results in the literature suggest that one can improve the bound given in the statement of Theorem \ref{theorem-semistable-reduction}. For example, in \cite{DM} it is shown that semistable reduction of $C$ and its Jacobian are the same thing if the residue field is perfect and presumably this is true for general residue fields as well. For an abelian variety we have semistable reduction if the action of Galois on the $\ell$-torsion is trivial for any $\ell \geq 3$ not equal to the residue characteristic. Thus we can presumably choose $\ell = 5$ in the formula (\ref{equation-bound}) for $B_g$ (but the proof would take a lot more work; if we ever need this we will make a precise statement and provide a proof here).


spaces-groupoids-remark-quotient-variant

A variant of the construction above would have been to sheafify the functor $$ \begin{matrix} (\textit{Spaces}/B)^{opp}_{fppf} & \longrightarrow & \textit{Sets}, \\ X & \longmapsto  & U(X)/\sim_X \end{matrix} $$ where now $\sim_X \subset U(X) \times U(X)$ is the equivalence relation generated by the image of $j : R(X) \to U(X) \times U(X)$. Here of course $U(X) = \Mor_B(X, U)$ and $R(X) = \Mor_B(X, R)$. In fact, the result would have been the same, via the identifications of (insert future reference in Topologies of Spaces here).


spaces-groupoids-remark-fundamental-square

In future chapters we will use the ambiguous notation where instead of writing $\mathcal{S}_X$ for the stack in sets associated to $X$ we simply write $X$. Using this notation the diagram of Lemma \ref{lemma-quotient-stack-2-arrow} becomes the familiar diagram $$ \xymatrix{ R \ar[r]_s \ar[d]_t & U \ar[d]^\pi \\ U \ar[r]^-\pi & [U/R] } $$ In the following sections we will show that this diagram has many good properties. In particular we will show that it is a $2$-fibre product (Section \ref{section-quotient-stack-2-cartesian}) and that it is close to being a $2$-coequalizer of $s$ and $t$ (Section \ref{section-quotient-stacks-2-coequalize}).


spaces-descent-remark-list-local-source-target

Using Lemma \ref{lemma-smooth-local-source-target} and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is smooth local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma \ref{lemma-smooth-local-source-target} is trivial to check, and we omit it. Here is the list: \begin{enumerate} \item flat, see Lemmas \ref{lemma-flat-fpqc-local-source} and \ref{lemma-descending-property-flat}, \item locally of finite presentation, see Lemmas \ref{lemma-locally-finite-presentation-fppf-local-source} and \ref{lemma-descending-property-locally-finite-presentation}, \item locally finite type, see Lemmas \ref{lemma-locally-finite-type-fppf-local-source} and \ref{lemma-descending-property-locally-finite-type}, \item universally open, see Lemmas \ref{lemma-universally-open-fppf-local-source} and \ref{lemma-descending-property-universally-open}, \item syntomic, see Lemmas \ref{lemma-syntomic-syntomic-local-source} and \ref{lemma-descending-property-syntomic}, \item smooth, see Lemmas \ref{lemma-smooth-smooth-local-source} and \ref{lemma-descending-property-smooth}, \item add more here as needed. \end{enumerate}


spaces-descent-remark-list-etale-smooth-local-source-target

Using Lemma \ref{lemma-etale-smooth-local-source-target} and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is etale local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma \ref{lemma-etale-smooth-local-source-target} is trivial to check, and we omit it. Here is the list: \begin{enumerate} \item \'etale, see Lemmas \ref{lemma-etale-etale-local-source} and \ref{lemma-descending-property-etale}, \item locally quasi-finite, see Lemmas \ref{lemma-locally-quasi-finite-etale-local-source} and \ref{lemma-descending-property-quasi-finite}, \item unramified, see Lemmas \ref{lemma-unramified-etale-local-source} and \ref{lemma-descending-property-unramified}, and \item add more here as needed. \end{enumerate} Of course any property listed in Remark \ref{remark-list-local-source-target} is a fortiori an example that could be listed here.


spaces-descent-remark-easier

Let $S$ be a scheme. Let $Y \to X$ be a morphism of algebraic spaces over $S$. Let $(V/Y, \varphi)$ be a descent datum relative to $Y \to X$. We may think of the isomorphism $\varphi$ as an isomorphism $$ (Y \times_X Y) \times_{\text{pr}_0, Y} V \longrightarrow (Y \times_X Y) \times_{\text{pr}_1, Y} V $$ of algebraic spaces over $Y \times_X Y$. So loosely speaking one may think of $\varphi$ as a map $\varphi : \text{pr}_0^*V \to \text{pr}_1^*V$\footnote{Unfortunately, we have chosen the ``wrong'' direction for our arrow here. In Definitions \ref{definition-descent-datum} and \ref{definition-descent-datum-for-family-of-morphisms} we should have the opposite direction to what was done in Definition \ref{definition-descent-datum-quasi-coherent} by the general principle that ``functions'' and ``spaces'' are dual.}. The cocycle condition then says that $\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.


spaces-descent-remark-easier-family

Let $S$ be a scheme. Let $\{X_i \to X\}_{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Let $(V_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to X\}$. We may think of the isomorphisms $\varphi_{ij}$ as isomorphisms $$ (X_i \times_X X_j) \times_{\text{pr}_0, X_i} V_i \longrightarrow (X_i \times_X X_j) \times_{\text{pr}_1, X_j} V_j $$ of algebraic spaces over $X_i \times_X X_j$. So loosely speaking one may think of $\varphi_{ij}$ as an isomorphism $\text{pr}_0^*V_i \to \text{pr}_1^*V_j$ over $X_i \times_X X_j$. The cocycle condition then says that $\text{pr}_{02}^*\varphi_{ik} = \text{pr}_{12}^*\varphi_{jk} \circ \text{pr}_{01}^*\varphi_{ij}$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.


decent-spaces-remark-recall

Before we give the proof of the next lemma let us recall some facts about \'etale morphisms of schemes: \begin{enumerate} \item An \'etale morphism is flat and hence generalizations lift along an \'etale morphism (Morphisms, Lemmas \ref{morphisms-lemma-etale-flat} and \ref{morphisms-lemma-generalizations-lift-flat}). \item An \'etale morphism is unramified, an unramified morphism is locally quasi-finite, hence fibres are discrete (Morphisms, Lemmas \ref{morphisms-lemma-flat-unramified-etale}, \ref{morphisms-lemma-unramified-quasi-finite}, and \ref{morphisms-lemma-quasi-finite-at-point-characterize}). \item A quasi-compact \'etale morphism is quasi-finite and in particular has finite fibres (Morphisms, Lemmas \ref{morphisms-lemma-quasi-finite-locally-quasi-compact} and \ref{morphisms-lemma-quasi-finite}). \item An \'etale scheme over a field $k$ is a disjoint union of spectra of finite separable field extension of $k$ (Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}). \end{enumerate} For a general discussion of \'etale morphisms, please see \'Etale Morphisms, Section \ref{etale-section-etale-morphisms}.


decent-spaces-remark-reasonable

Reasonable algebraic spaces are technically easier to work with than very reasonable algebraic spaces. For example, if $X \to Y$ is a quasi-compact \'etale surjective morphism of algebraic spaces and $X$ is reasonable, then so is $Y$, see Lemma \ref{lemma-descent-conditions} but we don't know if this is true for the property ``very reasonable''. Below we give another technical property enjoyed by reasonable algebraic spaces.


decent-spaces-remark-functoriality-henselian-local-ring

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \in |X|$ with image $y \in |Y|$. Choose an elementary \'etale neighbourhood $(V, v) \to (Y, y)$ (possible by Lemma \ref{lemma-decent-space-elementary-etale-neighbourhood}). Then $V \times_Y X$ is an algebraic space \'etale over $X$ which has a unique point $x'$ mapping to $x$ in $X$ and to $v$ in $V$. (Details omitted; use that all points can be represented by monomorphisms from spectra of fields.) Choose an elementary \'etale neighbourhood $(U, u) \to (V \times_Y X, x')$. Then we obtain the following commutative diagram $$ \xymatrix{ \Spec(\mathcal{O}_{X, \overline{x}}) \ar[r] \ar[d] & \Spec(\mathcal{O}_{X, x}^h) \ar[r] \ar[d] & \Spec(\mathcal{O}_{U, u}) \ar[r] \ar[d] & U \ar[r] \ar[d] & X \ar[d] \\ \Spec(\mathcal{O}_{Y, \overline{y}}) \ar[r] & \Spec(\mathcal{O}_{Y, y}^h) \ar[r] & \Spec(\mathcal{O}_{V, v}) \ar[r] & V \ar[r] & Y } $$ This comes from the identifications $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$, $\mathcal{O}_{X, x}^h = \mathcal{O}_{U, u}^h$, $\mathcal{O}_{Y, \overline{y}} = \mathcal{O}_{V, v}^{sh}$, $\mathcal{O}_{Y, y}^h = \mathcal{O}_{V, v}^h$ see in Lemma \ref{lemma-describe-henselian-local-ring} and Properties of Spaces, Lemma \ref{spaces-properties-lemma-describe-etale-local-ring} and the functoriality of the (strict) henselization discussed in Algebra, Sections \ref{algebra-section-ind-etale} and \ref{algebra-section-henselization}.


decent-spaces-remark-one-point-decent-scheme

We will see in Limits of Spaces, Lemma \ref{spaces-limits-lemma-reduction-scheme} that an algebraic space whose reduction is a scheme is a scheme.


decent-spaces-remark-very-reasonable

An informal description of the properties $(\beta)$, decent, reasonable, very reasonable was given in Section \ref{section-reasonable-decent}. A morphism has one of these properties if (very) loosely speaking the fibres of the morphism have the corresponding properties. Being decent is useful to prove things about specializations of points on $|X|$. Being reasonable is a bit stronger and technically quite easy to work with.


groupoids-remark-warning-group-scheme-geometrically-irreducible

Warning: The result of Lemma \ref{lemma-group-scheme-field-geometrically-irreducible} does not mean that every irreducible component of $G/k$ is geometrically irreducible. For example the group scheme $\mu_{3, \mathbf{Q}} = \Spec(\mathbf{Q}[x]/(x^3 - 1))$ over $\mathbf{Q}$ has two irreducible components corresponding to the factorization $x^3 - 1 = (x - 1)(x^2 + x + 1)$. The first factor corresponds to the irreducible component passing through the identity, and the second irreducible component is not geometrically irreducible over $\Spec(\mathbf{Q})$.


groupoids-remark-easy

If $G$ is a group scheme over a field, is there always a quasi-compact open and closed subgroup scheme? By Proposition \ref{proposition-connected-component} this question is only interesting if $G$ has infinitely many connected components (geometrically).


groupoids-remark-when-reduced

Any group scheme over a field of characteristic $0$ is reduced, see \cite[I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4]{Perrin-thesis} and also \cite[Proposition 4.2.8]{Perrin}. This was a question raised in \cite[page 80]{Oort}. We have seen in Lemma \ref{lemma-group-scheme-characteristic-zero-smooth} that this holds when the group scheme is locally of finite type.


groupoids-remark-reduced-smooth-not-true-general

Let $k$ be a field of characteristic $p > 0$. Let $\alpha \in k$ be an element which is not a $p$th power. The closed subgroup scheme $$ G = V(x^p + \alpha y^p) \subset \mathbf{G}_{a, k}^2 $$ is reduced and irreducible but not smooth (not even normal).


groupoids-remark-fun-with-torsors

Let $(G, m)$ be a group scheme over the scheme $S$. In this situation we have the following natural types of questions: \begin{enumerate} \item If $X \to S$ is a pseudo $G$-torsor and $X \to S$ is surjective, then is $X$ necessarily a $G$-torsor? \item Is every $\underline{G}$-torsor on $(\Sch/S)_{fppf}$ representable? In other words, does every $\underline{G}$-torsor come from a fppf $G$-torsor? \item Is every $G$-torsor an fppf (resp.\ smooth, resp.\ \'etale, resp.\ Zariski) torsor? \end{enumerate} In general the answers to these questions is no. To get a positive answer we need to impose additional conditions on $G \to S$. For example: If $S$ is the spectrum of a field, then the answer to (1) is yes because then $\{X \to S\}$ is a fpqc covering trivializing $X$. If $G \to S$ is affine, then the answer to (2) is yes (insert future reference here). If $G = \text{GL}_{n, S}$ then the answer to (3) is yes and in fact any $\text{GL}_{n, S}$-torsor is locally trivial (insert future reference here).


local-cohomology-remark-upshot

Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. The upshot of the discussion above is that $R\Gamma_T : D^+(A) \to D_T^+(A)$ is the right adjoint to the inclusion functor $D_T^+(A) \to D^+(A)$. If $\dim(A) < \infty$, then $R\Gamma_T : D(A) \to D_T(A)$ is the right adjoint to the inclusion functor $D_T(A) \to D(A)$. In both cases we have $$ H^i_T(K) = H^i(R\Gamma_T(K)) = R^iH^0_T(K) = \colim_{Z \subset T\text{ closed}} H^i_Z(K) $$ This follows by combining Lemmas \ref{lemma-adjoint}, \ref{lemma-adjoint-ext}, \ref{lemma-equal-plus}, and \ref{lemma-equal-full}.


local-cohomology-remark-closure

Let $j : U \to X$ be an open immersion of locally Noetherian schemes. Let $x \in U$. Let $i_x : W_x \to U$ be the integral closed subscheme with generic point $x$ and let $\overline{\{x\}}$ be the closure in $X$. Then we have a commutative diagram $$ \xymatrix{ W_x \ar[d]_{i_x} \ar[r]_{j'} & \overline{\{x\}} \ar[d]^i \\ U \ar[r]^j & X } $$ We have $j_*i_{x, *}\mathcal{O}_{W_x} = i_*j'_*\mathcal{O}_{W_x}$. As the left vertical arrow is a closed immersion we see that $j_*i_{x, *}\mathcal{O}_{W_x}$ is coherent if and only if $j'_*\mathcal{O}_{W_x}$ is coherent.


local-cohomology-remark-no-finiteness-pushforward

Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. If there exists an $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{x\}}$ such that $\dim(\mathcal{O}_{\overline{\{x\}}, z}) \leq 1$, then $j_*\mathcal{F}$ is not coherent. To prove this we can do a flat base change to the spectrum of $\mathcal{O}_{X, z}$. Let $X' = \overline{\{x\}}$. The assumption implies $\mathcal{O}_{X' \cap U} \subset \mathcal{F}$. Thus it suffices to see that $j_*\mathcal{O}_{X' \cap U}$ is not coherent. This is clear because $X' = \{x, z\}$, hence $j_*\mathcal{O}_{X' \cap U}$ corresponds to $\kappa(x)$ as an $\mathcal{O}_{X, z}$-module which cannot be finite as $x$ is not a closed point. \medskip\noindent In fact, the converse of Lemma \ref{lemma-sharp-finiteness-pushforward} holds true: given an open immersion $j : U \to X$ of integral Noetherian schemes and there exists a $z \in X \setminus U$ and an associated prime $\mathfrak p$ of the completion $\mathcal{O}_{X, z}^\wedge$ with $\dim(\mathcal{O}_{X, z}^\wedge/\mathfrak p) = 1$, then $j_*\mathcal{O}_U$ is not coherent. Namely, you can pass to the local ring, you can enlarge $U$ to the punctured spectrum, you can pass to the completion, and then the argument above gives the nonfiniteness.


local-cohomology-remark-astute-reader

The astute reader will have realized that we can get away with a slightly weaker condition on the formal fibres of the local rings of $A$. Namely, in the situation of Theorem \ref{theorem-finiteness} assume $A$ is universally catenary but make no assumptions on the formal fibres. Suppose we have an $n$ and we want to prove that $H^i_Z(M)$ are finite for $i \leq n$. Then the exact same proof shows that it suffices that $s_{A, I}(M) > n$ and that the formal fibres of local rings of $A$ are $(S_n)$. On the other hand, if we want to show that $H^s_Z(M)$ is not finite where $s = s_{A, I}(M)$, then our arguments prove this if the formal fibres are $(S_{s - 1})$.


local-cohomology-remark-higher-order-operators

We can upgrade Lemmas \ref{lemma-derivation} and \ref{lemma-etale-derivation} to include higher order differential operators. If we ever need this we will state and prove a precise lemma here.


local-cohomology-remark-better-bound

The paper \cite{AHS} shows, besides many other things, that if $A$ is local, then Proposition \ref{proposition-uniform-artin-rees} also holds with $e = t$ replaced by $e = \dim(A)$. Looking at Lemma \ref{lemma-cd-sequence-Koszul} it is natural to ask whether Proposition \ref{proposition-uniform-artin-rees} holds with $e = t$ replaced with $e = \text{cd}(A, I)$. We don't know.


local-cohomology-remark-strict-pro-isomorphism

Let $I$ be an ideal of a Noetherian ring $A$. Say $I = (f_1, \ldots, f_r)$. Denote $K_n^\bullet$ the Koszul complex on $f_1^n, \ldots, f_r^n$ as in More on Algebra, Situation \ref{more-algebra-situation-koszul} and denote $K_n \in D(A)$ the corresponding object. Let $M^\bullet$ be a bounded complex of finite $A$-modules and denote $M \in D(A)$ the corresponding object. Consider the following inverse systems in $D(A)$: \begin{enumerate} \item $M^\bullet/I^nM^\bullet$, i.e., the complex whose terms are $M^i/I^nM^i$, \item $M \otimes_A^\mathbf{L} A/I^n$, \item $M \otimes_A^\mathbf{L} K_n$, and \item $M \otimes_P^\mathbf{L} P/J^n$ (see below). \end{enumerate} All of these inverse systems are isomorphic as pro-objects: the isomorphism between (2) and (3) follows from More on Algebra, Lemma \ref{more-algebra-lemma-sequence-Koszul-complexes}. The isomorphism between (1) and (2) is given in More on Algebra, Lemma \ref{more-algebra-lemma-derived-completion-plain-completion}. For the last one, see below. \medskip\noindent However, we can ask if these isomorphisms of pro-systems are ``strict''; this terminology and question is related to the discussion in \cite[pages 61, 62]{quillenhomology}. Namely, given a category $\mathcal{C}$ we can define a ``strict pro-category'' whose objects are inverse systems $(X_n)$ and whose morphisms $(X_n) \to (Y_n)$ are given by tuples $(c, \varphi_n)$ consisting of a $c \geq 0$ and morphisms $\varphi_n : X_n \to Y_{n - c}$ for all $n \geq c$ satisfying an obvious compatibility condition and up to a certain equivalence (given essentially by increasing $c$). Then we ask whether the above inverse systems are isomorphic in this strict pro-category. \medskip\noindent This clearly cannot be the case for (1) and (3) even when $M = A[0]$. Namely, the system $H^0(K_n) = A/(f_1^n, \ldots, f_r^n)$ is not strictly pro-isomorphic in the category of modules to the system $A/I^n$ in general. For example, if we take $A = \mathbf{Z}[x_1, \ldots, x_r]$ and $f_i = x_i$, then $H^0(K_n)$ is not annihilated by $I^{r(n - 1)}$.\footnote{Of course, we can ask whether these pro-systems are isomorphic in a category whose objects are inverse systems and where maps are given by tuples $(r, c, \varphi_n)$ consisting of $r \geq 1$, $c \geq 0$ and maps $\varphi_n : X_{rn} \to Y_{n - c}$ for $n \geq c$.} \medskip\noindent It turns out that the results above show that the natural map from (2) to (1) discussed in More on Algebra, Lemma \ref{more-algebra-lemma-derived-completion-plain-completion} is a strict pro-isomorphism. We will sketch the proof. Using standard arguments involving stupid truncations, we first reduce to the case where $M^\bullet$ is given by a single finite $A$-module $M$ placed in degree $0$. Pick $N, c \geq 0$ as in Proposition \ref{proposition-uniform-artin-rees}. The proposition implies that for $n \geq N$ we get factorizations $$ M \otimes_A^\mathbf{L} A/I^n \to \tau_{\geq -t}(M \otimes_A^\mathbf{L} A/I^n) \to M \otimes_A^\mathbf{L} A/I^{n - c} $$ of the transition maps in the system (2). On the other hand, by More on Algebra, Lemma \ref{more-algebra-lemma-tor-strictly-pro-zero}, we can find another constant $c' = c'(M) \geq 0$ such that the maps $\text{Tor}_i^A(M, A/I^{n'}) \to \text{Tor}_i(M, A/I^{n' - c'})$ are zero for $i = 1, 2, \ldots, t$ and $n' \geq c'$. Then it follows from Derived Categories, Lemma \ref{derived-lemma-trick-vanishing-composition} that the map $$ \tau_{\geq -t}(M \otimes_A^\mathbf{L} A/I^{n + tc'}) \to \tau_{\geq -t}(M \otimes_A^\mathbf{L} A/I^n) $$ factors through $M \otimes_A^\mathbf{L}A/I^{n + tc'} \to M/I^{n + tc'}M$. Combined with the previous result we obtain a factorization $$ M \otimes_A^\mathbf{L}A/I^{n + tc'} \to M/I^{n + tc'}M \to M \otimes_A^\mathbf{L} A/I^{n - c} $$ which gives us what we want. If we ever need this result, we will carefully state it and provide a detailed proof. \medskip\noindent For number (4) suppose we have a Noetherian ring $P$, a ring homomorphism $P \to A$, and an ideal $J \subset P$ such that $I = JA$. By More on Algebra, Section \ref{more-algebra-section-derived-base-change} we get a functor $M \otimes_P^\mathbf{L} - : D(P) \to D(A)$ and we get an inverse system $M \otimes_P^\mathbf{L} P/J^n$ in $D(A)$ as in (4). If $P$ is Noetherian, then the system in (4) is pro-isomorphic to the system in (1) because we can compare with Koszul complexes. If $P \to A$ is finite, then the system (4) is strictly pro-isomorphic to the system (2) because the inverse system $A \otimes_P^\mathbf{L} P/J^n$ is strictly pro-isomorphic to the inverse system $A/I^n$ (by the discussion above) and because we have $$ M \otimes_P^\mathbf{L} P/J^n = M \otimes_A^\mathbf{L} (A \otimes_P^\mathbf{L} P/J^n) $$ by More on Algebra, Lemma \ref{more-algebra-lemma-derived-base-change}. \medskip\noindent A standard example in (4) is to take $P = \mathbf{Z}[x_1, \ldots, x_r]$, the map $P \to A$ sending $x_i$ to $f_i$, and $J = (x_1, \ldots, x_r)$. In this case one shows that $$ M \otimes_P^\mathbf{L} P/J^n = M \otimes_{A[x_1, \ldots, x_r]}^\mathbf{L} A[x_1, \ldots, x_r]/(x_1, \ldots, x_r)^n $$ and we reduce to one of the cases discussed above (although this case is strictly easier as $A[x_1, \ldots, x_r]/(x_1, \ldots, x_r)^n$ has tor dimension at most $r$ for all $n$ and hence the step using Proposition \ref{proposition-uniform-artin-rees} can be avoided). This case is discussed in the proof of \cite[Proposition 3.5.1]{BS}.


local-cohomology-remark-duals

Given a pair $(M, n)$ consisting of an integer $n \geq 0$ and a finite $A/I^n$-module $M$ we set $M^\vee = \Hom_{A/I^n}(M, A/I^n)$. Given a pair $(\mathcal{F}, n)$ consisting of an integer $n$ and a coherent $\mathcal{O}_{Y_n}$-module $\mathcal{F}$ we set $$ \mathcal{F}^\vee = \SheafHom_{\mathcal{O}_{Y_n}}(\mathcal{F}, \mathcal{O}_{Y_n}) $$ Given $(M, n)$ as above, there is a canonical map $$ can : p^*(M^\vee) \longrightarrow (p^*M)^\vee $$ Namely, if we choose a presentation $(A/I^n)^{\oplus s} \to (A/I^n)^{\oplus r} \to M \to 0$ then we obtain a presentation $\mathcal{O}_{Y_n}^{\oplus s} \to \mathcal{O}_{Y_n}^{\oplus r} \to p^*M \to 0$. Taking duals we obtain exact sequences $$ 0 \to M^\vee \to (A/I^n)^{\oplus r} \to (A/I^n)^{\oplus s} $$ and $$ 0 \to (p^*M)^\vee \to \mathcal{O}_{Y_n}^{\oplus r} \to \mathcal{O}_{Y_n}^{\oplus s} $$ Pulling back the first sequence by $p$ we find the desired map $can$. The construction of this map is functorial in the finite $A/I^n$-module $M$. The kernel and cokernel of $can$ are scheme theoretically supported on $Y_c$ if $M$ is an $(A, n, c)$-module. Namely, in that case for $a \in I^c$ the map $a : M \to M$ factors through a finite free $A/I^n$-module for which $can$ is an isomorphism. Hence $a$ annihilates the kernel and cokernel of $can$.


more-algebra-remark-relative-modules-over-fibre-product

In Situation \ref{situation-relative-module-over-fibre-product}. Assume $B' \to D'$ is of finite presentation and suppose we are given a $D'$-module $L'$. We claim there is a bijective correspondence between \begin{enumerate} \item surjections of $D'$-modules $L' \to Q'$ with $Q'$ of finite presentation over $D'$ and flat over $B'$, and \item pairs of surjections of modules $(L' \otimes_{D'} D \to Q_1, L' \otimes_{D'} C' \to Q_2)$ with \begin{enumerate} \item $Q_1$ of finite presentation over $D$ and flat over $B$, \item $Q_2$ of finite presentation over $C'$ and flat over $A'$, \item $Q_1 \otimes_D C = Q_2 \otimes_{C'} C$ as quotients of $L' \otimes_{D'} C$. \end{enumerate} \end{enumerate} The correspondence between these is given by $Q \mapsto (Q_1, Q_2)$ with $Q_1 = Q \otimes_{D'} D$ and $Q_2 = Q \otimes_{D'} C'$. And for the converse we use $Q = Q_1 \times_{Q_{12}} Q_2$ where $Q_{12}$ the common quotient $Q_1 \otimes_D C = Q_2 \otimes_{C'} C$ of $L' \otimes_{D'} C$. As quotient map we use $$ L' \longrightarrow (L' \otimes_{D'} D) \times_{(L' \otimes_{D'} C)} (L' \otimes_{D'} C') \longrightarrow Q_1 \times_{Q_{12}} Q_2 = Q $$ where the first arrow is surjective by Lemma \ref{lemma-module-over-fibre-product-bis} and the second by Lemma \ref{lemma-surjection-module-over-fibre-product}. The claim follows by Lemmas \ref{lemma-relative-flat-module-over-fibre-product} and \ref{lemma-relative-finitely-presented-module-over-fibre-product}.


more-algebra-remark-when-does-condition-hold

Let $R$ be a ring. When does $R$ satisfy the condition mentioned in Lemmas \ref{lemma-flat-finite-type-finite-presentation-local-module}, \ref{lemma-flat-finite-type-finite-presentation-local}, and \ref{lemma-flat-graded-finite-type-finite-presentation}? This holds if \begin{enumerate} \item $R$ is local, \item $R$ is Noetherian, \item $R$ is a domain, \item $R$ is a reduced ring with finitely many minimal primes, or \item $R$ has finitely many weakly associated primes, see Algebra, Lemma \ref{algebra-lemma-zero-at-weakly-ass-zero}. \end{enumerate} Thus these lemmas hold in all cases listed above.


more-algebra-remark-what-does-it-mean

The assertion of Lemma \ref{lemma-lift-fs} is quite strong. Namely, suppose that we have a diagram $$ \xymatrix{ & B \\ A \ar[r] & A' \ar[u] } $$ of local homomorphisms of Noetherian complete local rings where $A \to A'$ induces an isomorphism of residue fields $k = A/\mathfrak m_A = A'/\mathfrak m_{A'}$ and with $B \otimes_{A'} k$ formally smooth over $k$. Then we can extend this to a commutative diagram $$ \xymatrix{ C \ar[r] & B \\ A \ar[r] \ar[u] & A' \ar[u] } $$ of local homomorphisms of Noetherian complete local rings where $A \to C$ is formally smooth in the $\mathfrak m_C$-adic topology and where $C \otimes_A k \cong B \otimes_{A'} k$. Namely, pick $A \to C$ as in Lemma \ref{lemma-lift-fs} lifting $B \otimes_{A'} k$ over $k$. By formal smoothness we can find the arrow $C \to B$, see Lemma \ref{lemma-lift-continuous}. Denote $C \otimes_A^\wedge A'$ the completion of $C \otimes_A A'$ with respect to the ideal $C \otimes_A \mathfrak m_{A'}$. Note that $C \otimes_A^\wedge A'$ is a Noetherian complete local ring (see Algebra, Lemma \ref{algebra-lemma-completion-Noetherian}) which is flat over $A'$ (see Algebra, Lemma \ref{algebra-lemma-flat-module-powers}). We have moreover \begin{enumerate} \item $C \otimes_A^\wedge A' \to B$ is surjective, \item if $A \to A'$ is surjective, then $C \to B$ is surjective, \item if $A \to A'$ is finite, then $C \to B$ is finite, and \item if $A' \to B$ is flat, then $C \otimes_A^\wedge A' \cong B$. \end{enumerate} Namely, by Nakayama's lemma for nilpotent ideals (see Algebra, Lemma \ref{algebra-lemma-NAK}) we see that $C \otimes_A k \cong B \otimes_{A'} k$ implies that $C \otimes_A A'/\mathfrak m_{A'}^n \to B/\mathfrak m_{A'}^nB$ is surjective for all $n$. This proves (1). Parts (2) and (3) follow from part (1). Part (4) follows from Algebra, Lemma \ref{algebra-lemma-mod-injective}.


more-algebra-remark-G-does-not-survive-completion

Let $R$ be a G-ring and let $I \subset R$ be an ideal. In general it is not the case that the $I$-adic completion $R^\wedge$ is a G-ring. An example was given by Nishimura in \cite{Nishimura}. A generalization and, in some sense, clarification of this example can be found in the last section of \cite{Dumitrescu}.


more-algebra-remark-P-resolution

In fact, we can do better than Lemma \ref{lemma-K-flat-resolution}. Namely, we can find a quasi-isomorphism $P^\bullet \to M^\bullet$ where $P^\bullet$ is a complex of $R$-modules endowed with a filtration $$ 0 = F_{-1}P^\bullet \subset F_0P^\bullet \subset F_1P^\bullet \subset \ldots \subset P^\bullet $$ by subcomplexes such that \begin{enumerate} \item $P^\bullet = \bigcup F_pP^\bullet$, \item the inclusions $F_iP^\bullet \to F_{i + 1}P^\bullet$ are termwise split injections, \item the quotients $F_{i + 1}P^\bullet/F_iP^\bullet$ are isomorphic to direct sums of shifts $R[k]$ (as complexes, so differentials are zero). \end{enumerate} This will be shown in Differential Graded Algebra, Lemma \ref{dga-lemma-resolve}. Moreover, given such a complex we obtain a distinguished triangle $$ \bigoplus F_iP^\bullet \to \bigoplus F_iP^\bullet \to M^\bullet \to \bigoplus F_iP^\bullet[1] $$ in $D(R)$. Using this we can sometimes reduce statements about general complexes to statements about $R[k]$ (this of course only works if the statement is preserved under taking direct sums). More precisely, let $T$ be a property of objects of $D(R)$. Suppose that \begin{enumerate} \item if $K_i \in D(R)$, $i \in I$ is a family of objects with $T(K_i)$ for all $i \in I$, then $T(\bigoplus K_i)$, \item if $K \to L \to M \to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object, \item $T(R[k])$ holds for all $k$. \end{enumerate} Then $T$ holds for all objects of $D(R)$.


more-algebra-remark-warning-compute-base-change

Let $R \to A$ be a ring map, and let $N$ and $N'$ be $A$-modules. Denote $N_R$ and $N'_R$ the restriction of $N$ and $N'$ to $R$-modules, see Algebra, Section \ref{algebra-section-base-change}. In this situation, the objects $N_R \otimes_R^\mathbf{L} N'$ and $N \otimes_R^\mathbf{L} N'_R$ of $D(A)$ are in general not isomorphic! In other words, one has to pay careful attention as to which of the two sides is being used to provide the $A$-module structure. \medskip\noindent For a specific example, set $R = k[x, y]$, $A = R/(xy)$, $N = R/(x)$ and $N' = A = R/(xy)$. The resolution $0 \to R \xrightarrow{xy} R \to N'_R \to 0$ shows that $N \otimes_R^\mathbf{L} N'_R = N[1] \oplus N$ in $D(A)$. The resolution $0 \to R \xrightarrow{x} R \to N_R \to 0$ shows that $N_R \otimes_R^\mathbf{L} N'$ is represented by the complex $A \xrightarrow{x} A$. To see these two complexes are not isomorphic, one can show that the second complex is not isomorphic in $D(A)$ to the direct sum of its cohomology groups, or one can show that the first complex is not a perfect object of $D(A)$ whereas the second one is. Some details omitted.


more-algebra-remark-sign-explanation

In the yoga of super vector spaces the sign used in the proof of Lemma \ref{lemma-evaluate-and-more} above can be explained as follows. A super vector space is just a finite dimensional vector space $V$ which comes with a direct sum decomposition $V = V^+ \oplus V^-$. Here we think of the elements of $V^+$ as the even elements and the elements of $V^-$ as the odd ones. Given two super vector spaces $V$ and $W$ we set $$ (V \otimes W)^+ = (V^+ \otimes W^+) \oplus (V^- \otimes W^-) $$ and similarly for the odd part. In the category of super vector spaces the isomorphism $$ \psi : V \otimes W \longrightarrow W \otimes V $$ is defined to be the usual one, except that on the summand $V^- \otimes W^-$ we use the negative of the usual identification. In this way we obtain a symmetric monoidal category, see Categories, Section \ref{categories-section-monoidal}. An object $V$ of the category of super vector spaces has a left dual which we denote $V^\vee$ which comes equipped with an identity $\eta : \mathbf{1} \to V \otimes V^\vee$ and an evaluation map $\epsilon : V^\vee \otimes V \to \mathbf{1}$ which induce canonical isomorphisms $\Hom(V, W) = W \otimes V^\vee$ and $\Hom(V^\vee, U) = V \otimes U$, see Categories, Lemma \ref{categories-lemma-left-dual}. Given three super vector spaces $U$, $V$, $W$ we can try to construct the analogue $$ c : \Hom(V, W) \otimes U \longrightarrow \Hom(\Hom(U, V), W) $$ of the maps $c_{p, r, s}$ which occur in the lemma above. Using the formulae given above (which do not involve signs) this becomes a map $$ W \otimes V^\vee \otimes U \longrightarrow W \otimes (V \otimes U^\vee)^\vee = W \otimes (U^\vee)^\vee \otimes V^\vee $$ To find this arrow in a canonical fashion we need to do two things: \begin{enumerate} \item we need to use the commutativity constraint $\psi : V^\vee \otimes U \to U \otimes V^\vee$ which introduces a sign on $(V^\vee)^- \otimes U^-$, and \item we need to use the canonical isomorphism $U \to (U^\vee)^\vee$ which comes from the identification of $U^\vee$ as the {\bf right} dual of $U$ using $\psi$ as in Categories, Lemma \ref{categories-lemma-dual-symmetric}. This differs from the usual identification by $-1$ on the odd part of $U$. \end{enumerate} Part (1) explains the sign $(-1)^{qr}$ in the proof of the lemma and part (2) explains the sign $(-1)^r$ in the proof of the lemma.


more-algebra-remark-smoothness-ext-1-zero

The following two statements follow from Lemma \ref{lemma-ext-1-zero}, Algebra, Definition \ref{algebra-definition-smooth}, and Algebra, Proposition \ref{algebra-proposition-characterize-formally-smooth}. \begin{enumerate} \item A ring map $A \to B$ is smooth if and only if $A \to B$ is of finite presentation and $\Ext^1_B(\NL_{B/A}, N) = 0$ for every $B$-module $N$. \item A ring map $A \to B$ is formally smooth if and only if $\Ext^1_B(\NL_{B/A}, N) = 0$ for every $B$-module $N$. \end{enumerate}


more-algebra-remark-Rlim-cohomology

Consider the category $\mathbf{N}$ whose objects are natural numbers and whose morphisms are unique arrows $i \to j$ if $j \geq i$. Endow $\mathbf{N}$ with the chaotic topology (Sites, Example \ref{sites-example-indiscrete}) so that a sheaf $\mathcal{F}$ is the same thing as an inverse system $$ \mathcal{F}_1 \leftarrow \mathcal{F}_2 \leftarrow \mathcal{F}_3 \leftarrow \ldots $$ of sets over $\mathbf{N}$. Note that $\Gamma(\mathbf{N}, \mathcal{F}) = \lim \mathcal{F}_n$. For an inverse system of abelian groups $\mathcal{F}_n$ we have $$ R^p\lim \mathcal{F}_n = H^p(\mathbf{N}, \mathcal{F}) $$ because both sides are the higher right derived functors of $\mathcal{F} \mapsto \lim \mathcal{F}_n = H^0(\mathbf{N}, \mathcal{F})$. Thus the existence of $R\lim$ also follows from the general material in Cohomology on Sites, Sections \ref{sites-cohomology-section-cohomology-sheaves} and \ref{sites-cohomology-section-unbounded}.


more-algebra-remark-compare-derived-limit

Let $(K_n)$ be an inverse system of objects of $D(\textit{Ab})$. Let $K = R\lim K_n$ be a derived limit of this system (see Derived Categories, Section \ref{derived-section-derived-limit}). Such a derived limit exists because $D(\textit{Ab})$ has countable products (Derived Categories, Lemma \ref{derived-lemma-products}). By Lemma \ref{lemma-lift-to-system-complexes-Ab} we can also lift $(K_n)$ to an object $M$ of $D(\mathbf{N})$. Then $K \cong R\lim M$ where $R\lim$ is the functor (\ref{equation-Rlim}) because $R\lim M$ is also a derived limit of the system $(K_n)$ by Lemma \ref{lemma-distinguished-triangle-Rlim}. Thus, although there may be many isomorphism classes of lifts $M$ of the system $(K_n)$, the isomorphism type of $R\lim M$ is independent of the choice because it is isomorphic to the derived limit $K = R\lim K_n$ of the system. Thus we may apply results on $R\lim$ proved in this section to derived limits. For example, for every $p \in \mathbf{Z}$ there is a canonical short exact sequence $$ 0 \to R^1\lim H^{p - 1}(K_n) \to H^p(K) \to \lim H^p(K_n) \to 0 $$ because we may apply Lemma \ref{lemma-distinguished-triangle-Rlim} to $M$. This can also been seen directly, without invoking the existence of $M$, by applying the argument of the proof of Lemma \ref{lemma-distinguished-triangle-Rlim} to the (defining) distinguished triangle $K \to \prod K_n \to \prod K_n \to K[1]$.


more-algebra-remark-Rlim-cohomology-modules

This remark is a continuation of Remark \ref{remark-Rlim-cohomology}. A sheaf of rings on $\mathbf{N}$ is just an inverse system of rings $(A_n)$. A sheaf of modules over $(A_n)$ is exactly the same thing as an object of the category $\textit{Mod}(\mathbf{N}, (A_n))$ defined above. The derived functor $R\lim$ of Lemma \ref{lemma-compute-Rlim-modules} is simply $R\Gamma(\mathbf{N}, -)$ from the derived category of modules to the derived category of modules over the global sections of the structure sheaf. It is true in general that cohomology of groups and modules agree, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-modules-abelian-agree}.


more-algebra-remark-how-unique

With assumptions as in Lemma \ref{lemma-lift-to-system-complexes}. A priori there are many isomorphism classes of objects $M$ of $D(\textit{Mod}(\mathbf{N}, (A_n)))$ which give rise to the system $(K_n, \varphi_n)$ of the lemma. For each such $M$ we can consider the complex $R\lim M \in D(A)$ where $A = \lim A_n$. By Lemma \ref{lemma-distinguished-triangle-Rlim-modules} we see that $R\lim M$ is a derived limit of the inverse system $(K_n)$ of $D(A)$. Hence we see that the isomorphism class of $R\lim M$ in $D(A)$ is independent of the choices made in constructing $M$. In particular, we may apply results on $R\lim$ proved in this section to derived limits of inverse systems in $D(A)$. For example, for every $p \in \mathbf{Z}$ there is a canonical short exact sequence $$ 0 \to R^1\lim H^{p - 1}(K_n) \to H^p(R\lim K_n) \to \lim H^p(K_n) \to 0 $$ because we may apply Lemma \ref{lemma-distinguished-triangle-Rlim-modules} to $M$. This can also been seen directly, without invoking the existence of $M$, by applying the argument of the proof of Lemma \ref{lemma-distinguished-triangle-Rlim-modules} to the (defining) distinguished triangle $R\lim K_n \to \prod K_n \to \prod K_n \to (R\lim K_n)[1]$ of the derived limit.


more-algebra-remark-constructing-tensor-with-limits-functorially

Let $A$ be a ring. Let $(E_n)$ be an inverse system of objects of $D(A)$. We've seen above that a derived limit $R\lim E_n$ exists. Thus for every object $K$ of $D(A)$ also the derived limit $R\lim( K \otimes_A^\mathbf{L} E_n )$ exists. It turns out that we can construct these derived limits functorially in $K$ and obtain an exact functor $$ R\lim(- \otimes_A^\mathbf{L} E_n) : D(A) \longrightarrow D(A) $$ of triangulated categories. Namely, we first lift $(E_n)$ to an object $E$ of $D(\mathbf{N}, A)$, see Lemma \ref{lemma-lift-to-system-complexes}. (The functor will depend on the choice of this lift.) Next, observe that there is a ``diagonal'' or ``constant'' functor $$ \Delta : D(A) \longrightarrow D(\mathbf{N}, A) $$ mapping the complex $K^\bullet$ to the constant inverse system of complexes with value $K^\bullet$. Then we simply define $$ R\lim(K \otimes_A^\mathbf{L} E_n) = R\lim(\Delta(K)\otimes^\mathbf{L} E) $$ where on the right hand side we use the functor $R\lim$ of Lemma \ref{lemma-compute-Rlim-modules} and the functor $- \otimes^\mathbf{L} -$ of Lemma \ref{lemma-derived-tensor-product-systems}.


more-algebra-remark-glueing-data

In this remark we define a category of glueing data. Let $R \to S$ be a ring map. Let $f_1, \ldots, f_t \in R$ and $I = (f_1, \ldots, f_t)$. Consider the category $\text{Glue}(R \to S, f_1, \ldots, f_t)$ as the category whose \begin{enumerate} \item objects are systems $(M', M_i, \alpha_i, \alpha_{ij})$, where $M'$ is an $S$-module, $M_i$ is an $R_{f_i}$-module, $\alpha_i : (M')_{f_i} \to M_i \otimes_R S$ is an isomorphism, and $\alpha_{ij} : (M_i)_{f_j} \to (M_j)_{f_i}$ are isomorphisms such that \begin{enumerate} \item $\alpha_{ij} \circ \alpha_i = \alpha_j$ as maps $(M')_{f_if_j} \to (M_j)_{f_i}$, and \item $\alpha_{jk} \circ \alpha_{ij} = \alpha_{ik}$ as maps $(M_i)_{f_jf_k} \to (M_k)_{f_if_j}$ (cocycle condition). \end{enumerate} \item morphisms $(M', M_i, \alpha_i, \alpha_{ij}) \to (N', N_i, \beta_i, \beta_{ij})$ are given by maps $\varphi' : M' \to N'$ and $\varphi_i : M_i \to N_i$ compatible with the given maps $\alpha_i, \beta_i, \alpha_{ij}, \beta_{ij}$. \end{enumerate} There is a canonical functor $$ \text{Can} : \text{Mod}_R \longrightarrow \text{Glue}(R \to S, f_1, \ldots, f_t), \quad M \longmapsto (M \otimes_R S, M_{f_i}, \text{can}_i, \text{can}_{ij}) $$ where $\text{can}_i : (M \otimes_R S)_{f_i} \to M_{f_i} \otimes_R S$ and $\text{can}_{ij} : (M_{f_i})_{f_j} \to (M_{f_j})_{f_i}$ are the canonical isomorphisms. For any object $\mathbf{M} = (M', M_i, \alpha_i, \alpha_{ij})$ of the category $\text{Glue}(R \to S, f_1, \ldots, f_t)$ we define $$ H^0(\mathbf{M}) = \{(m', m_i) \mid \alpha_i(m') = m_i \otimes 1, \alpha_{ij}(m_i) = m_j\} $$ in other words defined by the exact sequence $$ 0 \to H^0(\mathbf{M}) \to M' \times \prod M_i \to \prod M'_{f_i} \times \prod (M_i)_{f_j} $$ similar to (\ref{equation-glueing-complex}). We think of $H^0(\mathbf{M})$ as an $R$-module. Thus we also get a functor $$ H^0 : \text{Glue}(R \to S, f_1, \ldots, f_t) \longrightarrow \text{Mod}_R $$ Our next goal is to show that the functors $\text{Can}$ and $H^0$ are sometimes quasi-inverse to each other.


more-algebra-remark-formal-glueing-algebras

The equivalences of Proposition \ref{proposition-equivalence}, Theorem \ref{theorem-formal-glueing}, and Proposition \ref{proposition-formal-glueing} preserve properties of modules. For example if $M$ corresponds to $\mathbf{M} = (M', M_i, \alpha_i, \alpha_{ij})$ then $M$ is finite, or finitely presented, or flat, or projective over $R$ if and only if $M'$ and $M_i$ have the corresponding property over $S$ and $R_{f_i}$. This follows from the fact that $R \to S \times \prod R_{f_i}$ is faithfully flat and descend and ascent of these properties along faithfully flat maps, see Algebra, Lemma \ref{algebra-lemma-descend-properties-modules} and Theorem \ref{algebra-theorem-ffdescent-projectivity}. These functors also preserve the $\otimes$-structures on either side. Thus, it defines equivalences of various categories built out of the pair $(\text{Mod}_R, \otimes)$, such as the category of algebras.


more-algebra-remark-topological-analogue

Given a differential manifold $X$ with a compact closed submanifold $Z$ having complement $U$, specifying a sheaf on $X$ is the same as specifying a sheaf on $U$, a sheaf on an unspecified tubular neighbourhood $T$ of $Z$ in $X$, and an isomorphism between the two resulting sheaves along $T \cap U$. Tubular neighbourhoods do not exist in algebraic geometry as such, but results such as Proposition \ref{proposition-equivalence}, Theorem \ref{theorem-formal-glueing}, and Proposition \ref{proposition-formal-glueing} allow us to work with formal neighbourhoods instead.


more-algebra-remark-not-descent

While $R \to R_f$ is always flat, $R \to R^\wedge$ is typically not flat unless $R$ is Noetherian (see Algebra, Lemma \ref{algebra-lemma-completion-flat} and the discussion in Examples, Section \ref{examples-section-nonflat}). Consequently, we cannot in general apply faithfully flat descent as discussed in Descent, Section \ref{descent-section-descent-modules} to the morphism $R \to R^\wedge \oplus R_f$. Moreover, even in the Noetherian case, the usual definition of a descent datum for this morphism refers to the ring $R^\wedge \otimes_R R^\wedge$, which we will avoid considering in this section.


more-algebra-remark-BL-special-case

Suppose that $f$ is a nonzerodivisor. Then Algebra, Lemma \ref{algebra-lemma-completion-differ-by-torsion} shows that $f$ is a nonzerodivisor in $R^\wedge$. Hence $(R, f)$ is a glueing pair.


more-algebra-remark-noetherian-case

If $R \to R^\wedge$ is flat, then for each positive integer $n$ tensoring the sequence $0 \to R[f^n] \to R \to R$ with $R^\wedge$ gives the sequence $0 \to R[f^n] \otimes_R R^\wedge \to R^\wedge \to R^\wedge$. Combined with Lemma \ref{lemma-torsion-completion} we conclude that $R[f^n] \to R^\wedge[f^n]$ is an isomorphism. Thus $(R, f)$ is a glueing pair. This holds in particular if $R$ is Noetherian, see Algebra, Lemma \ref{algebra-lemma-completion-flat}.


more-algebra-remark-glueable

Let $(R \to R', f)$ be a glueing pair and let $M$ be an $R$-module. Here are some  observations which can be used to determine whether $M$ is glueable for $(R \to R', f)$. \begin{enumerate} \item By Lemma \ref{lemma-same-f-torsion-module} we see that $M$ is glueable for $(R \to R^\wedge, f)$ if and only if $M[f^\infty] \to M \otimes_R R^\wedge$ is injective. This holds if $M[f] \to M^\wedge$ is injective, i.e., when $M[f] \cap \bigcap_{n = 1}^\infty f^n M = 0$. \item If $\text{Tor}_1^R(M, R'_f) = 0$, then $M$ is glueable for $(R \to R', f)$ (use Algebra, Lemma \ref{algebra-lemma-long-exact-sequence-tor}). This is equivalent to saying that $\text{Tor}_1^R(M, R')$ is $f$-power torsion. In particular, any flat $R$-module is glueable for $(R \to R', f)$. \item If $R \to R'$ is flat, then $\text{Tor}_1^R(M, R') = 0$ for every $R$-module so every $R$-module is glueable for $(R \to R', f)$. This holds in particular when $R$ is Noetherian and $R' = R^\wedge$, see Algebra, Lemma \ref{algebra-lemma-completion-flat} \end{enumerate}


more-algebra-remark-what-you-get-for-general-modules

Let $(R \to R', f)$ be a glueing pair. Let $M$ be an $R$-module that is not necessarily glueable for $(R \to R', f)$. Setting $M' = M \otimes_R R'$ and $M_1 = M_f$ we obtain the glueing datum $\text{Can}(M) = (M', M_1, \text{can})$. Then $\tilde M = H^0(M', M_1, \text{can})$ is an $R$-module that is glueable for $(R \to R', f)$ and the canonical map $M \to \tilde M$ gives isomorphisms $M \otimes_R R' \to \tilde M \otimes_R R'$ and $M_f \to \tilde M_f$, see Theorem \ref{theorem-BL-glueing}. From the exactness of the sequences $$ M \to (M \otimes_R R' )\oplus M_f \to M \otimes_R (R')_f  \to 0 $$ and $$ 0 \to \tilde M \to (\tilde M \otimes_R R') \oplus \tilde M_f \to \tilde M \otimes_R  (R')_f \to 0 $$ we conclude that the map $M \to \tilde M$ is surjective.


more-algebra-remark-compare-BL

In \cite{Beauville-Laszlo} it is assumed that $f$ is a nonzerodivisor in $R$ and $R' = R^\wedge$, which gives a glueing pair by Lemma \ref{lemma-same-f-torsion}. Even in this setting Theorem \ref{theorem-BL-glueing} says something new: the results of \cite{Beauville-Laszlo} only apply to modules on which $f$ is a nonzerodivisor (and hence glueable in our sense, see Lemma \ref{lemma-same-f-torsion-module}). Lemma \ref{lemma-BL-properties} also provides a slight extension of the results of  \cite{Beauville-Laszlo}: not only can we allow $M$ to have nonzero $f$-power torsion, we do not even require it to be glueable.


more-algebra-remark-derived-completion

Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. The left adjoint to the inclusion functor $D_{comp}(A, I) \to D(A)$ which exists by Lemma \ref{lemma-derived-completion} is called the {\it derived completion}. To indicate this we will say ``let $K^\wedge$ be the derived completion of $K$''. Please keep in mind that the unit of the adjunction is a functorial map $K \to K^\wedge$.


more-algebra-remark-Leta

Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Suppose that $M^\bullet$ is a complex of $A$-modules. By Lemma \ref{lemma-K-flat-resolution} we can choose a quasi-isomorphism $K^\bullet \to M^\bullet$ such that $K^\bullet$ is K-flat and consists of flat $A$-modules. In particular, $f$ is a nonzerodivisor on $K^i$ for all $i$ and we have $\eta_fK^\bullet$ defined above. In this situation we define $$ L\eta_f M^\bullet = \eta_fK^\bullet $$ This is independent of the choice of the K-flat resolution by Lemma \ref{lemma-eta-qis}. We obtain a functor $L\eta_f : D(A) \to D(A)$. Beware that this functor isn't exact, i.e., does not tranform distinguished triangles into distinguished triangles.


more-algebra-remark-eta-BZ

Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. For every $i$ set $\overline{M}^i = M^i/fM^i$. Denote $B^i \subset Z^i \subset \overline{M}^i$ the boundaries and cocycles for the differentials on the complex $\overline{M}^\bullet = M^\bullet \otimes_A A/fA$. We claim that there exists a commutative diagram $$ \xymatrix{ 0 \ar[r] & B^{i + 1} \ar[r] \ar@{=}[d] & B^{i + 1} \oplus B^i \ar[r] \ar[d]^{s, s'} & B^i \ar[r] \ar[d] & 0 \\ 0 \ar[r] & B^{i + 1} \ar[r]^-s & (\eta_fM)^i /f(\eta_fM)^i \ar[r]^-t & Z^i \ar[r] & 0 } $$ with exact rows. Here are the constructions of the maps \begin{enumerate} \item If $x \in (\eta_fM)^i$ then $x = f^ix'$ with $d^i(x') = 0$ in $\overline{M}^{i + 1}$. Hence we can define the map $t$ by sending $x$ to the class of $x'$. \item If $y \in M^{i + 1}$ has class $\overline{y}$ in $B^{i + 1} \subset \overline{M}^{i + 1}$ then we can write $y = fy' + d^i(x)$ for $y' \in M^{i + 1}$ and $x \in M^i$. Hence we can define the map $s$ sending $\overline{y}$ to the class of $f^{i + 1}x$ in $(\eta_fM)^i /f(\eta_fM)^i$; we omit the verification that this is well defined. \item If $x \in M^i$ has class $\overline{x}$ in $B^i \subset \overline{M}^i$ then we can write $x = fx' + d^{i - 1}(z)$ for $x' \in M^i$ and $z \in M^{i - 1}$. We define the map $s'$ by sending $\overline{x}$ to the class of $f^i d^{i - 1}(z)$ in $(\eta_fM)^i/f(\eta_fM)^i$. This is well defined because if $fx' + d^{i - 1}(z) = 0$, then $f^ix'$ is in $(\eta_fM)^i$ and consequently $f^id^{i - 1}(z)$ is in $f(\eta_fM)^i$. \end{enumerate} We omit the verification that the lower row in the displayed diagram is a short exact sequence of modules. It is immediately clear from these constructions that we have commutative diagrams $$ \xymatrix{ B^{i + 1} \oplus B^i \ar[d]^{s, s'} \ar[r] & B^{i + 2} \oplus B^{i + 1} \ar[d]^{s, s'} \\ (\eta_fM)^i /f(\eta_fM)^i \ar[r] & (\eta_fM)^{i + 1} /f(\eta_fM)^{i + 1} } $$ where the upper horizontal arrow is given by the identification of the summands $B^{i + 1}$ in source and target. In other words, we have found an acyclic subcomplex of $\eta_fM^\bullet / f(\eta_fM^\bullet) = \eta_fM^\bullet \otimes_A A/fA$ and the quotient by this subcomplex is a complex whose terms $Z^i/B^i$ are the cohomology modules of the complex $\overline{M}^\bullet = M^\bullet \otimes_A A/fA$.


more-algebra-remark-weird-systems

Let $I$ be an ideal of a Noetherian ring $A$. Set $A_n = A/I^n$ for $n \geq 1$. Consider the following category: \begin{enumerate} \item An object is a sequence $\{E_n\}_{n \geq 1}$ where $E_n$ is a finite $A_n$-module. \item A morphism $\{E_n\} \to \{E'_n\}$ is given by maps $$ \varphi_n : I^cE_n \longrightarrow E'_n/E'_n[I^c] \quad\text{for }n \geq c $$ where $E'_n[I^c]$ is the torsion submodule (Section \ref{section-torsion}) up to equivalence: we say $(c, \varphi_n)$ is the same as $(c + 1, \overline{\varphi}_n)$ where $\overline{\varphi}_n : I^{c + 1}E_n \longrightarrow E'_n/E'_n[I^{c + 1}]$ is the induced map. \end{enumerate} Composition of $(c, \varphi_n) : \{E_n\} \to \{E'_n\}$ and $(c', \varphi'_n) : \{E'_n\} \to \{E''_n\}$ is defined by the obvious compositions $$ I^{c + c'}E_n \to I^{c'}E'_n/E'_n[I^{c}] \to E''_n/E''_n[I^{c + c'}] $$ for $n \geq c + c'$. We omit the verification that this is a category.


more-algebra-remark-awkward

The awkwardness in the statement of Lemma \ref{lemma-dejong-kollar-kovacs} is partly due to the fact that there are no obvious maps between the modules $\Ext^i_{A_n}(M_n, N_n)$ for varying $n$. What we may conclude from the lemma is that there exists a $c \geq 0$ such that for $m \gg n \gg 0$ there are (canonical) maps $$ I^c\Ext^i_{A_n}(M_m, N_m)/I^n\Ext^i_{A_n}(M_m, N_m) \to \Ext^i_{A_n}(M_n, N_n)/\Ext^i_{A_n}(M_n, N_n)[I^c] $$ whose kernel and cokernel are annihilated by $I^c$. This is the (weak) sense in which we get a system of modules.


more-algebra-remark-finite-separable-extension

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite separable field extension. Let $B \subset L$ be the integral closure of $A$ in $L$. Picture: $$ \xymatrix{ B \ar[r] & L \\ A \ar[u] \ar[r] & K \ar[u] } $$ By Algebra, Lemma \ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension} the ring extension $A \subset B$ is finite, hence $B$ is Noetherian. By Algebra, Lemma \ref{algebra-lemma-integral-sub-dim-equal} the dimension of $B$ is $1$, hence $B$ is a Dedekind domain, see Algebra, Lemma \ref{algebra-lemma-characterize-Dedekind}. Let $\mathfrak m_1, \ldots, \mathfrak m_n$ be the maximal ideals of $B$ (i.e., the primes lying over $\mathfrak m_A$). We obtain extensions of discrete valuation rings $$ A \subset B_{\mathfrak m_i} $$ and hence ramification indices $e_i$ and residue degrees $f_i$. We have $$ [L : K] = \sum\nolimits_{i = 1, \ldots, n} e_i f_i $$ by Algebra, Lemma \ref{algebra-lemma-finite-extension-dim-1} applied to a uniformizer in $A$. We observe that $n = 1$ if $A$ is henselian (by Algebra, Lemma \ref{algebra-lemma-finite-over-henselian}), e.g.\ if $A$ is complete.


more-algebra-remark-tower-of-rings

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension. Let $\mathfrak m \subset B$ be a maximal ideal of the integral closure of $A$ in $L$. Let $$ P \subset I \subset D \subset G $$ be the wild inertia, inertia, decomposition group of $\mathfrak m$. Consider the diagram $$ \xymatrix{ \mathfrak m \ar@{-}[d] \ar@{-}[r] & \mathfrak m^P \ar@{-}[d] \ar@{-}[r] & \mathfrak m^I \ar@{-}[d] \ar@{-}[r] & \mathfrak m^D \ar@{-}[d] \ar@{-}[r] & A \cap \mathfrak m \ar@{-}[d] \\ B & B^P \ar[l] & B^I \ar[l] & B^D \ar[l] & A \ar[l] } $$ Observe that $B^P, B^I, B^D$ are the integral closures of $A$ in the fields $L^P$, $L^I$, $L^D$. Thus we also see that $B^P$ is the integral closure of $B^I$ in $L^P$ and so on. Observe that $\mathfrak m^P = \mathfrak m \cap B^P$, $\mathfrak m^I = \mathfrak m \cap B^I$, and $\mathfrak m^D = \mathfrak m \cap B^D$. Hence the top line of the diagram corresponds to the images of $\mathfrak m \in \Spec(B)$ under the induced maps of spectra. Having said all of this we have the following \begin{enumerate} \item the extension $L^I/L^D$ is Galois with group $D/I$, \item the extension $L^P/L^I$ is Galois with group $I_t = I/P$, \item the extension $L^P/L^D$ is Galois with group $D/P$, \item $\mathfrak m^I$ is the unique prime of $B^I$ lying over $\mathfrak m^D$, \item $\mathfrak m^P$ is the unique prime of $B^P$ lying over $\mathfrak m^I$, \item $\mathfrak m$ is the unique prime of $B$ lying over $\mathfrak m^P$, \item $\mathfrak m^P$ is the unique prime of $B^P$ lying over $\mathfrak m^D$, \item $\mathfrak m$ is the unique prime of $B$ lying over $\mathfrak m^I$, \item $\mathfrak m$ is the unique prime of $B$ lying over $\mathfrak m^D$, \item $A \to B^D_{\mathfrak m^D}$ is \'etale and induces a trivial residue field extension, \item $B^D_{\mathfrak m^D} \to B^I_{\mathfrak m^I}$ is \'etale and induces a Galois extension of residue fields with Galois group $D/I$, \item $A \to B^I_{\mathfrak m^I}$ is \'etale, \item $B^I_{\mathfrak m^I} \to B^P_{\mathfrak m^P}$ has ramification index $|I/P|$ prime to $p$ and induces a trivial residue field extension, \item $B^D_{\mathfrak m^D} \to B^P_{\mathfrak m^P}$ has ramification index $|I/P|$ prime to $p$ and induces a separable residue field extension, \item $A \to B^P_{\mathfrak m^P}$ has ramification index $|I/P|$ prime to $p$ and induces a separable residue field extension. \end{enumerate} Statements (1), (2), and (3) are immediate from Galois theory (Fields, Section \ref{fields-section-galois-theory}) and Lemma \ref{lemma-galois-inertia}. Statements (4) -- (9) are clear from Lemma \ref{lemma-galois}. Part (12) is Lemma \ref{lemma-inertial-invariants-unramified}. Since we have the factorization $A \to B^D_{\mathfrak m^D} \to B^I_{\mathfrak m^I}$ we obtain the \'etaleness in (10) and (11) as a consequence. The residue field extension in (10) must be trivial because it is separable and $D/I$ maps onto $\text{Aut}(\kappa(\mathfrak m)/\kappa_A)$ as shown in Lemma \ref{lemma-galois-galois}. The same argument provides the proof of the statement on residue fields in (11). To see (13), (14), and (15) it suffices to prove (13). By the above, the extension $L^P/L^I$ is Galois with a cyclic Galois group of order prime to $p$, the prime $\mathfrak m^P$ is the unique prime lying over $\mathfrak m^I$ and the action of $I/P$ on the residue field is trivial. Thus we can apply Lemma \ref{lemma-galois-inertia} to this extension and the discrete valuation ring $B^I_{\mathfrak m^I}$ to see that (13) holds.


more-algebra-remark-canonical-inertia-character

In order to use the inertia character $\theta : I \to \mu_e(\kappa(\mathfrak m))$ for infinite Galois extensions, it is convenient to scale it. Let $A, K, L, B, \mathfrak m, G, P, I, D, e, \theta$ be as in Lemma \ref{lemma-galois-inertia} and Definition \ref{definition-wild-inertia}. Then $e = q |I_t|$ with $q$ is a power of the characteristic $p$ of $\kappa(\mathfrak m)$ if positive or $1$ if zero. Note that $\mu_e(\kappa(\mathfrak m)) = \mu_{|I_t|}(\kappa(\mathfrak m))$ because the characteristic of $\kappa(\mathfrak m)$ is $p$. Consider the map $$ \theta_{can} = q\theta : I \longrightarrow \mu_{|I_t|}(\kappa(\mathfrak m)) $$ This map induces an isomorphism $\theta_{can} : I_t \to \mu_{|I_t|}(\kappa(\mathfrak m))$. We have $\theta_{can}(\tau \sigma \tau^{-1}) = \tau(\theta_{can}(\sigma))$ for $\tau \in D$ and $\sigma \in I$ by Lemma \ref{lemma-inertia-character}. Finally, if $M/L$ is an extension such that $M/K$ is Galois and $\mathfrak m'$ is a prime of the integral closure of $A$ in $M$ lying over $\mathfrak m$, then we get the commutative diagram $$ \xymatrix{ I' \ar[r]_-{\theta'_{can}} \ar[d] & \mu_{|I'_t|}(\kappa(\mathfrak m')) \ar[d]^{(-)^{|I'_t|/|I_t|}} \\ I \ar[r]^-{\theta_{can}} & \mu_{|I_t|}(\kappa(\mathfrak m)) } $$ by Lemma \ref{lemma-compare-inertia}.


more-algebra-remark-construction

Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Let $K \subset K_1$ be a finite extension of fields. Let $A_1 \subset K_1$ be the integral closure of $A$ in $K_1$. On the other hand, let $L_1 = (L \otimes_K K_1)_{red}$. Then $L_1$ is a nonempty finite product of finite field extensions of $L$. Let $B_1$ be the integral closure of $B$ in $L_1$. We obtain compatible commutative diagrams $$ \vcenter{ \xymatrix{ L \ar[r] & L_1 \\ K \ar[u] \ar[r] & K_1 \ar[u] } } \quad\text{and}\quad \vcenter{ \xymatrix{ B \ar[r] & B_1 \\ A \ar[u] \ar[r] & A_1 \ar[u] } } $$ In this situation we have the following \begin{enumerate} \item By Algebra, Lemma \ref{algebra-lemma-integral-closure-Dedekind} the ring $A_1$ is a Dedekind domain and $B_1$ is a finite product of Dedekind domains. \item Note that $L \otimes_K K_1 = (B \otimes_A A_1)_\pi$ where $\pi \in A$ is a uniformizer and that $\pi$ is a nonzerodivisor on $B \otimes_A A_1$.  Thus the ring map $B \otimes_A A_1 \to B_1$ is integral with kernel consisting of nilpotent elements. Hence $\Spec(B_1) \to \Spec(B \otimes_A A_1)$ is surjective on spectra (Algebra, Lemma \ref{algebra-lemma-integral-overring-surjective}). The map $\Spec(B \otimes_A A_1) \to \Spec(A_1)$ is surjective as $A_1/\mathfrak m_A A_1 \to B/\mathfrak m_AB \otimes_{\kappa_A} A_1/\mathfrak m_A A_1$ is an injective ring map with $A_1/\mathfrak m_A A_1$ Artinian. We conclude that $\Spec(B_1) \to \Spec(A_1)$ is surjective. \item Let $\mathfrak m_i$, $i = 1, \ldots n$ with $n \geq 1$ be the maximal ideals of $A_1$. For each $i = 1, \ldots, n$ let $\mathfrak m_{ij}$, $j = 1, \ldots, m_i$ with $m_i \geq 1$ be the maximal ideals of $B_1$ lying over $\mathfrak m_i$. We obtain diagrams $$ \xymatrix{ B \ar[r] & (B_1)_{\mathfrak m_{ij}} \\ A \ar[u] \ar[r] & (A_1)_{\mathfrak m_i} \ar[u] } $$ of extensions of discrete valuation rings. \item If $A$ is henselian (for example complete), then $A_1$ is a discrete valuation ring, i.e., $n = 1$. Namely, $A_1$ is a union of finite extensions of $A$ which are domains, hence local by Algebra, Lemma \ref{algebra-lemma-finite-over-henselian}. \item If $B$ is henselian (for example complete), then $B_1$ is a product of discrete valuation rings, i.e., $m_i = 1$ for $i = 1, \ldots, n$. \item If $K \subset K_1$ is purely inseparable, then $A_1$ and $B_1$ are both discrete valuation rings, i.e., $n = 1$ and $m_1 = 1$. This is true because for every $b \in B_1$ a $p$-power power of $b$ is in $B$, hence $B_1$ can only have one maximal ideal. \item If $K \subset K_1$ is finite separable, then $L_1 = L \otimes_K K_1$ and is a finite product of finite separable extensions too. Hence $A \subset A_1$ and $B \subset B_1$ are finite by Algebra, Lemma \ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension}. \item If $A$ is Nagata, then $A \subset A_1$ is finite. \item If $B$ is Nagata, then $B \subset B_1$ is finite. \end{enumerate}


more-algebra-remark-determinant-as-socle

Let $R$ be a ring. Let $M$ be a finite projective $R$-module. Then we can consider the graded commutative $R$-algebra exterior algebra $\wedge^*_R(M)$ on $M$ over $R$. A formula for $\det(M)$ is that $\det(M) \subset \wedge^*_R(M)$ is the annihilator of $M \subset \wedge^*_R(M)$. This is sometimes useful as it does not refer to the decomposition of $R$ into a product. Of course, to prove this satisfies the desired properties one has to either decompose $R$ into a product (as above), or one has to look at the localizations at primes of $R$.


etale-remark-technicality-needed

The result on ``reducedness'' does not hold with a weaker definition of \'etale local ring maps $A \to B$ where one drops the assumption that $B$ is essentially of finite type over $A$. Namely, it can happen that a Noetherian local domain $A$ has nonreduced completion $A^\wedge$, see Examples, Section \ref{examples-section-local-completion-nonreduced}. But the ring map $A \to A^\wedge$ is flat, and $\mathfrak m_AA^\wedge$ is the maximal ideal of $A^\wedge$ and of course $A$ and $A^\wedge$ have the same residue fields. This is why it is important to consider this notion only for ring extensions which are essentially of finite type (or essentially of finite presentation if $A$ is not Noetherian).


crystalline-remark-completed-affine-site

In Situation \ref{situation-affine} we denote $\text{Cris}^\wedge(C/A)$ the category whose objects are pairs $(B \to C, \delta)$ such that \begin{enumerate} \item $B$ is a $p$-adically complete $A$-algebra, \item $B \to C$ is a surjection of $A$-algebras, \item $\delta$ is a divided power structure on $\Ker(B \to C)$, \item $A \to B$ is a homomorphism of divided power rings. \end{enumerate} Morphisms are defined as in Definition \ref{definition-affine-thickening}. Then $\text{Cris}(C/A) \subset \text{Cris}^\wedge(C/A)$ is the full subcategory consisting of those $B$ such that $p$ is nilpotent in $B$. Conversely, any object $(B \to C, \delta)$ of $\text{Cris}^\wedge(C/A)$ is equal to the limit $$ (B \to C, \delta) = \lim_e (B/p^eB \to C, \delta) $$ where for $e \gg 0$ the object $(B/p^eB \to C, \delta)$ lies in $\text{Cris}(C/A)$, see Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}. In particular, we see that $\text{Cris}^\wedge(C/A)$ is a full subcategory of the category of pro-objects of $\text{Cris}(C/A)$, see Categories, Remark \ref{categories-remark-pro-category}.


crystalline-remark-filtration-differentials

Let $A \to B$ be a ring map and let $(J, \delta)$ be a divided power structure on $B$. The universal module $\Omega_{B/A, \delta}$ comes with a little bit of extra structure, namely the $B$-submodule $N$ of $\Omega_{B/A, \delta}$ generated by $\text{d}_{B/A, \delta}(J)$. In terms of the isomorphism given in Lemma \ref{lemma-diagonal-and-differentials} this corresponds to the image of $K \cap J(1)$ in $\Omega_{B/A, \delta}$. Consider the $A$-algebra $D = B \oplus \Omega^1_{B/A, \delta}$ with ideal $\bar J = J \oplus N$ and divided powers $\bar \delta$ as in the proof of the lemma. Then $(D, \bar J, \bar \delta)$ is a divided power ring and the two maps $B \to D$ given by $b \mapsto b$ and $b \mapsto b + \text{d}_{B/A, \delta}(b)$ are homomorphisms of divided power rings over $A$. Moreover, $N$ is the smallest submodule of $\Omega_{B/A, \delta}$ such that this is true.


crystalline-remark-divided-powers-de-rham-complex

Let $A \to B$ be a ring map and let $(J, \delta)$ be a divided power structure on $B$. Set $\Omega_{B/A, \delta}^i = \wedge^i_B \Omega_{B/A, \delta}$ where $\Omega_{B/A, \delta}$ is the target of the universal divided power $A$-derivation $\text{d} = \text{d}_{B/A} : B \to \Omega_{B/A, \delta}$. Note that $\Omega_{B/A, \delta}$ is the quotient of $\Omega_{B/A}$ by the $B$-submodule generated by the elements $\text{d}\delta_n(x) - \delta_{n - 1}(x)\text{d}x$ for $x \in J$. We claim Algebra, Lemma \ref{algebra-lemma-de-rham-complex} applies. To see this it suffices to verify the elements $\text{d}\delta_n(x) - \delta_{n - 1}(x)\text{d}x$ of $\Omega_B$ are mapped to zero in $\Omega^2_{B/A, \delta}$. We observe that $$ \text{d}(\delta_{n - 1}(x)) \wedge \text{d}x = \delta_{n - 2}(x) \text{d}x \wedge \text{d}x = 0 $$ in $\Omega^2_{B/A, \delta}$ as desired. Hence we obtain a {\it divided power de Rham complex} $$ \Omega^0_{B/A, \delta} \to \Omega^1_{B/A, \delta} \to \Omega^2_{B/A, \delta} \to \ldots $$ which will play an important role in the sequel.


crystalline-remark-connection

Let $A \to B$ be a ring map. Let $\Omega_{B/A} \to \Omega$ be a quotient satisfying the assumptions of Algebra, Lemma \ref{algebra-lemma-de-rham-complex}. Let $M$ be a $B$-module. A {\it connection} is an additive map $$ \nabla : M \longrightarrow M \otimes_B \Omega $$ such that $\nabla(bm) = b \nabla(m) + m \otimes \text{d}b$ for $b \in B$ and $m \in M$. In this situation we can define maps $$ \nabla : M \otimes_B \Omega^i \longrightarrow M \otimes_B \Omega^{i + 1} $$ by the rule $\nabla(m \otimes \omega) = \nabla(m) \wedge \omega + m \otimes \text{d}\omega$. This works because if $b \in B$, then \begin{align*} \nabla(bm \otimes \omega) - \nabla(m \otimes b\omega) & = \nabla(bm) \wedge \omega + bm \otimes \text{d}\omega - \nabla(m) \wedge b\omega - m \otimes \text{d}(b\omega) \\ & = b\nabla(m) \wedge \omega + m \otimes \text{d}b \wedge \omega + bm \otimes \text{d}\omega \\ & \ \ \ \ \ \ - b\nabla(m) \wedge \omega - bm \otimes \text{d}(\omega) - m \otimes \text{d}b \wedge \omega = 0 \end{align*} As is customary we say the connection is {\it integrable} if and only if the composition $$ M \xrightarrow{\nabla} M \otimes_B \Omega^1 \xrightarrow{\nabla} M \otimes_B \Omega^2 $$ is zero. In this case we obtain a complex $$ M \xrightarrow{\nabla} M \otimes_B \Omega^1 \xrightarrow{\nabla} M \otimes_B \Omega^2 \xrightarrow{\nabla} M \otimes_B \Omega^3 \xrightarrow{\nabla} M \otimes_B \Omega^4 \to \ldots $$ which is called the de Rham complex of the connection.


crystalline-remark-base-change-connection

Consider a commutative diagram of rings $$ \xymatrix{ B \ar[r]_\varphi & B' \\ A \ar[u] \ar[r] & A' \ar[u] } $$ Let $\Omega_{B/A} \to \Omega$ and $\Omega_{B'/A'} \to \Omega'$ be quotients satisfying the assumptions of Algebra, Lemma \ref{algebra-lemma-de-rham-complex}. Assume there is a map $\varphi : \Omega \to \Omega'$ which fits into a commutative diagram $$ \xymatrix{ \Omega_{B/A} \ar[r] \ar[d] & \Omega_{B'/A'} \ar[d] \\ \Omega \ar[r]^{\varphi} & \Omega' } $$ where the top horizontal arrow is the canonical map $\Omega_{B/A} \to \Omega_{B'/A'}$ induced by $\varphi : B \to B'$. In this situation, given any pair $(M, \nabla)$ where $M$ is a $B$-module and $\nabla : M \to M \otimes_B \Omega$ is a connection we obtain a {\it base change} $(M \otimes_B B', \nabla')$ where $$ \nabla' : M \otimes_B B' \longrightarrow (M \otimes_B B') \otimes_{B'} \Omega' = M \otimes_B \Omega' $$ is defined by the rule $$ \nabla'(m \otimes b') = \sum m_i \otimes b'\text{d}\varphi(b_i) + m \otimes \text{d}b'  $$ if $\nabla(m) = \sum m_i \otimes \text{d}b_i$. If $\nabla$ is integrable, then so is $\nabla'$, and in this case there is a canonical map of de Rham complexes (Remark \ref{remark-connection}) \begin{equation} \label{equation-base-change-map-complexes} M \otimes_B \Omega^\bullet \longrightarrow (M \otimes_B B') \otimes_{B'} (\Omega')^\bullet = M \otimes_B (\Omega')^\bullet \end{equation} which maps $m \otimes \eta$ to $m \otimes \varphi(\eta)$.


crystalline-remark-functoriality-big-cris

Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Set $S_0 = V(\mathcal{I})$ and $S'_0 = V(\mathcal{I}')$. Let $$ \xymatrix{ X \ar[r]_f \ar[d] & Y \ar[d] \\ S_0 \ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $Y$. Then we get a continuous and cocontinuous functor $$ \text{CRIS}(X/S) \longrightarrow \text{CRIS}(Y/S') $$ by letting $(U, T, \delta)$ correspond to $(U, T, \delta)$ with $U \to X \to Y$ as the $S'$-morphism from $U$ to $Y$. Hence we get a morphism of topoi $$ f_{\text{CRIS}} : (X/S)_{\text{CRIS}} \longrightarrow (Y/S')_{\text{CRIS}} $$ see Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.


crystalline-remark-compare-big-zariski

In Situation \ref{situation-global}. The functor (\ref{equation-forget}) is cocontinuous (details omitted) and commutes with products and fibred products (Lemma \ref{lemma-divided-power-thickening-fibre-products}). Hence we obtain a morphism of topoi $$ U_{X/S} : (X/S)_{\text{CRIS}} \longrightarrow \Sh((\Sch/X)_{Zar}) $$ from the big crystalline topos of $X/S$ to the big Zariski topos of $X$. See Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.


crystalline-remark-big-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 morphisms of ringed topoi $$ \xymatrix{ (X/S)_{\text{CRIS}} \ar[r]_{f_{\text{CRIS}}} \ar[d]_{U_{X/S}} & (S_0/S)_{\text{CRIS}} \ar[d]^{U_{S_0/S}} \\ \Sh((\Sch/X)_{Zar}) \ar[r]^{f_{big}} & \Sh((\Sch/S_0)_{Zar}) \ar[rd] \\ & & \Sh((\Sch/S)_{Zar}) } $$ by Remark \ref{remark-functoriality-big-cris}. We think of the composition $(X/S)_{\text{CRIS}} \to \Sh((\Sch/S)_{Zar})$ as the structure morphism of the big crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism $$ (X/S)_{\text{CRIS}} \longrightarrow \Sh((\Sch/S)_{Zar}) $$ is defined as we can take the lower route through the diagram above. Thus it is the morphism of topoi corresponding to the cocontinuous functor $\text{CRIS}(X/S) \to (\Sch/S)_{Zar}$ given by the rule $(U, T, \delta)/S \mapsto U/S$, see Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.


crystalline-remark-compatibilities-big-cris

The morphisms defined above satisfy numerous compatibilities. For example, in the situation of Remark \ref{remark-functoriality-big-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-functoriality-cris

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 $$ \xymatrix{ X \ar[r]_f \ar[d] & Y \ar[d] \\ S_0 \ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $Y$. By analogy with Topologies, Lemma \ref{topologies-lemma-morphism-big-small} we define $$ f_{\text{cris}} : (X/S)_{\text{cris}} \longrightarrow (Y/S')_{\text{cris}} $$ by the formula $f_{\text{cris}} = \pi_Y \circ f_{\text{CRIS}} \circ i_X$ where $i_X$ and $\pi_Y$ are as in Lemma \ref{lemma-compare-big-small} for $X$ and $Y$ and where $f_{\text{CRIS}}$ is as in Remark \ref{remark-functoriality-big-cris}.


crystalline-remark-compare-zariski

In Situation \ref{situation-global}. The functor (\ref{equation-forget-small}) is continuous, cocontinuous, and commutes with products and fibred products. Hence we obtain a morphism of topoi $$ u_{X/S} : (X/S)_{\text{cris}} \longrightarrow \Sh(X_{Zar}) $$ relating the small crystalline topos of $X/S$ with the small Zariski topos of $X$. See Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.