spaces-simplicial-lemma-describe-functoriality

Let $f : Y \to X$ be a morphism of simplicial spaces. In terms of the description of sheaves in Lemma \ref{lemma-describe-sheaves-simplicial-site} the morphism $f_{Zar}$ of Lemma \ref{lemma-simplicial-space-site-functorial} can be described as follows. \begin{enumerate} \item If $\mathcal{G}$ is a sheaf on $Y$, then $(f_{Zar, *}\mathcal{G})_n = f_{n, *}\mathcal{G}_n$. \item If $\mathcal{F}$ is a sheaf on $X$, then $(f_{Zar}^{-1}\mathcal{F})_n = f_n^{-1}\mathcal{F}_n$. \end{enumerate}


spaces-simplicial-lemma-restriction-to-components

Let $X$ be a simplicial space. The functor $X_{n, Zar} \to X_{Zar}$, $U \mapsto U$ is continuous and cocontinuous. The associated morphism of topoi $g_n : \Sh(X_n) \to \Sh(X_{Zar})$ satisfies \begin{enumerate} \item $g_n^{-1}$ associates to the sheaf $\mathcal{F}$ on $X$ the sheaf $\mathcal{F}_n$ on $X_n$, \item $g_n^{-1} : \Sh(X_{Zar}) \to \Sh(X_n)$ has a left adjoint $g^{Sh}_{n!}$, \item $g^{Sh}_{n!}$ commutes with finite connected limits, \item $g_n^{-1} : \textit{Ab}(X_{Zar}) \to \textit{Ab}(X_n)$ has a left adjoint $g_{n!}$, and \item $g_{n!}$ is exact. \end{enumerate}


spaces-simplicial-lemma-restriction-injective-to-component

Let $X$ be a simplicial space. If $\mathcal{I}$ is an injective abelian sheaf on $X_{Zar}$, then $\mathcal{I}_n$ is an injective abelian sheaf on $X_n$.


spaces-simplicial-lemma-augmentation

Let $Y$ be a simplicial space and let $a : Y \to X$ be an augmentation (Simplicial, Definition \ref{simplicial-definition-augmentation}). Let $a_n : Y_n \to X$ be the corresponding morphisms of topological spaces. There is a canonical morphism of topoi $$ a : \Sh(Y_{Zar}) \to \Sh(X) $$ with the following properties: \begin{enumerate} \item $a^{-1}\mathcal{F}$ is the sheaf restricting to $a_n^{-1}\mathcal{F}$ on $Y_n$, \item $a_m \circ Y(\varphi) = a_n$ for all $\varphi : [m] \to [n]$, \item $a \circ g_n = a_n$ as morphisms of topoi with $g_n$ as in Lemma \ref{lemma-restriction-to-components}, \item $a_*\mathcal{G}$ for $\mathcal{G} \in \Sh(Y_{Zar})$ is the equalizer of the two maps $a_{0, *}\mathcal{G}_0 \to a_{1, *}\mathcal{G}_1$. \end{enumerate}


spaces-simplicial-lemma-simplicial-resolution-Z

Let $X$ be a simplicial topological space. The complex of abelian presheaves on $X_{Zar}$ $$ \ldots \to \mathbf{Z}_{X_2} \to \mathbf{Z}_{X_1} \to \mathbf{Z}_{X_0} $$ with boundary $\sum (-1)^i d^n_i$ is a resolution of the constant presheaf $\mathbf{Z}$.


spaces-simplicial-lemma-simplicial-sheaf-cohomology

Let $X$ be a simplicial topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_1^{p, q} = H^q(X_p, \mathcal{F}_p) $$ converging to $H^{p + q}(X_{Zar}, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$.


spaces-simplicial-lemma-simplicial-site-site

Let $\mathcal{C}$ be a simplicial object in the category of sites. With notation as above we construct a site $\mathcal{C}_{total}$ as follows. \begin{enumerate} \item An object of $\mathcal{C}_{total}$ is an object $U$ of $\mathcal{C}_n$ for some $n$, \item a morphism $(\varphi, f) : U \to V$ of $\mathcal{C}_{total}$ is given by a map $\varphi : [m] \to [n]$ with $U \in \Ob(\mathcal{C}_n)$, $V \in \Ob(\mathcal{C}_m)$ and a morphism $f : U \to u_\varphi(V)$ of $\mathcal{C}_n$, and \item a covering $\{(\text{id}, f_i) :  U_i \to U\}$ in $\mathcal{C}_{total}$ is given by an $n$ and a covering $\{f_i : U_i \to U\}$ of $\mathcal{C}_n$. \end{enumerate}


spaces-simplicial-lemma-simplicial-cocontinuous-site

Let $\mathcal{C}$ be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. With notation as above, assume the functors $u_\varphi : \mathcal{C}_n \to \mathcal{C}_m$ have property $P$ of Sites, Remark \ref{sites-remark-cartesian-cocontinuous}. Then we can construct a site $\mathcal{C}_{total}$ as follows. \begin{enumerate} \item An object of $\mathcal{C}_{total}$ is an object $U$ of $\mathcal{C}_n$ for some $n$, \item a morphism $(\varphi, f) : U \to V$ of $\mathcal{C}_{total}$ is given by a map $\varphi : [m] \to [n]$ with $U \in \Ob(\mathcal{C}_n)$, $V \in \Ob(\mathcal{C}_m)$ and a morphism $f : u_\varphi(U) \to V$ of $\mathcal{C}_m$, and \item a covering $\{(\text{id}, f_i) :  U_i \to U\}$ in $\mathcal{C}_{total}$ is given by an $n$ and a covering $\{f_i : U_i \to U\}$ of $\mathcal{C}_n$. \end{enumerate}


spaces-simplicial-lemma-describe-sheaves-simplicial-site-site

In Situation \ref{situation-simplicial-site} there is an equivalence of categories between \begin{enumerate} \item $\Sh(\mathcal{C}_{total})$, and \item the category of systems $(\mathcal{F}_n, \mathcal{F}(\varphi))$ described above. \end{enumerate} In particular, the topos $\Sh(\mathcal{C}_{total})$ only depends on the topoi $\Sh(\mathcal{C}_n)$ and the morphisms of topoi $f_\varphi$.


spaces-simplicial-lemma-restriction-to-components-site

In Situation \ref{situation-simplicial-site} the functor $\mathcal{C}_n \to \mathcal{C}_{total}$, $U \mapsto U$ is continuous and cocontinuous. The associated morphism of topoi $g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$ satisfies \begin{enumerate} \item $g_n^{-1}$ associates to the sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ the sheaf $\mathcal{F}_n$ on $\mathcal{C}_n$, \item $g_n^{-1} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}_n)$ has a left adjoint $g^{Sh}_{n!}$, \item for $\mathcal{G}$ in $\Sh(\mathcal{C}_n)$ the restriction of $g_{n!}^{Sh}\mathcal{G}$ to $\mathcal{C}_m$ is $\coprod\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}$, \item $g_{n!}^{Sh}$ commutes with finite connected limits, \item $g_n^{-1} : \textit{Ab}(\mathcal{C}_{total}) \to \textit{Ab}(\mathcal{C}_n)$ has a left adjoint $g_{n!}$, \item for $\mathcal{G}$ in $\textit{Ab}(\mathcal{C}_n)$ the restriction of $g_{n!}\mathcal{G}$ to $\mathcal{C}_m$ is $\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}$, and \item $g_{n!}$ is exact. \end{enumerate}


spaces-simplicial-lemma-restriction-injective-to-component-site

\begin{slogan} An injective abelian sheaf on a simplicial site is injective on each component \end{slogan} In Situation \ref{situation-simplicial-site}. If $\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_n$ is injective in $\textit{Ab}(\mathcal{C}_n)$. If $\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_n^\bullet$ is K-injective in $\textit{Ab}(\mathcal{C}_n)$.


spaces-simplicial-lemma-augmentation-site

In Situation \ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark \ref{remark-augmentation-site}. Then $a_0$ induces \begin{enumerate} \item a morphism of topoi $a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D})$ for all $n \geq 0$, \item a morphism of topoi $a : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})$ \end{enumerate} such that \begin{enumerate} \item for all $\varphi : [m] \to [n]$ we have $a_m \circ f_\varphi = a_n$, \item if $g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$ is as in Lemma \ref{lemma-restriction-to-components-site}, then $a \circ g_n = a_n$, and \item $a_*\mathcal{F}$ for $\mathcal{F} \in \Sh(\mathcal{C}_{total})$ is the equalizer of the two maps $a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1$. \end{enumerate}


spaces-simplicial-lemma-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}. Let $h$ be a morphism between simplicial sites as in Remark \ref{remark-morphism-simplicial-sites}. Then we obtain a morphism of topoi $$ h_{total} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total}) $$ and commutative diagrams $$ \xymatrix{ \Sh(\mathcal{C}_n) \ar[d]_{g_n} \ar[r]_{h_n} & \Sh(\mathcal{C}'_n) \ar[d]^{g'_n} \\ \Sh(\mathcal{C}_{total}) \ar[r]^{h_{total}} & \Sh(\mathcal{C}'_{total}) } $$ Moreover, we have $(g'_n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_n^{-1}$.


spaces-simplicial-lemma-direct-image-morphism-simplicial-sites

With notation and hypotheses as in Lemma \ref{lemma-morphism-simplicial-sites}. For $K \in D(\mathcal{C}_{total})$ we have $(g'_n)^{-1}Rh_{total, *}K = Rh_{n, *}g_n^{-1}K$.


spaces-simplicial-lemma-morphism-augmentation-simplicial-sites

Let $\mathcal{C}_n, f_\varphi, u_\varphi, \mathcal{D}, a_0$, $\mathcal{C}'_n, f'_\varphi, u'_\varphi, \mathcal{D}', a'_0$, and $h_n$, $n \geq -1$ be as in Remark \ref{remark-morphism-augmentation-simplicial-sites}. Then we obtain a commutative diagram $$ \xymatrix{ \Sh(\mathcal{C}_{total}) \ar[d]_a \ar[r]_{h_{total}} & \Sh(\mathcal{C}'_{total}) \ar[d]^{a'} \\ \Sh(\mathcal{D}) \ar[r]^{h_{-1}} & \Sh(\mathcal{D}') } $$


spaces-simplicial-lemma-restriction-module-to-components-site

In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. There is a canonical morphism of ringed topoi $g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{total}), \mathcal{O})$ agreeing with the morphism $g_n$ of Lemma \ref{lemma-restriction-to-components-site} on underlying topoi. The functor $g_n^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_n)$ has a left adjoint $g_{n!}$. For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_n)$-modules the restriction of $g_{n!}\mathcal{G}$ to $\mathcal{C}_m$ is $$ \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G} $$ where $f_\varphi : (\Sh(\mathcal{C}_m), \mathcal{O}_m) \to (\Sh(\mathcal{C}_n), \mathcal{O}_n)$ is the morphism of ringed topoi agreeing with the previously defined $f_\varphi$ on topoi and using the map $\mathcal{O}(\varphi) : f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ on sheaves of rings.


spaces-simplicial-lemma-restriction-injective-to-component-limp

In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then $\mathcal{I}_n$ is a totally acyclic sheaf on $\mathcal{C}_n$.


spaces-simplicial-lemma-exactness-g-shriek-modules

With assumptions as in Lemma \ref{lemma-restriction-module-to-components-site} the functor $g_{n!} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O})$ is exact if the maps $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ are flat for all $\varphi : [n] \to [m]$.


spaces-simplicial-lemma-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 $\mathcal{C}'_{total}$. Let $(h, h^\sharp)$ be a morphism between simplicial sites as in Remark \ref{remark-morphism-simplicial-sites-modules}. Then we obtain a morphism of ringed topoi $$ h_{total} : (\Sh(\mathcal{C}_{total}, \mathcal{O}) \to (\Sh(\mathcal{C}'_{total}), \mathcal{O}') $$ and commutative diagrams $$ \xymatrix{ (\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[d]_{g_n} \ar[r]_{h_n} & (\Sh(\mathcal{C}'_n), \mathcal{O}'_n) \ar[d]^{g'_n} \\ (\Sh(\mathcal{C}_{total}), \mathcal{O}) \ar[r]^{h_{total}} & (\Sh(\mathcal{C}'_{total}), \mathcal{O}') } $$ of ringed topoi where $g_n$ and $g'_n$ are as in Lemma \ref{lemma-restriction-module-to-components-site}. Moreover, we have $(g'_n)^* \circ h_{total, *} = h_{n, *} \circ g_n^*$ as functor $\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}'_n)$.


spaces-simplicial-lemma-direct-image-morphism-simplicial-sites-modules

With notation and hypotheses as in Lemma \ref{lemma-morphism-simplicial-sites-modules}. For $K \in D(\mathcal{O})$ we have $(g'_n)^*Rh_{total, *}K = Rh_{n, *}g_n^*K$.


spaces-simplicial-lemma-simplicial-resolution-Z-site

In Situation \ref{situation-simplicial-site} and with notation as above there is a complex $$ \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z} $$ of abelian sheaves on $\mathcal{C}_{total}$ which forms a resolution of the constant sheaf with value $\mathbf{Z}$ on $\mathcal{C}_{total}$.


spaces-simplicial-lemma-cech-complex

In Situation \ref{situation-simplicial-site}. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}_{total}$ there is a canonical complex $$ 0 \to \Gamma(\mathcal{C}_{total}, \mathcal{F}) \to \Gamma(\mathcal{C}_0, \mathcal{F}_0) \to \Gamma(\mathcal{C}_1, \mathcal{F}_1) \to \Gamma(\mathcal{C}_2, \mathcal{F}_2) \to \ldots $$ which is exact in degrees $-1, 0$ and exact everywhere if $\mathcal{F}$ is injective.


spaces-simplicial-lemma-simplicial-sheaf-cohomology-site

In Situation \ref{situation-simplicial-site}. For $K$ in $D^+(\mathcal{C}_{total})$ there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} $$ converging to $H^{p + q}(\mathcal{C}_{total}, K)$. This spectral sequence is functorial in $K$.


spaces-simplicial-lemma-sanity-check

Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}. Let $U \in \Ob(\mathcal{C}_n)$. Let $\mathcal{F} \in \textit{Ab}(\mathcal{C}_{total})$. Then $H^p(U, \mathcal{F}) = H^p(U, g_n^{-1}\mathcal{F})$ where on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$.


spaces-simplicial-lemma-simplicial-resolution-augmentation

In Situation \ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark \ref{remark-augmentation-site}. For any abelian sheaf $\mathcal{G}$ on $\mathcal{D}$  there is an exact complex $$ \ldots \to g_{2!}(a_2^{-1}\mathcal{G}) \to g_{1!}(a_1^{-1}\mathcal{G}) \to g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G} \to 0 $$ of abelian sheaves on $\mathcal{C}_{total}$.


spaces-simplicial-lemma-augmentation-cech-complex

In Situation \ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark \ref{remark-augmentation-site}. For an abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ there is a canonical complex $$ 0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots $$ on $\mathcal{D}$ which is exact in degrees $-1, 0$ and exact everywhere if $\mathcal{F}$ is injective.


spaces-simplicial-lemma-augmentation-spectral-sequence

In Situation \ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark \ref{remark-augmentation-site}. For any $K$ in $D^+(\mathcal{C}_{total})$ there is a spectral sequence  $(E_r, d_r)_{r \geq 0}$ with $$ E_1^{p, q} = R^qa_{p, *} K_p,\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} $$ converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.


spaces-simplicial-lemma-simplicial-resolution-ringed

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. There is a complex $$ \ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0 $$ of $\mathcal{O}$-modules which forms a resolution of $\mathcal{O}$. Here $g_{n!}$ is as in Lemma \ref{lemma-restriction-module-to-components-site}.


spaces-simplicial-lemma-cech-complex-modules

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. There is a canonical complex $$ 0 \to \Gamma(\mathcal{C}_{total}, \mathcal{F}) \to \Gamma(\mathcal{C}_0, \mathcal{F}_0) \to \Gamma(\mathcal{C}_1, \mathcal{F}_1) \to \Gamma(\mathcal{C}_2, \mathcal{F}_2) \to \ldots $$ which is exact in degrees $-1, 0$ and exact everywhere if $\mathcal{F}$ is an injective $\mathcal{O}$-module.


spaces-simplicial-lemma-simplicial-module-cohomology-site

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings. For $K$ in $D^+(\mathcal{O})$ there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} $$ converging to $H^{p + q}(\mathcal{C}_{total}, K)$. This spectral sequence is functorial in $K$.


spaces-simplicial-lemma-sanity-check-modules

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings. Let $U \in \Ob(\mathcal{C}_n)$. Let $\mathcal{F} \in \textit{Mod}(\mathcal{O})$. Then $H^p(U, \mathcal{F}) = H^p(U, g_n^*\mathcal{F})$ where on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$.


spaces-simplicial-lemma-flat-augmentation-modules

With notation as above. The morphism $a : (\Sh(\mathcal{C}_{total}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ is flat if and only if $a_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ is flat for $n \geq 0$.


spaces-simplicial-lemma-simplicial-resolution-augmentation-modules

With notation as above. For a $\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$ there is an exact complex $$ \ldots \to g_{2!}(a_2^*\mathcal{G}) \to g_{1!}(a_1^*\mathcal{G}) \to g_{0!}(a_0^*\mathcal{G}) \to a^*\mathcal{G} \to 0 $$ of sheaves of $\mathcal{O}$-modules on $\mathcal{C}_{total}$. Here $g_{n!}$ is as in Lemma \ref{lemma-restriction-module-to-components-site}.


spaces-simplicial-lemma-augmentation-cech-complex-modules

With notation as above. For an $\mathcal{O}$-module $\mathcal{F}$ on $\mathcal{C}_{total}$ there is a canonical complex $$ 0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots $$ of $\mathcal{O}_\mathcal{D}$-modules which is exact in degrees $-1, 0$. If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the complex is exact in all degrees and remains exact on applying the functor $\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, -)$ for any $\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$.


spaces-simplicial-lemma-check-cartesian-module

In Situation \ref{situation-simplicial-site}. \begin{enumerate} \item A sheaf $\mathcal{F}$ of sets or abelian groups is cartesian if and only if the maps $(f_{\delta^n_j})^{-1}\mathcal{F}_{n - 1} \to \mathcal{F}_n$ are isomorphisms. \item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only if the maps $(f_{\delta^n_j})^{-1}K_{n - 1} \to K_n$ are isomorphisms. \item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$ a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules is cartesian if and only if the maps $(f_{\delta^n_j})^*\mathcal{F}_{n - 1} \to \mathcal{F}_n$ are isomorphisms. \item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$ an object $K$ of $D(\mathcal{O})$ is cartesian if and only if the maps $L(f_{\delta^n_j})^*K_{n - 1} \to K_n$ are isomorphisms. \item Add more here. \end{enumerate}


spaces-simplicial-lemma-augmentation-cartesian-module

In Situation \ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark \ref{remark-augmentation-site}. \begin{enumerate} \item The pullback $a^{-1}\mathcal{G}$ of a sheaf of sets or abelian groups on $\mathcal{D}$ is cartesian. \item The pullback $a^{-1}K$ of an object $K$ of $D(\mathcal{D})$ is cartesian. \end{enumerate} Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{O}_\mathcal{D}$ a sheaf of rings on $\mathcal{D}$ and $a^\sharp : \mathcal{O}_\mathcal{D} \to a_*\mathcal{O}$ a morphism as in Section \ref{section-cohomology-augmentation-ringed-simplicial-sites}. \begin{enumerate} \item[(3)] The pullback $a^*\mathcal{F}$ of a sheaf of $\mathcal{O}_\mathcal{D}$-modules is cartesian. \item[(4)] The derived pullback $La^*K$ of an object $K$ of $D(\mathcal{O}_\mathcal{D})$ is cartesian. \end{enumerate}


spaces-simplicial-lemma-characterize-cartesian

In Situation \ref{situation-simplicial-site}. The category of cartesian sheaves of sets (resp.\ abelian groups) is equivalent to the category of pairs $(\mathcal{F}, \alpha)$ where $\mathcal{F}$ is a sheaf of sets (resp.\ abelian groups) on $\mathcal{C}_0$ and $$ \alpha : (f_{\delta_1^1})^{-1}\mathcal{F} \longrightarrow (f_{\delta_0^1})^{-1}\mathcal{F} $$ is an isomorphism of sheaves of sets (resp.\ abelian groups) on $\mathcal{C}_1$ such that $(f_{\delta^2_1})^{-1}\alpha = (f_{\delta^2_0})^{-1}\alpha \circ (f_{\delta^2_2})^{-1}\alpha$ as maps of sheaves on $\mathcal{C}_2$.


spaces-simplicial-lemma-characterize-cartesian-modules

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. The category of cartesian $\mathcal{O}$-modules is equivalent to the category of pairs $(\mathcal{F}, \alpha)$ where $\mathcal{F}$ is a $\mathcal{O}_0$-module and $$ \alpha : (f_{\delta_1^1})^*\mathcal{F} \longrightarrow (f_{\delta_0^1})^*\mathcal{F} $$ is an isomorphism of $\mathcal{O}_1$-modules such that $(f_{\delta^2_1})^*\alpha = (f_{\delta^2_0})^*\alpha \circ (f_{\delta^2_2})^*\alpha$ as $\mathcal{O}_2$-module maps.


spaces-simplicial-lemma-Serre-subcat-cartesian-modules

In Situation \ref{situation-simplicial-site}. \begin{enumerate} \item The full subcategory of cartesian abelian sheaves forms a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{total})$. Colimits of systems of cartesian abelian sheaves are cartesian. \item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that the morphisms $$ f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1}) $$ are flat. The full subcategory of cartesian $\mathcal{O}$-modules forms a weak Serre subcategory of $\textit{Mod}(\mathcal{O})$. Colimits of systems of cartesian $\mathcal{O}$-modules are cartesian. \end{enumerate}


spaces-simplicial-lemma-derived-cartesian-shriek

In Situation \ref{situation-simplicial-site}. \begin{enumerate} \item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only the canonical map $$ g_{n!}K_n \longrightarrow g_{n!}\mathbf{Z} \otimes^\mathbf{L}_\mathbf{Z} K $$ is an isomorphism for all $n$. \item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that the morphisms $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ are flat for all $\varphi : [n] \to [m]$. Then an object $K$ of $D(\mathcal{O})$ is cartesian if and only if the canonical map $$ g_{n!}K_n \longrightarrow g_{n!}\mathcal{O}_n \otimes^\mathbf{L}_\mathcal{O} K $$ is an isomorphism for all $n$. \end{enumerate}


spaces-simplicial-lemma-quasi-coherent-sheaf

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Then $\mathcal{F}$ is quasi-coherent in the sense of Modules on Sites, Definition \ref{sites-modules-definition-site-local} if and only if $\mathcal{F}$ is cartesian and $\mathcal{F}_n$ is a quasi-coherent $\mathcal{O}_n$-module for all $n$.


spaces-simplicial-lemma-cartesian-objects-derived

In Situation \ref{situation-simplicial-site}. If $K \in D(\mathcal{C}_{total})$ is an object, then $(K_n, K(\varphi))$ is a simplicial system of the derived category. If $K$ is cartesian, so is the system.


spaces-simplicial-lemma-abelian-postnikov

In Situation \ref{situation-simplicial-site}. Let $K$ be an object of $D(\mathcal{C}_{total})$. Set $$ X_n = (g_{n!}\mathbf{Z}) \otimes^\mathbf{L}_\mathbf{Z} K \quad\text{and}\quad Y_n = (g_{n!}\mathbf{Z} \to \ldots \to g_{0!}\mathbf{Z})[-n] \otimes^\mathbf{L}_\mathbf{Z} K $$ as objects of $D(\mathcal{C}_{total})$ where the maps are as in Lemma \ref{lemma-simplicial-resolution-Z-site}. With the evident canonical maps $Y_n \to X_n$ and $Y_0 \to Y_1[1] \to Y_2[2] \to \ldots$ we have \begin{enumerate} \item the distinguished triangles $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$ define a Postnikov system (Derived Categories, Definition \ref{derived-definition-postnikov-system}) for $\ldots \to X_2 \to X_1 \to X_0$, \item $K = \text{hocolim} Y_n[n]$ in $D(\mathcal{C}_{total})$. \end{enumerate}


spaces-simplicial-lemma-nullity-cartesian-objects-derived

In Situation \ref{situation-simplicial-site}. If $K, K' \in D(\mathcal{C}_{total})$. Assume \begin{enumerate} \item $K$ is cartesian, \item $\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and \item $\Hom(K_i[i + 1], K'_i) = 0$ for $i \geq 0$. \end{enumerate} Then any map $K \to K'$ which induces the zero map $K_0 \to K'_0$ is zero.


spaces-simplicial-lemma-hom-cartesian-objects-derived

In Situation \ref{situation-simplicial-site}. If $K, K' \in D(\mathcal{C}_{total})$. Assume \begin{enumerate} \item $K$ is cartesian, \item $\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$. \end{enumerate} Then any map $\{K_n \to K'_n\}$ between the associated simplicial systems  of $K$ and $K'$ comes from a map $K \to K'$ in $D(\mathcal{C}_{total})$.


spaces-simplicial-lemma-cartesian-object-derived-from-simplicial

In Situation \ref{situation-simplicial-site}. Let $(K_n, K_\varphi)$ be a simplicial system of the derived category. Assume \begin{enumerate} \item $(K_n, K_\varphi)$ is cartesian, \item $\Hom(K_i[t], K_i) = 0$ for $i \geq 0$ and $t > 0$. \end{enumerate} Then there exists a cartesian object $K$ of $D(\mathcal{C}_{total})$ whose associated simplicial system is isomorphic to $(K_n, K_\varphi)$.


spaces-simplicial-lemma-cartesian-objects-derived-modules

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K \in D(\mathcal{O})$ is an object, then $(K_n, K(\varphi))$ is a simplicial system of the derived category of modules. If $K$ is cartesian, so is the system.


spaces-simplicial-lemma-modules-postnikov

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $K$ be an object of $D(\mathcal{C}_{total})$. Set $$ X_n = (g_{n!}\mathcal{O}_n) \otimes^\mathbf{L}_\mathcal{O} K \quad\text{and}\quad Y_n = (g_{n!}\mathcal{O}_n \to \ldots \to g_{0!}\mathcal{O}_0)[-n] \otimes^\mathbf{L}_\mathcal{O} K $$ as objects of $D(\mathcal{O})$ where the maps are as in Lemma \ref{lemma-simplicial-resolution-Z-site}. With the evident canonical maps $Y_n \to X_n$ and $Y_0 \to Y_1[1] \to Y_2[2] \to \ldots$ we have \begin{enumerate} \item the distinguished triangles $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$ define a Postnikov system (Derived Categories, Definition \ref{derived-definition-postnikov-system}) for $\ldots \to X_2 \to X_1 \to X_0$, \item $K = \text{hocolim} Y_n[n]$ in $D(\mathcal{O})$. \end{enumerate}


spaces-simplicial-lemma-nullity-cartesian-modules-derived

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume \begin{enumerate} \item $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ is flat for $\varphi : [m] \to [n]$, \item $K$ is cartesian, \item $\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and \item $\Hom(K_i[i + 1], K'_i) = 0$ for $i \geq 0$. \end{enumerate} Then any map $K \to K'$ which induces the zero map $K_0 \to K'_0$ is zero.


spaces-simplicial-lemma-hom-cartesian-modules-derived

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume \begin{enumerate} \item $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ is flat for $\varphi : [m] \to [n]$, \item $K$ is cartesian, \item $\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$. \end{enumerate} Then any map $\{K_n \to K'_n\}$ between the associated simplicial systems  of $K$ and $K'$ comes from a map $K \to K'$ in $D(\mathcal{O})$.


spaces-simplicial-lemma-cartesian-module-derived-from-simplicial

In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $(K_n, K_\varphi)$ be a simplicial system of the derived category of modules. Assume \begin{enumerate} \item $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ is flat for $\varphi : [m] \to [n]$, \item $(K_n, K_\varphi)$ is cartesian, \item $\Hom(K_i[t], K_i) = 0$ for $i \geq 0$ and $t > 0$. \end{enumerate} Then there exists a cartesian object $K$ of $D(\mathcal{O})$ whose associated simplicial system is isomorphic to $(K_n, K_\varphi)$.


spaces-simplicial-lemma-push-pull-localization

Let $\mathcal{C}$ be a site and $K$ in $\text{SR}(\mathcal{C})$. For $\mathcal{F}$ in $\Sh(\mathcal{C})$ we have $$ j_*j^{-1}\mathcal{F} = \SheafHom(F(K)^\#, \mathcal{F}) $$ where $F$ is as in Hypercoverings, Definition \ref{hypercovering-definition-SR-F}.


spaces-simplicial-lemma-localize-compare

Let $\mathcal{C}$ be a site. \begin{enumerate} \item For $K$ in $\text{SR}(\mathcal{C})$ the functor $j_!$ gives an equivalence $\Sh(\mathcal{C}/K) \to \Sh(\mathcal{C})/F(K)^\#$ where $F$ is as in Hypercoverings, Definition \ref{hypercovering-definition-SR-F}. \item The functor $j^{-1} : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}/K)$ corresponds via the identification of (1) with $\mathcal{F} \mapsto (\mathcal{F} \times F(K)^\# \to F(K)^\#)$. \item For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $f^{-1}$ corresponds via the identifications of (1) to the functor $\Sh(\mathcal{C})/F(L)^\# \to \Sh(\mathcal{C})/F(K)^\#$, $(\mathcal{G} \to F(L)^\#) \mapsto (\mathcal{G} \times_{F(L)^\#} F(K)^\# \to F(K)^\#)$. \end{enumerate}


spaces-simplicial-lemma-localize-injective

Let $\mathcal{C}$ be a site. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j^{-1}$ sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves.


spaces-simplicial-lemma-augmentation-simplicial-semi-representable

Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\text{SR}(\mathcal{C})$. The localization functor $j_0 : \mathcal{C}/K_0 \to \mathcal{C}$ defines an augmentation $a_0 : \Sh(\mathcal{C}/K_0) \to \Sh(\mathcal{C})$, as in case (B) of Remark \ref{remark-augmentation-site}. The corresponding morphisms of topoi $$ a_n : \Sh(\mathcal{C}/K_n) \longrightarrow \Sh(\mathcal{C}),\quad a : \Sh((\mathcal{C}/K)_{total}) \longrightarrow \Sh(\mathcal{C}) $$ of Lemma \ref{lemma-augmentation-site} are equal to the morphisms of topoi associated to the continuous and cocontinuous localization functors $j_n : \mathcal{C}/K_n \to \mathcal{C}$ and $j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}$.


spaces-simplicial-lemma-comparison

With assumption and notation as in Lemma \ref{lemma-augmentation-simplicial-semi-representable} we have the following properties: \begin{enumerate} \item there is a functor $a^{Sh}_! : \Sh((\mathcal{C}/K)_{total}) \to \Sh(\mathcal{C})$ left adjoint to $a^{-1} : \Sh(\mathcal{C}) \to \Sh((\mathcal{C}/K)_{total})$, \item there is a functor $a_! : \textit{Ab}((\mathcal{C}/K)_{total}) \to \textit{Ab}(\mathcal{C})$ left adjoint to $a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})$, \item the functor $a^{-1}$ associates to $\mathcal{F}$ in $\Sh(\mathcal{C})$ the sheaf on $(\mathcal{C}/K)_{total}$ wich in degree $n$ is equal to $a_n^{-1}\mathcal{F}$, \item the functor $a_*$ associates to $\mathcal{G}$ in $\textit{Ab}((\mathcal{C}/K)_{total})$ the equalizer of the two maps $j_{0, *}\mathcal{G}_0 \to j_{1, *}\mathcal{G}_1$, \end{enumerate}


spaces-simplicial-lemma-sanity-check-simplicial-semi-representable

Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\text{SR}(\mathcal{C})$. Let $U/U_{n, i}$ be an object of $\mathcal{C}/K_n$. Let $\mathcal{F} \in \textit{Ab}((\mathcal{C}/K)_{total})$. Then $$ H^p(U, \mathcal{F}) = H^p(U, \mathcal{F}_{n, i}) $$ where \begin{enumerate} \item on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$, and \item on the right hand side $\mathcal{F}_{n, i}$ is the $i$th component of the sheaf $\mathcal{F}_n$ on $\mathcal{C}/K_n$ in the decomposition $\Sh(\mathcal{C}/K_n) = \prod \Sh(\mathcal{C}/U_{n, i})$ of Section \ref{section-semi-representable}. \end{enumerate}


spaces-simplicial-lemma-hypercovering-descent-sheaves

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then \begin{enumerate} \item $a^{-1} : \Sh(\mathcal{C}) \to \Sh((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets, \item $a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups. \end{enumerate} In both cases $a_*$ provides the quasi-inverse functor.


spaces-simplicial-lemma-hypercovering-cech-complex

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. The {\v C}ech complex of Lemma \ref{lemma-augmentation-cech-complex} associated to $a^{-1}\mathcal{F}$ $$ a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F} \to a_{2, *}a_2^{-1}\mathcal{F} \to \ldots $$ is equal to the complex $\SheafHom(s(\mathbf{Z}_{F(K)}^\#), \mathcal{F})$. Here $s(\mathbf{Z}_{F(K)}^\#)$ is as in Hypercoverings, Definition \ref{hypercovering-definition-homology}.


spaces-simplicial-lemma-hypercovering-descent-bounded-abelian

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. For $E \in D(\mathcal{C})$ the map $$ E \longrightarrow Ra_*a^{-1}E $$ is an isomorphism.


spaces-simplicial-lemma-compare-cohomology-hypercovering

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then we have a canonical isomorphism $$ R\Gamma(\mathcal{C}, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}E) $$ for $E \in D(\mathcal{C})$.


spaces-simplicial-lemma-hypercovering-equivalence-bounded

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence $$ D^+(\mathcal{C}) \longrightarrow D_\mathcal{A}^+((\mathcal{C}/K)_{total}) $$ with quasi-inverse $Ra_*$.


spaces-simplicial-lemma-hypercovering-descent-modules

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. With notation as above $$ a^* : \textit{Mod}(\mathcal{O}_\mathcal{C}) \to \textit{Mod}(\mathcal{O}) $$ is fully faithful with essential image the cartesian $\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse.


spaces-simplicial-lemma-hypercovering-descent-bounded-modules

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. For $E \in D(\mathcal{O}_\mathcal{C})$ the map $$ E \longrightarrow Ra_*La^*E $$ is an isomorphism.


spaces-simplicial-lemma-compare-cohomology-hypercovering-modules

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. Then we have a canonical isomorphism $$ R\Gamma(\mathcal{C}, E) = R\Gamma((\mathcal{C}/K)_{total}, La^*E) $$ for $E \in D(\mathcal{O}_\mathcal{C})$.


spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence $$ D^+(\mathcal{O}_\mathcal{C}) \longrightarrow D_\mathcal{A}^+(\mathcal{O}) $$ with quasi-inverse $Ra_*$.


spaces-simplicial-lemma-hypercovering-X-descent-sheaves

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $K$ be a hypercovering of $X$. Then \begin{enumerate} \item $a^{-1} : \Sh(\mathcal{C}/X) \to \Sh((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets, \item $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups. \end{enumerate} In both cases $a_*$ provides the quasi-inverse functor.


spaces-simplicial-lemma-hypercovering-X-descent-bounded-abelian

Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$. Let $K$ be a hypercovering of $X$. For $E \in D(\mathcal{C}/X)$ the map $$ E \longrightarrow Ra_*a^{-1}E $$ is an isomorphism.


spaces-simplicial-lemma-compare-cohomology-hypercovering-X

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $K$ be a hypercovering of $X$. Then we have a canonical isomorphism $$ R\Gamma(X, E) = R\Gamma((\mathcal{C}/K)_{total}, a^{-1}E) $$ for $E \in D(\mathcal{C}/X)$.


spaces-simplicial-lemma-hypercovering-X-equivalence-bounded

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $K$ be a hypercovering of $X$. Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence $$ D^+(\mathcal{C}/X) \longrightarrow D_\mathcal{A}^+((\mathcal{C}/K)_{total}) $$ with quasi-inverse $Ra_*$.


spaces-simplicial-lemma-hypercovering-X-descent-modules

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. With notation as above $$ a^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}) $$ is fully faithful with essential image the cartesian $\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse.


spaces-simplicial-lemma-hypercovering-X-descent-bounded-modules

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. For $E \in D(\mathcal{O}_X)$ the map $$ E \longrightarrow Ra_*La^*E $$ is an isomorphism.


spaces-simplicial-lemma-compare-cohomology-hypercovering-X-modules

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. Then we have a canonical isomorphism $$ R\Gamma(X, E) = R\Gamma((\mathcal{C}/K)_{total}, La^*E) $$ for $E \in D(\mathcal{O}_\mathcal{C})$.


spaces-simplicial-lemma-hypercovering-X-equivalence-bounded-modules

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence $$ D^+(\mathcal{O}_X) \longrightarrow D_\mathcal{A}^+(\mathcal{O}) $$ with quasi-inverse $Ra_*$.


spaces-simplicial-lemma-hypercovering-X-simple-descent-sheaves

Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. Then \begin{enumerate} \item $a^{-1} : \Sh(\mathcal{C}/X) \to \Sh((\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of sets, \item $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups. \end{enumerate} In both cases $a_*$ provides the quasi-inverse functor.


spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-abelian

Let $\mathcal{C}$ be a site with fibre product and $X \in \Ob(\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. For $E \in D(\mathcal{C}/X)$ the map $$ E \longrightarrow Ra_*a^{-1}E $$ is an isomorphism.


spaces-simplicial-lemma-sr-when-fibre-products

Let $U$ be a simplicial object of a site $\mathcal{C}$ with fibre products. \begin{enumerate} \item $\mathcal{C}/U$ has the structure of a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites, \item the construction of Lemma \ref{lemma-simplicial-site-site} applied to the structure in (1) reproduces the site $(\mathcal{C}/U)_{total}$ above, \item if $a : U \to X$ is an augmentation, then $a_0 : \mathcal{C}/U_0 \to \mathcal{C}/X$ is an augmentation as in Remark \ref{remark-augmentation-site} part (A) and gives the same morphism of topoi $a : \Sh((\mathcal{C}/U)_{total}) \to \Sh(\mathcal{C}/X)$ as the one above. \end{enumerate}


spaces-simplicial-lemma-hypercovering-X-simple-descent-modules

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\mathcal{C}$. With notation as above $$ a^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}) $$ is fully faithful with essential image the cartesian $\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse.


spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-modules

Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\mathcal{C}$. For $E \in D(\mathcal{O}_X)$ the map $$ E \longrightarrow Ra_*La^*E $$ is an isomorphism.


spaces-simplicial-lemma-hypercovering-equivalence-modules

Let $(\mathcal{C}, \mathcal{O}_\mathcal{C})$ be a ringed site. Assume given weak Serre subcategories $\mathcal{A}_U \subset \textit{Mod}(\mathcal{O}_U)$ satisfying conditions (\ref{item-restriction}), (\ref{item-local}), and (\ref{item-bounded-dimension}) above. Assume $\mathcal{C}$ has equalizers and fibre products and let $K$ be a hypercovering. Let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in Remark \ref{remark-augmentation-ringed}. Let $\mathcal{A}_{total} \subset \textit{Mod}(\mathcal{O})$ denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules $\mathcal{F}$ whose restriction $\mathcal{F}_n$ is in $\mathcal{A}_{K_n}$ for all $n$ (as defined above). Then the functor $La^*$ defines an equivalence $$ D_\mathcal{A}(\mathcal{O}_\mathcal{C}) \longrightarrow D_{\mathcal{A}_{total}}(\mathcal{O}) $$ with quasi-inverse $Ra_*$.


spaces-simplicial-lemma-prepare-bbd-glueing

In Situation \ref{situation-locally-given}. Assume negative self-exts of $E_U$ in $D(\mathcal{O}_{u(U)})$ are zero. Let $L$ be a simplicial object of $\text{SR}(\mathcal{B})$. Consider the simplicial object $K = u(L)$ of $\text{SR}(\mathcal{C})$ and let $((\mathcal{C}/K)_{total}, \mathcal{O})$ be as in Remark \ref{remark-augmentation-ringed}. There exists a cartesian object $E$ of $D(\mathcal{O})$ such that writing $L_n = \{U_{n, i}\}_{i \in I_n}$ the restriction of $E$ to $D(\mathcal{O}_{\mathcal{C}/u(U_{n, i})})$ is $E_{U_{n, i}}$ compatibly (see proof for details). Moreover, $E$ is unique up to unique isomorphism.


spaces-simplicial-lemma-bbd-glueing

In Situation \ref{situation-locally-given}. Assume \begin{enumerate} \item $\mathcal{C}$ has equalizers and fibre products, \item there is a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ such that \begin{enumerate} \item $\mathcal{D}$ has equalizers and fibre products and $u$ commutes with them, \item $\mathcal{B}$ is a full subcategory of $\mathcal{D}$ and $u : \mathcal{B} \to \mathcal{C}$ is the restriction of $u$, \item every object of $\mathcal{D}$ has a covering whose members are objects of $\mathcal{B}$, \end{enumerate} \item all negative self-exts of $E_U$ in $D(\mathcal{O}_{u(U)})$ are zero, and \item there exists a $t \in \mathbf{Z}$ such that $H^i(E_U) = 0$ for $i < t$ and $U \in \Ob(\mathcal{B})$. \end{enumerate} Then there exists a solution unique up to unique isomorphism.


spaces-simplicial-lemma-compare-simplicial-objects

Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. There is a commutative diagram $$ \xymatrix{ \Sh((\textit{LC}_{qc}/U)_{total}) \ar[r]_-h \ar[d]_{a_{qc}} & \Sh(U_{Zar}) \ar[d]^a \\ \Sh(\textit{LC}_{qc}/X) \ar[r]^-{h_{-1}} & \Sh(X) } $$ where the left vertical arrow is defined in Section \ref{section-hypercovering} and the right vertical arrow is defined in Lemma \ref{lemma-augmentation}.


spaces-simplicial-lemma-descent-sheaves-for-proper-hypercovering

Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $a : U \to X$ gives a proper hypercovering of $X$, then $$ a^{-1} : \Sh(X) \to \Sh(U_{Zar}) \quad\text{and}\quad a^{-1} : \textit{Ab}(X) \to \textit{Ab}(U_{Zar}) $$ are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \Sh(U_{Zar}) \to \Sh(X)$ is as in Lemma \ref{lemma-augmentation}.


spaces-simplicial-lemma-cohomological-descent-for-proper-hypercovering

Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $a : U \to X$ gives a proper hypercovering of $X$, then for $K \in D^+(X)$ $$ K \to Ra_*(a^{-1}K) $$ is an isomorphism where $a : \Sh(U_{Zar}) \to \Sh(X)$ is as in Lemma \ref{lemma-augmentation}.


spaces-simplicial-lemma-compute-via-proper-hypercovering

Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $U$ is a proper hypercovering of $X$, then $$ R\Gamma(X, K) = R\Gamma(U_{Zar}, a^{-1}K) $$ for $K \in D^+(X)$ where $a : \Sh(U_{Zar}) \to \Sh(X)$ is as in Lemma \ref{lemma-augmentation}.


spaces-simplicial-lemma-characterize-cartesian-schemes

Let $X$ be a simplicial scheme. The category of simplicial schemes cartesian over $X$ is equivalent to the category of pairs $(V, \varphi)$ where $V$ is a scheme over $X_0$ and $$ \varphi : V \times_{X_0, d^1_1} X_1 \longrightarrow X_1 \times_{d^1_0, X_0} V $$ is an isomorphism over $X_1$ such that $(s_0^0)^*\varphi = \text{id}_V$ and such that $$ (d^2_1)^*\varphi = (d^2_0)^*\varphi \circ (d^2_2)^*\varphi $$ as morphisms of schemes over $X_2$.


spaces-simplicial-lemma-cartesian-over

Let $f : X \to S$ be a morphism of schemes. Let $\pi : Y \to (X/S)_\bullet$ be a cartesian morphism of simplicial schemes. Set $V = Y_0$ considered as a scheme over $X$. The morphisms $d^1_0, d^1_1 : Y_1 \to Y_0$ and the morphism $\pi_1 : Y_1 \to X \times_S X$ induce isomorphisms $$ \xymatrix{ V \times_S X & & Y_1 \ar[ll]_-{(d^1_1, \text{pr}_1 \circ \pi_1)} \ar[rr]^-{(\text{pr}_0 \circ \pi_1, d^1_0)} & & X \times_S V. } $$ Denote $\varphi : V \times_S X \to X \times_S V$ the resulting isomorphism. Then the pair $(V, \varphi)$ is a descent datum relative to $X \to S$.


spaces-simplicial-lemma-cartesian-equivalent-descent-datum

Let $f : X \to S$ be a morphism of schemes. The construction $$ \begin{matrix} \text{category of cartesian } \\ \text{schemes over } (X/S)_\bullet \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/S \end{matrix} $$ of Lemma \ref{lemma-cartesian-over} is an equivalence of categories.


spaces-simplicial-lemma-pullback-cartesian-module

Let $f : V \to U$ be a morphism of simplicial schemes. Given a quasi-coherent module $\mathcal{F}$ on $U_{Zar}$ the pullback $f^*\mathcal{F}$ is a quasi-coherent module on $V_{Zar}$.


spaces-simplicial-lemma-pushforward-cartesian-module

Let $f : V \to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^n_j : U_n \to U_{n - 1}$ are flat and the morphisms $V_n \to U_n$ are quasi-compact and quasi-separated. For a quasi-coherent module $\mathcal{G}$ on $V_{Zar}$ the pushforward $f_*\mathcal{G}$ is a quasi-coherent module on $U_{Zar}$.


spaces-simplicial-lemma-groupoid-simplicial

Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. There exists a simplicial scheme $X$ over $S$ with the following properties \begin{enumerate} \item $X_0 = U$, $X_1 = R$, $X_2 = R \times_{s, U, t} R$, \item $s_0^0 = e : X_0 \to X_1$, \item $d^1_0 = s : X_1 \to X_0$, $d^1_1 = t : X_1 \to X_0$, \item $s_0^1 = (e \circ t, 1) : X_1 \to X_2$, $s_1^1 = (1, e \circ t) : X_1 \to X_2$, \item $d^2_0 = \text{pr}_1 : X_2 \to X_1$, $d^2_1 = c : X_2 \to X_1$, $d^2_2 = \text{pr}_0$, and \item $X = \text{cosk}_2 \text{sk}_2 X$. \end{enumerate} For all $n$ we have $X_n = R \times_{s, U, t} \ldots \times_{s, U, t} R$ with $n$ factors. The map $d^n_j : X_n \to X_{n - 1}$ is given on functors of points by $$ (r_1, \ldots, r_n) \longmapsto (r_1, \ldots, c(r_j, r_{j + 1}), \ldots, r_n) $$ for $1 \leq j \leq n - 1$ whereas $d^n_0(r_1, \ldots, r_n) = (r_2, \ldots, r_n)$ and $d^n_n(r_1, \ldots, r_n) = (r_1, \ldots, r_{n - 1})$.


spaces-simplicial-lemma-equivalence-relation

Let $f : X \to S$ be a morphism of schemes. Let $\pi : Y \to (X/S)_\bullet$ be a cartesian morphism of simplicial schemes, see Definitions \ref{definition-cartesian-morphism} and \ref{definition-fibre-products-simplicial-scheme}. Then the morphism $$ j = (d^1_1, d^1_0) : Y_1 \to Y_0 \times_S Y_0 $$ defines an equivalence relation on $Y_0$ over $S$, see Groupoids, Definition \ref{groupoids-definition-equivalence-relation}.


spaces-simplicial-lemma-equivalence-classes-points

Let $X \to S$ be a morphism of schemes. Suppose $Y \to (X/S)_\bullet$ is a cartesian morphism of simplicial schemes. For $y \in Y_0$ a point define $$ T_y = \{y' \in Y_0 \mid \exists\ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\} $$ as a subset of $Y_0$. Then $y \in T_y$ and $T_y \cap T_{y'} \not = \emptyset \Rightarrow T_y = T_{y'}$.


spaces-simplicial-lemma-quasi-compact

Let $X \to S$ be a morphism of schemes. Suppose $Y \to (X/S)_\bullet$ is a cartesian morphism of simplicial schemes. Let $y \in Y_0$ be a point. If $X \to S$ is quasi-compact, then $$ T_y = \{y' \in Y_0 \mid \exists\ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\} $$ is a quasi-compact subset of $Y_0$.


spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. There is a commutative diagram $$ \xymatrix{ \Sh((\textit{Spaces}/U)_{fppf, total}) \ar[r]_-h \ar[d]_{a_{fppf}} & \Sh(U_\etale) \ar[d]^a \\ \Sh((\textit{Spaces}/X)_{fppf}) \ar[r]^-{h_{-1}} & \Sh(X_\etale) } $$ where the left vertical arrow is defined in Section \ref{section-hypercovering} and the right vertical arrow is defined in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-descent-sheaves-for-fppf-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then $$ a^{-1} : \Sh(X_\etale) \to \Sh(U_\etale) \quad\text{and}\quad a^{-1} : \textit{Ab}(X_\etale) \to \textit{Ab}(U_\etale) $$ are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-cohomological-descent-for-fppf-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then for $K \in D^+(X_\etale)$ $$ K \to Ra_*(a^{-1}K) $$ is an isomorphism. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-compute-via-fppf-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then $$ R\Gamma(X_\etale, K) = R\Gamma(U_\etale, a^{-1}K) $$ for $K \in D^+(X_\etale)$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale-modules

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. There is a commutative diagram $$ \xymatrix{ (\Sh((\textit{Spaces}/U)_{fppf, total}), \mathcal{O}_{big, total}) \ar[r]_-h \ar[d]_{a_{fppf}} & (\Sh(U_\etale), \mathcal{O}_U) \ar[d]^a \\ (\Sh((\textit{Spaces}/X)_{fppf}), \mathcal{O}_{big}) \ar[r]^-{h_{-1}} & (\Sh(X_\etale), \mathcal{O}_X) } $$ of ringed topoi where the left vertical arrow is defined in Section \ref{section-hypercovering-modules} and the right vertical arrow is defined in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-descent-qcoh-for-fppf-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then $$ a^* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U) $$ is an equivalence fully faithful with quasi-inverse given by $a_*$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-cohomological-descent-qcoh-for-fppf-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then for $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module the map $$ \mathcal{F} \to Ra_*(a^*\mathcal{F}) $$ is an isomorphism. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Assume $a : U \to X$ is an fppf hypercovering of $X$. Then $\QCoh(\mathcal{O}_U)$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_U)$ and $$ a^* : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(\mathcal{O}_U) $$ is an equivalence of categories with quasi-inverse given by $Ra_*$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-compute-via-fppf-hypercovering-modules

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then $$ R\Gamma(X_\etale, K) = R\Gamma(U_\etale, a^*K) $$ for $K \in D_\QCoh(\mathcal{O}_X)$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-fppf-neg-ext-zero-hom

Let $X$ be an algebraic space over a scheme $S$. Let $K, E \in D_\QCoh(\mathcal{O}_X)$. Let $a : U \to X$ be an fppf hypercovering. Assume that for all $n \geq 0$ we have $$ \Ext_{\mathcal{O}_{U_n}}^i(La_n^*K, La_n^*E) = 0 \text{ for } i < 0 $$ Then we have \begin{enumerate} \item $\Ext_{\mathcal{O}_X}^i(K, E) = 0$ for $i < 0$, and \item there is an exact sequence $$ 0 \to \Hom_{\mathcal{O}_X}(K, E) \to \Hom_{\mathcal{O}_{U_0}}(La_0^*K, La_0^*E) \to \Hom_{\mathcal{O}_{U_1}}(La_1^*K, La_1^*E) $$ \end{enumerate}


spaces-simplicial-lemma-fppf-glue-neg-ext-zero

Let $X$ be an algebraic space over a scheme $S$. Let $a : U \to X$ be an fppf hypercovering. Suppose given $K_0 \in D_\QCoh(U_0)$ and an isomorphism $$ \alpha : L(f_{\delta_1^1})^*K_0 \longrightarrow L(f_{\delta_0^1})^*K_0 $$ satisfying the cocycle condition on $U_1$. Set $\tau^n_i : [0] \to [n]$, $0 \mapsto i$ and set $K_n = Lf_{\tau^n_n}^*K_0$. Assume $\Ext^i_{\mathcal{O}_{U_n}}(K_n, K_n) = 0$ for $i < 0$. Then there exists an object $K \in D_\QCoh(\mathcal{O}_X)$ and an isomorphism $La_0^*K \to K$ compatible with $\alpha$.


spaces-simplicial-lemma-compare-simplicial-objects-ph-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. There is a commutative diagram $$ \xymatrix{ \Sh((\textit{Spaces}/U)_{ph, total}) \ar[r]_-h \ar[d]_{a_{ph}} & \Sh(U_\etale) \ar[d]^a \\ \Sh((\textit{Spaces}/X)_{ph}) \ar[r]^-{h_{-1}} & \Sh(X_\etale) } $$ where the left vertical arrow is defined in Section \ref{section-hypercovering} and the right vertical arrow is defined in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-descent-sheaves-for-ph-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then $$ a^{-1} : \Sh(X_\etale) \to \Sh(U_\etale) \quad\text{and}\quad a^{-1} : \textit{Ab}(X_\etale) \to \textit{Ab}(U_\etale) $$ are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-cohomological-descent-for-ph-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then for $K \in D^+(X_\etale)$ $$ K \to Ra_*(a^{-1}K) $$ is an isomorphism. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


spaces-simplicial-lemma-compute-via-ph-hypercovering

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then $$ R\Gamma(X_\etale, K) = R\Gamma(U_\etale, a^{-1}K) $$ for $K \in D^+(X_\etale)$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


examples-stacks-lemma-quasi-coherent-strongly-cartesian

A morphism $(f, \varphi) : (Y, \mathcal{G}) \to (X, \mathcal{F})$ of $\QCohstack$ is strongly cartesian if and only if the map $\varphi$ induces an isomorphism $f^*\mathcal{F} \to \mathcal{G}$.


examples-stacks-lemma-stack-of-finite-type-quasi-coherent-sheaves

The functor $p_{fg} : \QCohstack_{fg} \to (\Sch/S)_{fppf}$ satisfies conditions (1), (2) and (3) of Stacks, Definition \ref{stacks-definition-stack}.


examples-stacks-lemma-finite-type

Let $(X, \mathcal{O}_X)$ be a ringed space. \begin{enumerate} \item The category of finite type $\mathcal{O}_X$-modules has a set of isomorphism classes. \item The category of finite type quasi-coherent $\mathcal{O}_X$-modules has a set of isomorphism classes. \end{enumerate}


examples-stacks-lemma-spaces-strongly-cartesian

A morphism $(f, g) : X/U \to Y/V$ of $\Spacesstack$ is strongly cartesian if and only if the map $f$ induces an isomorphism $X \to U \times_{g, V} Y$.


examples-stacks-lemma-pre-stack-of-spaces

The functor $p : \Spacesstack \to (\Sch/S)_{fppf}$ satisfies conditions (1) and (2) of Stacks, Definition \ref{stacks-definition-stack}.


examples-stacks-lemma-stack-of-finite-type-spaces

The functor $p_{ft} : \Spacesstack_{ft} \to (\Sch/S)_{fppf}$ satisfies the conditions (1), (2) and (3) of Stacks, Definition \ref{stacks-definition-stack}.


examples-stacks-lemma-torsors-sheaf-stack-in-groupoids

Up to a replacement as in Stacks, Remark \ref{stacks-remark-stack-make-small} the functor $$ p : \mathcal{G}\textit{-Torsors} \longrightarrow (\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\Sch/S)_{fppf}$.


examples-stacks-lemma-torsors-stack-in-groupoids

Up to a replacement as in Stacks, Remark \ref{stacks-remark-stack-make-small} the functor $$ p : G\textit{-Torsors} \longrightarrow (\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\Sch/S)_{fppf}$.


examples-stacks-lemma-compare-torsors

Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Denote $\mathcal{G}$, resp.\ $\mathcal{B}$ the algebraic space $G$, resp.\ $B$ seen as a sheaf on $(\Sch/S)_{fppf}$. The functor $$ G\textit{-Torsors} \longrightarrow \mathcal{G}/\mathcal{B}\textit{-Torsors} $$ which associates to a triple $(U, b, X)$ the triple $(U, b, \mathcal{X})$ where $\mathcal{X}$ is $X$ viewed as a sheaf is an equivalence of stacks in groupoids over $(\Sch/S)_{fppf}$.


examples-stacks-lemma-picard-stack

Up to a replacement as in Stacks, Remark \ref{stacks-remark-stack-make-small} the functor $$ \Picardstack_{X/B} \longrightarrow (\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\Sch/S)_{fppf}$.


examples-stacks-lemma-faithful-hilbert

The $1$-morphism $\mathcal{H}_d(\mathcal{X}/\mathcal{Y}) \to \mathcal{H}_d(\mathcal{X})$ is faithful.


examples-stacks-proposition-equal-quotient-stacks

In Situation \ref{situation-quotient-stack} there exists a canonical equivalence $$ [X/G] \longrightarrow [[X/G]] $$ of stacks in groupoids over $(\Sch/S)_{fppf}$.


models-theorem-semistable-reduction

\begin{reference} \cite[Corollary 2.7]{DM} \end{reference} Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$. Then there exists an extension of discrete valuation rings $R \subset R'$ which induces a finite separable extension of fraction fields $K \subset K'$ such that $C_{K'}$ has semistable reduction. More precisely, we have the following \begin{enumerate} \item If the genus of $C$ is zero, then there exists a degree $2$ separable extension $K'/K$ such that $C_{K'} \cong \mathbf{P}^1_{K'}$ and hence $C_{K'}$ is isomorphic to the generic fibre of the smooth projective scheme $\mathbf{P}^1_{R'}$ over the integral closure $R'$ of $R$ in $K'$. \item If the genus of $C$ is one, then there exists a finite separable extension $K'/K$ such that $C_{K'}$ has semistable reduction over $R'_\mathfrak m$ for every maximal ideal $\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$. Moreover, the special fibre of the (unique) minimal model of $C_{K'}$ over $R'_\mathfrak m$ is either a smooth genus one curve or a cycle of rational curves. \item If the genus $g$ of $C$ is greater than one, then there exists a finite separable extension $K'/K$ of degree at most $B_g$ (\ref{equation-bound}) such that $C_{K'}$ has semistable reduction over $R'_\mathfrak m$ for every maximal ideal $\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$. \end{enumerate}


models-lemma-recurring

\begin{reference} \cite[Theorem I]{Taussky} \end{reference} Let $A = (a_{ij})$ be a complex $n \times n$ matrix. \begin{enumerate} \item If $|a_{ii}| > \sum_{j \not = i} |a_{ij}|$ for each $i$, then $\det(A)$ is nonzero. \item If there exists a real vector $m = (m_1, \ldots, m_n)$ with $m_i > 0$ such that $|a_{ii} m_i| > \sum_{j \not = i} |a_{ij}m_j|$ for each $i$, then $\det(A)$ is nonzero. \end{enumerate}


models-lemma-recurring-real

Let $A = (a_{ij})$ be a real $n \times n$ matrix with $a_{ij} \geq 0$ for $i \not = j$. Let $m = (m_1, \ldots, m_n)$ be a real vector with $m_i > 0$. For $I \subset \{1, \ldots, n\}$ let $x_I \in \mathbf{R}^n$ be the vector whose $i$th coordinate is $m_i$ if $i \in I$ and $0$ otherwise. If \begin{equation} \label{equation-ineq} -a_{ii}m_i \geq \sum\nolimits_{j \not = i} a_{ij}m_j \end{equation} for each $i$, then $\Ker(A)$ is the vector space spanned by the vectors $x_I$ such that \begin{enumerate} \item $a_{ij} = 0$ for $i \in I$, $j \not \in I$, and \item equality holds in (\ref{equation-ineq}) for $i \in I$. \end{enumerate}


models-lemma-recurring-symmetric-real

Let $A = (a_{ij})$ be a symmetric real $n \times n$ matrix with $a_{ij} \geq 0$ for $i \not = j$. Let $m = (m_1, \ldots, m_n)$ be a real vector with $m_i > 0$. Assume \begin{enumerate} \item $Am = 0$, \item there is no proper nonempty subset $I \subset \{1, \ldots, n\}$ such that $a_{ij} = 0$ for $i \in I$ and $j \not \in I$. \end{enumerate} Then $x^t A x \leq 0$ with equality if and only if $x = qm$ for some $q \in \mathbf{R}$.


models-lemma-orthogonal-direct-sum

Let $L$ be a finite free $\mathbf{Z}$-module endowed with an integral symmetric bilinear positive definite form $\langle\ ,\ \rangle : L \times L \to \mathbf{Z}$. Let $A \subset L$ be a submodule with $L/A$ torsion free. Set $B = \{b \in L \mid \langle a, b\rangle = 0,\ \forall a \in A\}$. Then we have injective maps $$ A^\#/A \leftarrow L/(A \oplus B) \rightarrow B^\#/B $$ whose cokernels are quotients of $L^\#/L$. Here $A^\# = \{a' \in A \otimes \mathbf{Q} \mid \langle a, a'\rangle \in \mathbf{Z},\ \forall a \in A\}$ and similarly for $B$ and $L$.


models-lemma-coker

Let $L_0$, $L_1$ be a finite free $\mathbf{Z}$-modules endowed with integral symmetric bilinear positive definite forms $\langle\ ,\ \rangle : L_i \times L_i \to \mathbf{Z}$. Let $\text{d} : L_0 \to L_1$ and $\text{d}^* : L_1 \to L_0$ be adjoint. If $\langle\ ,\ \rangle$ on $L_0$ is unimodular, then there is an isomorphism $$ \Phi : \Coker(\text{d}^*\text{d})_{torsion} \longrightarrow \Im(\text{d})^\#/\Im(\text{d}) $$ with notation as in Lemma \ref{lemma-orthogonal-direct-sum}.


models-lemma-recurring-symmetric-integer

Let $A = (a_{ij})$ be a symmetric $n \times n$ integer matrix with $a_{ij} \geq 0$ for $i \not = j$. Let $m = (m_1, \ldots, m_n)$ be an integer vector with $m_i > 0$. Assume \begin{enumerate} \item $Am = 0$, \item there is no proper nonempty subset $I \subset \{1, \ldots, n\}$ such that $a_{ij} = 0$ for $i \in I$ and $j \not \in I$. \end{enumerate} Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$. Then for $\ell$ a prime number coprime with all $a_{ij}$ and $m_i$ we have $$ \dim_{\mathbf{F}_\ell}(\Coker(A)[\ell]) \leq 1 - n + e $$


models-lemma-genus

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. Then the expression $$ g = 1 + \sum m_i(w_i(g_i - 1) - \frac{1}{2} a_{ii}) $$ is an integer.


models-lemma-irreducible

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. If $n = 1$, then $a_{11} = 0$ and $g = 1 + m_1w_1(g_1 - 1)$. Moreover, we can classify all such numerical types as follows \begin{enumerate} \item If $g < 0$, then $g_1 = 0$ and there are finitely many possible numerical types of genus $g$ with $n = 1$ corresponding to factorizations $m_1w_1 = 1 - g$. \item If $g = 0$, then $m_1 = 1$, $w_1 = 1$, $g_1 = 0$ as in Lemma \ref{lemma-genus-zero}. \item If $g = 1$, then we conclude $g_1 = 1$ but $m_1, w_1$ can be arbitrary positive integers; this is case (\ref{item-one}) of Lemma \ref{lemma-genus-one}. \item If $g > 1$, then $g_1 > 1$ and there are finitely many possible numerical types of genus $g$ with $n = 1$ corresponding to factorizations $m_1w_1(g_1 - 1) = g - 1$. \end{enumerate}


models-lemma-diagonal-negative

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. If $n > 1$, then $a_{ii} < 0$ for all $i$.


models-lemma-minus-one

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Assume $n > 1$. If $i$ is such that the contribution $m_i(w_i(g_i - 1) - \frac{1}{2} a_{ii})$ to the genus $g$ is $< 0$, then $g_i = 0$ and $a_{ii} = -w_i$.


models-lemma-contract

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. Assume $n$ is a $(-1)$-index. Then there is a numerical type $T'$ given by $n', m'_i, a'_{ij}, w'_i, g'_i$ with \begin{enumerate} \item $n' = n - 1$, \item $m'_i = m_i$, \item $a'_{ij} = a_{ij} - a_{in}a_{jn}/a_{nn}$, \item $w'_i = w_i/2$ if $a_{in}/w_n$ even and $a_{in}/w_i$ odd and $w'_i = w_i$ else, \item $g'_i = \frac{w_i}{w'_i}(g_i - 1) + 1 + \frac{a_{in}^2 - w_na_{in}}{2w'_iw_n}$. \end{enumerate} Moreover, we have $g = g'$.


models-lemma-top-genus

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$. Then the expression $g_{top} = 1 - n + e$ is $\geq 0$.


models-lemma-non-irreducible-minimal-type-genus-at-least-one

If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type with $n > 1$, then $g \geq 1$.


models-lemma-genus-nonnegative

If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type with $n > 1$, then $g \geq g_{top}$.


models-lemma-minus-two

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Assume $n > 1$. If $i$ is such that the contribution $m_i(w_i(g_i - 1) - \frac{1}{2} a_{ii})$ to the genus $g$ is $0$, then $g_i = 0$ and $a_{ii} = -2w_i$.


models-lemma-picard-T-and-A

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. Then $\Pic(T) \subset \Coker(A)$ where $A = (a_{ij})$.


models-lemma-contract-picard-group

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. Assume $n$ is a $(-1)$-index. Let $T'$ be the numerical type constructed in Lemma \ref{lemma-contract}. There exists an injective map $$ \Pic(T) \to \Pic(T') $$ whose cokernel is an elementary abelian $2$-group.


models-lemma-two-by-two

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet } $$ If $n > 2$, then given a pair $i, j$ of $(-2)$-indices with $a_{ij} > 0$, then up to ordering we have the $m$'s, $a$'s, $w$'s \begin{enumerate} \item \label{item-A2} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w \\ w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \end{matrix} \right) $$ with $w$ arbitrary and $2m_1 \geq m_2$ and $2m_2 \geq m_1$, or \item \label{item-B2} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w \\ 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \end{matrix} \right) $$ with $w$ arbitrary and $m_1 \geq m_2$ and $2m_2 \geq m_1$, or \item \label{item-G2} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \end{matrix} \right), \quad \left( \begin{matrix} -2w & 3w \\ 3w & -6w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 3w \end{matrix} \right) $$ with $w$ arbitrary and $2m_1 \geq 3m_2$ and $2m_2 \geq m_1$. \end{enumerate}


models-lemma-three-by-three

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet } $$ If $n > 3$, then given a triple $i, j, k$ of $(-2)$-indices with at least two $a_{ij}, a_{ik}, a_{jk}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \begin{enumerate} \item \label{item-A3} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 \\ w & -2w & w \\ 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2$, or \item \label{item-C3} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 \\ w & -2w & 2w \\ 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ 2w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + 2m_3$, $2m_3 \geq m_2$, or \item \label{item-B3} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 \\ 2w & -4w & 2w \\ 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $m_3 \geq m_2$. \end{enumerate}


models-lemma-four-by-four

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet } $$ If $n > 4$, then given four $(-2)$-indices $i, j, k, l$ with $a_{ij}, a_{jk}, a_{kl}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \begin{enumerate} \item \label{item-A4} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 \\ w & -2w & w & 0 \\ 0 & w & -2w & w \\ 0 & 0 & w & -2w  \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4$, and $2m_4 \geq m_3$, or \item \label{item-C4} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 \\ w & -2w & w & 0 \\ 0 & w & -2w & 2w \\ 0 & 0 & 2w & -4w  \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ 2w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + 2m_4$, and $2m_4 \geq m_3$, or \item \label{item-B4} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 \\ 2w & -4w & 2w & 0 \\ 0 & 2w & -4w & 2w \\ 0 & 0 & 2w & -2w  \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4$, and $m_4 \geq m_3$, or \item \label{item-F4} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 \\ w & -2w & 2w & 0 \\ 0 & 2w & -4w & 2w \\ 0 & 0 & 2w & -4w  \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ 2w \\ 2w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + 2m_3$, $2m_3 \geq m_2 + m_4$, and $2m_4 \geq m_3$. \end{enumerate}


models-lemma-D4

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \\ & \bullet } $$ If $n > 4$, then given four $(-2)$-indices $i, j, k, l$ with $a_{ij}, a_{ik}, a_{il}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \begin{enumerate} \item \label{item-D4} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w & w \\ w & -2w & 0 & 0 \\ w & 0 & -2w & 0 \\ w & 0 & 0 & -2w  \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2 + m_3 + m_4$, $2m_2 \geq m_1$, $2m_3 \geq m_1$, $2m_4 \geq m_1$. Observe that this implies $m_1 \geq \max(m_2, m_3, m_4)$. \end{enumerate}


models-lemma-five-by-five

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet } $$ If $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$ with $a_{hi}, a_{ij}, a_{jk}, a_{kl}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \begin{enumerate} \item \label{item-A5} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & w \\ 0 & 0 & 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4$, $2m_4 \geq m_3 + m_5$, and $2m_5 \geq m_4$, or \item \label{item-C5} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & 2w \\ 0 & 0 & 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ 2w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + 2m_4$, $2m_4 \geq m_3 + m_5$, and $2m_5 \geq m_4$, or \item \label{item-B5} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -4w & 2w \\ 0 & 0 & 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ 2w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4$, $2m_4 \geq m_3 + m_5$, and $m_4 \geq m_3$. \end{enumerate}


models-lemma-fourfold

Nonexistence of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[ld] \ar@{-}[r] \ar@{-}[d] & \bullet \\ \bullet & \bullet } $$ If $n > 5$, there do {\bf not} exist five $(-2)$-indices $h$, $i$, $j$, $k$ with $a_{hi} > 0$, $a_{hj} > 0$, $a_{hk} > 0$, and $a_{hl} > 0$.


models-lemma-D5

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \\ & & \bullet } $$ If $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$ with $a_{hi}, a_{ij}, a_{jk}, a_{jl}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \begin{enumerate} \item \label{item-D5} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & w \\ 0 & 0 & w & -2w & 0 \\ 0 & 0 & w & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4 + m_5$, $2m_4 \geq m_3$, and $2m_5 \geq m_3$. \end{enumerate}


models-lemma-long

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{..}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet } $$ Let $t > 5$ and $n > t$. Then given $t$ distinct $(-2)$-indices $i_1, \ldots, i_t$ such that $a_{i_ji_{j + 1}}$ is nonzero for $j = 1, \ldots, t - 1$, then up to reversing the order of these indices we have the $a$'s and $w$'s \begin{enumerate} \item \label{item-An} are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_t} = w$, $a_{i_ji_{j + 1}} = w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or \item \label{item-Cn} are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_{t - 1}} = w$, $w_{j_t} = 2w$, $a_{i_ji_{j + 1}} = w$ for $j < t - 1$, $a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or \item \label{item-Bn} are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_{t - 1}} = 2w$, $w_{j_t} = w$, $a_{i_ji_{j + 1}} = 2w$, and $a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$. \end{enumerate}


models-lemma-Dn

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{..}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \\ & & & \bullet } $$ Let $t > 4$ and $n > t + 1$. Then given $t + 1$ distinct $(-2)$-indices $i_1, \ldots, i_{t + 1}$ such that $a_{i_ji_{j + 1}}$ is nonzero for $j = 1, \ldots, t - 1$ and $a_{i_{t - 1}i_{t + 1}}$ is nonzero, then we have the $a$'s and $w$'s \begin{enumerate} \item \label{item-Dn} are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_{t + 1}} = w$, $a_{i_ji_{j + 1}} = w$ for $j = 1, \ldots, t - 1$, $a_{i_{t - 1}i_{t + 1}} = w$ and $a_{i_ji_k} = 0$ for other pairs $(j, k)$ with $j > k$. \end{enumerate}


models-lemma-E6

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \\ & & \bullet } $$ Let $n > 6$. Then given $6$ distinct $(-2)$-indices $i_1, \ldots, i_6$ such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{36}$ are nonzero, then we have the $m$'s, $a$'s, and $w$'s \begin{enumerate} \item \label{item-E6} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & w \\ 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & w & 0 & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4 + m_6$, $2m_4 \geq m_3 + m_5$, $2m_5 \geq m_3$, and $2m_6 \geq m_3$. \end{enumerate}


models-lemma-double-triple

Nonexistence of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{..}[r] \ar@{-}[d] & \bullet \ar@{-}[d] \ar@{-}[r] & \bullet \\ & \bullet & \bullet } $$ Assume $t \geq 4$ and $n > t + 2$. There do {\bf not} exist $t + 2$ distinct $(-2)$-indices $i_0, \ldots, i_{t + 1}$ such that $a_{i_ji_{j + 1}} > 0$ for $j = 1, \ldots, t - 1$ and $a_{i_0i_2} > 0$ and $a_{i_{t - 1}i_{t + 1}} > 0$.


models-lemma-E6-completed

Nonexistence of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \\ & & \bullet \ar@{-}[d] \\ & & \bullet } $$ Assume $n > 7$. There do {\bf not} exist $7$ distinct $(-2)$-indices $f, g, h, i, j, k, l$ such that $a_{fg}, a_{gh}, a_{ij}, a_{jh}, a_{kl}, a_{lh}$ are nonzero.


models-lemma-E7

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \\ & & & \bullet } $$ Let $n > 7$. Then given $7$ distinct $(-2)$-indices $i_1, \ldots, i_7$ such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{47}$ are nonzero, then we have the $m$'s, $a$'s, and $w$'s \begin{enumerate} \item \label{item-E7} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & w \\ 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & w & 0 & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4$, $2m_4 \geq m_3 + m_5 + m_7$, $2m_5 \geq m_4 + m_6$, $2m_6 \geq m_5$, and $2m_7 \geq m_4$. \end{enumerate}


models-lemma-E8

Classification of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \\ & & & & \bullet } $$ Let $n > 8$. Then given $8$ distinct $(-2)$-indices $i_1, \ldots, i_8$ such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{65}, a_{57}$ are nonzero, then we have the $m$'s, $a$'s, and $w$'s \begin{enumerate} \item \label{item-E8} are given by $$ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & 0 & 0 \\ 0 & 0 & 0 & w & -2w & w & 0 & w \\ 0 & 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right) $$ with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4$, $2m_4 \geq m_3 + m_5$, $2m_5 \geq m_4 + m_6 + m_8$, $2m_6 \geq m_5 + m_7$, $2m_7 \geq m_6$, and $2m_8 \geq m_5$. \end{enumerate}


models-lemma-E7-completed

Nonexistence of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \\ & & & \bullet } $$ Assume $n > 8$. There do {\bf not} exist $8$ distinct $(-2)$-indices $e, f, g, h, i, j, k, l$ such that $a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$ are nonzero.


models-lemma-E8-completed

Nonexistence of proper subgraphs of the form $$ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \\ & & & & & \bullet } $$ Assume $n > 9$. There do {\bf not} exist $9$ distinct $(-2)$-indices $d, e, f, g, h, i, j, k, l$ such that $a_{de}, a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$ are nonzero.


models-lemma-genus-zero

The only minimal numerical type of genus zero is $n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 1$, $g_1 = 0$.


models-lemma-genus-one

The minimal numerical types of genus one are up to equivalence \begin{enumerate} \item \label{item-one} $n = 1$, $a_{11} = 0$, $g_1 = 1$, $m_1, w_1 \geq 1$ arbitrary, \item \label{item-two-cycle} $n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w \\ 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-up4} $n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 4w \\ 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-three-cycle} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w \\ w & -2w & w \\ w & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-equal-up3} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 \\ w & -2w & 3w \\ 0 & 3w & -6w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ 3w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-equal-down3} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 3m \end{matrix} \right), \quad \left( \begin{matrix} -6w & 3w & 0 \\ 3w & -6w & 3w \\ 0 & 3w & -2w \end{matrix} \right), \quad \left( \begin{matrix} 3w \\ 3w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-up-up} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 \\ 2w & -4w & 4w \\ 0 & 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-up-down} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 \\ 2w & -4w & 2w \\ 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-down-up} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 \\ 2w & -2w & 2w \\ 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-four-cycle} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & w \\ w & -2w & w & 0 \\ 0 & w & -2w & w \\ w & 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-up-equal-up} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 \\ 2w & -4w & 2w & 0 \\ 0 & 2w & -4w & 4w \\ 0 & 0 & 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-up-equal-down} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 \\ 2w & -4w & 2w & 0 \\ 0 & 2w & -4w & 2w \\ 0 & 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-down-equal-up} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 \\ 2w & -2w & w & 0 \\ 0 & w & -2w & 2w \\ 0 & 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-triple-with-up} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w & 2w \\ w & -2w & 0 & 0 \\ w & 0 & -2w & 0 \\ 2w & 0 & 0 & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-triple-with-down} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 2w & 2w \\ 2w & -4w & 0 & 0 \\ 2w & 0 & -4w & 0 \\ 2w & 0 & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-five-cycle} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & w \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & w \\ w & 0 & 0 & w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-equal-equal-up-equal} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 3m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & 2w & 0 \\ 0 & 0 & 2w & -4w & 2w \\ 0 & 0 & 0 & 2w & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ 2w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-equal-equal-down-equal} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -2w & w \\ 0 & 0 & 0 & w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-up-equal-equal-up} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -4w & 4w \\ 0 & 0 & 0 & 4w & -8w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-up-equal-equal-down} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -4w & 2w \\ 0 & 0 & 0 & 2w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-down-equal-equal-up} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & 2w \\ 0 & 0 & 0 & 2w & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-quadruple} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w & w & w \\ w & -2w & 0 & 0 & 0 \\ w & 0 & -2w & 0 & 0 \\ w & 0 & 0 & -2w & 0 \\ w & 0 & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-triple-extended-up} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 2m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & w \\ 0 & 0 & w & -2w & 0 \\ 0 & 0 & w & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-triple-extended-down} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} 2m \\ 2m \\ 2m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 2w \\ 0 & 0 & 2w & -4w & 0 \\ 0 & 0 & 2w & 0 & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-n-cycle} $n \geq 6$ and we have an $n$-cycle generalizing (\ref{item-five-cycle}): \begin{enumerate} \item $m_1 = \ldots = m_n = m$, \item $a_{12} = \ldots = a_{(n - 1) n} = w$, $a_{1n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \item $w_1 = \ldots = w_n = w$ \end{enumerate} with $w$ and $m$ arbitrary, \item \label{item-up-chain-equal-up} $n \geq 6$ and we have a chain generalizing (\ref{item-up-equal-equal-up}): \begin{enumerate} \item $m_1 = \ldots = m_{n - 1} = 2m$, $m_n = m$, \item $a_{12} = \ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 1) n} = 4w$, and for other $i < j$ we have $a_{ij} = 0$, \item $w_1 = w$, $w_2 = \ldots = w_{n - 1} = 2w$, $w_n = 4w$ \end{enumerate} with $w$ and $m$ arbitrary, \item \label{item-up-chain-equal-down} $n \geq 6$ and we have a chain generalizing (\ref{item-up-equal-equal-down}): \begin{enumerate} \item $m_1 = \ldots = m_n = m$, \item $a_{12} = \ldots = a_{(n - 1) n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \item $w_1 = w$, $w_2 = \ldots = w_{n - 1} = 2w$, $w_n = w$ \end{enumerate} with $w$ and $m$ arbitrary, \item \label{item-down-chain-equal-up} $n \geq 6$ and we have a chain generalizing (\ref{item-down-equal-equal-up}): \begin{enumerate} \item $m_1 = w$, $w_2 = \ldots = m_{n - 1} = 2m$, $m_n = m$, \item $a_{12} = 2w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 1) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$, \item $w_1 = 2w$, $w_2 = \ldots = w_{n - 1} = w$, $w_n = 2w$ \end{enumerate} with $w$ and $m$ arbitrary, \item \label{item-Dn-extended-up} $n \geq 6$ and we have a type generalizing (\ref{item-triple-extended-up}): \begin{enumerate} \item $m_1 = m$, $m_2 = \ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$, \item $a_{12} = 2w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \item $w_1 = 2w$, $w_2 = \ldots = w_n = w$ \end{enumerate} with $w$ and $m$ arbitrary, \item \label{item-Dn-extended-down} $n \geq 6$ and we have a type generalizing (\ref{item-triple-extended-down}): \begin{enumerate} \item $m_1 = \ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$, \item $a_{12} = \ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 2) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$, \item $w_1 = w$, $w_2 = \ldots = w_n = 2w$ \end{enumerate} with $w$ and $m$ arbitrary, \item \label{item-double-triple} $n \geq 6$ and we have a type generalizing (\ref{item-quadruple}): \begin{enumerate} \item $m_1 = m_2 = m$, $m_3 = \ldots = m_{n - 2} = 2m$, $m_{n - 1} = m_n = m$, \item $a_{13} = w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \item $w_1 = \ldots = w_n = w$, \end{enumerate} with $w$ and $m$ arbitrary, \item \label{item-E6-completed} $n = 7$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 3m \\ m \\ 2m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & w & -2w & 0 & w & 0 & w \\ 0 & 0 & 0 & -2w & w & 0 & 0 \\ 0 & 0 & w & w & -2w & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -2w & w \\ 0 & 0 & w & 0 & 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-E7-completed} $n = 8$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 3m \\ 2m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & 0 & w \\ 0 & 0 & 0 & w & -2w & w & 0 & 0 \\ 0 & 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & w & 0 & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary, \item \label{item-E8-completed} $n = 9$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 5m \\ 6m \\ 4m \\ 2m \\ 3m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & w & -2w & w & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & w & -2w & w & 0 & w \\ 0 & 0 & 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) $$ with $w$ and $m$ arbitrary. \end{enumerate}


models-lemma-bound-neighbours

Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Given $i, j$ with $a_{ij} > 0$ we have $m_ia_{ij} \leq m_j|a_{jj}|$ and $m_iw_i \leq m_j|a_{jj}|$.


models-lemma-bound-heart

Fix $g \geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$ with $n > 1$ we have \begin{enumerate} \item the set $J \subset \{1, \ldots, n\}$ of non-$(-2)$-indices has at most $2g - 2$ elements, \item for $j \in J$ we have $g_j < g$, \item for $j \in J$ we have $m_j|a_{jj}| \leq 6g - 6$, and \item for $j \in J$ and $i \in \{1, \ldots, n\}$ we have $m_ia_{ij} \leq 6g - 6$. \end{enumerate}


models-lemma-bound-wm

Fix $g \geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$ we have $m_i|a_{ij}| \leq 768g$.


models-lemma-closure-is-model

Let $V_1 \to V_2$ be a closed immersion of algebraic schemes over $K$. If $X_2$ is a model for $V_2$, then the scheme theoretic image of $V_1 \to X_2$ is a model for $V_1$.