descent-lemma-pullback-selfmap

Let $X \to S$ be a morphism of schemes. Let $f : X \to X$ be a selfmap of $X$ over $S$. In this case pullback by $f$ is isomorphic to the identity functor on the category of descent data relative to $X \to S$.


descent-lemma-fpqc-refinement-coverings-fully-faithful

Let $S$ be a scheme. Let $\mathcal{U} = \{U_i \to S\}_{i \in I}$, and $\mathcal{V} = \{V_j \to S\}_{j \in J}$, be families of morphisms with target $S$. Let $\alpha : I \to J$, $\text{id} : S \to S$ and $g_i : U_i \to V_{\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \ref{sites-definition-morphism-coverings}. Assume that for each $j \in J$ the family $\{g_i : U_i \to V_j\}_{\alpha(i) = j}$ is an fpqc covering of $V_j$. Then the pullback functor $$ \text{descent data relative to } \mathcal{V} \longrightarrow \text{descent data relative to } \mathcal{U} $$ of Lemma \ref{lemma-pullback-family} is fully faithful.


descent-lemma-Zariski-refinement-coverings-equivalence

Let $S$ be a scheme. Let $\mathcal{U} = \{U_i \to S\}_{i \in I}$, and $\mathcal{V} = \{V_j \to S\}_{j \in J}$, be families of morphisms with target $S$. Let $\alpha : I \to J$, $\text{id} : S \to S$ and $g_i : U_i \to V_{\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \ref{sites-definition-morphism-coverings}. Assume that for each $j \in J$ the family $\{g_i : U_i \to V_j\}_{\alpha(i) = j}$ is a Zariski covering (see Topologies, Definition \ref{topologies-definition-zariski-covering}) of $V_j$. Then the pullback functor $$ \text{descent data relative to } \mathcal{V} \longrightarrow \text{descent data relative to } \mathcal{U} $$ of Lemma \ref{lemma-pullback-family} is an equivalence of categories. In particular, the category of schemes over $S$ is equivalent to the category of descent data relative to any Zariski covering of $S$.


descent-lemma-refine-coverings-fully-faithful

Let $S$ be a scheme. Let $\mathcal{U} = \{U_i \to S\}_{i \in I}$, and $\mathcal{V} = \{V_j \to S\}_{j \in J}$, be fpqc-coverings of $S$. If $\mathcal{U}$ is a refinement of $\mathcal{V}$, then the pullback functor $$ \text{descent data relative to } \mathcal{V} \longrightarrow \text{descent data relative to } \mathcal{U} $$ is fully faithful. In particular, the category of schemes over $S$ is identified with a full subcategory of the category of descent data relative to any fpqc-covering of $S$.


descent-lemma-effective-for-fpqc-is-local-upstairs

Let $X \to S$ be a surjective, quasi-compact, flat morphism of schemes. Let $(V, \varphi)$ be a descent datum relative to $X/S$. Suppose that for all $v \in V$ there exists an open subscheme $v \in W \subset V$ such that $\varphi(W \times_S X) \subset X \times_S W$ and such that the descent datum $(W, \varphi|_{W \times_S X})$ is effective. Then $(V, \varphi)$ is effective.


descent-lemma-descending-types-morphisms

Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{fpqc, fppf, \etale, smooth, syntomic\}$. Suppose that \begin{enumerate} \item $\mathcal{P}$ is stable under any base change (see Schemes, Definition \ref{schemes-definition-preserved-by-base-change}), \item if $Y_j \to V_j$, $j = 1, \ldots, m$ have $\mathcal{P}$, then so does $\coprod Y_j \to \coprod V_j$, and \item for any surjective morphism of affines $X \to S$ which is flat, flat of finite presentation, \'etale, smooth or syntomic depending on whether $\tau$ is fpqc, fppf, \'etale, smooth, or syntomic, any descent datum $(V, \varphi)$ relative to $X$ over $S$ such that $\mathcal{P}$ holds for $V \to X$ is effective. \end{enumerate} Then morphisms of type $\mathcal{P}$ satisfy descent for $\tau$-coverings.


descent-lemma-affine

Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering, see Topologies, Definition \ref{topologies-definition-fpqc-covering}. Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to S\}$. If each morphism $V_i \to X_i$ is affine, then the descent datum is effective.


descent-lemma-closed-immersion

Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering, see Topologies, Definition \ref{topologies-definition-fpqc-covering}. Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to S\}$. If each morphism $V_i \to X_i$ is a closed immersion, then the descent datum is effective.


descent-lemma-quasi-affine

Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering, see Topologies, Definition \ref{topologies-definition-fpqc-covering}. Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to S\}$. If each morphism $V_i \to X_i$ is quasi-affine, then the descent datum is effective.


descent-lemma-descent-data-sheaves

Let $\tau \in \{Zariski, fppf, \etale, smooth, syntomic\}$\footnote{The fact that fpqc is missing is not a typo. See discussion in Topologies, Section \ref{topologies-section-fpqc}.}. Let $\Sch_\tau$ be a big $\tau$-site. Let $S \in \Ob(\Sch_\tau)$. Let $\{S_i \to S\}_{i \in I}$ be a covering in the site $(\Sch/S)_\tau$. There is an equivalence of categories $$ \left\{ \begin{matrix} \text{descent data }(X_i, \varphi_{ii'})\text{ such that}\\ \text{each }X_i \in \Ob((\Sch/S)_\tau) \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{sheaves }F\text{ on }(\Sch/S)_\tau\text{ such that}\\ \text{each }h_{S_i} \times F\text{ is representable} \end{matrix} \right\}. $$ Moreover, \begin{enumerate} \item the objects representing $h_{S_i} \times F$ on the right hand side correspond to the schemes $X_i$ on the left hand side, and \item the sheaf $F$ is representable if and only if the corresponding descent datum $(X_i, \varphi_{ii'})$ is effective. \end{enumerate}


descent-proposition-descent-module

\begin{slogan} Effective descent for modules along faithfully flat ring maps. \end{slogan} Let $R \to A$ be a faithfully flat ring map. Then \begin{enumerate} \item any descent datum on modules with respect to $R \to A$ is effective, \item the functor $M \mapsto (A \otimes_R M, can)$ from $R$-modules to the category of descent data is an equivalence, and \item the inverse functor is given by $(N, \varphi) \mapsto H^0(s(N_\bullet))$. \end{enumerate}


descent-proposition-fpqc-descent-quasi-coherent

Let $S$ be a scheme. Let $\mathcal{U} = \{\varphi_i : U_i \to S\}$ be an fpqc covering, see Topologies, Definition \ref{topologies-definition-fpqc-covering}. Any descent datum on quasi-coherent sheaves for $\mathcal{U} = \{U_i \to S\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.


descent-proposition-same-cohomology-quasi-coherent

\begin{slogan} Cohomology of quasi-coherent sheaves is the same no matter which topology you use. \end{slogan} Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. \begin{enumerate} \item There is a canonical isomorphism $$ H^q(S, \mathcal{F}) = H^q((\Sch/S)_\tau, \mathcal{F}^a). $$ \item There are canonical isomorphisms $$ H^q(S, \mathcal{F}) = H^q(S_{Zar}, \mathcal{F}^a) = H^q(S_\etale, \mathcal{F}^a). $$ \end{enumerate}


descent-proposition-equivalence-quasi-coherent

Let $S$ be a scheme. Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. \begin{enumerate} \item The functor $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence of categories $$ \QCoh(\mathcal{O}_S) \longrightarrow \QCoh((\Sch/S)_\tau, \mathcal{O}) $$ between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the big $\tau$ site of $S$. \item Let $\tau = \etale$, or $\tau = Zariski$. The functor $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence of categories $$ \QCoh(\mathcal{O}_S) \longrightarrow \QCoh(S_\tau, \mathcal{O}) $$ between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the small $\tau$ site of $S$. \end{enumerate}


descent-proposition-equivalence-quasi-coherent-functorial

Let $f : T \to S$ be a morphism of schemes. \begin{enumerate} \item The equivalences of categories of Proposition \ref{proposition-equivalence-quasi-coherent} are compatible with pullback. More precisely, we have $f^*(\mathcal{G}^a) = (f^*\mathcal{G})^a$ for any quasi-coherent sheaf $\mathcal{G}$ on $S$. \item The equivalences of categories of Proposition \ref{proposition-equivalence-quasi-coherent} part (1) are {\bf not} compatible with pushforward in general. \item If $f$ is quasi-compact and quasi-separated, and $\tau \in \{Zariski, \etale\}$ then $f_*$ and $f_{small, *}$ preserve quasi-coherent sheaves and the diagram $$ \xymatrix{ \QCoh(\mathcal{O}_T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^a} & & \QCoh(\mathcal{O}_S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^a} \\ \QCoh(T_\tau, \mathcal{O}) \ar[rr]^{f_{small, *}} & & \QCoh(S_\tau, \mathcal{O}) } $$ is commutative, i.e., $f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a$. \end{enumerate}


simplicial-theorem-dold-kan

Let $\mathcal{A}$ be an abelian category. The functor $N$ induces an equivalence of categories $$ N : \text{Simp}(\mathcal{A}) \longrightarrow \text{Ch}_{\geq 0}(\mathcal{A}) $$


simplicial-lemma-face-degeneracy

Any morphism in $\Delta$ can be written as a composition of the morphisms $\delta^n_j$ and $\sigma^n_j$.


simplicial-lemma-relations-face-degeneracy

The morphisms $\delta^n_j$ and $\sigma^n_j$ satisfy the following relations. \begin{enumerate} \item If $0 \leq i < j \leq n + 1$, then $\delta^{n + 1}_j \circ \delta^n_i = \delta^{n + 1}_i \circ \delta^n_{j - 1}$. In other words the diagram $$ \xymatrix{ & [n] \ar[rd]^{\delta^{n + 1}_j} & \\ [n - 1] \ar[ru]^{\delta^n_i} \ar[rd]_{\delta^n_{j - 1}} & & [n + 1] \\ & [n] \ar[ru]_{\delta^{n + 1}_i} & } $$ commutes. \item If $0 \leq i < j \leq n - 1$, then $\sigma^{n - 1}_j \circ \delta^n_i = \delta^{n - 1}_i \circ \sigma^{n - 2}_{j - 1}$. In other words the diagram $$ \xymatrix{ & [n] \ar[rd]^{\sigma^{n - 1}_j} & \\ [n - 1] \ar[ru]^{\delta^n_i} \ar[rd]_{\sigma^{n - 2}_{j - 1}} & & [n - 1] \\ & [n - 2] \ar[ru]_{\delta^{n - 1}_i} & } $$ commutes. \item If $0 \leq j \leq n - 1$, then $\sigma^{n - 1}_j \circ \delta^n_j = \text{id}_{[n - 1]}$ and $\sigma^{n - 1}_j \circ \delta^n_{j + 1} = \text{id}_{[n - 1]}$. In other words the diagram $$ \xymatrix{ & [n] \ar[rd]^{\sigma^{n - 1}_j} & \\ [n - 1] \ar[ru]^{\delta^n_j} \ar[rd]_{\delta^n_{j + 1}} \ar[rr]^{\text{id}_{[n - 1]}} & & [n - 1] \\ & [n] \ar[ru]_{\sigma^{n - 1}_j} & } $$ commutes. \item If $0 < j + 1 < i \leq n$, then $\sigma^{n - 1}_j \circ \delta^n_i = \delta^{n - 1}_{i - 1} \circ \sigma^{n - 2}_j$. In other words the diagram $$ \xymatrix{ & [n] \ar[rd]^{\sigma^{n - 1}_j} & \\ [n - 1] \ar[ru]^{\delta^n_i} \ar[rd]_{\sigma^{n - 2}_j} & & [n - 1] \\ & [n - 2] \ar[ru]_{\delta^{n - 1}_{i - 1}} & } $$ commutes. \item If $0 \leq i \leq j \leq n - 1$, then $\sigma^{n - 1}_j \circ \sigma^n_i = \sigma^{n - 1}_i \circ \sigma^n_{j + 1}$. In other words the diagram $$ \xymatrix{ & [n] \ar[rd]^{\sigma^{n - 1}_j} & \\ [n + 1] \ar[ru]^{\sigma^n_i} \ar[rd]_{\sigma^n_{j + 1}} & & [n - 1] \\ & [n] \ar[ru]_{\sigma^{n - 1}_i} & } $$ commutes. \end{enumerate}


simplicial-lemma-face-degeneracy-category

The category $\Delta$ is the universal category with objects $[n]$, $n \geq 0$ and morphisms $\delta^n_j$ and $\sigma^n_j$ such that (a) every morphism is a composition of these morphisms, (b) the relations listed in Lemma \ref{lemma-relations-face-degeneracy} are satisfied, and (c) any relation among the morphisms is a consequence of those relations.


simplicial-lemma-characterize-simplicial-object

Let $\mathcal{C}$ be a category. \begin{enumerate} \item Given a simplicial object $U$ in $\mathcal{C}$ we obtain a sequence of objects $U_n = U([n])$ endowed with the morphisms $d^n_j = U(\delta^n_j) : U_n \to U_{n-1}$ and $s^n_j = U(\sigma^n_j) : U_n \to U_{n + 1}$. These morphisms satisfy the opposites of the relations displayed in Lemma \ref{lemma-relations-face-degeneracy}, namely \begin{enumerate} \item If $0 \leq i < j \leq n + 1$, then $d^n_i \circ d^{n + 1}_j = d^n_{j - 1} \circ d^{n + 1}_i$. \item If $0 \leq i < j \leq n - 1$, then $d^n_i \circ s^{n - 1}_j = s^{n - 2}_{j - 1} \circ d^{n - 1}_i$. \item If $0 \leq j \leq n - 1$, then $\text{id} = d^n_j \circ s^{n - 1}_j = d^n_{j + 1} \circ s^{n - 1}_j$. \item If $0 < j + 1 < i \leq n$, then $d^n_i \circ s^{n - 1}_j = s^{n - 2}_j \circ d^{n - 1}_{i - 1}$. \item If $0 \leq i \leq j \leq n - 1$, then $s^n_i \circ s^{n - 1}_j = s^n_{j + 1} \circ s^{n - 1}_i$. \end{enumerate} \item Conversely, given a sequence of objects $U_n$ and morphisms $d^n_j$, $s^n_j$ satisfying (1)(a) -- (e) there exists a unique simplicial object $U$ in $\mathcal{C}$ such that $U_n = U([n])$, $d^n_j = U(\delta^n_j)$, and $s^n_j = U(\sigma^n_j)$. \item A morphism between simplicial objects $U$ and $U'$ is given by a family of morphisms $U_n \to U'_n$ commuting with the morphisms $d^n_j$ and $s^n_j$. \end{enumerate}


simplicial-lemma-si-injective

Let $\mathcal{C}$ be a category. Let $U$ be a simplicial object of $\mathcal{C}$. Each of the morphisms $s^n_i : U_n \to U_{n + 1}$ has a left inverse. In particular $s^n_i$ is a monomorphism.


simplicial-lemma-di-injective

Let $\mathcal{C}$ be a category. Let $U$ be a cosimplicial object of $\mathcal{C}$. Each of the morphisms $\delta^n_i : U_{n - 1} \to U_n$ has a left inverse. In particular $\delta^n_i$ is a monomorphism.


simplicial-lemma-product

If $U$ and $V$ are simplicial objects in the category $\mathcal{C}$, and if $U \times V$ exists, then we have $$ \Mor(W, U \times V) = \Mor(W, U) \times \Mor(W, V) $$ for any third simplicial object $W$ of $\mathcal{C}$.


simplicial-lemma-fibre-product

If $U, V, W$ are simplicial objects in the category $\mathcal{C}$, and if $a : V \to U$, $b : W \to U$ are morphisms and if $V \times_U W$ exists, then we have $$ \Mor(T, V \times_U W) = \Mor(T, V) \times_{\Mor(T, U)} \Mor(T, W) $$ for any fourth simplicial object $T$ of $\mathcal{C}$.


simplicial-lemma-push-out

If $U, V, W$ are simplicial objects in the category $\mathcal{C}$, and if $a : U \to V$, $b : U \to W$ are morphisms and if $V\amalg_U W$ exists, then we have $$ \Mor(V\amalg_U W, T) = \Mor(V, T) \times_{\Mor(U, T)} \Mor(W, T) $$ for any fourth simplicial object $T$ of $\mathcal{C}$.


simplicial-lemma-product-cosimplicial-objects

If $U$ and $V$ are cosimplicial objects in the category $\mathcal{C}$, and if $U \times V$ exists, then we have $$ \Mor(W, U \times V) = \Mor(W, U) \times \Mor(W, V) $$ for any third cosimplicial object $W$ of $\mathcal{C}$.


simplicial-lemma-fibre-product-cosimplicial-objects

If $U, V, W$ are cosimplicial objects in the category $\mathcal{C}$, and if $a : V \to U$, $b : W \to U$ are morphisms and if $V \times_U W$ exists, then we have $$ \Mor(T, V \times_U W) = \Mor(T, V) \times_{\Mor(T, U)} \Mor(T, W) $$ for any fourth cosimplicial object $T$ of $\mathcal{C}$.


simplicial-lemma-simplex-map

Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. There is a canonical bijection $$ \Mor(\Delta[n], U) \longrightarrow U_n $$ which maps a morphism $\varphi$ to the value of $\varphi$ on the unique nondegenerate $n$-simplex of $\Delta[n]$.


simplicial-lemma-product-degenerate

Let $U$, $V$ be simplicial sets. Let $a, b \geq 0$ be integers. Assume every $n$-simplex of $U$ is degenerate if $n > a$. Assume every $n$-simplex of $V$ is degenerate if $n > b$. Then every $n$-simplex of $U \times V$ is degenerate if $n > a + b$.


simplicial-lemma-check-product-with-simplicial-set

Let $\mathcal{C}$ be a category such that the coproduct of any two objects of $\mathcal{C}$ exists. Let $U$ be a simplicial set. Let $V$ be a simplicial object of $\mathcal{C}$. Assume that each $U_n$ is finite nonempty. The functor $W \mapsto \Mor_{\text{Simp}(\mathcal{C})}(U \times V, W)$ is canonically isomorphic to the functor which maps $W$ to the set in Equation (\ref{equation-functor-product-with-simplicial-set}).


simplicial-lemma-back-to-U

Let $\mathcal{C}$ be a category such that the coproduct of any two objects of $\mathcal{C}$ exists. Let us temporarily denote $\textit{FSSets}$ the category of simplicial sets all of whose components are finite nonempty. \begin{enumerate} \item The rule $(U, V) \mapsto U \times V$ defines a functor $\textit{FSSets} \times \text{Simp}(\mathcal{C}) \to \text{Simp}(\mathcal{C})$. \item For every $U$, $V$ as above there is a canonical map of simplicial objects $$ U \times V \longrightarrow V $$ defined by taking the identity on each component of $(U \times V)_n = \coprod_u V_n$. \end{enumerate}


simplicial-lemma-morphism-from-coproduct

With $X$ and $k$ as above. For any simplicial object $V$ of $\mathcal{C}$ we have the following canonical bijection $$ \Mor_{\text{Simp}(\mathcal{C})}(X \times \Delta[k], V) \longrightarrow \Mor_\mathcal{C}(X, V_k). $$ which maps $\gamma$ to the restriction of the morphism $\gamma_k$ to the component corresponding to $\text{id}_{[k]}$. Similarly, for any $n \geq k$, if $W$ is an $n$-truncated simplicial object of $\mathcal{C}$, then we have $$ \Mor_{\text{Simp}_n(\mathcal{C})}(\text{sk}_n(X \times \Delta[k]), W) = \Mor_\mathcal{C}(X, W_k). $$


simplicial-lemma-morphism-into-product

With $X$, $k$ and $U$ as above. \begin{enumerate} \item For any simplicial object $V$ of $\mathcal{C}$ we have the following canonical bijection $$ \Mor_{\text{Simp}(\mathcal{C})}(V, U) \longrightarrow \Mor_\mathcal{C}(V_k, X). $$ wich maps $\gamma$ to the morphism $\gamma_k$ composed with the projection onto the factor corresponding to $\text{id}_{[k]}$. \item Similarly, if $W$ is an $k$-truncated simplicial object of $\mathcal{C}$, then we have $$ \Mor_{\text{Simp}_k(\mathcal{C})}(W, \text{sk}_k U) = \Mor_\mathcal{C}(W_k, X). $$ \item The object $U$ constructed above is an incarnation of $\Hom(C[k], X)$ where $C[k]$ is the cosimplicial set from Example \ref{example-simplex-cosimplicial-set}. \end{enumerate}


simplicial-lemma-exists-hom-0-from-simplicial-set

Assume the category $\mathcal{C}$ has coproducts of any two objects and countable limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \geq 0$. Let $V$ be a simplicial object of $\mathcal{C}$. Then the functor \begin{eqnarray*} \mathcal{C}^{opp} & \longrightarrow & \textit{Sets} \\ X & \longmapsto & \Mor_{\text{Simp}(\mathcal{C})}(X \times U, V) \end{eqnarray*} is representable.


simplicial-lemma-exists-hom-0-from-simplicial-set-finite

Assume the category $\mathcal{C}$ has coproducts of any two objects and finite limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \geq 0$. Assume that all $n$-simplices of $U$ are degenerate for all $n \gg 0$. Let $V$ be a simplicial object of $\mathcal{C}$. Then the functor \begin{eqnarray*} \mathcal{C}^{opp} & \longrightarrow & \textit{Sets} \\ X & \longmapsto & \Mor_{\text{Simp}(\mathcal{C})}(X \times U, V) \end{eqnarray*} is representable.


simplicial-lemma-exists-hom-from-simplicial-set-finite

Assume the category $\mathcal{C}$ has coproducts of any two objects and finite limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \geq 0$. Assume that all $n$-simplices of $U$ are degenerate for all $n \gg 0$. Let $V$ be a simplicial object of $\mathcal{C}$. Then $\Hom(U, V)$ exists, moreover we have the expected equalities $$ \Hom(U, V)_n = \Hom(U \times \Delta[n], V)_0. $$


simplicial-lemma-hom-from-coprod

Assume the category $\mathcal{C}$ has coproducts of any two objects and finite limits. Let $a : U \to V$, $b : U \to W$ be morphisms of simplicial sets. Assume $U_n, V_n, W_n$ finite nonempty for all $n \geq 0$. Assume that all $n$-simplices of $U, V, W$ are degenerate for all $n \gg 0$. Let $T$ be a simplicial object of $\mathcal{C}$. Then $$ \Hom(V, T) \times_{\Hom(U, T)} \Hom(W, T) = \Hom(V \amalg_U W, T) $$ In other words, the fibre product on the left hand side is represented by the Hom object on the right hand side.


simplicial-lemma-splitting-simplicial-sets

Let $U$ be a simplicial set. Then $U$ has a unique splitting with $N(U_m)$ equal to the set of nondegenerate $m$-simplices.


simplicial-lemma-injective-map-simplicial-sets

Let $f : U \to V$ be a morphism of simplicial sets. Suppose that (a) the image of every nondegenerate simplex of $U$ is a nondegenerate simplex of $V$ and (b) the restriction of $f$ to a map from the set of nondegenerate simplices of $U$ to the set of nondegenerate simplices of $V$ is injective. Then $f_n$ is injective for all $n$. Same holds with ``injective'' replaced by ``surjective'' or ``bijective''.


simplicial-lemma-simplicial-set-n-skel-sub

Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. The rule $$ U'_m = \bigcup\nolimits_{\varphi : [m] \to [i], \ i\leq n} \Im(U(\varphi)) $$ defines a sub simplicial set $U' \subset U$ with $U'_i = U_i$ for $i \leq n$. Moreover, all $m$-simplices of $U'$ are degenerate for all $m > n$.


simplicial-lemma-splitting-simplicial-groups

Let $U$ be a simplicial abelian group. Then $U$ has a splitting obtained by taking $N(U_0) = U_0$ and for $m \geq 1$ taking $$ N(U_m) = \bigcap\nolimits_{i = 0}^{m - 1} \Ker(d^m_i). $$ Moreover, this splitting is functorial on the category of simplicial abelian groups.


simplicial-lemma-splitting-abelian-category

Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object in $\mathcal{A}$. Then $U$ has a splitting obtained by taking $N(U_0) = U_0$ and for $m \geq 1$ taking $$ N(U_m) = \bigcap\nolimits_{i = 0}^{m - 1} \Ker(d^m_i). $$ Moreover, this splitting is functorial on the category of simplicial objects of $\mathcal{A}$.


simplicial-lemma-injective-map-simplicial-abelian

\begin{slogan} The Dold-Kan normalization functor reflects injectivity, surjectivity, and isomorphy. \end{slogan} Let $\mathcal{A}$ be an abelian category. Let $f : U \to V$ be a morphism of simplicial objects of $\mathcal{A}$. If the induced morphisms $N(f)_i : N(U)_i \to N(V)_i$ are injective for all $i$, then $f_i$ is injective for all $i$. Same holds with ``injective'' replaced with ``surjective'', or ``isomorphism''.


simplicial-lemma-simplicial-abelian-n-skel-sub

Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\mathcal{A}$. Let $n \geq 0$ be an integer. The rule $$ U'_m = \sum\nolimits_{\varphi : [m] \to [i], \ i\leq n} \Im(U(\varphi)) $$ defines a sub simplicial object $U' \subset U$ with $U'_i = U_i$ for $i \leq n$. Moreover, $N(U'_m) = 0$ for all $m > n$.


simplicial-lemma-existence-cosk

If the category $\mathcal{C}$ has finite limits, then $\text{cosk}_m$ functors exist for all $m$. Moreover, for any $m$-truncated simplicial object $U$ the simplicial object $\text{cosk}_mU$ is described by the formula $$ (\text{cosk}_mU)_n = \lim_{(\Delta/[n])_{\leq m}^{opp}} U(n) $$ and for $\varphi : [n] \to [n']$ the map $\text{cosk}_mU(\varphi)$ comes from the identification $U(n') \circ \overline{\varphi} = U(n)$ above via Categories, Lemma \ref{categories-lemma-functorial-limit}.


simplicial-lemma-trivial-cosk

Let $\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. For $n \leq m$ the limit $\lim_{(\Delta/[n])_{\leq m}^{opp}} U(n)$ exists and is canonically isomorphic to $U_n$.


simplicial-lemma-recover-cosk

Let $\mathcal{C}$ be a category with finite limits. Let $U$ be an $n$-truncated simplicial object of $\mathcal{C}$. The morphism $\text{sk}_n \text{cosk}_n U \to U$ is an isomorphism.


simplicial-lemma-formula-limit

Let $n$ be an integer $\geq 1$. Let $U$ be a $n$-truncated simplicial object of $\mathcal{C}$. Consider the contravariant functor from $\mathcal{C}$ to $\textit{Sets}$ which associates to an object $T$ the set $$ \{ (f_0, \ldots, f_{n + 1}) \in \Mor_\mathcal{C}(T, U_n) \mid d^n_{j - 1} \circ f_i = d^n_i \circ f_j \ \forall\ 0\leq i < j\leq n + 1\} $$ If this functor is representable by some object $U_{n + 1}$ of $\mathcal{C}$, then $$ U_{n + 1} = \lim_{(\Delta/[n + 1])_{\leq n}^{opp}} U(n) $$


simplicial-lemma-work-out

Let $n$ be an integer $\geq 1$. Let $U$ be a $n$-truncated simplicial object of $\mathcal{C}$. Consider the contravariant functor from $\mathcal{C}$ to $\textit{Sets}$ which associates to an object $T$ the set $$ \{ (f_0, \ldots, f_{n + 1}) \in \Mor_\mathcal{C}(T, U_n) \mid d^n_{j - 1} \circ f_i = d^n_i \circ f_j \ \forall\ 0\leq i < j\leq n + 1\} $$ If this functor is representable by some object $U_{n + 1}$ of $\mathcal{C}$, then there exists an $(n + 1)$-truncated simplicial object $\tilde U$, with $\text{sk}_n \tilde U = U$ and $\tilde U_{n + 1} = U_{n + 1}$ such that the following adjointness holds $$ \Mor_{\text{Simp}_{n + 1}(\mathcal{C})}(V, \tilde U) = \Mor_{\text{Simp}_n(\mathcal{C})}(\text{sk}_nV, U) $$


simplicial-lemma-cosk-product

Let $U$, $V$ be $n$-truncated simplicial objects of a category $\mathcal{C}$. Then $$ \text{cosk}_n (U \times V) = \text{cosk}_nU \times \text{cosk}_nV $$ whenever the left and right hand sides exist.


simplicial-lemma-cosk-fibre-product

Assume $\mathcal{C}$ has fibre products. Let $U \to V$ and $W \to V$ be morphisms of $n$-truncated simplicial objects of the category $\mathcal{C}$. Then $$ \text{cosk}_n (U \times_V W) = \text{cosk}_nU \times_{\text{cosk}_n V} \text{cosk}_nW $$ whenever the left and right hand side exist.


simplicial-lemma-cosk-above-object

Let $\mathcal{C}$ be a category with finite limits. Let $X \in \Ob(\mathcal{C})$. The functor $\mathcal{C}/X \to \mathcal{C}$ commutes with the coskeleton functors $\text{cosk}_k$ for $k \geq 1$.


simplicial-lemma-simplex-cosk

The canonical map $\Delta[n] \to \text{cosk}_1 \text{sk}_1 \Delta[n]$ is an isomorphism.


simplicial-lemma-augmentation-howto

Let $\mathcal{C}$ be a category. Let $X \in \Ob(\mathcal{C})$. Let $U$ be a simplicial object of $\mathcal{C}$. To give an augmentation of $U$ towards $X$ is the same as giving a morphism $\epsilon_0 : U_0 \to X$ such that $\epsilon_0 \circ d^1_0 = \epsilon_0 \circ d^1_1$.


simplicial-lemma-cosk-minus-one

Let $\mathcal{C}$ be a category with fibred products. Let $f : Y\to X$ be a morphism of $\mathcal{C}$. Let $U$ be the simplicial object of $\mathcal{C}$ whose $n$th term is the $(n + 1)$fold fibred product $Y \times_X Y \times_X \ldots \times_X Y$. See Example \ref{example-fibre-products-simplicial-object}. For any simplicial object $V$ of $\mathcal{C}$ we have \begin{align*} \Mor_{\text{Simp}(\mathcal{C})}(V, U) & = \Mor_{\text{Simp}_1(\mathcal{C})}(\text{sk}_1 V, \text{sk}_1 U) \\ & = \{g_0 : V_0 \to Y \mid f \circ g_0 \circ d^1_0 = f \circ g_0 \circ d^1_1\} \end{align*} In particular we have $U = \text{cosk}_1 \text{sk}_1 U$.


simplicial-lemma-left-adjoint-exists

Let $\mathcal{C}$ be a category which has finite colimits. The functors $i_{m!}$ exist for all $m$. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. The simplicial object $i_{m!}U$ is described by the formula $$ (i_{m!}U)_n = \colim_{([n]/\Delta)_{\leq m}^{opp}} U(n) $$ and for $\varphi : [n] \to [n']$ the map $i_{m!}U(\varphi)$ comes from the identification $U(n) \circ \underline{\varphi} = U(n')$ above via Categories, Lemma \ref{categories-lemma-functorial-colimit}.


simplicial-lemma-recovering-U

Let $\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. For any $n \leq m$ the colimit $$ \colim_{([n]/\Delta)_{\leq m}^{opp}} U(n) $$ exists and is equal to $U_n$.


simplicial-lemma-recovering-U-for-real

Let $\mathcal{C}$ be a category which has finite colimits. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. The map $U \to \text{sk}_m i_{m!}U$ is an isomorphism.


simplicial-lemma-imshriek-sets

If $U$ is an $m$-truncated simplicial set and $n > m$ then all $n$-simplices of $i_{m!}U$ are degenerate.


simplicial-lemma-n-skeleton-sets

Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. The morphism $i_{n!} \text{sk}_n U \to U$ identifies $i_{n!} \text{sk}_n U$ with the simplicial set $U' \subset U$ defined in Lemma \ref{lemma-simplicial-set-n-skel-sub}.


simplicial-lemma-glue-simplex

Let $U \subset V$ be simplicial sets. Suppose $n \geq 0$ and $x \in V_n$, $x \not \in U_n$ are such that \begin{enumerate} \item $V_i = U_i$ for $i < n$, \item $V_n = U_n \cup \{x\}$, \item any $z \in V_j$, $z \not \in U_j$ for $j > n$ is degenerate. \end{enumerate} Let $\Delta[n] \to V$ be the unique morphism mapping the nondegenerate $n$-simplex of $\Delta[n]$ to $x$. In this case the diagram $$ \xymatrix{ \Delta[n] \ar[r] & V \\ i_{(n - 1)!} \text{sk}_{n - 1} \Delta[n] \ar[r] \ar[u] & U \ar[u] } $$ is a pushout diagram.


simplicial-lemma-add-simplices

Let $U \subset V$ be simplicial sets, with $U_n, V_n$ finite nonempty for all $n$. Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then there exists a sequence of sub simplicial sets $$ U = W^0 \subset W^1 \subset W^2 \subset \ldots W^r = V $$ such that Lemma \ref{lemma-glue-simplex} applies to each of the inclusions $W^i \subset W^{i + 1}$.


simplicial-lemma-imshriek-abelian

Let $\mathcal{A}$ be an abelian category Let $U$ be an $m$-truncated simplicial object of $\mathcal{A}$. For $n > m$ we have $N(i_{m!}U)_n = 0$.


simplicial-lemma-n-skeleton-abelian

Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\mathcal{A}$. Let $n \geq 0$ be an integer. The morphism $i_{n!} \text{sk}_n U \to U$ identifies $i_{n!} \text{sk}_n U$ with the simplicial subobject $U' \subset U$ defined in Lemma \ref{lemma-simplicial-abelian-n-skel-sub}.


simplicial-lemma-cosk-shriek

Let $\mathcal{C}$ be a category with finite coproducts and finite limits. Let $V$ be a simplicial object of $\mathcal{C}$. In this case $$ (\text{cosk}_n \text{sk}_n V)_{n + 1} = \Hom(i_{n !}\text{sk}_n \Delta[n + 1], V)_0. $$


simplicial-lemma-abelian

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item The categories $\text{Simp}(\mathcal{A})$ and $\text{CoSimp}(\mathcal{A})$ are abelian. \item A morphism of (co)simplicial objects $f : A \to B$ is injective if and only if each $f_n : A_n \to B_n$ is injective. \item A morphism of (co)simplicial objects $f : A \to B$ is surjective if and only if each $f_n : A_n \to B_n$ is surjective. \item A sequence of (co)simplicial objects $$ A \xrightarrow{f} B \xrightarrow{g} C $$ is exact at $B$ if and only if each sequence $$ A_i \xrightarrow{f_i} B_i \xrightarrow{g_i} C_i $$ is exact at $B_i$. \end{enumerate}


simplicial-lemma-eilenberg-maclane-object

With $A$, $k$ and $U$ as above, so $U_i = 0$, $i < k$ and $U_k = A$. \begin{enumerate} \item Given a $k$-truncated simplicial object $V$ we have $$ \Mor(U, V) = \{ f : A \to V_k \mid d^k_i \circ f = 0, \ i = 0, \ldots, k \} $$ and $$ \Mor(V, U) = \{ f : V_k \to A \mid f \circ s^{k - 1}_i = 0, \ i = 0, \ldots, k - 1 \}. $$ \item The object $i_{k!} U$ has $n$th term equal to $\bigoplus_\alpha A$ where $\alpha$ runs over all surjective morphisms $\alpha : [n] \to [k]$. \item For any $\varphi : [m] \to [n]$ the map $i_{k!} U(\varphi)$ is described as the mapping $\bigoplus_\alpha A \to \bigoplus_{\alpha'} A$ which maps to component corresponding to $\alpha : [n] \to [k]$ to zero if $\alpha \circ \varphi$ is not surjective and by the identity to the component corresponding to $\alpha \circ \varphi$ if it is surjective. \item The object $\text{cosk}_k U$ has $n$th term equal to $\bigoplus_\beta A$, where $\beta$ runs over all injective morphisms $\beta : [k] \to [n]$. \item For any $\varphi : [m] \to [n]$ the map $\text{cosk}_k U(\varphi)$ is described as the mapping $\bigoplus_\beta A \to \bigoplus_{\beta'} A$ which maps to component corresponding to $\beta : [k] \to [n]$ to zero if $\beta$ does not factor through $\varphi$ and by the identity to each of the components corresponding to $\beta'$ such that $\beta = \varphi \circ \beta'$ if it does. \item The canonical map $ c : i_{k !} U \to \text{cosk}_k U $ in degree $n$ has $(\alpha, \beta)$ coefficient $A \to A$ equal to zero if $\alpha \circ \beta$ is not the identity and equal to $\text{id}_A$ if it is. \item The canonical map $ c : i_{k !} U \to \text{cosk}_k U $ is injective. \end{enumerate}


simplicial-lemma-extension

Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer $\geq 0$. Consider the simplicial object $E$ defined by the following rules \begin{enumerate} \item $E_n = \bigoplus_\alpha A$, where the sum is over $\alpha : [n] \to [k + 1]$ whose image is either $[k]$ or $[k + 1]$. \item Given $\varphi : [m] \to [n]$ the map $E_n \to E_m$ maps the summand corresponding to $\alpha$ via $\text{id}_A$ to the summand corresponding to $\alpha \circ \varphi$, provided $\Im(\alpha \circ \varphi)$ is equal to $[k]$ or $[k + 1]$. \end{enumerate} Then there exists a short exact sequence $$ 0 \to K(A, k) \to E \to K(A, k + 1) \to 0 $$ which is term by term split exact.


simplicial-lemma-s-exact

The functor $s$ is exact.


simplicial-lemma-homology-extension

Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer. Let $E$ be the object described in Lemma \ref{lemma-extension}. Then the complex $s(E)$ is acyclic.


simplicial-lemma-homology-eilenberg-maclane

Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer. We have $H_i(s(K(A, k))) = A$ if $i = k$ and $0$ else.


simplicial-lemma-map-associated-complexes

Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\mathcal{A}$. The canonical map $N(U_n) \to U_n$ gives rise to a morphism of complexes $N(U) \to s(U)$.


simplicial-lemma-N-K

Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer. We have $N(K(A, k))_i = A$ if $i = k$ and $0$ else.


simplicial-lemma-decompose-associated-complexes

Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\mathcal{A}$. The canonical morphism of chain complexes $N(U) \to s(U)$ is split. In fact, $$ s(U) = N(U) \oplus D(U) $$ for some complex $D(U)$. The construction $U \mapsto D(U)$ is functorial.


simplicial-lemma-N-exact

The functor $N$ is exact.


simplicial-lemma-quasi-isomorphism

Let $\mathcal{A}$ be an abelian category. Let $V$ be a simplicial object of $\mathcal{A}$. The canonical morphism of chain complexes $N(V) \to s(V)$ is a quasi-isomorphism. In other words, the complex $D(V)$ of Lemma \ref{lemma-decompose-associated-complexes} is acyclic.


simplicial-lemma-S-N

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $N : \mathcal{A} \to \mathcal{B}$, and $S : \mathcal{B} \to \mathcal{A}$ be functors. Suppose that \begin{enumerate} \item the functors $S$ and $N$ are exact, \item there is an isomorphism $g : N \circ S \to \text{id}_\mathcal{B}$ to the identity functor of $\mathcal{B}$, \item $N$ is faithful, and \item $S$ is essentially surjective. \end{enumerate} Then $S$ and $N$ are quasi-inverse equivalences of categories.


simplicial-lemma-relations-homotopy

In the situation above, we have the following relations: \begin{enumerate} \item We have $h_{n, 0} = b_n$ and $h_{n, n + 1} = a_n$. \item We have $d^n_j \circ h_{n, i} = h_{n - 1, i - 1} \circ d^n_j$ for $i > j$. \item We have $d^n_j \circ h_{n, i} = h_{n - 1, i} \circ d^n_j$ for $i \leq j$. \item We have $s^n_j \circ h_{n, i} = h_{n + 1, i + 1} \circ s^n_j$ for $i > j$. \item We have $s^n_j \circ h_{n, i} = h_{n + 1, i} \circ s^n_j$ for $i \leq j$. \end{enumerate} Conversely, given a system of maps $h_{n, i}$ satisfying the properties listed above, then these define a morphism $h$ which is a homotopy from $a$ to $b$.


simplicial-lemma-products-homotopy

Let $\mathcal{C}$ be a category. Let $T$ be a set. For $t \in T$ let $X_t$, $Y_t$ be simplicial objects of $\mathcal{C}$. Assume $X = \prod_{t \in T} X_t$ and $Y = \prod_{t \in T} Y_t$ exist. \begin{enumerate} \item If $X_t$ and $Y_t$ are homotopy equivalent for all $t \in T$ and $T$ is finite, then $X$ and $Y$ are homotopy equivalent. \end{enumerate} For $t \in T$ let $a_t, b_t : X_t \to Y_t$ be morphisms. Set $a = \prod a_t : X \to Y$ and $b = \prod b_t : X \to Y$. \begin{enumerate} \item[(2)] If there exists a homotopy from $a_t$ to $b_t$ for all $t \in T$, then there exists a homotopy from $a$ to $b$. \item[(3)] If $T$ is finite and $a_t, b_t : X_t \to Y_t$ for $t \in T$ are homotopic, then $a$ and $b$ are homotopic. \end{enumerate}


simplicial-lemma-homotopy-s-N

Let $\mathcal{A}$ be an additive category. Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{A}$. If $a$, $b$ are homotopic, then $s(a), s(b) : s(U) \to s(V)$ are homotopic maps of chain complexes. If $\mathcal{A}$ is abelian, then also $N(a), N(b) : N(U) \to N(V)$ are homotopic maps of chain complexes.


simplicial-lemma-homotopy-equivalence-s-N

Let $\mathcal{A}$ be an additive category. Let $a : U \to V$ be a morphism of simplicial objects of $\mathcal{A}$. If $a$ is a homotopy equivalence, then $s(a) : s(U) \to s(V)$ is a homotopy equivalence of chain complexes. If in addition $\mathcal{A}$ is abelian, then also $N(a) : N(U) \to N(V)$ is a homotopy equivalence of chain complexes.


simplicial-lemma-compare-homotopies

Let $\mathcal{C}$ be a category. Suppose that $U$ and $V$ are two cosimplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms of cosimplicial objects. Recall that $U$, $V$ correspond to simplicial objects $U'$, $V'$ of $\mathcal{C}^{opp}$. Moreover $a, b$ correspond to morphisms $a', b' : V' \to U'$. The following are equivalent \begin{enumerate} \item There exists a homotopy $h = \{h_{n, \alpha}\}$ from $a$ to $b$ as in Remark \ref{remark-homotopy-cosimplicial-better}. \item There exists a homotopy $h = \{h_{n, i}\}$ from $a'$ to $b'$ as in Remark \ref{remark-homotopy-better}. \end{enumerate} Thus $a$ is homotopic to $b$ as in Remark \ref{remark-homotopy-cosimplicial-better} if and only if $a'$ is homotopic to $b'$ as in Remark \ref{remark-homotopy-better}.


simplicial-lemma-functorial-homotopy

\begin{slogan} Functors preserve homotopic morphisms of (co)simplicial objects. \end{slogan} Let $\mathcal{C}, \mathcal{C}', \mathcal{D}, \mathcal{D}'$ be categories. With terminology as in Remarks \ref{remark-homotopy-cosimplicial-better} and \ref{remark-homotopy-better}. \begin{enumerate} \item Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{D}$. Let $F : \mathcal{D} \to \mathcal{D}'$ be a covariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(U) \to F(V)$ of simplicial objects. \item Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{C}$. Let $F : \mathcal{C} \to \mathcal{C}'$ be a covariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(U) \to F(V)$ of cosimplicial objects. \item Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{D}$. Let $F : \mathcal{D} \to \mathcal{C}$ be a contravariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(V) \to F(U)$ of cosimplicial objects. \item Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{C}$. Let $F : \mathcal{C} \to \mathcal{D}$ be a contravariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(V) \to F(U)$ of simplicial objects. \end{enumerate}


simplicial-lemma-homotopy-s-Q

\begin{slogan} The (cosimplicial) Dold-Kan functor carries homotopic maps to homotopic maps. \end{slogan} Let $\mathcal{A}$ be an additive category. Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{A}$. If $a$, $b$ are homotopic, then $s(a), s(b) : s(U) \to s(V)$ are homotopic maps of cochain complexes. If in addition $\mathcal{A}$ is abelian, then $Q(a), Q(b) : Q(U) \to Q(V)$ are homotopic maps of cochain complexes.


simplicial-lemma-homotopy-equivalence-s-Q

Let $\mathcal{A}$ be an additive category. Let $a : U \to V$ be a morphism of cosimplicial objects of $\mathcal{A}$. If $a$ is a homotopy equivalence, then $s(a) : s(U) \to s(V)$ is a homotopy equivalence of chain complexes. If in addition $\mathcal{A}$ is abelian, then also $Q(a) : Q(U) \to Q(V)$ is a homotopy equivalence of chain complexes.


simplicial-lemma-represent-homotopy

Let $\mathcal{A}$ be an abelian category. Let $A$ be a chain complex. Consider the covariant functor $$ B \longmapsto \{ (a, b, h) \mid a, b : A \to B\text{ and }h\text{ a homotopy between }a, b \} $$ There exists a chain complex $\diamond A$ such that $\Mor_{\text{Ch}(\mathcal{A})}(\diamond A, -)$ is isomorphic to the displayed functor. The construction $A \mapsto \diamond A$ is functorial.


simplicial-lemma-map-into-diamond

Let $\mathcal{A}$ be an abelian category. Let $$ 0 \to A \oplus A \to B \to C \to 0 $$ be a short exact sequence of chain complexes of $\mathcal{A}$. Suppose given in addition morphisms $s_n : C_n \to B_n$ splitting the associated short exact sequence in degree $n$. Let $\delta(s) : C \to (A \oplus A)[-1] = A[-1] \oplus A[-1]$ be the associated morphism of complexes, see Homology, Lemma \ref{homology-lemma-ses-termwise-split}. If $\delta(s)$ factors through the morphism $(1, -1) : A[-1] \to A[-1] \oplus A[-1]$, then there is a unique morphism $B \to \diamond A$ fitting into a commutative diagram $$ \xymatrix{ 0 \ar[r] & A \oplus A \ar[d] \ar[r] & B \ar[r] \ar[d] & C \ar[d] \ar[r] & 0 \\ 0 \ar[r] & A \oplus A \ar[r] & \diamond A \ar[r] & A[-1] \ar[r] & 0 } $$ where the vertical maps are compatible with the splittings $s_n$ and the splittings of $\diamond A_n \to A[-1]_n$ as well.


simplicial-lemma-trivial-kan

Let $f : X \to Y$ be a trivial Kan fibration of simplicial sets. For any solid commutative diagram $$ \xymatrix{ Z \ar[r]_b \ar[d] & X \ar[d] \\ W \ar[r]^a \ar@{-->}[ru] & Y } $$ of simplicial sets with $Z \to W$ (termwise) injective a dotted arrow exists making the diagram commute.


simplicial-lemma-trivial-kan-base-change

Let $f : X \to Y$ be a trivial Kan fibration of simplicial sets. Let $Y' \to Y$ be a morphism of simplicial sets. Then $X \times_Y Y' \to Y'$ is a trivial Kan fibration.


simplicial-lemma-trivial-kan-composition

The composition of two trivial Kan fibrations is a trivial Kan fibration.


simplicial-lemma-limit-trivial-kan

Let $\ldots \to U^2 \to U^1 \to U^0$ be a sequence of trivial Kan fibrations. Let $U = \lim U^t$ defined by taking $U_n = \lim U_n^t$. Then $U \to U^0$ is a trivial Kan fibration.


simplicial-lemma-product-trivial-kan

\begin{slogan} Products of trivial Kan fibrations are trivial Kan fibrations. \end{slogan} Let $X_i \to Y_i$ be a set of trivial Kan fibrations. Then $\prod X_i \to \prod Y_i$ is a trivial Kan fibration.


simplicial-lemma-filtered-colimit-trivial-kan

A filtered colimit of trivial Kan fibrations is a trivial Kan fibration.


simplicial-lemma-trivial-kan-homotopy

Let $f : X \to Y$ be a trivial Kan fibration of simplicial sets. Then $f$ is a homotopy equivalence.


simplicial-lemma-kan-composition

The composition of two Kan fibrations is a Kan fibration.


simplicial-lemma-limit-kan

Let $\ldots \to U^2 \to U^1 \to U^0$ be a sequence of Kan fibrations. Let $U = \lim U^t$ defined by taking $U_n = \lim U_n^t$. Then $U \to U^0$ is a Kan fibration.


simplicial-lemma-product-kan

Let $X_i \to Y_i$ be a set of Kan fibrations. Then $\prod X_i \to \prod Y_i$ is a Kan fibration.


simplicial-lemma-simplicial-group-kan

Let $X$ be a simplicial group. Then $X$ is a Kan complex.


simplicial-lemma-qis-simplicial-abelian-groups

Let $f : X \to Y$ be a homomorphism of simplicial abelian groups which is termwise surjective and induces a quasi-isomorphism on associated chain complexes. Then $f$ is a trivial Kan fibration of simplicial sets.


simplicial-lemma-homotopy-equivalence

Let $f : X \to Y$ be a map of simplicial abelian groups. If $f$ is a homotopy equivalence of simplicial sets, then $f$ induces a quasi-isomorphism of associated chain complexes.


simplicial-lemma-cosk-minus-one-equivalence

Let $A$, $B$ be sets, and that $f : A \to B$ is a map. Consider the simplicial set $U$ with $n$-simplices $$ A \times_B A \times_B \ldots \times_B A\ (n + 1 \text{ factors)}. $$ see Example \ref{example-fibre-products-simplicial-object}. If $f$ is surjective, the morphism $U \to B$ where $B$ indicates the constant simplicial set with value $B$ is a trivial Kan fibration.


simplicial-lemma-godement

In Example \ref{example-godement} if $$ 1_Y = (d \star 1_Y) \circ s = (1_Y \star d) \circ s \quad\text{and}\quad (s \star 1) \circ s = (1 \star s) \circ s $$ then $X = (X_n, d^n_j, s^n_j)$ is a simplicial object in the category of endofunctors of $\mathcal{C}$ and $d : X_0 = Y \to \text{id}_\mathcal{C}$ defines an augmentation.


simplicial-lemma-godement-section

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$ be as in Example \ref{example-godement-functorial}. Given a transformation of functors $h_0 : G \circ F \to G \circ Y \circ F$ such that $$ 1_{G \circ F} = (1_G \star d \star 1_F) \circ h_0 $$ Then there is a morphism $h : G \circ F \to G \circ X \circ F$ of simplicial objects such that $\epsilon \circ h = \text{id}$ where $\epsilon : G \circ X \circ F \to G \circ F$ is the augmentation.


simplicial-lemma-godement-two-maps

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$ be as in Example \ref{example-godement-functorial}. Let $F' : \mathcal{A} \to \mathcal{C}$ and $G' : \mathcal{C} \to \mathcal{B}$ be two functors. Let $(a_n) : G \circ X \to G' \circ X$ be a morphism of simplicial objects compatible via augmentations with $a : G \to G'$. Let $(b_n) : X \circ F \to X \circ F'$ be a morphism of simplicial objects compatible via augmentations with $b : F \to F'$. Then the two maps $$ a \star (b_n), (a_n) \star b : G \circ X \circ F \to G' \circ X \circ F' $$ are homotopic.


simplicial-lemma-godement-before-after

Let $\mathcal{C}$, $Y$, $d$, $s$ be as in Example \ref{example-godement} satisfying the equations of Lemma \ref{lemma-godement}. Let $f : \text{id}_\mathcal{C} \to \text{id}_\mathcal{C}$ be an endomorphism of the identity functor. Then $f \star 1_X, 1_X \star f : X \to X$ are maps of simplicial objects compatible with $f$ via the augmentation $\epsilon : X \to \text{id}_\mathcal{C}$. Moreover, $f \star 1_X$ and $1_X \star f$ are homotopic.


simplicial-lemma-standard-simplicial

In Situation \ref{situation-adjoint-functors} the system $X = (X_n, d^n_j, s^n_j)$ is a simplicial object of $\text{Fun}(\mathcal{A}, \mathcal{A})$ and $\epsilon_0$ defines an augmentation $\epsilon$ from $X$ to the constant simplicial object with value $X_{-1} = \text{id}_\mathcal{A}$.


simplicial-lemma-standard-simplicial-homotopy

In Situation \ref{situation-adjoint-functors} the maps $$ 1_V \star \epsilon : V \circ X \to V, \quad\text{and}\quad \epsilon \star 1_U : X \circ U \to U $$ are homotopy equivalences.


discriminant-lemma-dominate-factorizations

Let $A \to B$ be a quasi-finite ring map. Given two factorizations $A \to B' \to B$ and $A \to B'' \to B$ with $A \to B'$ and $A \to B''$ finite and $\Spec(B) \to \Spec(B')$ and $\Spec(B) \to \Spec(B'')$ open immersions, there exists an $A$-subalgebra $B''' \subset B$ finite over $A$ such that $\Spec(B) \to \Spec(B''')$ an open immersion and $B' \to B$ and $B'' \to B$ factor through $B'''$.


discriminant-lemma-dualizing-well-defined

The module (\ref{equation-dualizing}) is well defined, i.e., independent of the choice of the factorization.


discriminant-lemma-localize-dualizing

Let $A \to B$ be a quasi-finite map of Noetherian rings. \begin{enumerate} \item If $A \to B$ factors as $A \to A_f \to B$ for some $f \in A$, then $\omega_{B/A} = \omega_{B/A_f}$. \item If $g \in B$, then $(\omega_{B/A})_g = \omega_{B_g/A}$. \item If $f \in A$, then $\omega_{B_f/A_f} = (\omega_{B/A})_f$. \end{enumerate}


discriminant-lemma-bc-map-dualizing

The base change map (\ref{equation-bc-dualizing}) is independent of the choice of the factorization $A \to B' \to B$. Given ring maps $A \to A_1 \to A_2$ the composition of the base change maps for $A \to A_1$ and $A_1 \to A_2$ is the base change map for $A \to A_2$.


discriminant-lemma-dualizing-flat-base-change

If $A \to A_1$ is flat, then the base change map (\ref{equation-bc-dualizing}) induces an isomorphism $\omega_{B/A} \otimes_B B_1 \to \omega_{B_1/A_1}$.


discriminant-lemma-dualizing-composition

Let $A \to B \to C$ be quasi-finite homomorphisms of Noetherian rings. There is a canonical map $\omega_{B/A} \otimes_B \omega_{C/B} \to \omega_{C/A}$.


discriminant-lemma-dualizing-product

Let $A \to B$ and $A \to C$ be quasi-finite maps of Noetherian rings. Then $\omega_{B \times C/A} = \omega_{B/A} \times \omega_{C/A}$ as modules over $B \times C$.


discriminant-lemma-dualizing-associated-primes

Let $A \to B$ be a quasi-finite homomorphism of Noetherian rings. Then $\text{Ass}_B(\omega_{B/A})$ is the set of primes of $B$ lying over associated primes of $A$.


discriminant-lemma-dualizing-base-flat-flat

Let $A \to B$ be a flat quasi-finite homomorphism of Noetherian rings. Then $\omega_{B/A}$ is a flat $A$-module.


discriminant-lemma-dualizing-base-change-of-flat

If $A \to B$ is flat, then the base change map (\ref{equation-bc-dualizing}) induces an isomorphism $\omega_{B/A} \otimes_B B_1 \to \omega_{B_1/A_1}$.


discriminant-lemma-discriminant

Let $\pi : X \to Y$ be a morphism of schemes which is finite locally free. Then $\pi$ is \'etale if and only if its discriminant is empty.


discriminant-lemma-trace-unique

Let $A \to B$ be a flat quasi-finite map of Noetherian rings. Then there is at most one trace element in $\omega_{B/A}$.


discriminant-lemma-finite-flat-trace

Let $A \to B$ be a finite flat map of Noetherian rings. Then $\text{Trace}_{B/A} \in \omega_{B/A}$ is the trace element.


discriminant-lemma-trace-base-change

Let $A \to B$ be a flat quasi-finite map of Noetherian rings. Let $\tau \in \omega_{B/A}$ be a trace element. \begin{enumerate} \item If $A \to A_1$ is a map with $A_1$ Noetherian, then with $B_1 = A_1 \otimes_A B$ the image of $\tau$ in $\omega_{B_1/A_1}$ is a trace element. \item If $A = R_f$, then $\tau$ is a trace element in $\omega_{B/R}$. \item If $g \in B$, then the image of $\tau$ in $\omega_{B_g/A}$ is a trace element. \item If $B = B_1 \times B_2$, then $\tau$ maps to a trace element in both $\omega_{B_1/A}$ and $\omega_{B_2/A}$. \end{enumerate}


discriminant-lemma-glue-trace

Let $A \to B$ be a flat quasi-finite map of Noetherian rings. Let $g_1, \ldots, g_m \in B$ be elements generating the unit ideal. Let $\tau \in \omega_{B/A}$ be an element whose image in $\omega_{B_{g_i}/A}$ is a trace element for $A \to B_{g_i}$. Then $\tau$ is a trace element.


discriminant-lemma-dualizing-tau

Let $A \to B$ be a flat quasi-finite map of Noetherian rings. There exists a trace element $\tau \in \omega_{B/A}$.


discriminant-lemma-tau-nonzero

Let $k$ be a field and let $A$ be a finite $k$-algebra. Assume $A$ is local with residue field $k'$. The following are equivalent \begin{enumerate} \item $\text{Trace}_{A/k}$ is nonzero, \item $\tau_{A/k} \in \omega_{A/k}$ is nonzero, and \item $k'/k$ is separable and $\text{length}_A(A)$ is prime to the characteristic of $k$. \end{enumerate}


discriminant-lemma-noether-different-product

Let $A \to B_i$, $i = 1, 2$ be ring maps. Set $B = B_1 \times B_2$. \begin{enumerate} \item The annihilator $J$ of $\Ker(B \otimes_A B \to B)$ is $J_1 \times J_2$ where $J_i$ is the annihilator of $\Ker(B_i \otimes_A B_i \to B_i)$. \item The Noether different $\mathfrak{D}$ of $B$ over $A$ is $\mathfrak{D}_1 \times \mathfrak{D}_2$, where $\mathfrak{D}_i$ is the Noether different of $B_i$ over $A$. \end{enumerate}


discriminant-lemma-noether-different-base-change

Let $A \to B$ be a finite type ring map. Let $A \to A'$ be a flat ring map. Set $B' = B \otimes_A A'$. \begin{enumerate} \item The annihilator $J'$ of $\Ker(B' \otimes_{A'} B' \to B')$ is $J \otimes_A A'$ where $J$ is the annihilator of $\Ker(B \otimes_A B \to B)$. \item The Noether different $\mathfrak{D}'$ of $B'$ over $A'$ is $\mathfrak{D}B'$, where $\mathfrak{D}$ is the Noether different of $B$ over $A$. \end{enumerate}


discriminant-lemma-noether-different-localization

Let $A \to B' \to B$ be ring maps with $A \to B'$ of finite type and $B' \to B$ inducing an open immersion of spectra. \begin{enumerate} \item The annihilator $J$ of $\Ker(B \otimes_A B \to B)$ is $J' \otimes_{B'} B$ where $J'$ is the annihilator of $\Ker(B' \otimes_A B' \to B')$. \item The Noether different $\mathfrak{D}$ of $B$ over $A$ is $\mathfrak{D}'B$, where $\mathfrak{D}'$ is the Noether different of $B'$ over $A$. \end{enumerate}


discriminant-lemma-noether-pairing-compatibilities

Let $A \to B$ be a quasi-finite homomorphism of Noetherian rings. \begin{enumerate} \item If $A \to A'$ is a flat map of Noetherian rings, then $$ \xymatrix{ \omega_{B/A} \times J \ar[r] \ar[d] & B \ar[d]  \\ \omega_{B'/A'} \times J' \ar[r] & B' } $$ is commutative where notation as in Lemma \ref{lemma-noether-different-base-change} and horizontal arrows are given by (\ref{equation-pairing-noether}). \item If $B = B_1 \times B_2$, then $$ \xymatrix{ \omega_{B/A} \times J \ar[r] \ar[d] & B \ar[d]  \\ \omega_{B_i/A} \times J_i \ar[r] & B_i } $$ is commutative for $i = 1, 2$ where notation as in Lemma \ref{lemma-noether-different-product} and horizontal arrows are given by (\ref{equation-pairing-noether}). \end{enumerate}


discriminant-lemma-noether-pairing-flat-quasi-finite

Let $A \to B$ be a flat quasi-finite homomorphism of Noetherian rings. The pairing of Remark \ref{remark-construction-pairing} induces an isomorphism $J \to \Hom_B(\omega_{B/A}, B)$.


discriminant-lemma-noether-different-flat-quasi-finite

Let $A \to B$ be a flat quasi-finite homomorphism of Noetherian rings. The diagram $$ \xymatrix{ J \ar[rr] \ar[rd]_\mu & & \Hom_B(\omega_{B/A}, B) \ar[ld]^{\varphi \mapsto \varphi(\tau_{B/A})} \\ & B } $$ commutes where the horizontal arrow is the isomorphism of Lemma \ref{lemma-noether-pairing-flat-quasi-finite}. Hence the Noether different of $B$ over $A$ is the image of the map $\Hom_B(\omega_{B/A}, B) \to B$.


discriminant-lemma-kahler-different-complete-intersection

Let $A$ be a ring. Let $n \geq 1$ and $f_1, \ldots, f_n \in A[x_1, \ldots, x_n]$. Set $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$. The K\"ahler different of $B$ over $A$ is the ideal of $B$ generated by $\det(\partial f_i/\partial x_j)$.


discriminant-lemma-dedekind-different-ideal

Assume the Dedekind different of $A \to B$ is defined. Consider the statements \begin{enumerate} \item $A \to B$ is flat, \item $A$ is a normal ring, \item $\text{Trace}_{L/K}(B) \subset A$, \item $1 \in \mathcal{L}_{B/A}$, and \item the Dedekind different $\mathfrak{D}_{B/A}$ is an ideal of $B$. \end{enumerate} Then we have (1) $\Rightarrow$ (3), (2) $\Rightarrow$ (3), (3) $\Leftrightarrow$ (4), and (4) $\Rightarrow$ (5).


discriminant-lemma-dedekind-complementary-module

If the Dedekind different of $A \to B$ is defined, then there is a canonical isomorphism $\mathcal{L}_{B/A} \to \omega_{B/A}$.


discriminant-lemma-flat-dedekind-complementary-module-trace

If the Dedekind different of $A \to B$ is defined and $A \to B$ is flat, then \begin{enumerate} \item the canonical isomorphism $\mathcal{L}_{B/A} \to \omega_{B/A}$ sends $1 \in \mathcal{L}_{B/A}$ to the trace element $\tau_{B/A} \in \omega_{B/A}$, and \item the Dedekind different is $\mathfrak{D}_{B/A} = \{b \in B \mid b\omega_{B/A} \subset B\tau_{B/A}\}$. \end{enumerate}


discriminant-lemma-flat-gorenstein-agree-noether

Let $f : Y \to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$ be affine open subschemes with $f(V) \subset U$. If $\omega_{Y/X}|_V$ is invertible, i.e., if $\omega_{B/A}$ is an invertible $B$-module, then $$ \mathfrak{D}_f|_V = \widetilde{\mathfrak{D}} $$ as coherent ideal sheaves on $V$ where $\mathfrak{D} \subset B$ is the Noether different of $B$ over $A$.


discriminant-lemma-base-change-different

Consider a cartesian diagram of Noetherian schemes $$ \xymatrix{ Y' \ar[d]_{f'} \ar[r] & Y \ar[d]^f \\ X' \ar[r]^g & X } $$ with $f$ flat and quasi-finite. Let $R \subset Y$, resp.\ $R' \subset Y'$ be the closed subscheme cut out by the different $\mathfrak{D}_f$, resp.\ $\mathfrak{D}_{f'}$. Then $Y' \to Y$ induces a bijective closed immersion $R' \to R \times_Y Y'$. If $g$ is flat or if $\omega_{Y/X}$ is invertible, then $R' = R \times_Y Y'$.


discriminant-lemma-norm-different-in-discriminant

Let $f : Y \to X$ be a finite flat morphism of Noetherian schemes. Then $\text{Norm}_f : f_*\mathcal{O}_Y \to \mathcal{O}_X$ maps $f_*\mathfrak{D}_f$ into the ideal sheaf of the discriminant $D_f$.


discriminant-lemma-different-ramification

Let $f : Y \to X$ be a flat quasi-finite morphism of Noetherian schemes. The closed subscheme $R \subset Y$ defined by the different $\mathfrak{D}_f$ is exactly the set of points where $f$ is not \'etale (equivalently not unramified).


discriminant-lemma-norm-different-is-discriminant

Let $f : Y \to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $R \subset Y$ be the closed subscheme defined by $\mathfrak{D}_f$. \begin{enumerate} \item If $\omega_{Y/X}$ is invertible, then $R$ is a locally principal closed subscheme of $Y$. \item If $\omega_{Y/X}$ is invertible and $f$ is finite, then the norm of $R$ is the discriminant $D_f$ of $f$. \item If $\omega_{Y/X}$ is invertible and $f$ is \'etale at the associated points of $Y$, then $R$ is an effective Cartier divisor and there is an isomorphism $\mathcal{O}_Y(R) = \omega_{Y/X}$. \end{enumerate}


discriminant-lemma-syntomic-quasi-finite

Let $f : Y \to X$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is locally quasi-finite and syntomic, \item $f$ is locally quasi-finite, flat, and a local complete intersection morphism, \item $f$ is locally quasi-finite, flat, locally of finite presentation, and the fibres of $f$ are local complete intersections, \item $f$ is locally quasi-finite and for every $y \in Y$ there are affine opens $y \in V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$ with $f(V) \subset U$ an integer $n$ and $h, f_1, \ldots, f_n \in A[x_1, \ldots, x_n]$ such that $B = A[x_1, \ldots, x_n, 1/h]/(f_1, \ldots, f_n)$, \item for every $y \in Y$ there are affine opens $y \in V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$ with $f(V) \subset U$ such that $A \to B$ is a relative global complete intersection of the form $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$, \item $f$ is locally quasi-finite, flat, locally of finite presentation, and $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and \item $f$ is flat, locally of finite presentation, $\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$, \end{enumerate}


discriminant-lemma-characterize-invertible

Invertibility of the relative dualizing module. \begin{enumerate} \item If $A \to B$ is a quasi-finite flat homomorphism of Noetherian rings, then $\omega_{B/A}$ is an invertible $B$-module if and only if $\omega_{B \otimes_A \kappa(\mathfrak p)/\kappa(\mathfrak p)}$ is an invertible $B \otimes_A \kappa(\mathfrak p)$-module for all primes $\mathfrak p \subset A$. \item If $Y \to X$ is a quasi-finite flat morphism of Noetherian schemes, then $\omega_{Y/X}$ is invertible if and only if $\omega_{Y_x/x}$ is invertible for all $x \in X$. \end{enumerate}


discriminant-lemma-dim-zero-global-complete-intersection-over-field

Let $k$ be a field. Let $B = k[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$ be a global complete intersection over $k$ of dimension $0$. Then $\omega_{B/k}$ is invertible.


discriminant-lemma-dualizing-syntomic-quasi-finite

Let $f : Y \to X$ be a morphism of locally Noetherian schemes. If $f$ satisfies the equivalent conditions of Lemma \ref{lemma-syntomic-quasi-finite} then $\omega_{Y/X}$ is an invertible $\mathcal{O}_Y$-module.


discriminant-lemma-universal-quasi-finite-syntomic-etale

With notation as in Example \ref{example-universal-quasi-finite-syntomic} the schemes $X_{n, d}$ and $Y_{n, d}$ are regular and irreducible, the morphism $Y_{n, d} \to X_{n, d}$ is locally quasi-finite and syntomic, and there is a dense open subscheme $V \subset Y_{n, d}$ such that $Y_{n, d} \to X_{n, d}$ restricts to an \'etale morphism $V \to X_{n, d}$.


discriminant-lemma-locally-comes-from-universal

Let $f : Y \to X$ be a morphism of schemes. If $f$ satisfies the equivalent conditions of Lemma \ref{lemma-syntomic-quasi-finite} then for every $y \in Y$ there exist $n, d$ and a commutative diagram $$ \xymatrix{ Y \ar[d] & V \ar[d] \ar[l] \ar[r] & Y_{n, d} \ar[d] \\ X & U \ar[l] \ar[r] & X_{n, d} } $$ where $U \subset X$ and $V \subset Y$ are open, where $Y_{n, d} \to X_{n, d}$ is as in Example \ref{example-universal-quasi-finite-syntomic}, and where the square on the right hand side is cartesian.


discriminant-lemma-syntomic-finite

Let $f : Y \to X$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is finite and syntomic, \item $f$ is finite, flat, and a local complete intersection morphism, \item $f$ is finite, flat, locally of finite presentation, and the fibres of $f$ are local complete intersections, \item $f$ is finite and for every $x \in X$ there is an affine open $x \in U = \Spec(A) \subset X$ an integer $n$ and $f_1, \ldots, f_n \in A[x_1, \ldots, x_n]$ such that $f^{-1}(U)$ is isomorphic to the spectrum of $A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$, \item $f$ is finite, flat, locally of finite presentation, and $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and \item $f$ is finite, flat, locally of finite presentation, and $\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$, \end{enumerate}


discriminant-lemma-universal-finite-syntomic

With notation as in Example \ref{example-universal-finite-syntomic} there is an open subscheme $U_d \subset X_d$ with the following property: a morphism of schemes $X \to X_d$ factors through $U_d$ if and only if $Y_d \times_{X_d} X \to X$ is syntomic.


discriminant-lemma-universal-finite-syntomic-smooth

With notation as in Example \ref{example-universal-finite-syntomic} and $U_d$ as in Lemma \ref{lemma-universal-finite-syntomic} then $U_d$ is smooth over $\Spec(\mathbf{Z})$.


discriminant-lemma-universal-finite-syntomic-etale

With notation as in Example \ref{example-universal-finite-syntomic} consider the open subscheme $U'_d \subset X_d$ over which $\pi_d$ is \'etale. Then $U'_d$ is a dense subset of the open $U_d$ of Lemma \ref{lemma-universal-finite-syntomic}


discriminant-lemma-locally-comes-from-universal-finite

Let $f : Y \to X$ be a morphism of schemes. If $f$ satisfies the equivalent conditions of Lemma \ref{lemma-syntomic-finite} then for every $x \in X$ there exist a $d$ and a commutative diagram $$ \xymatrix{ Y \ar[d] & V \ar[d] \ar[l] \ar[r] & V_d \ar[d] \ar[r] & Y_d \ar[d]^{\pi_d}\\ X & U \ar[l] \ar[r] & U_d \ar[r] & X_d } $$ with the following properties \begin{enumerate} \item $U \subset X$ is open and $V = f^{-1}(U)$, \item $\pi_d : Y_d \to X_d$ is as in Example \ref{example-universal-finite-syntomic}, \item $U_d \subset X_d$ is as in Lemma \ref{lemma-universal-finite-syntomic} and $V_d = \pi_d^{-1}(U_d) \subset Y_d$, \item where the middle square is cartesian. \end{enumerate}


discriminant-lemma-tate

\begin{reference} \cite[Appendix]{Mazur-Roberts} \end{reference} Let $A \to P$ be a ring map. Let $f_1, \ldots, f_n \in P$ be a Koszul regular sequence. Assume $B = P/(f_1, \ldots, f_n)$ is flat over $A$. Let $g_1, \ldots, g_n \in P \otimes_A B$ be a Koszul regular sequence generating the kernel of the multiplication map $P \otimes_A B \to B$. Write $f_i \otimes 1 = \sum g_{ij} g_j$. Then the annihilator of $\Ker(B \otimes_A B \to B)$ is a principal ideal generated by the image of $\det(g_{ij})$.


discriminant-lemma-quasi-finite-complete-intersection

Let $A$ be a ring. Let $n \geq 1$ and $h, f_1, \ldots, f_n \in A[x_1, \ldots, x_n]$. Set $B = A[x_1, \ldots, x_n, 1/h]/(f_1, \ldots, f_n)$. Assume that $B$ is quasi-finite over $A$. Then \begin{enumerate} \item $B$ is flat over $A$ and $A \to B$ is a relative local complete intersection, \item the annihilator $J$ of $I = \Ker(B \otimes_A B \to B)$ is free of rank $1$ over $B$, \item the Noether different of $B$ over $A$ is generated by $\det(\partial f_i/\partial x_j)$ in $B$. \end{enumerate}


discriminant-lemma-different-syntomic-quasi-finite

Let $f : Y \to X$ be a morphism of Noetherian schemes. If $f$ satisfies the equivalent conditions of Lemma \ref{lemma-syntomic-quasi-finite} then the different $\mathfrak{D}_f$ of $f$ is the K\"ahler different of $f$.


discriminant-lemma-different-quasi-finite-complete-intersection

Let $A$ be a ring. Let $n \geq 1$ and $h, f_1, \ldots, f_n \in A[x_1, \ldots, x_n]$. Set $B = A[x_1, \ldots, x_n, 1/h]/(f_1, \ldots, f_n)$. Assume that $B$ is quasi-finite over $A$. Then there is an isomorphism $B \to \omega_{B/A}$ mapping $\det(\partial f_i/\partial x_j)$ to $\tau_{B/A}$.


discriminant-lemma-discriminant-quasi-finite-morphism-smooth

Let $S$ be a Noetherian scheme. Let $X$, $Y$ be smooth schemes of relative dimension $n$ over $S$. Let $f : Y \to X$ be a quasi-finite morphism over $S$. Then $f$ is flat and the closed subscheme $R \subset Y$ cut out by the different of $f$ is the locally principal closed subscheme cut out by $$ \wedge^n(\text{d}f) \in \Gamma(Y, (f^*\Omega^n_{X/S})^{\otimes -1} \otimes_{\mathcal{O}_Y} \Omega^n_{Y/S}) $$ If $f$ is \'etale at the associated points of $Y$, then $R$ is an effective Cartier divisor and $$ f^*\Omega^n_{X/S} \otimes_{\mathcal{O}_Y} \mathcal{O}(R) = \Omega^n_{Y/S} $$ as invertible sheaves on $Y$.


discriminant-lemma-explain-condition

Let $A \to B$ be a map of Noetherian rings. Consider the conditions \begin{enumerate} \item nonzerodivisors of $A$ map to nonzerodivisors of $B$, \item (1) holds and $Q(A) \to Q(A) \otimes_A B$ is flat, \item $A \to B_\mathfrak q$ is flat for every $\mathfrak q \in \text{Ass}(B)$, \item (3) holds and $A \to B_\mathfrak q$ is flat for every $\mathfrak q$ lying over an element in $\text{Ass}(A)$. \end{enumerate} Then we have the following implications $$ \xymatrix{ (1) & (2) \ar@{=>}[l] \ar@{=>}[d] \\ (3) \ar@{=>}[u] & (4) \ar@{=>}[l] } $$ If going up holds for $A \to B$ then (2) and (4) are equivalent.


discriminant-lemma-agree-dedekind

Assume the Dedekind different is defined for $A \to B$. Set $X = \Spec(A)$ and $Y = \Spec(B)$. The generalization of Remark \ref{remark-different-generalization} applies to the morphism $f : Y \to X$ if and only if $1 \in \mathcal{L}_{B/A}$ (e.g., if $A$ is normal, see Lemma \ref{lemma-dedekind-different-ideal}). In this case $\mathfrak{D}_{B/A}$ is an ideal of $B$ and we have $$ \mathfrak{D}_f = \widetilde{\mathfrak{D}_{B/A}} $$ as coherent ideal sheaves on $Y$.


discriminant-lemma-compare-dualizing

Let $f : Y \to X$ be a quasi-finite separated morphism of Noetherian schemes. For every pair of affine opens $\Spec(B) = V \subset Y$, $\Spec(A) = U \subset X$ with $f(V) \subset U$ there is an isomorphism $$ H^0(V, f^!\mathcal{O}_Y) = \omega_{B/A} $$ where $f^!$ is as in Duality for Schemes, Section \ref{duality-section-upper-shriek}. These isomorphisms are compatible with restriction maps and define a canonical isomorphism $H^0(f^!\mathcal{O}_X) = \omega_{Y/X}$ with $\omega_{Y/X}$ as in Remark \ref{remark-relative-dualizing-for-quasi-finite}. Similarly, if $f : Y \to X$ is a quasi-finite morphism of schemes of finite type over a Noetherian base $S$ endowed with a dualizing complex $\omega_S^\bullet$, then $H^0(f_{new}^!\mathcal{O}_X) = \omega_{Y/X}$.


discriminant-lemma-compare-trace

Let $f : Y \to X$ be a finite flat morphism of Noetherian schemes. The map $$ \text{Trace}_f : f_*\mathcal{O}_Y \longrightarrow \mathcal{O}_X $$ of Section \ref{section-discriminant} corresponds to a map $\mathcal{O}_Y \to f^!\mathcal{O}_X$. Denote $\tau_{Y/X} \in H^0(Y, f^!\mathcal{O}_X)$ the image of $1$. Via the isomorphism $H^0(f^!\mathcal{O}_X) = \omega_{X/Y}$ of Lemma \ref{lemma-compare-dualizing} this agrees with the construction in Remark \ref{remark-relative-dualizing-for-flat-quasi-finite}.


discriminant-lemma-gorenstein-quasi-finite

Let $f : Y \to X$ be a quasi-finite morphism of Noetherian schemes. The following are equivalent \begin{enumerate} \item $f$ is Gorenstein, \item $f$ is flat and the fibres of $f$ are Gorenstein, \item $f$ is flat and $\omega_{Y/X}$ is invertible (Remark \ref{remark-relative-dualizing-for-quasi-finite}), \item for every $y \in Y$ there are affine opens $y \in V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$ with $f(V) \subset U$ such that $A \to B$ is flat and $\omega_{B/A}$ is an invertible $B$-module. \end{enumerate}


discriminant-proposition-tate-map

There exists a unique rule that to every locally quasi-finite syntomic morphism of locally Noetherian schemes $Y \to X$ assigns an isomorphism $$ c_{Y/X} : \det(\NL_{Y/X}) \longrightarrow \omega_{Y/X} $$ satisfying the following two properties \begin{enumerate} \item the section $\delta(\NL_{Y/X})$ is mapped to $\tau_{Y/X}$, and \item the rule is compatible with restriction to opens and with base change. \end{enumerate}


stacks-limits-lemma-limit-preserving-objects

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ is limit preserving (Definition \ref{definition-limit-preserving}), then $f$ is limit preserving on objects (Criteria for Representability, Section \ref{criteria-section-limit-preserving}).


stacks-limits-lemma-base-change-limit-preserving

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p : \mathcal{X} \to \mathcal{Y}$ is limit preserving, then so is the base change $p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$ of $p$ by $q$.


stacks-limits-lemma-composition-limit-preserving

Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ and $q$ are limit preserving, then so is the composition $q \circ p$.


stacks-limits-lemma-representable-by-spaces-limit-preserving

Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \begin{enumerate} \item $p$ is limit preserving, \item $p$ is limit preserving on objects, and \item $p$ is locally of finite presentation (see Algebraic Stacks, Definition \ref{algebraic-definition-relative-representable-property}). \end{enumerate}


stacks-limits-lemma-limit-preserving-diagonal

Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The following are equivalent \begin{enumerate} \item the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is limit preserving, and \item for every directed limit $U = \lim U_i$ of affine schemes over $S$ the functor $$ \colim \mathcal{X}_{U_i} \longrightarrow \mathcal{X}_U \times_{\mathcal{Y}_U} \colim \mathcal{Y}_{U_i} $$ is fully faithful. \end{enumerate} In particular, if $p$ is limit preserving, then $\Delta$ is too.


stacks-limits-lemma-locally-finite-presentation-limit-preserving

Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. If $\mathcal{X} \to S$ is locally of finite presentation, then $\mathcal{X}$ is limit preserving in the sense of Artin's Axioms, Definition \ref{artin-definition-limit-preserving} (equivalently: the morphism $\mathcal{X} \to S$ is limit preserving).


stacks-limits-lemma-eventually-separated

In Situation \ref{situation-descent} assume that $\mathcal{X}_0 \to Y_0$ is a morphism from algebraic stack to $Y_0$. Assume $\mathcal{X}_0$ is quasi-compact and quasi-separated. If $Y \times_{Y_0} \mathcal{X}_0 \to Y$ is separated, then $Y_i \times_{Y_0} \mathcal{X}_0 \to Y_i$ is separated for all sufficiently large $i \in I$.


stacks-limits-lemma-descend-a-stack-down

Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system of algebraic spaces over $I$. Assume \begin{enumerate} \item the morphisms $f_{ii'} : X_i \to X_{i'}$ are affine, \item the spaces $X_i$ are quasi-compact and quasi-separated. \end{enumerate} Let $X = \lim X_i$. If $\mathcal{X}$ is an algebraic stack of finite presentation over $X$, then there exists an $i \in I$ and an algebraic stack $\mathcal{X}_i$ of finite presentation over $X_i$ with $\mathcal{X} \cong \mathcal{X}_i \times_{X_i} X$ as algebraic stacks over $X$.


stacks-limits-lemma-finite-type-closed-in-finite-presentation

Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. Assume: \begin{enumerate} \item $f$ is of finite type and quasi-separated, \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : \mathcal{X}' \to Y$ and a closed immersion $\mathcal{X} \to \mathcal{X}'$ of algebraic stacks over $Y$.


stacks-limits-lemma-separated-closed-in-finite-presentation

Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. Assume: \begin{enumerate} \item $f$ is of finite type and separated, \item $Y$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a separated morphism of finite presentation $f' : \mathcal{X}' \to Y$ and a closed immersion $\mathcal{X} \to \mathcal{X}'$ of algebraic stacks over $Y$.


stacks-limits-proposition-characterize-locally-finite-presentation

\begin{reference} This is a special case of \cite[Lemma 2.3.15]{Emerton-Gee} \end{reference} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is limit preserving, \item $f$ is limit preserving on objects, and \item $f$ is locally of finite presentation. \end{enumerate}


limits-lemma-directed-inverse-system-affine-schemes-has-limit

Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.  If all the schemes $S_i$ are affine, then the limit $S = \lim_i S_i$ exists in the category of schemes. In fact $S$ is affine and $S = \Spec(\colim_i R_i)$ with $R_i = \Gamma(S_i, \mathcal{O})$.


limits-lemma-directed-inverse-system-has-limit

Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. If all the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, then the limit $S = \lim_i S_i$ exists in the category of schemes. Moreover, \begin{enumerate} \item each of the morphisms $f_i : S \to S_i$ is affine, \item for an element $0 \in I$ and any open subscheme $U_0 \subset S_0$ we have $$ f_0^{-1}(U_0) = \lim_{i \geq 0} f_{i0}^{-1}(U_0) $$ in the category of schemes. \end{enumerate}


limits-lemma-scheme-over-limit

Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume all the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, Let $S = \lim_i S_i$. Let $0 \in I$. Suppose that $T$ is a scheme over $S_0$. Then $$ T \times_{S_0} S = \lim_{i \geq 0} T \times_{S_0} S_i $$


limits-lemma-infinite-product

\begin{slogan} Infinite products of affine schemes exist and are affine. \end{slogan} Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_i : T_i \to S$ be an affine morphism. Then the product $T = \prod T_i$ exists in the category of schemes over $S$. In fact, we have $$ T = \lim_{\{i_1, \ldots, i_n\} \subset I} T_{i_1} \times_S \ldots \times_S T_{i_n} $$ and the projection morphisms $T \to T_{i_1} \times_S \ldots \times_S T_{i_n}$ are affine.


limits-lemma-infinite-product-surjective

Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_i : T_i \to S$ be a surjective affine morphism. Then the product $T = \prod T_i$ in the category of schemes over $S$ (Lemma \ref{lemma-infinite-product}) maps surjectively to $S$.


limits-lemma-infinite-product-integral

Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_i : T_i \to S$ be an integral morphism. Then the product $T = \prod T_i$ in the category of schemes over $S$ (Lemma \ref{lemma-infinite-product}) is integral over $S$.


limits-lemma-inverse-limit-sets

Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Then $S_{set} = \lim_i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$.


limits-lemma-inverse-limit-top

\begin{reference} \cite[IV, Proposition 8.2.9]{EGA} \end{reference} Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Then $S_{top} = \lim_i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$.


limits-lemma-limit-nonempty

Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). If all the schemes $S_i$ are nonempty and quasi-compact, then the limit $S = \lim_i S_i$ is nonempty.


limits-lemma-inverse-limit-irreducibles

Let $S = \lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \ref{lemma-directed-inverse-system-has-limit}). Let $s \in S$ with images $s_i \in S_i$. Then \begin{enumerate} \item $s = \lim s_i$ as schemes, i.e., $\kappa(s) = \colim \kappa(s_i)$, \item $\overline{\{s\}} = \lim \overline{\{s_i\}}$ as sets, and \item $\overline{\{s\}} = \lim \overline{\{s_i\}}$ as schemes where $\overline{\{s\}}$ and $\overline{\{s_i\}}$ are endowed with the reduced induced scheme structure. \end{enumerate}


limits-lemma-topology-limit

In Situation \ref{situation-descent}. \begin{enumerate} \item We have $S_{set} = \lim_i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$. \item We have $S_{top} = \lim_i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$. \item If $s, s' \in S$ and $s'$ is not a specialization of $s$ then for some $i \in I$ the image $s'_i \in S_i$ of $s'$ is not a specialization of the image $s_i \in S_i$ of $s$. \item Add more easy facts on topology of $S$ here. (Requirement: whatever is added should be easy in the affine case.) \end{enumerate}


limits-lemma-limit-closed-nonempty

In Situation \ref{situation-descent}. Suppose for each $i$ we are given a nonempty closed subset $Z_i \subset S_i$ with $f_{ii'}(Z_i) \subset Z_{i'}$. Then there exists a point $s \in S$ with $f_i(s) \in Z_i$ for all $i$.


limits-lemma-limit-fibre-product-empty

In Situation \ref{situation-descent}. Suppose we are given an $i$ and a morphism $T \to S_i$ such that \begin{enumerate} \item $T \times_{S_i} S = \emptyset$, and \item $T$ is quasi-compact. \end{enumerate} Then $T \times_{S_i} S_{i'} = \emptyset$ for all sufficiently large $i'$.


limits-lemma-limit-contained-in-constructible

In Situation \ref{situation-descent}. Suppose we are given an $i$ and a locally constructible subset $E \subset S_i$ such that $f_i(S) \subset E$. Then $f_{i'i}(S_{i'}) \subset E$ for all sufficiently large $i'$.


limits-lemma-descend-opens

In Situation \ref{situation-descent} we have the following: \begin{enumerate} \item Given any quasi-compact open $V \subset S = \lim_i S_i$ there exists an $i \in I$ and a quasi-compact open $V_i \subset S_i$ such that $f_i^{-1}(V_i) = V$. \item Given $V_i \subset S_i$ and $V_{i'} \subset S_{i'}$ quasi-compact opens such that $f_i^{-1}(V_i) = f_{i'}^{-1}(V_{i'})$ there exists an index $i'' \geq i, i'$ such that $f_{i''i}^{-1}(V_i) = f_{i''i'}^{-1}(V_{i'})$. \item If $V_{1, i}, \ldots, V_{n, i} \subset S_i$ are quasi-compact opens and $S = f_i^{-1}(V_{1, i}) \cup \ldots \cup f_i^{-1}(V_{n, i})$ then $S_{i'} = f_{i'i}^{-1}(V_{1, i}) \cup \ldots \cup f_{i'i}^{-1}(V_{n, i})$ for some $i' \geq i$. \end{enumerate}


limits-lemma-limit-quasi-affine

In Situation \ref{situation-descent} if $S$ is quasi-affine, then for some $i_0 \in I$ the schemes $S_i$ for $i \geq i_0$ are quasi-affine.


limits-lemma-limit-affine

In Situation \ref{situation-descent} if $S$ is affine, then for some $i_0 \in I$ the schemes $S_i$ for $i \geq i_0$ are affine.


limits-lemma-limit-separated

In Situation \ref{situation-descent} if $S$ is separated, then for some $i_0 \in I$ the schemes $S_i$ for $i \geq i_0$ are separated.


limits-lemma-limit-ample

In Situation \ref{situation-descent} let $\mathcal{L}_0$ be an invertible sheaf of modules on $S_0$. If the pullback $\mathcal{L}$ to $S$ is ample, then for some $i \in I$ the pullback $\mathcal{L}_i$ to $S_i$ is ample.


limits-lemma-finite-type-eventually-closed

Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Let $Y \to X$ be a morphism of schemes over $S$. \begin{enumerate} \item If $Y \to X$ is a closed immersion, $X_i$ quasi-compact, and $Y$ locally of finite type over $S$, then $Y \to X_i$ is a closed immersion for $i$ large enough. \item If $Y \to X$ is an immersion, $X_i$ quasi-separated, $Y \to S$ locally of finite type, and $Y$ quasi-compact, then $Y \to X_i$ is an immersion for $i$ large enough. \item If $Y \to X$ is an isomorphism, $X_i$ quasi-compact, $X_i \to S$ locally of finite type, the transition morphisms $X_{i'} \to X_i$ are closed immersions, and $Y \to S$ is locally of finite presentation, then $Y \to X_i$ is an isomorphism for $i$ large enough. \end{enumerate}


limits-lemma-eventually-separated

Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item $X \to S$ separated. \end{enumerate} Then $X_i \to S$ is separated for all $i$ large enough.


limits-lemma-eventually-affine

Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item $X \to S$ affine. \end{enumerate} Then $X_i \to S$ is affine for $i$ large enough.


limits-lemma-eventually-finite

Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item the transition morphisms $X_{i'} \to X_i$ are finite, \item $X_i \to S$ locally of finite type \item $X \to S$ integral. \end{enumerate} Then $X_i \to S$ is finite for $i$ large enough.


limits-lemma-eventually-closed-immersion

Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \begin{enumerate} \item $S$ quasi-compact and quasi-separated, \item $X_i$ quasi-compact and quasi-separated, \item the transition morphisms $X_{i'} \to X_i$ are closed immersions, \item $X_i \to S$ locally of finite type \item $X \to S$ a closed immersion. \end{enumerate} Then $X_i \to S$ is a closed immersion for $i$ large enough.


limits-lemma-quasi-affine-finite-type-over-Z

Let $W$ be a quasi-affine scheme of finite type over $\mathbf{Z}$. Suppose $W \to \Spec(R)$ is an open immersion into an affine scheme. There exists a finite type $\mathbf{Z}$-algebra $A \subset R$ which induces an open immersion $W \to \Spec(A)$. Moreover, $R$ is the directed colimit of such subalgebras.


limits-lemma-diagram

Suppose given a cartesian diagram of rings $$ \xymatrix{ B \ar[r]_s & R \\ B'\ar[u] \ar[r] & R' \ar[u]_t } $$ Let $W' \subset \Spec(R')$ be an open of the form $W' = D(f_1) \cup \ldots \cup D(f_n)$ such that $t(f_i) = s(g_i)$ for some $g_i \in B$ and $B_{g_i} \cong R_{s(g_i)}$. Then $B' \to R'$ induces an open immersion of $W'$ into $\Spec(B')$.


limits-lemma-approximate

Let $S$ be a quasi-compact and quasi-separated scheme. Let $V \subset S$ be a quasi-compact open. Let $I$ be a directed set and let $(V_i, f_{ii'})$ be an inverse system of schemes over $I$ with affine transition maps, with each $V_i$ of finite type over $\mathbf{Z}$, and with $V = \lim V_i$. Then there exist \begin{enumerate} \item a directed set $J$, \item an inverse system of schemes $(S_j, g_{jj'})$ over $J$, \item an order preserving map $\alpha : J \to I$, \item open subschemes $V'_j \subset S_j$, and \item isomorphisms $V'_j \to V_{\alpha(j)}$ \end{enumerate} such that \begin{enumerate} \item the transition morphisms $g_{jj'} : S_j \to S_{j'}$ are affine, \item each $S_j$ is of finite type over $\mathbf{Z}$, \item $g_{jj'}^{-1}(V'_{j'}) = V'_j$, \item $S = \lim S_j$ and $V = \lim V'_j$, and \item the diagrams $$ \vcenter{ \xymatrix{ V \ar[d] \ar[rd] \\ V'_j \ar[r] & V_{\alpha(j)} } } \quad\text{and}\quad \vcenter{ \xymatrix{ V'_j \ar[r] \ar[d] & V_{\alpha(j)} \ar[d] \\ V'_{j'} \ar[r] & V_{\alpha(j')} } } $$ are commutative. \end{enumerate}


limits-lemma-surjection-is-enough

Let $f : X \to S$ be a morphism of schemes. If for every directed limit $T = \lim_{i \in I} T_i$ of affine schemes over $S$ the map $$ \colim \Mor_S(T_i, X) \longrightarrow \Mor_S(T, X) $$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition \ref{proposition-characterize-locally-finite-presentation} parts (2) and (3) it suffices to check surjectivity of the map.


limits-lemma-relative-approximation

Let $f : X \to S$ be a morphism of schemes. Assume that \begin{enumerate} \item $X$ is quasi-compact and quasi-separated, and \item $S$ is quasi-separated. \end{enumerate} Then $X = \lim X_i$ is a limit of a directed system of schemes $X_i$ of finite presentation over $S$ with affine transition morphisms over $S$.


limits-lemma-integral-limit-finite-and-finite-presentation

Let $X \to S$ be an integral morphism with $S$ quasi-compact and quasi-separated. Then $X = \lim X_i$ with $X_i \to S$ finite and of finite presentation.


limits-lemma-descend-affine-finite-presentation

Notation and assumptions as in Situation \ref{situation-descent-property}. If $f$ is affine, then there exists an index $i \geq 0$ such that $f_i$ is affine.


limits-lemma-descend-finite-finite-presentation

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a finite morphism, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is finite.


limits-lemma-descend-unramified

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is unramified, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is unramified.


limits-lemma-descend-closed-immersion-finite-presentation

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a closed immersion, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is a closed immersion.


limits-lemma-descend-separated-finite-presentation

Notation and assumptions as in Situation \ref{situation-descent-property}. If $f$ is separated, then $f_i$ is separated for some $i \geq 0$.


limits-lemma-descend-flat-finite-presentation

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is flat, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is flat for some $i \geq 0$.


limits-lemma-descend-finite-locally-free

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is finite locally free (of degree $d$), \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is finite locally free (of degree $d$) for some $i \geq 0$.


limits-lemma-descend-smooth

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is smooth, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is smooth for some $i \geq 0$.


limits-lemma-descend-etale

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is \'etale, \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is \'etale for some $i \geq 0$.


limits-lemma-descend-isomorphism

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is an isomorphism, and \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is an isomorphism for some $i \geq 0$.


limits-lemma-descend-open-immersion

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is an open immersion, and \item $f_0$ is locally of finite presentation, \end{enumerate} then $f_i$ is an open immersion for some $i \geq 0$.


limits-lemma-descend-monomorphism

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a monomorphism, and \item $f_0$ is locally of finite type, \end{enumerate} then $f_i$ is a monomorphism for some $i \geq 0$.


limits-lemma-descend-surjective

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is surjective, and \item $f_0$ is locally of finite presentation, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is surjective.


limits-lemma-descend-syntomic

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is syntomic, and \item $f_0$ is locally of finite presentation, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is syntomic.


limits-lemma-locally-finite-type-in-finite-presentation

Let $f : X \to S$ be a morphism of schemes. Assume: \begin{enumerate} \item The morphism $f$ is locally of finite type. \item The scheme $X$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to S$ and an immersion $X \to X'$ of schemes over $S$.


limits-lemma-finite-type-closed-in-finite-presentation

Let $f : X \to S$ be a morphism of schemes. Assume: \begin{enumerate} \item The morphism $f$ is of locally of finite type. \item The scheme $X$ is quasi-compact and quasi-separated, and \item The scheme $S$ is quasi-separated. \end{enumerate} Then there exists a morphism of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$.


limits-lemma-closed-is-limit-closed-and-finite-presentation

Let $X \to Y$ be a closed immersion of schemes. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \lim X_i$ of schemes over $Y$ where $X_i \to Y$ is a closed immersion of finite presentation.


limits-lemma-finite-type-is-limit-finite-presentation

Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item The morphism $f$ is of locally of finite type. \item The scheme $X$ is quasi-compact and quasi-separated, and \item The scheme $S$ is quasi-separated. \end{enumerate} Then $X = \lim X_i$ where the $X_i \to S$ are of finite presentation, the $X_i$ are quasi-compact and quasi-separated, and the transition morphisms $X_{i'} \to X_i$ are closed immersions (which implies that $X \to X_i$ are closed immersions for all $i$).


limits-lemma-finite-in-finite-and-finite-presentation

Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is finite, and \item $S$ quasi-compact and quasi-separated. \end{enumerate} Then $X$ is a directed limit $X = \lim X_i$ where the transition maps are closed immersions and the objects $X_i$ are finite and of finite presentation over $S$.


limits-lemma-descend-finite-presentation

Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume \begin{enumerate} \item the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, \item the schemes $S_i$ are quasi-compact and quasi-separated. \end{enumerate} Let $S = \lim_i S_i$. Then we have the following: \begin{enumerate} \item For any morphism of finite presentation $X \to S$ there exists an index $i \in I$ and a morphism of finite presentation $X_i \to S_i$ such that $X \cong X_{i, S}$ as schemes over $S$. \item Given an index $i \in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$, and a morphism $\varphi : X_{i, S} \to Y_{i, S}$ over $S$, there exists an index $i' \geq i$ and a morphism $\varphi_{i'} : X_{i, S_{i'}} \to Y_{i, S_{i'}}$ whose base change to $S$ is $\varphi$. \item Given an index $i \in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$ and a pair of morphisms $\varphi_i, \psi_i : X_i \to Y_i$ whose base changes $\varphi_{i, S} = \psi_{i, S}$ are equal, there exists an index $i' \geq i$ such that $\varphi_{i, S_{i'}} = \psi_{i, S_{i'}}$. \end{enumerate} In other words, the category of schemes of finite presentation over $S$ is the colimit over $I$ of the categories of schemes of finite presentation over $S_i$.


limits-lemma-descend-modules-finite-presentation

Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume \begin{enumerate} \item all the morphisms $f_{ii'} : S_i \to S_{i'}$ are affine, \item all the schemes $S_i$ are quasi-compact and quasi-separated. \end{enumerate} Let $S = \lim_i S_i$. Then we have the following: \begin{enumerate} \item For any sheaf of $\mathcal{O}_S$-modules $\mathcal{F}$ of finite presentation there exists an index $i \in I$ and a sheaf of $\mathcal{O}_{S_i}$-modules of finite presentation $\mathcal{F}_i$ such that $\mathcal{F} \cong f_i^*\mathcal{F}_i$. \item Suppose given an index $i \in I$, sheaves of $\mathcal{O}_{S_i}$-modules $\mathcal{F}_i$, $\mathcal{G}_i$ of finite presentation and a morphism $\varphi : f_i^*\mathcal{F}_i \to f_i^*\mathcal{G}_i$ over $S$. Then there exists an index $i' \geq i$ and a morphism $\varphi_{i'} : f_{i'i}^*\mathcal{F}_i \to f_{i'i}^*\mathcal{G}_i$ whose base change to $S$ is $\varphi$. \item Suppose given an index $i \in I$, sheaves of $\mathcal{O}_{S_i}$-modules $\mathcal{F}_i$, $\mathcal{G}_i$ of finite presentation and a pair of morphisms $\varphi_i, \psi_i : \mathcal{F}_i \to \mathcal{G}_i$. Assume that the base changes are equal: $f_i^*\varphi_i = f_i^*\psi_i$. Then there exists an index $i' \geq i$ such that $f_{i'i}^*\varphi_i = f_{i'i}^*\psi_i$. \end{enumerate} In other words, the category of modules of finite presentation over $S$ is the colimit over $I$ of the categories modules of finite presentation over $S_i$.


limits-lemma-descend-invertible-modules

Let $S = \lim S_i$ be the limit of a directed system of quasi-compact and quasi-separated schemes $S_i$ with affine transition morphisms. Then \begin{enumerate} \item any finite locally free $\mathcal{O}_S$-module is the pullback of a finite locally free $\mathcal{O}_{S_i}$-module for some $i$, \item any invertible $\mathcal{O}_S$-module is the pullback of an invertible $\mathcal{O}_{S_i}$-module for some $i$, and \item any finite type quasi-coherent ideal $\mathcal{I} \subset \mathcal{O}_S$ is of the form $\mathcal{I}_i \cdot \mathcal{O}_S$ for some $i$ and some finite type quasi-coherent ideal $\mathcal{I}_i \subset \mathcal{O}_{S_i}$. \end{enumerate}


limits-lemma-descend-module-flat-finite-presentation

With notation and assumptions as in Lemma \ref{lemma-descend-finite-presentation}. Let $i \in I$. Suppose that $\varphi_i : X_i \to Y_i$ is a morphism of schemes of finite presentation over $S_i$ and that $\mathcal{F}_i$ is a quasi-coherent $\mathcal{O}_{X_i}$-module of finite presentation. If the pullback of $\mathcal{F}_i$ to $X_i \times_{S_i} S$ is flat over $Y_i \times_{S_i} S$, then there exists an index $i' \geq i$ such that the pullback of $\mathcal{F}_i$ to $X_i \times_{S_i} S_{i'}$ is flat over $Y_i \times_{S_i} S_{i'}$.


limits-lemma-descend-finite-presentation-variant

For a scheme $T$ denote $\mathcal{C}_T$ the full subcategory of schemes $W$ over $T$ such that $W$ is quasi-compact and quasi-separated and such that the structure morphism $W \to T$ is locally of finite presentation. Let $S = \lim S_i$ be a directed limit of schemes with affine transition morphisms. Then there is an equivalence of categories $$ \colim \mathcal{C}_{S_i} \longrightarrow \mathcal{C}_S $$ given by the base change functors.


limits-lemma-affine

\begin{slogan} A scheme, admitting a finite surjective map from an affine scheme, is affine. \end{slogan} Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is surjective and finite, and assume that $X$ is affine. Then $S$ is affine.


limits-lemma-affines-glued-in-closed-affine

Let $X$ be a scheme which is set theoretically the union of finitely many affine closed subschemes. Then $X$ is affine.


limits-lemma-ample-on-reduction

Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then $i^*\mathcal{L}$ is ample on $Z$, if and only if $\mathcal{L}$ is ample on $X$.


limits-lemma-thickening-quasi-affine

Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.


limits-lemma-ample-profinite-set-in-principal-affine

Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Assume we have morphisms of schemes $$ \Spec(k) \leftarrow \Spec(A) \to W \subset X $$ where $k$ is a field, $A$ is an integral $k$-algebra, $W$ is open in $X$. Then there exists an $n > 0$ and a section $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $X_s$ is affine, $X_s \subset W$, and $\Spec(A) \to W$ factors through $X_s$


limits-lemma-chow-finite-type

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a separated morphism of finite type. Then there exists an $n \geq 0$ and a diagram $$ \xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & } $$ where $X' \to \mathbf{P}^n_S$ is an immersion, and $\pi : X' \to X$ is proper and surjective.


limits-lemma-eventually-proper

\begin{slogan} If the base change of a scheme to a limit is proper, then already the base change is proper at a finite level. \end{slogan} Assumptions and notation as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is proper, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i$ such that $f_i$ is proper.


limits-lemma-proper-limit-of-proper-finite-presentation

Let $f : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then $X = \lim X_i$ is a directed limit of schemes $X_i$ proper and of finite presentation over $S$ such that all transition morphisms and the morphisms $X \to X_i$ are closed immersions.