flat-lemma-compactifications-cofiltered

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a compactifyable scheme over $S$. \begin{enumerate} \item[(a)] The category of compactifications of $X$ over $S$ is cofiltered. \item[(b)] The full subcategory consisting of compactifications $j : X \to \overline{X}$ such that $j(X)$ is dense and scheme theoretically dense in $\overline{X}$ is initial (Categories, Definition \ref{categories-definition-initial}). \item[(c)] If $f : \overline{X}' \to \overline{X}$ is a morphism of compactifications of $X$ such that $j'(X)$ is dense in $\overline{X}'$, then $f^{-1}(j(X)) = j'(X)$. \end{enumerate}


flat-lemma-compactifyable

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes over $S$ with $Y$ separated and of finite type over $S$ and $X$ compactifyable over $S$. Then $X$ has a compactification over $Y$.


flat-lemma-right-multiplicative-system

Let $S$ be a quasi-compact and quasi-separated scheme. The collection of morphisms $(u, \overline{u}) : (X', \overline{X}') \to (X, \overline{X})$ such that $u$ is an isomorphism forms a right multiplicative system (Categories, Definition \ref{categories-definition-multiplicative-system}) of arrows in the category of compactifications.


flat-lemma-check-separated

Let $X \to S$ be a morphism of schemes. If $X = U \cup V$ is an open cover such that $U \to S$ and $V \to S$ are separated and $U \cap V \to U \times_S V$ is closed, then $X \to S$ is separated.


flat-lemma-separate-disjoint-locally-closed-by-blowing-up

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. \begin{enumerate} \item If $Z_1, Z_2 \subset X$ are closed subschemes of finite presentation such that $Z_1 \cap Z_2 \cap U = \emptyset$, then there exists a $U$-admissible blowing up $X' \to X$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint. \item If $T_1, T_2 \subset U$ are disjoint constructible closed subsets, then there is a $U$-admissible blowing up $X' \to X$ such that the closures of $T_1$ and $T_2$ are disjoint. \end{enumerate}


flat-lemma-blowup-iso-along

Let $f : X \to Y$ be a proper morphism of quasi-compact and quasi-separated schemes. Let $V \subset Y$ be a quasi-compact open and $U = f^{-1}(V)$. Let $T \subset V$ be a closed subset such that $f|_U : U \to V$ is an isomorphism over an open neighbourhood of $T$ in $V$. Then there exists a $V$-admissible blowing up $Y' \to Y$ such that the strict transform $f' : X' \to Y'$ of $f$ is an isomorphism over an open neighbourhood of the closure of $T$ in $Y'$.


flat-lemma-find-common-blowups

Let $S$ be a quasi-compact and quasi-separated scheme. Let $U \to X_1$ and $U \to X_2$ be open immersions of schemes over $S$ and assume $U$, $X_1$, $X_2$ of finite type and separated over $S$. Then there exists a commutative diagram $$ \xymatrix{ X_1' \ar[d] \ar[r] & X & X_2' \ar[l] \ar[d] \\ X_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & X_2 } $$ of schemes over $S$ where $X_i' \to X_i$ is a $U$-admissible blowup, $X_i' \to X$ is an open immersion, and $X$ is separated and finite type over $S$.


flat-lemma-replaced-by-strict-transform

Let $X \to S$ and $Y \to S$ be morphisms of schemes. Let $U \subset X$ be an open subscheme. Let $V \to X \times_S Y$ be a quasi-compact morphism whose composition with the first projection maps into $U$. Let $Z \subset X \times_S Y$ be the scheme theoretic image of $V \to X \times_S Y$. Let $X' \to X$ be a $U$-admissible blowup. Then the scheme theoretic image of $V \to X' \times_S Y$ is the strict transform of $Z$ with respect to the blowing up.


flat-lemma-compactification-dominates

Let $S$ be a quasi-compact and quasi-separated scheme. Let $U$ be a scheme of finite type and separated over $S$. Let $V \subset U$ be a quasi-compact open. If $V$ has a compactification $V \subset Y$ over $S$, then there exists a $V$-admissible blowing up $Y' \to Y$ and an open $V \subset V' \subset Y'$ such that $V \to U$ extends to a proper morphism $V' \to U$.


flat-lemma-two-compactifications

Let $S$ be a Noetherian scheme. Let $U$ be a scheme of finite type and separated over $S$. Let $U = U_1 \cup U_2$ be opens such that $U_1$ and $U_2$ have compactifications over $S$ and such that $U_1 \cap U_2$ is dense in $U$. Then $U$ has a compactification over $S$.


flat-lemma-equivalence-h-v-locally-finite-presentation

Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of schemes with fixed target with $f_i$ locally of finite presentation for all $i$. The following are equivalent \begin{enumerate} \item $\{X_i \to X\}$ is a ph covering, and \item $\{X_i \to X\}$ is a V covering. \end{enumerate}


flat-lemma-approximate-h-cover

Let $X$ be an affine scheme. Let $\{X_i \to X\}_{i \in I}$ be an h covering. Then there exists a surjective proper morphism $$ Y \longrightarrow X $$ of finite presentation (!) and a finite affine open covering $Y = \bigcup_{j = 1, \ldots, m} Y_j$ such that $\{Y_j \to X\}_{j = 1, \ldots, m}$ refines $\{X_i \to X\}_{i \in I}$.


flat-lemma-zariski-h

An fppf covering is a h covering. Hence syntomic, smooth, \'etale, and Zariski coverings are h coverings as well.


flat-lemma-surjective-proper-finite-presentation-h

Let $f : Y \to X$ be a surjective proper morphism of schemes which is of finite presentation. Then $\{Y \to X\}$ is an h covering.


flat-lemma-h

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is an h covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is an h covering and for each $i$ we have an h covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is an h covering. \item If $\{T_i \to T\}_{i\in I}$ is an h covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is an h covering. \end{enumerate}


flat-lemma-h-induced

Let $\Sch_h$ be a big h site as in Definition \ref{definition-big-h-site}. Let $T \in \Ob(\Sch_h)$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary h covering of $T$. \begin{enumerate} \item There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_h$ which refines $\{T_i \to T\}_{i \in I}$. \item If $\{T_i \to T\}_{i \in I}$ is a standard h covering, then it is tautologically equivalent to a covering of $\Sch_h$. \item If $\{T_i \to T\}_{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\Sch_h$. \end{enumerate}


flat-lemma-characterize-sheaf-h

Let $\mathcal{F}$ be a presheaf on $(\Sch/S)_h$. Then $\mathcal{F}$ is a sheaf if and only if \begin{enumerate} \item $\mathcal{F}$ satisfies the sheaf condition for Zariski coverings, and \item if $f : V \to U$ is proper, surjective, and of finite presentation, then $\mathcal{F}(U)$ maps bijectively to the equalizer of the two maps $\mathcal{F}(V) \to \mathcal{F}(V \times_U V)$. \end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \begin{enumerate} \item[(2')] the sheaf property for $\{V \to U\}$ as in (2) with $U$ affine. \end{enumerate}


flat-lemma-morphism-big-h

Let $\Sch_h$ be a big h site. Let $f : T \to S$ be a morphism in $\Sch_h$. The functor $$ u : (\Sch/T)_h \longrightarrow (\Sch/S)_h, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_h \longrightarrow (\Sch/T)_h, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_h) \longrightarrow \Sh((\Sch/S)_h) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


flat-lemma-limit-h-topology

Let $T$ be an affine scheme which is written as a limit $T = \lim_{i \in I} T_i$ of a directed inverse system of affine schemes. \begin{enumerate} \item Let $\mathcal{V} = \{V_j \to T\}_{j = 1, \ldots, m}$ be a standard h covering of $T$, see Definition \ref{definition-standard-h}. Then there exists an index $i$ and a standard h covering $\mathcal{V}_i = \{V_{i, j} \to T_i\}_{j = 1, \ldots, m}$ whose base change $T \times_{T_i} \mathcal{V}_i$ to $T$ is isomorphic to $\mathcal{V}$. \item Let $\mathcal{V}_i$, $\mathcal{V}'_i$ be a pair of standard h coverings of $T_i$. If $f : T \times_{T_i} \mathcal{V}_i \to T \times_{T_i} \mathcal{V}'_i$ is a morphism of coverings of $T$, then there exists an index $i' \geq i$ and a morphism $f_{i'} : T_{i'} \times_{T_i} \mathcal{V} \to T_{i'} \times_{T_i} \mathcal{V}'_i$ whose base change to $T$ is $f$. \item If $f, g : \mathcal{V} \to \mathcal{V}'_i$ are morphisms of standard h coverings of $T_i$ whose base changes $f_T, g_T$ to $T$ are equal then there exists an index $i' \geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$. \end{enumerate} In other words, the category of standard h coverings of $T$ is the colimit over $I$ of the categories of standard h coverings of $T_i$.


flat-lemma-extend-sheaf-h

Let $S$ be a scheme contained in a big site $\Sch_h$. Let $F : (\Sch/S)_h^{opp} \to \textit{Sets}$ be an h sheaf satisfying property (b) of Topologies, Lemma \ref{topologies-lemma-extend} with $\mathcal{C} = (\Sch/S)_h$. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all h coverings and is limit preserving (Limits, Remark \ref{limits-remark-limit-preserving}).


flat-lemma-base-change-almost-blow-up

Consider an almost blow up square (\ref{equation-almost-blow-up-square}). Let $Y \to X$ be any morphism. Then the base change $$ \xymatrix{ Y \times_X E \ar[d] \ar[r] & Y \times_X X' \ar[d] \\ Y \times_X Z \ar[r] & Y } $$ is an almost blow up square too.


flat-lemma-shrink-almost-blow-up

Consider an almost blow up square (\ref{equation-almost-blow-up-square}). Let $W \to X'$ be a closed immersion of finite presentation. The following are equivalent \begin{enumerate} \item $X' \setminus E$ is scheme theoretically contained in $W$, \item the blowup $X''$ of $X$ in $Z$ is scheme theoretically contained in $W$, \item the diagram $$ \xymatrix{ E \cap W \ar[d] \ar[r] & W \ar[d] \\ Z \ar[r] & X } $$ is an almost blow up square. Here $E \cap W$ is the scheme theoretic intersection. \end{enumerate}


flat-lemma-blow-up-limit-almost-blow-up

Consider an almost blow up square (\ref{equation-almost-blow-up-square}) with $X$ quasi-compact and quasi-separated. Then the blowup $X''$ of $X$ in $Z$ can be written as $$ X'' = \lim X'_i $$ where the limit is over the directed system of closed subschemes $X'_i \subset X'$ of finite presentation satisfying the equivalent conditions of Lemma \ref{lemma-shrink-almost-blow-up}.


flat-lemma-almost-blow-up-square

Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \subset X$ be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Then there exists an almost blow up square as in (\ref{equation-almost-blow-up-square}).


flat-lemma-almost-blow-up-unique

Let $X$ be a quasi-compact and quasi-separated scheme and let $Z \subset X$ be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Suppose given almost blow up squares (\ref{equation-almost-blow-up-square}) $$ \xymatrix{ E_k \ar[r] \ar[d] & X_k' \ar[d] \\ Z \ar[r] & X } $$ for $k = 1, 2$, then there exists an almost blow up square $$ \xymatrix{ E \ar[r] \ar[d] & X' \ar[d] \\ Z \ar[r] & X } $$ and closed immersions $i_k : X' \to X'_k$ over $X$ with $E = i_k^{-1}(E_k)$.


flat-lemma-flat-after-almost-blowing-up

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme of finite presentation over $Y$. Let $V \subset Y$ be a quasi-compact open such that $X_V \to V$ is flat. Then there exist a commutative diagram $$ \xymatrix{ E \ar[ddd] \ar[rd] & & & D \ar[lll] \ar[ddd] \ar[ld] \\ & Y' \ar[d] & X' \ar[l] \ar[d] \\ & Y & X \ar[l] \\ Z \ar[ru] & & & T \ar[lll] \ar[lu] } $$ whose right and left hand squares are almost blow up squares, whose lower and top squares are cartesian, such that $Z \cap V = \emptyset$, and such that $X' \to Y'$ is flat (and of finite presentation).


flat-lemma-blow-up-square-h

Let $\mathcal{F}$ be a sheaf on one of the sites $(\Sch/S)_h$ constructed in Definition \ref{definition-big-small-h}. Then for any almost blow up square (\ref{equation-almost-blow-up-square}) in the category $(\Sch/S)_h$ the diagram $$ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } $$ is cartesian in the category of sets.


flat-lemma-thickening-h

Let $\mathcal{F}$ be a sheaf on one of the sites $(\Sch/S)_h$ constructed in Definition \ref{definition-big-small-h}. Let $X \to X'$ be a morphism of $(\Sch/S)_h$ which is a thickening and of finite presentation. Then $\mathcal{F}(X') \to \mathcal{F}(X)$ is bijective.


flat-lemma-refine-check-h

Let $\mathcal{F}$ be a presheaf on one of the sites $(\Sch/S)_h$ constructed in Definition \ref{definition-big-small-h}. Then $\mathcal{F}$ is a sheaf if and only if the following conditions are satisfied \begin{enumerate} \item $\mathcal{F}$ is a sheaf for the Zariski topology, \item given a morphism $f : X \to Y$ of $(\Sch/S)_h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then $\mathcal{F}(Y)$ is the equalizer of the two maps $\mathcal{F}(X) \to \mathcal{F}(X \times_Y X)$, \item $\mathcal{F}$ turns an almost blow up square as in Example \ref{example-one-generator} in the category $(\Sch/S)_h$ into a cartesian diagram of sets, and \item $\mathcal{F}$ turns an almost blow up square as in Example \ref{example-two-generators} in the category $(\Sch/S)_h$ into a cartesian diagram of sets. \end{enumerate}


flat-lemma-refine-check-h-stack

Let $p : \mathcal{S} \to (\Sch/S)_h$ be a category fibred in groupoids. Then $\mathcal{S}$ is a stack in groupoids if and only if the following conditions are satisfied \begin{enumerate} \item $\mathcal{S}$ is a stack in groupoids for the Zariski topology, \item given a morphism $f : X \to Y$ of $(\Sch/S)_h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then $$ \mathcal{S}_Y \longrightarrow \mathcal{S}_X \times_{\mathcal{S}_{X \times_Y X}} \mathcal{S}_X $$ is an equivalence of categories, \item for an almost blow up square as in Example \ref{example-one-generator} or \ref{example-two-generators} in the category $(\Sch/S)_h$ the functor $$ \mathcal{S}_X \longrightarrow \mathcal{S}_Z \times_{\mathcal{S}_E} \mathcal{S}_{X'} $$ is an equivalence of categories. \end{enumerate}


flat-lemma-funny-blow-up

Let $Z, X, X', E$ be an almost blow up square as in Example \ref{example-two-generators}. Then $H^p(X', \mathcal{O}_{X'}) = 0$ for $p > 0$ and $\Gamma(X, \mathcal{O}_X) \to \Gamma(X', \mathcal{O}_{X'})$ is a surjective map of rings whose kernel is an ideal of square zero.


flat-lemma-h-sheaf-colim-F

Let $p$ be a prime number. Let $S$ be a scheme over $\mathbf{F}_p$. Let $(\Sch/S)_h$ be a site as in Definition \ref{definition-big-small-h}. There is a unique sheaf $\mathcal{F}$ on $(\Sch/S)_h$ such that $$ \mathcal{F}(X) = \colim_F \Gamma(X, \mathcal{O}_X) $$ for any quasi-compact and quasi-separated object $X$ of $(\Sch/S)_h$.


flat-lemma-weak-normalization-ph-sheaf

Let $(\Sch/S)_{ph}$ be a site as in Topologies, Definition \ref{topologies-definition-big-small-ph}. The rule $$ X \longmapsto \Gamma(X^{awn}, \mathcal{O}_{X^{awn}}) $$ is a sheaf on $(\Sch/S)_{ph}$.


flat-lemma-weak-normalization-h-sheaf

Let $S$ be a scheme. Choose a site $(\Sch/S)_h$ as in Definition \ref{definition-big-small-h}. The rule $$ X \longmapsto \Gamma(X^{awn}, \mathcal{O}_{X^{awn}}) $$ is the sheafification of the ``structure sheaf'' $\mathcal{O}$ on $(\Sch/S)_h$. Similarly for the ph topology.


flat-lemma-perfect-weankly-normal

Let $p$ be a prime number. An $\mathbf{F}_p$-algebra $A$ is absolutely weakly normal if and only if it is perfect.


flat-lemma-char-p

Let $p$ be a prime number. \begin{enumerate} \item If $A$ is an $\mathbf{F}_p$-algebra, then $\colim_F A = A^{awn}$. \item If $S$ is a scheme over $\mathbf{F}_p$, then the h sheafification of $\mathcal{O}$ sends a quasi-compact and quasi-separated $X$ to $\colim_F \Gamma(X, \mathcal{O}_X)$. \end{enumerate}


flat-lemma-colim-F-Vect

Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated scheme over $\mathbf{F}_p$. The category $\colim_F \textit{Vect}(S)$ is equivalent to the category of finite locally free modules over the sheaf of rings $\colim_F \mathcal{O}_S$ on $S$.


flat-lemma-vector-bundle-I

Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$ in characteristic $p$ as in Example \ref{example-one-generator}. Then the functor $$ \colim_F \textit{Vect}(X) \longrightarrow \colim_F \textit{Vect}(Z) \times_{\colim_F \textit{Vect}(E)} \colim_F \textit{Vect}(X') $$ is an equivalence.


flat-lemma-vector-bundle-II

Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$ in characteristic $p$ as in Example \ref{example-two-generators}. Then the functor $$ G : \colim_F \textit{Vect}(X) \longrightarrow \colim_F \textit{Vect}(Z) \times_{\colim_F \textit{Vect}(E)} \colim_F \textit{Vect}(X') $$ is an equivalence.


flat-lemma-trivial-fibres-dvr

Let $f : X \to S$ be a proper morphism with geometrically connected fibres where $S$ is the spectrum of a discrete valuation ring. Denote $\eta \in S$ the generic point and denote $X_n \subset X$ the closed subscheme cutout by the $n$th power of a uniformizer on $S$. Then there exists an integer $n$ such that the following is true: any finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ such that $\mathcal{E}|_{X_\eta}$ and $\mathcal{E}|_{X_n}$ are free, is free.


flat-lemma-trivial-over-dvrs

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $f$ is flat and proper and $\mathcal{O}_S = f_*\mathcal{O}_X$, \item $S$ is a normal Noetherian scheme, \item the pullback of $\mathcal{E}$ to $X \times_S \Spec(\mathcal{O}_{S, s})$ is free for every codimension $1$ point $s \in S$. \end{enumerate} Then $\mathcal{E}$ is isomorphic to the pullback of a finite locally free $\mathcal{O}_S$-module.


flat-lemma-fitting-ideals-complex

Let $X$ be a scheme. Let $E \in D(\mathcal{O}_X)$ be pseudo-coherent. For every $p, k \in \mathbf{Z}$ there is an finite type quasi-coherent sheaf of ideals $\text{Fit}_{p, k}(E) \subset \mathcal{O}_X$ with the following property: for $U \subset X$ open such that $E|_U$ is isomorphic to $$ \ldots \to \mathcal{O}_U^{\oplus n_{b - 2}} \xrightarrow{d_{b - 2}} \mathcal{O}_U^{\oplus n_{b - 1}} \xrightarrow{d_{b - 1}} \mathcal{O}_U^{\oplus n_b} \to 0 \to \ldots $$ the restriction $\text{Fit}_{p, k}(E)|_U$ is generated by the minors of the matrix of $d_p$ of size $$ - k + n_{p + 1} - n_{p + 2} + \ldots + (-1)^{b - p + 1} n_b $$ Convention: the ideal generated by $r \times r$-minors is $\mathcal{O}_U$ if $r \leq 0$ and the ideal generated by $r \times r$-minors where $r > \min(n_p, n_{p + 1})$ is zero.


flat-lemma-blowup-complex

Let $X$ be a scheme. Let $E \in D(\mathcal{O}_X)$ be perfect. Let $U \subset X$ be a scheme theoretically dense open subscheme such that $H^i(E|_U)$ is finite locally free of constant rank $r_i$ for all $i \in \mathbf{Z}$. Then there exists a $U$-admissible blowup $b : X' \to X$ such that $H^i(Lb^*E)$ is a perfect $\mathcal{O}_{X'}$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$.


flat-lemma-blowup-complex-integral

Let $X$ be an integral scheme. Let $E \in D(\mathcal{O}_X)$ be perfect. Then there exists a nonempty open $U \subset X$ such that $H^i(E|_U)$ is finite locally free of constant rank $r_i$ for all $i \in \mathbf{Z}$ and there exists a $U$-admissible blowup $b : X' \to X$ such that $H^i(Lb^*E)$ is a perfect $\mathcal{O}_{X'}$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$.


flat-lemma-blowup-pd1-derived

Let $X$ be a scheme. Let $\mathcal{F}$ be a perfect $\mathcal{O}_X$-module of tor dimension $\leq 1$. For any blowup $b : X' \to X$ we have $Lb^*\mathcal{F} = b^*\mathcal{F}$ and $b^*\mathcal{F}$ is a perfect $\mathcal{O}_X$-module of tor dimension $\leq 1$.


flat-lemma-eta-stalks

Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ is $\mathcal{I}$-torsion free for all $i$. \begin{enumerate} \item For $x \in X$ choose a generator $f \in \mathcal{I}_x$. Then the stalk of $\eta_\mathcal{I}\mathcal{F}^\bullet$ is canonically isomorphic to $\eta_f\mathcal{F}^\bullet_x$. \item If the $\mathcal{F}^i$ are quasi-coherent $\mathcal{O}_X$-modules, then so are the $(\eta_\mathcal{I}\mathcal{F})^i$ and in this case if $U = \Spec(A) \subset X$ is affine open and $D \cap U = V(f)$, then $\eta_f(\mathcal{F}^\bullet(U))$ is canonically isomorphic to $(\eta_\mathcal{I}\mathcal{F}^\bullet)(U)$. \end{enumerate}


flat-lemma-eta-first-property

Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ is $\mathcal{I}$-torsion free for all $i$. There is a canonical isomorphism $$ \mathcal{I}^{\otimes i} \otimes_{\mathcal{O}_X} \left( H^i(\mathcal{F}^\bullet)/H^i(\mathcal{F}^\bullet)[\mathcal{I}] \right) \longrightarrow H^i(\eta_\mathcal{I}\mathcal{F}^\bullet) $$ of cohomology sheaves.


flat-lemma-eta-qis

Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ be a map of complexes of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ and $\mathcal{G}^i$ are $\mathcal{I}$-torsion free for all $i$. Then the induced map $\eta_\mathcal{I}\mathcal{F}^\bullet \to \eta_\mathcal{I}\mathcal{G}^\bullet$ is a quasi-isomorphism too.


flat-lemma-eta-tensor-invertible

Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ is $\mathcal{I}$-torsion free for all $i$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $\eta_\mathcal{I}(\mathcal{F}^\bullet \otimes \mathcal{L}) = (\eta_\mathcal{I}\mathcal{F}^\bullet) \otimes \mathcal{L}$.


flat-lemma-complex-and-divisor-blowup

In Situation \ref{situation-complex-and-divisor} let $b : X' \to X$ be the blowing up of the product of the ideals $\mathcal{J}_i$ from Remark \ref{remark-complex-and-divisor-ideal}. Denote $D' = b^{-1}D$ with ideal sheaf $\mathcal{I}' \subset \mathcal{O}_{X'}$. Then $$ \mathcal{Q}^\bullet = \eta_{\mathcal{I}'}b^*\mathcal{E}^\bullet $$ is a bounded complex of finite locally free $\mathcal{O}_{X'}$-modules.


flat-lemma-complex-and-divisor-eta-pull

In Situation \ref{situation-complex-and-divisor} let $f : Y \to X$ be a morphism of schemes such that the inverse image $f^{-1}D$ is an effective Cartier divisor with ideal sheaf $\mathcal{J}$. Assume $\mathcal{J}_i$ as in Remark \ref{remark-complex-and-divisor-ideal} is invertible for all $i$. Then $f^*(\eta_\mathcal{I}\mathcal{E}^\bullet) = \eta_\mathcal{J}(f^*\mathcal{E}^\bullet)$.


flat-lemma-complex-and-divisor-blowup-base-change

In Situation \ref{situation-complex-and-divisor} let $f : Y \to X$ be a morphism of schemes such that the inverse image $f^{-1}D$ is an effective Cartier divisor. Let $X' \to X$ and $\mathcal{Q}^\bullet$, resp.\ $Y' \to Y$ and $\mathcal{Q}_{Y'}^\bullet$, be as constructed in Lemma \ref{lemma-complex-and-divisor-blowup} for $D \subset X$ and $\mathcal{E}^\bullet$, resp.\ $f^{-1}D \subset Y$ and $f^*\mathcal{E}^\bullet$. Then $Y'$ is the strict transform of $Y$ with respect to $X' \to X$ and $\mathcal{Q}_{Y'}^\bullet = (Y' \to X')^*\mathcal{Q}^\bullet$.


flat-lemma-complex-and-divisor-blowup-good

In Situation \ref{situation-complex-and-divisor} let $U \subset X$ be the maximal open subscheme over which the cohomology sheaves of $\mathcal{E}^\bullet$ are locally free. Then the blowing up $b : X' \to X$ of Lemma \ref{lemma-complex-and-divisor-blowup} is an isomorphism over $U$.


flat-lemma-complex-and-divisor-blowup-T

In Situation \ref{situation-complex-and-divisor}. Let $b : X' \to X$, $D' \subset X'$, and $\mathcal{Q}^\bullet$ be as in Lemma \ref{lemma-complex-and-divisor-blowup}. Let $U \subset X$ be as in Lemma \ref{lemma-complex-and-divisor-blowup-good}. Then there exists a closed immersion $T \to D'$ of finite presentation with $D' \cap b^{-1}(U) \subset T$ scheme theoretically such that $\mathcal{Q}^\bullet|_T$ has finite locally free cohomology sheaves.


flat-lemma-complex-and-divisor-blowup-T-ranks

In Situation \ref{situation-complex-and-divisor}. Let $b : X' \to X$, $D' \subset X'$, and $\mathcal{Q}^\bullet$ be as in Lemma \ref{lemma-complex-and-divisor-blowup}. Given integers $\rho_i \geq 0$ almost all zero, let $U' \subset X$ be the maximal open subscheme where $H^i(\mathcal{E}^\bullet)$ is finite locally free of rank $\rho_i$ for all $i$. Let $T \subset D'$ be as in Lemma \ref{lemma-complex-and-divisor-blowup-T}. Then there exists an open and closed subscheme $T' \subset T$ containing $D' \cap b^{-1}(U')$ scheme theoretically such that $\mathcal{Q}^\bullet|_{T'}$ has finite locally free cohomology sheaves $H^i(\mathcal{Q}^\bullet|_{T'})$ of rank $\rho_i$.


flat-lemma-complex-and-divisor-derived

Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor. Let $E \in D(\mathcal{O}_X)$ be a perfect object. Let $U \subset X$ be the maximal open over which the cohomology sheaves $H^i(E)$ are locally free. There exists a proper morphism $b : X' \longrightarrow X$ and an object $Q \in D(\mathcal{O}_{X'})$ with the following properties \begin{enumerate} \item $D' = b^{-1}D$ is an effective Cartier divisor, \item $Q = L\eta_{\mathcal{I}'}Lb^*E$ where $\mathcal{I}'$ is the ideal sheaf of $D'$, \item $Q$ is a perfect object of $D(\mathcal{O}_{X'})$, \item there exists a closed immersion $T \to D'$ of finite presentation with $D' \cap b^{-1}(U) \subset T$ scheme theoretically such that $Q|_T$ has finite locally free cohomology sheaves, \item for any open subscheme $V \subset X$ such that $E|_V$ can be represented by a bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_V$-modules, the base changes of $X' \to X$, $Q$, $D'$, and $T$ to $V$ are given by the constructions of Lemmas \ref{lemma-complex-and-divisor-blowup} and \ref{lemma-complex-and-divisor-blowup-T}. \end{enumerate}


flat-lemma-graph-construction

The construction above has the following properties: \begin{enumerate} \item $b$ is an isomorphism over $\mathbf{P}^1_U \cup \mathbf{A}^1_X$, \item the restriction of $\mathcal{Q}^\bullet$ to $\mathbf{A}^1_X$ is equal to the pullback of $\mathcal{E}^\bullet$, \item there exists a closed immersion $T \to W_\infty$ of finite presentation such that $\infty(U) \subset T$ scheme theoretically and such that $\mathcal{Q}^\bullet|_T$ has finite locally free cohomology sheaves. \end{enumerate}


flat-proposition-existence-complete-at-x

Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \in X$ be a point with image $s \in S$. There exists a commutative diagram $$ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] } $$ of pointed schemes such that the horizontal arrows are elementary \'etale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a complete d\'evissage at $x$.


flat-proposition-finite-type-flat-at-point

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat at $x$ over $S$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and an open subscheme $$ V \subset X \times_S \Spec(\mathcal{O}_{S', s'}) $$ which contains the unique point of $X \times_S \Spec(\mathcal{O}_{S', s'})$ mapping to $x$ such that the pullback of $\mathcal{F}$ to $V$ is an $\mathcal{O}_V$-module of finite presentation and flat over $\mathcal{O}_{S', s'}$.


flat-proposition-finite-presentation-flat-at-point

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in X$ with image $s \in S$. Assume that \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{F}$ is of finite presentation, and \item $\mathcal{F}$ is flat at $x$ over $S$. \end{enumerate} Then there exists a commutative diagram of pointed schemes $$ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] } $$ whose horizontal arrows are elementary \'etale neighbourhoods such that $X'$, $S'$ are affine and such that $\Gamma(X', g^*\mathcal{F})$ is a projective $\Gamma(S', \mathcal{O}_{S'})$-module.


flat-proposition-flat-finite-type-finite-presentation-domain

Let $R$ be a domain. Let $R \to S$ be a ring map of finite type. Let $M$ be a finite $S$-module. \begin{enumerate} \item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra. \item If $M$ is flat as an $R$-module, then $M$ is finitely presented as an $S$-module. \end{enumerate}


flat-proposition-finite-presentation-flat-pure-is-projective

Let $f : X \to S$ be an affine, finitely presented morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite presentation, flat over $S$. Then the following are equivalent \begin{enumerate} \item $f_*\mathcal{F}$ is locally projective on $S$, and \item $\mathcal{F}$ is pure relative to $S$. \end{enumerate} In particular, given a ring map $A \to B$ of finite presentation and a finitely presented $B$-module $N$ flat over $A$ we have: $N$ is projective as an $A$-module if and only if $\widetilde{N}$ on $\Spec(B)$ is pure relative to $\Spec(A)$.


flat-proposition-check-h

Let $\mathcal{F}$ be a presheaf on one of the sites $(\Sch/S)_h$ constructed in Definition \ref{definition-big-small-h}. Then $\mathcal{F}$ is a sheaf if and only if the following conditions are satisfied \begin{enumerate} \item $\mathcal{F}$ is a sheaf for the Zariski topology, \item given a morphism $f : X \to Y$ of $(\Sch/S)_h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then $\mathcal{F}(Y)$ is the equalizer of the two maps $\mathcal{F}(X) \to \mathcal{F}(X \times_Y X)$, \item given an almost blow up square (\ref{equation-almost-blow-up-square}) with $X$ affine in the category $(\Sch/S)_h$ the diagram $$ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } $$ is cartesian in the category of sets. \end{enumerate}


flat-proposition-h-descent-vector-bundles-p

Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$. Then the category fibred in groupoids $$ p : \mathcal{S} \longrightarrow (\Sch/S)_h $$ whose fibre category over $U$ is the category of finite locally free $\colim_F \mathcal{O}_U$-modules over $U$ is a stack in groupoids. Moreover, if $U$ is quasi-compact and quasi-separated, then $\mathcal{S}_U$ is $\colim_F \textit{Vect}(U)$.


curves-theorem-curves-rational-maps

Let $k$ be a field. The following categories are canonically equivalent \begin{enumerate} \item The category of finitely generated field extensions $K/k$ of transcendence degree $1$. \item The category of curves and dominant rational maps. \item The category of normal projective curves and nonconstant morphisms. \item The category of nonsingular projective curves and nonconstant morphisms. \item The category of regular projective curves and nonconstant morphisms. \item The category of normal proper curves and nonconstant morphisms. \end{enumerate}


curves-lemma-extend-over-dvr

Let $k$ be a field. Let $X$ be a curve and $Y$ a proper variety. Let $U \subset X$ be a nonempty open and let $f : U \to Y$ be a morphism. If $x \in X$ is a closed point such that $\mathcal{O}_{X, x}$ is a discrete valuation ring, then there exists an open $U \subset U' \subset X$ containing $x$ and a morphism of varieties $f' : U' \to Y$ extending $f$.


curves-lemma-extend-over-normal-curve

Let $k$ be a field. Let $X$ be a normal curve and $Y$ a proper variety. The set of rational maps from $X$ to $Y$ is the same as the set of morphisms $X \to Y$.


curves-lemma-flat

Let $k$ be a field. Let $f : X \to Y$ be a nonconstant morphism of curves over $k$. If $Y$ is normal, then $f$ is flat.


curves-lemma-finite

Let $k$ be a field. Let $f : X \to Y$ be a morphism of schemes over $k$. Assume \begin{enumerate} \item $Y$ is separated over $k$, \item $X$ is proper of dimension $\leq 1$ over $k$, \item $f(Z)$ has at least two points for every irreducible component $Z \subset X$ of dimension $1$. \end{enumerate} Then $f$ is finite.


curves-lemma-smooth-models

Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$. For a field extension $K/k$ denote $Y_K$ a nonsingular projective model of $(X_K)_{red}$. \begin{enumerate} \item If $X$ is proper, then $Y_K$ is the normalization of $X_K$. \item There exists $K/k$ finite purely inseparable such that $Y_K$ is smooth. \item Whenever $Y_K$ is smooth\footnote{Or even geometrically reduced.} we have $H^0(Y_K, \mathcal{O}_{Y_K}) = K$. \item Given a commutative diagram $$ \xymatrix{ \Omega & K' \ar[l] \\ K \ar[u] & k \ar[l] \ar[u] } $$ of fields such that $Y_K$ and $Y_{K'}$ are smooth, then $Y_\Omega = (Y_K)_\Omega = (Y_{K'})_\Omega$. \end{enumerate}


curves-lemma-linear-series

Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$. Let $(\mathcal{L}, V)$ be a $\mathfrak g^r_d$ on $X$. Then there exists a morphism $$ \varphi : X \longrightarrow \mathbf{P}^r_k = \text{Proj}(k[T_0, \ldots, T_r]) $$ of varieties over $k$ and a map $\alpha : \varphi^*\mathcal{O}_{\mathbf{P}^r_k}(1) \to \mathcal{L}$ such that $\varphi^*T_0, \ldots, \varphi^*T_r$ are sent to a basis of $V$ by $\alpha$.


curves-lemma-linear-series-trivial-existence

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. If $X$ has a $\mathfrak g^r_d$, then $X$ has a $\mathfrak g^s_d$ for all $0 \leq s \leq r$.


curves-lemma-g1d

Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$. Let $(\mathcal{L}, V)$ be a $\mathfrak g^1_d$ on $X$. Then the morphism $\varphi : X \to \mathbf{P}^1_k$ of Lemma \ref{lemma-linear-series} either \begin{enumerate} \item is nonconstant and has degree $\leq d$, or \item factors through a closed point of $\mathbf{P}^1_k$ and in this case $H^0(X, \mathcal{O}_X) \not = k$. \end{enumerate}


curves-lemma-grd-inequalities

Let $k$ be a field. Let $X$ be a proper curve over $k$ with $H^0(X, \mathcal{O}_X) = k$. If $X$ has a $\mathfrak g^r_d$, then $r \leq d$. If equality holds, then $H^1(X, \mathcal{O}_X) = 0$, i.e., the genus of $X$ (Definition \ref{definition-genus}) is $0$.


curves-lemma-duality-dim-1

Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. There exists a dualizing complex $\omega_X^\bullet$ with the following properties \begin{enumerate} \item $H^i(\omega_X^\bullet)$ is nonzero only for $i = -1, 0$, \item $\omega_X = H^{-1}(\omega_X^\bullet)$ is a coherent Cohen-Macaulay module whose support is the irreducible components of dimension $1$, \item for $x \in X$ closed, the module $H^0(\omega_{X, x}^\bullet)$ is nonzero if and only if either \begin{enumerate} \item $\dim(\mathcal{O}_{X, x}) = 0$ or \item $\dim(\mathcal{O}_{X, x}) = 1$ and $\mathcal{O}_{X, x}$ is not Cohen-Macaulay, \end{enumerate} \item for $K \in D_\QCoh(\mathcal{O}_X)$ there are functorial isomorphisms\footnote{This property characterizes $\omega_X^\bullet$ in $D_\QCoh(\mathcal{O}_X)$ up to unique isomorphism by the Yoneda lemma. Since $\omega_X^\bullet$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ in fact it suffices to consider $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$.} $$ \Ext^i_X(K, \omega_X^\bullet) = \Hom_k(H^{-i}(X, K), k) $$ compatible with shifts and distinguished triangles, \item there are functorial isomorphisms $\Hom(\mathcal{F}, \omega_X) = \Hom_k(H^1(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$, \item if $X \to \Spec(k)$ is smooth of relative dimension $1$, then $\omega_X \cong \Omega_{X/k}$. \end{enumerate}


curves-lemma-duality-dim-1-CM

Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $1$. The module $\omega_X$ of Lemma \ref{lemma-duality-dim-1} has the following properties \begin{enumerate} \item $\omega_X$ is a dualizing module on $X$ (Duality for Schemes, Section \ref{duality-section-dualizing-module}), \item $\omega_X$ is a coherent Cohen-Macaulay module whose support is $X$, \item there are functorial isomorphisms $\Ext^i_X(K, \omega_X[1]) = \Hom_k(H^{-i}(X, K), k)$ compatible with shifts for $K \in D_\QCoh(X)$, \item there are functorial isomorphisms $\Ext^{1 + i}(\mathcal{F}, \omega_X) = \Hom_k(H^{-i}(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$. \end{enumerate}


curves-lemma-sanity-check-duality

Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $\omega_X^\bullet$ and $\omega_X$ be as in Lemma \ref{lemma-duality-dim-1}. \begin{enumerate} \item If $X \to \Spec(k)$ factors as $X \to \Spec(k') \to \Spec(k)$ for some field $k'$, then $\omega_X^\bullet$ and $\omega_X$ satisfy properties (4), (5), (6) with $k$ replaced with $k'$. \item If $K/k$ is a field extension, then the pullback of $\omega_X^\bullet$ and $\omega_X$ to the base change $X_K$ are as in  Lemma \ref{lemma-duality-dim-1} for the morphism $X_K \to \Spec(K)$. \end{enumerate}


curves-lemma-closed-immersion-dim-1-CM

Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $i : Y \to X$ be a closed immersion. Let $\omega_X^\bullet$, $\omega_X$, $\omega_Y^\bullet$, $\omega_Y$ be as in Lemma \ref{lemma-duality-dim-1}. Then \begin{enumerate} \item $\omega_Y^\bullet = R\SheafHom(\mathcal{O}_Y, \omega_X^\bullet)$, \item $\omega_Y = \SheafHom(\mathcal{O}_Y, \omega_X)$ and $i_*\omega_Y = \SheafHom_{\mathcal{O}_X}(i_*\mathcal{O}_Y, \omega_X)$. \end{enumerate}


curves-lemma-closed-subscheme-reduced-gorenstein

Let $X$ be a proper scheme over a field $k$ which is Gorenstein, reduced, and equidimensional of dimension $1$. Let $i : Y \to X$ be a reduced closed subscheme equidimensional of dimension $1$. Let $j : Z \to X$ be the scheme theoretic closure of $X \setminus Y$. Then \begin{enumerate} \item $Y$ and $Z$ are Cohen-Macaulay, \item if $\mathcal{I} \subset \mathcal{O}_X$, resp.\ $\mathcal{J} \subset \mathcal{O}_X$ is the ideal sheaf of $Y$, resp.\ $Z$ in $X$, then $$ \mathcal{I} = i_*\mathcal{I}' \quad\text{and}\quad \mathcal{J} = j_*\mathcal{J}' $$ where $\mathcal{I}' \subset \mathcal{O}_Z$, resp.\ $\mathcal{J}' \subset \mathcal{O}_Y$ is the ideal sheaf of $Y \cap Z$ in $Z$, resp.\ $Y$, \item $\omega_Y = \mathcal{J}'(i^*\omega_X)$ and $i_*(\omega_Y) = \mathcal{J}\omega_X$, \item $\omega_Z = \mathcal{I}'(i^*\omega_X)$ and $i_*(\omega_Z) = \mathcal{I}\omega_X$, \item we have the following short exact sequences \begin{align*} 0 \to \omega_X \to i_*i^*\omega_X \oplus j_*j^*\omega_X \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to i_*\omega_Y \to \omega_X \to j_*j^*\omega_X \to 0 \\ 0 \to j_*\omega_Z \to \omega_X \to i_*i^*\omega_X \to 0 \\ 0 \to i_*\omega_Y \oplus j_*\omega_Z \to \omega_X \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to \omega_Y \to i^*\omega_X \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to \omega_Z \to j^*\omega_X \to \mathcal{O}_{Y \cap Z} \to 0 \end{align*} \end{enumerate} Here $\omega_X$, $\omega_Y$, $\omega_Z$ are as in Lemma \ref{lemma-duality-dim-1}.


curves-lemma-euler

Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. With $\omega_X^\bullet$ and $\omega_X$ as in Lemma \ref{lemma-duality-dim-1} we have $$ \chi(X, \mathcal{O}_X) = \chi(X, \omega_X^\bullet) $$ If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then $$ \chi(X, \mathcal{O}_X) = - \chi(X, \omega_X) $$


curves-lemma-rr

Let $X$ be a proper scheme over a field $k$ which is Gorenstein and equidimensional of dimension $1$. Let $\omega_X$ be as in Lemma \ref{lemma-duality-dim-1}. Then \begin{enumerate} \item $\omega_X$ is an invertible $\mathcal{O}_X$-module, \item $\deg(\omega_X) = -2\chi(X, \mathcal{O}_X)$, \item for a locally free $\mathcal{O}_X$-module $\mathcal{E}$ of constant rank we have $$ \chi(X, \mathcal{E}) = \deg(\mathcal{E}) - \textstyle{\frac{1}{2}} \text{rank}(\mathcal{E}) \deg(\omega_X) $$ and $\dim_k(H^i(X, \mathcal{E})) = \dim_k(H^{1 - i}(X, \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \omega_X))$ for all $i \in \mathbf{Z}$. \end{enumerate}


curves-lemma-automatic

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Then $X$ is connected, Cohen-Macaulay, and equidimensional of dimension $1$.


curves-lemma-vanishing

In Situation \ref{situation-Cohen-Macaulay-curve}. Given an exact sequence $$ \omega_X \to \mathcal{F} \to \mathcal{Q} \to 0 $$ of coherent $\mathcal{O}_X$-modules with $H^1(X, \mathcal{Q}) = 0$ (for example if $\dim(\text{Supp}(\mathcal{Q})) = 0$), then either $H^1(X, \mathcal{F}) = 0$ or $\mathcal{F} = \omega_X \oplus \mathcal{Q}$.


curves-lemma-vanishing-twist

In Situation \ref{situation-Cohen-Macaulay-curve}. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module which is globally generated and not isomorphic to $\mathcal{O}_X$. Then $H^1(X, \omega_X \otimes \mathcal{L}) = 0$.


curves-lemma-globally-generated

In Situation \ref{situation-Cohen-Macaulay-curve}. Given an exact sequence $$ \omega_X \to \mathcal{F} \to \mathcal{Q} \to 0 $$ of coherent $\mathcal{O}_X$-modules with $\dim(\text{Supp}(\mathcal{Q})) = 0$ and $\dim_k H^0(X, \mathcal{Q}) \geq 2$ and such that there is no nonzero submodule $\mathcal{Q}' \subset \mathcal{F}$ such that $\mathcal{Q}' \to \mathcal{Q}$ is injective. Then the submodule of $\mathcal{F}$ generated by global sections surjects onto $\mathcal{Q}$.


curves-lemma-tensor-omega-with-globally-generated-invertible

In Situation \ref{situation-Cohen-Macaulay-curve}. Let $\mathcal{L}$ be a very ample invertible $\mathcal{O}_X$-module with $\deg(\mathcal{L}) \geq 2$. Then $\omega_X \otimes_{\mathcal{O}_X} \mathcal{L}$ is globally generated.


curves-lemma-criterion-very-ample

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $\mathcal{L}$ has a regular global section, \item $H^1(X, \mathcal{L}) = 0$, and \item $\mathcal{L}$ is ample. \end{enumerate} Then $\mathcal{L}^{\otimes 6}$ is very ample on $X$ over $k$.


curves-lemma-genus-base-change

Let $k'/k$ be a field extension. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Then $X_{k'}$ is a proper scheme over $k'$ having dimension $1$ and $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$. Moreover the genus of $X_{k'}$ is equal to the genus of $X$.


curves-lemma-genus-gorenstein

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. If $X$ is Gorenstein, then $$ \deg(\omega_X) = 2g - 2 $$ where $g$ is the genus of $X$ and $\omega_X$ is as in Lemma \ref{lemma-duality-dim-1}.


curves-lemma-genus-smooth

Let $X$ be a smooth proper curve over a field $k$ with $H^0(X, \mathcal{O}_X) = k$. Then $$ \dim_k H^0(X, \Omega_{X/k}) = g \quad\text{and}\quad \deg(\Omega_{X/k}) = 2g - 2 $$ where $g$ is the genus of $X$.


curves-lemma-genus-plane-curve

Let $Z \subset \mathbf{P}^2_k$ be as in Lemma \ref{lemma-equation-plane-curve} and let $I(Z) = (F)$ for some $F \in k[T_0, T_1, T_2]$. Then $H^0(Z, \mathcal{O}_Z) = k$ and the genus of $Z$ is $(d - 1)(d - 2)/2$ where $d = \deg(F)$.


curves-lemma-smooth-plane-curve-point-over-separable

Let $Z \subset \mathbf{P}^2_k$ be as in Lemma \ref{lemma-equation-plane-curve} and let $I(Z) = (F)$ for some $F \in k[T_0, T_1, T_2]$. If $Z \to \Spec(k)$ is smooth in at least one point and $k$ is infinite, then there exists a closed point $z \in Z$ contained in the smooth locus such that $\kappa(z)/k$ is finite separable of degree at most $d$.


curves-lemma-genus-zero-pic

Let $X$ be a proper curve over a field $k$ with $H^0(X, \mathcal{O}_X) = k$. If $X$ has genus $0$, then every invertible $\mathcal{O}_X$-module $\mathcal{L}$ of degree $0$ is trivial.


curves-lemma-genus-zero-positive-degree

Let $X$ be a proper curve over a field $k$ with $H^0(X, \mathcal{O}_X) = k$. Assume $X$ has genus $0$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module of degree $d > 0$. Then we have \begin{enumerate} \item $\dim_k H^0(X, \mathcal{L}) = d + 1$ and $\dim_k H^1(X, \mathcal{L}) = 0$, \item $\mathcal{L}$ is very ample and defines a closed immersion into $\mathbf{P}^d_k$. \end{enumerate}


curves-lemma-genus-zero

Let $X$ be a proper curve over a field $k$ with $H^0(X, \mathcal{O}_X) = k$. If $X$ is Gorenstein and has genus $0$, then $X$ is isomorphic to a plane curve of degree $2$.


curves-lemma-generically-etale

\begin{slogan} A morphism of smooth curves is separable iff it is etale almost everywhere \end{slogan} Let $k$ be a field. Let $f : X \to Y$ be a morphism of smooth curves over $k$. The following are equivalent \begin{enumerate} \item $\text{d}f : f^*\Omega_{Y/k} \to \Omega_{X/k}$ is nonzero, \item $\Omega_{X/Y}$ is supported on a proper closed subset of $X$, \item there exists a nonempty open $U \subset X$ such that $f|_U : U \to Y$ is unramified, \item there exists a nonempty open $U \subset X$ such that $f|_U : U \to Y$ is \'etale, \item the extension $k(Y) \subset k(X)$ of function fields is finite separable. \end{enumerate}


curves-lemma-rh

Let $f : X \to Y$ be a morphism of smooth proper curves over a field $k$ which satisfies the equivalent conditions of Lemma \ref{lemma-generically-etale}. If $k = H^0(Y, \mathcal{O}_Y) = H^0(X, \mathcal{O}_X)$ and $X$ and $Y$ have genus $g_X$ and $g_Y$, then $$ 2g_X - 2 = (2g_Y - 2) \deg(f) + \deg(R) $$ where $R \subset X$ is the effective Cartier divisor cut out by the different of $f$.


curves-lemma-uniformizer-works

Let $X \to \Spec(k)$ be smooth of relative dimension $1$ at a closed point $x \in X$. If $\kappa(x)$ is separable over $k$, then for any uniformizer $s$ in the discrete valuation ring $\mathcal{O}_{X, x}$ the element $\text{d}s$ freely generates $\Omega_{X/k, x}$ over $\mathcal{O}_{X, x}$.


curves-lemma-purely-inseparable

Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If the extension $k(Y) \subset k(X)$ of function fields is purely inseparable, then there exists a factorization $$ X = X_0 \to X_1 \to \ldots \to X_n = Y $$ such that each $X_i$ is a proper nonsingular curve and $X_i \to X_{i + 1}$ is a degree $p$ morphism with $k(X_{i + 1}) \subset k(X_i)$ inseparable.


curves-lemma-inseparable-deg-p-smooth

Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If $X$ is smooth and $k(Y) \subset k(X)$ is inseparable of degree $p$, then there is a unique isomorphism $Y = X^{(p)}$ such that $f$ is $F_{X/k}$.


curves-lemma-purely-inseparable-smooth

Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If $X$ is smooth and $k(Y) \subset k(X)$ is purely inseparable, then there is a unique $n \geq 0$ and a unique isomorphism $Y = X^{(p^n)}$ such that $f$ is the $n$-fold relative Frobenius of $X/k$.


curves-lemma-inseparable-linear-system

Let $k$ be a field of characteristic $p > 0$. Let $X$ be a smooth proper curve over $k$. Let $(\mathcal{L}, V)$ be a $\mathfrak g^r_d$ with $r \geq 1$. Then one of the following two is true \begin{enumerate} \item there exists a $\mathfrak g^1_d$ whose corresponding morphism $X \to \mathbf{P}^1_k$ (Lemma \ref{lemma-linear-series}) is generically \'etale (i.e., is as in Lemma \ref{lemma-generically-etale}), or \item there exists a $\mathfrak g^r_{d'}$ on $X^{(p)}$ where $d' \leq d/p$. \end{enumerate}


curves-lemma-no-in-between-over-k

Let $k$ be an algebraically closed field. Let $k \subset A$ be a ring extension such that $A$ has exactly two $k$-sub algebras, then either $A = k \times k$ or $A = k[\epsilon]$.


curves-lemma-factor-almost-isomorphism

Let $k$ be an algebraically closed field. Let $f : X' \to X$ be a finite morphism algebraic $k$-schemes such that $\mathcal{O}_X \subset f_*\mathcal{O}_{X'}$ and such that $f$ is an isomorphism away from a finite set of points. Then there is a factorization $$ X' = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X $$ such that each $X_i \to X_{i - 1}$ is either the glueing of two points or the squishing of a tangent vector (see Examples \ref{example-glue-points} and \ref{example-squish-tangent-vector}).


curves-lemma-glue-points

Let $k$ be an algebraically closed field. If $f : X' \to X$ is the glueing of two points $a, b$ as in Example \ref{example-glue-points}, then there is an exact sequence $$ k^* \to \Pic(X) \to \Pic(X') \to 0 $$ The first map is zero if $a$ and $b$ are on different connected components of $X'$ and injective if $X'$ is proper and $a$ and $b$ are on the same connected component of $X'$.


curves-lemma-squish-tangent-vector

Let $k$ be an algebraically closed field. If $f : X' \to X$ is the squishing of a tangent vector $\vartheta$ as in Example \ref{example-squish-tangent-vector}, then there is an exact sequence $$ (k, +) \to \Pic(X) \to \Pic(X') \to 0 $$ and the first map is injective if $X'$ is proper and reduced.


curves-lemma-multicross-algebra

Let $k$ be a separably closed field. Let $A$ be a $1$-dimensional reduced Nagata local $k$-algebra with residue field $k$. Then $$ \delta\text{-invariant }A \geq \text{number of branches of }A - 1 $$ If equality holds, then $A^\wedge$ is as in (\ref{equation-multicross}).


curves-lemma-multicross

Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic $1$-dimensional $k$-scheme. Let $x \in X$. The following are equivalent \begin{enumerate} \item $x$ defines a multicross singularity, \item the $\delta$-invariant of $X$ at $x$ is the number of branches of $X$ at $x$ minus $1$, \item there is a sequence of morphisms $U_n \to U_{n - 1} \to \ldots \to U_0 = U \subset X$ where $U$ is an open neighbourhood of $x$, where $U_n$ is nonsingular, and where each $U_i \to U_{i - 1}$ is the glueing of two points as in Example \ref{example-glue-points}. \end{enumerate}


curves-lemma-torsion-picard-smooth-projective

Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. \begin{enumerate} \item If $n \geq 1$ is invertible in $k$, then $\Pic(X)[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2g}$. \item If the characteristic of $k$ is $p > 0$, then there exists an integer $0 \leq f \leq g$ such that $\Pic(X)[p^m] \cong (\mathbf{Z}/p^m\mathbf{Z})^{\oplus f}$ for all $m \geq 1$. \end{enumerate}


curves-lemma-torsion-picard-becomes-visible

Let $k$ be a field. Let $n$ be prime to the characteristic of $k$. Let $X$ be a smooth proper curve over $k$ with $H^0(X, \mathcal{O}_X) = k$ and of genus $g$. \begin{enumerate} \item If $g = 1$ then there exists a finite separable extension $k'/k$ such that $X_{k'}$ has a $k'$-rational point and $\Pic(X_{k'})[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2}$. \item If $g \geq 2$ then there exists a finite separable extension $k'/k$ with $[k' : k] \leq (2g - 2)(n^{2g})!$ such that $X_{k'}$ has a $k'$-rational point and $\Pic(X_{k'})[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2g}$. \end{enumerate}


curves-lemma-bound-geometric-genus

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Then $$ g_{geom}(X/k) = \sum\nolimits_{C \subset X} g_{geom}(C/k) $$ where the sum is over irreducible components $C \subset X$ of dimension $1$.


curves-lemma-genus-goes-down

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $f : Y \to X$ be a finite morphism such that there exists a dense open $U \subset X$ over which $f$ is a closed immersion. Then $$ \dim_k H^1(X, \mathcal{O}_X) \geq \dim_k H^1(Y, \mathcal{O}_Y) $$


curves-lemma-genus-normalization

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. If $X' \to X$ is a birational proper morphism, then $$ \dim_k H^1(X, \mathcal{O}_X) \geq \dim_k H^1(X', \mathcal{O}_{X'}) $$ If $X$ is reduced, $H^0(X, \mathcal{O}_X) \to H^0(X', \mathcal{O}_{X'})$ is surjective, and equality holds, then $X' = X$.


curves-lemma-bound-geometric-genus-curve

Let $k$ be a field. Let $C$ be a proper curve over $k$. Set $\kappa = H^0(C, \mathcal{O}_C)$. Then $$ [\kappa : k]_s \dim_\kappa H^1(C, \mathcal{O}_C) \geq g_{geom}(C/k) $$


curves-lemma-reduced-quotient-regular-ring-dim-2

Let $(A, \mathfrak m)$ be a regular local ring of dimension $2$. Let $I \subset \mathfrak m$ be an ideal. \begin{enumerate} \item If $A/I$ is reduced, then $I = (0)$, $I = \mathfrak m$, or $I = (f)$ for some nonzero $f \in \mathfrak m$. \item If $A/I$ has depth $1$, then $I = (f)$ for some nonzero $f \in \mathfrak m$.  \end{enumerate}


curves-lemma-nodal-algebraic

Let $k$ be a field. Let $(A, \mathfrak m, \kappa)$ be a Noetherian local $k$-algebra. The following are equivalent \begin{enumerate} \item $\kappa/k$ is separable, $A$ is reduced, $\dim_\kappa(\mathfrak m/\mathfrak m^2) = 2$, and there exists a nondegenerate $q \in \text{Sym}^2_\kappa(\mathfrak m/\mathfrak m^2)$ which maps to zero in $\mathfrak m^2/\mathfrak m^3$, \item $\kappa/k$ is separable, $\text{depth}(A) = 1$, $\dim_\kappa(\mathfrak m/\mathfrak m^2) = 2$, and there exists a nondegenerate $q \in \text{Sym}^2_\kappa(\mathfrak m/\mathfrak m^2)$ which maps to zero in $\mathfrak m^2/\mathfrak m^3$, \item $\kappa/k$ is separable, $A^\wedge \cong \kappa[[x, y]]/(ax^2 + bxy + cy^2)$ as a $k$-algebra where $ax^2 + bxy + cy^2$ is a nondegenerate quadratic form over $\kappa$. \end{enumerate}


curves-lemma-2-branches-delta-1

Let $k$ be a field. Let $(A, \mathfrak m, \kappa)$ be a Nagata local $k$-algebra. The following are equivalent \begin{enumerate} \item $k \to A$ is as in Lemma \ref{lemma-nodal-algebraic}, \item $\kappa/k$ is separable, $A$ is reduced of dimension $1$, the $\delta$-invariant of $A$ is $1$, and $A$ has $2$ geometric branches. \end{enumerate} If this holds, then the integral closure $A'$ of $A$ in its total ring of fractions has either $1$ or $2$ maximal ideals $\mathfrak m'$ and the extensions $\kappa(\mathfrak m')/k$ are separable.


curves-lemma-fitting-ideal-well-defined

Let $k$ be a field. Let $A = k[[x_1, \ldots, x_n]]$. Let $I = (f_1, \ldots, f_m) \subset A$ be an ideal. For any $r \geq 0$ the ideal in $A/I$ generated by the $r \times r$-minors of the matrix $(\partial f_j/\partial x_i)$ is independent of the choice of the generators of $I$ or the regular system of parameters $x_1, \ldots, x_n$ of $A$.


curves-lemma-fitting-ideal

Let $k$ be a field. Let $A = k[[x_1, \ldots, x_n]]$. Let $I = (f_1, \ldots, f_m) \subset \mathfrak m_A$ be an ideal. The following are equivalent \begin{enumerate} \item $k \to A/I$ is as in Lemma \ref{lemma-nodal-algebraic}, \item $A/I$ is reduced and the $(n - 1) \times (n - 1)$ minors of the matrix $(\partial f_j/\partial x_i)$ generate $I + \mathfrak m_A$, \item $\text{depth}(A/I) = 1$ and the $(n - 1) \times (n - 1)$ minors of the matrix $(\partial f_j/\partial x_i)$ generate $I + \mathfrak m_A$. \end{enumerate}


curves-lemma-nodal

Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme. Let $x \in X$ be a closed point. The following are equivalent \begin{enumerate} \item $x$ is a node, \item $k \to \mathcal{O}_{X, x}$ is as in Lemma \ref{lemma-nodal-algebraic}, \item any $\overline{x} \in X_{\overline{k}}$ mapping to $x$ defines a nodal singularity, \item $\kappa(x)/k$ is separable, $\mathcal{O}_{X, x}$ is reduced, and the first Fitting ideal of $\Omega_{X/k}$ generates $\mathfrak m_x$ in $\mathcal{O}_{X, x}$, \item $\kappa(x)/k$ is separable, $\text{depth}(\mathcal{O}_{X, x}) = 1$, and the first Fitting ideal of $\Omega_{X/k}$ generates $\mathfrak m_x$ in $\mathcal{O}_{X, x}$, \item $\kappa(x)/k$ is separable and $\mathcal{O}_{X, x}$ is reduced, has $\delta$-invariant $1$, and has $2$ geometric branches. \end{enumerate}


curves-lemma-split-node

Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme. Let $x \in X$ be a closed point. The following are equivalent \begin{enumerate} \item $x$ is a split node, \item $x$ is a node and there are exactly two points $x_1, x_2$ of the normalization $X^\nu$ lying over $x$ with $k = \kappa(x_1) = \kappa(x_2)$, \item $\mathcal{O}_{X, x}^\wedge \cong k[[x, y]]/(xy)$ as a $k$-algebra, and \item add more here. \end{enumerate}


curves-lemma-node-field-extension

Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic $k$-scheme of dimension $1$. Let $y \in X_K$ be a point with image $x \in X$. The following are equivalent \begin{enumerate} \item $x$ is a closed point of $X$ and a node, and \item $y$ is a closed point of $Y$ and a node. \end{enumerate}


curves-lemma-node-etale-local

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of dimension $1$. Let $Y \to X$ be an \'etale morphism. Let $y \in Y$ be a point with image $x \in X$. The following are equivalent \begin{enumerate} \item $x$ is a closed point of $X$ and a node, and \item $y$ is a closed point of $Y$ and a node. \end{enumerate}


curves-lemma-node-over-separable-extension

Let $k'/k$ be a finite separable field extension. Let $X$ be a locally algebraic $k'$-scheme of dimension $1$. Let $x \in X$ be a closed point. The following are equivalent \begin{enumerate} \item $x$ is a node, and \item $x$ is a node when $X$ viewed as a locally algebraic $k$-scheme. \end{enumerate}


curves-lemma-nodal-lci

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$. The following are equivalent \begin{enumerate} \item the singularities of $X$ are at-worst-nodal, and \item $X$ is a local complete intersection over $k$ and the closed subscheme $Z \subset X$ cut out by the first fitting ideal of $\Omega_{X/k}$ is unramified over $k$. \end{enumerate}


curves-lemma-facts-about-nodal-curves

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$ whose singularities are at-worst-nodal. Then $X$ is Gorenstein and geometrically reduced.


curves-lemma-closed-subscheme-nodal-curve

Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$ whose singularities are at-worst-nodal. If $Y \subset X$ is a reduced closed subscheme equidimensional of dimension $1$, then \begin{enumerate} \item the singularities of $Y$ are at-worst-nodal, and \item if $Z \subset X$ is the scheme theoretic closure of $X \setminus Y$, then \begin{enumerate} \item the scheme theoretic intersection $Y \cap Z$ is the disjoint union of spectra of finite separable extensions of $k$, \item each point of $Y \cap Z$ is a node of $X$, and \item $Y \to \Spec(k)$ is smooth at every point of $Y \cap Z$. \end{enumerate} \end{enumerate}


curves-lemma-nodal-family

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is flat, locally of finite presentation, every nonempty fibre $X_s$ is equidimensional of dimension $1$, and $X_s$ has at-worst-nodal singularities, and \item $f$ is syntomic of relative dimension $1$ and the closed subscheme $\text{Sing}(f) \subset X$ defined by the first Fitting ideal of $\Omega_{X/S}$ is unramified over $S$. \end{enumerate}


curves-lemma-smooth-relative-dimension-1

A smooth morphism of relative dimension $1$ is at-worst-nodal of relative dimension $1$.


curves-lemma-base-change-nodal-family

Let $f : X \to S$ be at-worst-nodal of relative dimension $1$. Then the same is true for any base change of $f$.


curves-lemma-locus-where-nodal

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Then there is a maximal open subscheme $U \subset X$ such that $f|_U : U \to S$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $U$ commutes with arbitrary base change.


curves-lemma-nodal-family-precompose-etale

Let $f : X \to S$ be at-worst-nodal of relative dimension $1$. If $Y \to X$ is an \'etale morphism, then the composition $g : Y \to S$ is at-worst-nodal of relative dimension $1$.


curves-lemma-nodal-family-postcompose-etale

Let $S' \to S$ be an \'etale morphism of schemes. Let $f : X \to S'$ be at-worst-nodal of relative dimension $1$. Then the composition $g : X \to S$ is at-worst-nodal of relative dimension $1$.


curves-lemma-nodal-family-etale-local-source

Let $f : X \to S$ be a morphism of schemes. Let $\{U_i \to X\}$ be an \'etale covering. The following are equivalent \begin{enumerate} \item $f$ is at-worst-nodal of relative dimension $1$, \item each $U_i \to S$ is at-worst-nodal of relative dimension $1$. \end{enumerate} In other words, being at-worst-nodal of relative dimension $1$ is \'etale local on the source.


curves-lemma-descend-nodal-family

Let $S = \lim S_i$ be a limit of a directed system of schemes with affine transition morphisms. Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_i : X_i \to Y_i$ be the base change of $f_0$ to $S_i$ and let $f : X \to Y$ be the base change of $f_0$ to $S$. If \begin{enumerate} \item $f$ is at-worst-nodal of relative dimension $1$, and \item $f_0$ is locally of finite presentation, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is at-worst-nodal of relative dimension $1$.


curves-lemma-formal-local-structure-nodal-family

Let $f : T \to S$ be a morphism of schemes. Let $t \in T$ with image $s \in S$. Assume \begin{enumerate} \item $f$ is flat at $t$, \item $\mathcal{O}_{S, s}$ is Noetherian, \item $f$ is locally of finite type, \item $t$ is a split node of the fibre $T_s$. \end{enumerate} Then there exists an $h \in \mathfrak m_s^\wedge$ and an isomorphism $$ \mathcal{O}_{T, t}^\wedge \cong \mathcal{O}_{S, s}^\wedge[[x, y]]/(xy - h) $$ of $\mathcal{O}_{S, s}^\wedge$-algebras.


curves-lemma-etale-local-structure-nodal-family

Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is at-worst-nodal of relative dimension $1$. Let $x \in X$ be a point which is a singular point of the fibre $X_s$. Then there exists a commutative diagram of schemes $$ \xymatrix{ X \ar[d] & U \ar[rr] \ar[l] \ar[rd] & & W \ar[r] \ar[ld] & \Spec(\mathbf{Z}[u, v, a]/(uv - a)) \ar[d] \\ S & & V \ar[ll] \ar[rr] & & \Spec(\mathbf{Z}[a]) } $$ with $X \leftarrow U$, $S \leftarrow V$, and $U \to W$ \'etale morphisms, and with the right hand square cartesian, such that there exists a point $u \in U$ mapping to $x$ in $X$.


curves-lemma-h1-nonzero-degree-leq-2g-2

In Situation \ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Z \subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. If $H^1(X, \mathcal{I}\mathcal{L})$ is nonzero, then $$ \deg(\mathcal{L}) \leq 2g - 2 + \deg(Z) $$ with strict inequality unless $\mathcal{I}\mathcal{L} \cong \omega_X$.


curves-lemma-degree-more-than-2g-2

\begin{reference} \cite[Lemma 2]{Jongmin} \end{reference} In Situation \ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Z \subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. If $\deg(\mathcal{L}) > 2g - 2 + \deg(Z)$, then $H^1(X, \mathcal{I}\mathcal{L}) = 0$ and one of the following possibilities occurs \begin{enumerate} \item $H^0(X, \mathcal{I}\mathcal{L}) \not = 0$, or \item $g = 0$ and $\deg(\mathcal{L}) = \deg(Z) - 1$. \end{enumerate} In case (2) if $Z = \emptyset$, then $X \cong \mathbf{P}^1_k$ and $\mathcal{L}$ corresponds to $\mathcal{O}_{\mathbf{P}^1}(-1)$.


curves-lemma-degree-more-than-2g

\begin{reference} \cite[Lemma 3]{Jongmin} \end{reference} In Situation \ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. If $\deg(\mathcal{L}) \geq 2g$, then $\mathcal{L}$ is globally generated.


curves-lemma-vanishing-on-gorenstein

\begin{reference} Weak version of \cite[Lemma 4]{Jongmin} \end{reference} Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and of dimension $1$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Z \subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. If $H^1(X, \mathcal{I}\mathcal{L}) \not = 0$, then there exists a reduced connected closed subscheme $Y \subset X$ of dimension $1$ such that $$ \deg(\mathcal{L}|_Y) \leq -2\chi(Y, \mathcal{O}_Y) + \deg(Z \cap Y) $$ where $Z \cap Y$ is the scheme theoretic intersection.


curves-lemma-global-generation

Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and of dimension $1$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume that for every reduced connected closed subscheme $Y \subset X$ of dimension $1$ we have $$ \deg(\mathcal{L}|_Y) \geq 2\dim_k H^1(Y, \mathcal{O}_Y) $$ Then $\mathcal{L}$ is globally generated.


curves-lemma-rational-tail-negative

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \subset X$ be a rational tail (Example \ref{example-rational-tail}). Then $\deg(\omega_X|_C) < 0$.


curves-lemma-rational-tail-field-extension

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \subset X$ be a rational tail (Example \ref{example-rational-tail}). For any field extension $K/k$ the base change $C_K \subset X_K$ is a finite disjoint union of rational tails.


curves-lemma-no-rational-tail

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. If $X$ does not have a rational tail (Example \ref{example-rational-tail}), then for every reduced connected closed subscheme $Y \subset X$, $Y \not = X$ of dimension $1$ we have $\deg(\omega_X|_Y) \geq \dim_k H^1(Y, \mathcal{O}_Y)$.


curves-lemma-no-rational-tail-semiample-genus-geq-2

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Assume $X$ does not have a rational tail (Example \ref{example-rational-tail}). If \begin{enumerate} \item the genus of $X$ is $0$, then $X$ is isomorphic to an irreducible plane conic and $\omega_X^{\otimes -1}$ is very ample, \item the genus of $X$ is $1$, then $\omega_X \cong \mathcal{O}_X$, \item the genus of $X$ is $\geq 2$, then $\omega_X^{\otimes m}$ is globally generated for $m \geq 2$. \end{enumerate}


curves-lemma-contracting-rational-tails

Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Consider a sequence $$ X = X_0 \to X_1 \to \ldots \to X_n = X' $$ of contractions of rational tails (Example \ref{example-rational-tail}) until none are left. Then \begin{enumerate} \item if the genus of $X$ is $0$, then $X'$ is an irreducible plane conic, \item if the genus of $X$ is $1$, then $\omega_{X'} \cong \mathcal{O}_X$, \item if the genus of $X$ is $> 1$, then $\omega_{X'}^{\otimes m}$ is globally generated for $m \geq 2$. \end{enumerate} If the genus of $X$ is $\geq 1$, then the morphism $X \to X'$ is independent of choices and formation of this morphism commutes with base field extensions.


curves-lemma-rational-bridge-zero

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \subset X$ be a rational bridge (Example \ref{example-rational-bridge}). Then $\deg(\omega_X|_C) = 0$.


curves-lemma-rational-bridge-field-extension

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \subset X$ be a rational bridge (Example \ref{example-rational-bridge}). For any field extension $K/k$ the base change $C_K \subset X_K$ is a finite disjoint union of rational bridges.


curves-lemma-rational-bridge-canonical

Let $c : X \to Y$ be the contraction of a rational bridge (Example \ref{example-rational-bridge}). Then $c^*\omega_Y \cong \omega_X$.


curves-lemma-no-rational-bridge-ample-genus-geq-2

Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Assume \begin{enumerate} \item the singularities of $X$ are at-worst-nodal, \item $X$ does not have a rational tail (Example \ref{example-rational-tail}), \item $X$ does not have a rational bridge (Example \ref{example-rational-bridge}), \item the genus $g$ of $X$ is $\geq 2$. \end{enumerate} Then $\omega_X$ is ample.


curves-lemma-contracting-rational-bridges

Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \mathcal{O}_X) = k$ having genus $g \geq 2$. Assume the singularities of $X$ are at-worst-nodal and that $X$ has no rational tails. Consider a sequence $$ X = X_0 \to X_1 \to \ldots \to X_n = X' $$ of contractions of rational bridges (Example \ref{example-rational-bridge}) until none are left. Then $\omega_{X'}$ ample. The morphism $X \to X'$ is independent of choices and formation of this morphism commutes with base field extensions.


curves-lemma-contract-gorenstein-canonical

Let $k$ be a field. Let $c : X \to Y$ be a morphism of proper schemes over $k$ Assume \begin{enumerate} \item $\mathcal{O}_Y = c_*\mathcal{O}_X$ and $R^1c_*\mathcal{O}_X = 0$, \item $X$ and $Y$ are reduced, Gorenstein, and have dimension $1$, \item $\exists\ m \in \mathbf{Z}$ with $H^1(X, \omega_X^{\otimes m}) = 0$ and $\omega_X^{\otimes m}$ generated by global sections. \end{enumerate} Then $c^*\omega_Y \cong \omega_X$.


curves-lemma-characterize-contraction-to-stable

Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \mathcal{O}_X) = k$ having genus $g \geq 2$. Assume the singularities of $X$ are at-worst-nodal. There is a unique morphism (up to unique isomorphism) $$ c : X \longrightarrow Y $$ of schemes over $k$ having the following properties: \begin{enumerate} \item $Y$ is proper over $k$, $\dim(Y) = 1$, the singularities of $Y$ are at-worst-nodal, \item $\mathcal{O}_Y = c_*\mathcal{O}_X$ and $R^1c_*\mathcal{O}_X = 0$, and \item $\omega_Y$ is ample on $Y$. \end{enumerate}


curves-lemma-tricanonical

Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \mathcal{O}_X) = k$ having genus $g \geq 2$. Assume the singularities of $X$ are at-worst-nodal and $\omega_X$ is ample. Then $\omega_X^{\otimes 3}$ is very ample and $H^1(X, \omega_X^{\otimes 3}) = 0$.


curves-lemma-smooth-vector-fields

Let $k$ be an algebraically closed field. Let $X$ be a smooth, proper, connected curve over $k$. Let $g$ be the genus of $X$. \begin{enumerate} \item If $g \geq 2$, then $\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)$ is zero, \item if $g = 1$ and $D \in \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)$ is nonzero, then $D$ does not fix any closed point of $X$, and \item if $g = 0$ and $D \in \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)$ is nonzero, then $D$ fixes at most $2$ closed points of $X$. \end{enumerate}


curves-lemma-nodal-vector-fields

Let $k$ be an algebraically closed field. Let $X$ be an at-worst-nodal, proper, connected $1$-dimensional scheme over $k$. Let $\nu : X^\nu \to X$ be the normalization. Let $S \subset X^\nu$ be the set of points where $\nu$ is not an isomorphism. Then $$ \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = \{D' \in \text{Der}_k(\mathcal{O}_{X^\nu}, \mathcal{O}_{X^\nu}) \mid D' \text{ fixes every }x^\nu \in S\} $$


curves-lemma-stable-vector-fields

Let $k$ be an algebraically closed field. Let $X$ be an at-worst-nodal, proper, connected $1$-dimensional scheme over $k$. Assume the genus of $X$ is at least $2$ and that $X$ has no rational tails or bridges. Then $\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = 0$.


curves-proposition-projective-line

Let $k$ be a field. Let $X$ be a proper curve over $k$. The following are equivalent \begin{enumerate} \item $X \cong \mathbf{P}^1_k$, \item $X$ is smooth and geometrically irreducible over $k$, $X$ has genus $0$, and $X$ has an invertible module of odd degree, \item $X$ is geometrically integral over $k$, $X$ has genus $0$, $X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree, \item $H^0(X, \mathcal{O}_X) = k$, $X$ has genus $0$, $X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree, \item $X$ is geometrically integral over $k$, $X$ has genus $0$, and $X$ has an invertible $\mathcal{O}_X$-module of degree $1$, \item $H^0(X, \mathcal{O}_X) = k$, $X$ has genus $0$, and $X$ has an invertible $\mathcal{O}_X$-module of degree $1$, \item $H^1(X, \mathcal{O}_X) = 0$ and $X$ has an invertible $\mathcal{O}_X$-module of degree $1$, \item $H^1(X, \mathcal{O}_X) = 0$ and $X$ has closed points $x_1, \ldots, x_n$ such that $\mathcal{O}_{X, x_i}$ is normal and $\gcd([\kappa(x_i) : k]) = 1$, and \item add more here. \end{enumerate}


curves-proposition-unwind-morphism-smooth

Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper smooth curves over $k$. Then we can factor $f$ as $$ X \longrightarrow X^{(p^n)} \longrightarrow Y $$ where $X^{(p^n)} \to Y$ is a nonconstant morphism of proper smooth curves inducing a separable field extension $k(X^{(p^n)})/k(Y)$, we have $$ X^{(p^n)} = X \times_{\Spec(k), F_{\Spec(k)}^n} \Spec(k), $$ and $X \to X^{(p^n)}$ is the $n$-fold relative frobenius of $X$.


curves-proposition-torsion-picard-reduced-proper

Let $k$ be an algebraically closed field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and has dimension $1$. Let $g$ be the genus of $X$ and let $g_{geom}$ be the sum of the geometric genera of the irreducible components of $X$. For any prime $\ell$ different from the characteristic of $k$ we have $$ \dim_{\mathbf{F}_\ell} \Pic(X)[\ell] \leq g + g_{geom} $$ and equality holds if and only if all the singularities of $X$ are multicross.


etale-cohomology-theorem-sheafification

Let $\mathcal{C}$ be a site and $\mathcal{F}$ a presheaf on $\mathcal{C}$. \begin{enumerate} \item The rule $$ U \mapsto \mathcal{F}^+(U) := \colim_{\mathcal{U} \text{ covering of }U} \check H^0(\mathcal{U}, \mathcal{F}) $$ is a presheaf. And the colimit is a directed one. \item There is a canonical map of presheaves $\mathcal{F} \to \mathcal{F}^+$. \item If $\mathcal{F}$ is a separated presheaf then $\mathcal{F}^+$ is a sheaf and the map in (2) is injective. \item $\mathcal{F}^+$ is a separated presheaf. \item $\mathcal{F}^\# = (\mathcal{F}^+)^+$ is a sheaf, and the canonical map induces a functorial isomorphism $$ \Hom_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, \mathcal{G}) = \Hom_{\Sh(\mathcal{C})}(\mathcal{F}^\#, \mathcal{G}) $$ for any $\mathcal{G} \in \Sh(\mathcal{C})$. \end{enumerate}


etale-cohomology-theorem-descent-quasi-coherent

If $\mathcal{V} = \{T_i \to T\}_{i\in I}$ is an fpqc covering, then all descent data for quasi-coherent sheaves with respect to $\mathcal{V}$ are effective.


etale-cohomology-theorem-quasi-coherent

Let $S$ be a scheme. Let $\mathcal{C}$ be a site. Assume that \begin{enumerate} \item the underlying category $\mathcal{C}$ is a full subcategory of $\Sch/S$, \item any Zariski covering of $T \in \Ob(\mathcal{C})$ can be refined by a covering of $\mathcal{C}$, \item $S/S$ is an object of $\mathcal{C}$, \item every covering of $\mathcal{C}$ is an fpqc covering of schemes. \end{enumerate} Then the presheaf $\mathcal{O}$ is a sheaf on $\mathcal{C}$ and any quasi-coherent $\mathcal{O}$-module on $(\mathcal{C}, \mathcal{O})$ is of the form $\mathcal{F}^a$ for some quasi-coherent sheaf $\mathcal{F}$ on $S$.


etale-cohomology-theorem-cech-ss

Let $\mathcal{C}$ be a site. For any covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$ of $U \in \Ob(\mathcal{C})$ and any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ there is a spectral sequence $$ E_2^{p, q} = \check H^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) \Rightarrow H^{p+q}(U, \mathcal{F}), $$ where $\underline{H}^q(\mathcal{F})$ is the abelian presheaf $V \mapsto H^q(V, \mathcal{F})$.


etale-cohomology-theorem-zariski-fpqc-quasi-coherent

Let $S$ be a scheme and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_S$-module. Let $\mathcal{C}$ be either $(\Sch/S)_\tau$ for $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$ or $S_\etale$. Then $$ H^p(S, \mathcal{F}) = H^p_\tau(S, \mathcal{F}^a) $$ for all $p \geq 0$ where \begin{enumerate} \item the left hand side indicates the usual cohomology of the sheaf $\mathcal{F}$ on the underlying topological space of the scheme $S$, and \item the right hand side indicates cohomology of the abelian sheaf $\mathcal{F}^a$ (see Proposition \ref{proposition-quasi-coherent-sheaf-fpqc}) on the site $\mathcal{C}$. \end{enumerate}


etale-cohomology-theorem-picard-group

For any scheme $X$ we have canonical identifications \begin{align*} H_{fppf}^1(X, \mathbf{G}_m) & = H^1_{syntomic}(X, \mathbf{G}_m) \\ & = H^1_{smooth}(X, \mathbf{G}_m) \\ & = H_\etale^1(X, \mathbf{G}_m) \\ & = H^1_{Zar}(X, \mathbf{G}_m) \\ & = \Pic(X) \\ & = H^1(X, \mathcal{O}_X^*) \end{align*}


etale-cohomology-theorem-exactness-stalks

Let $S$ be a scheme. A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective (resp.\ surjective) if and only if the map on stalks $a_{\overline{s}} : \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$ is injective (resp.\ surjective) for all geometric points of $S$. A sequence of abelian sheaves on $S_\etale$ is exact if and only if it is exact on all stalks at geometric points of $S$.


etale-cohomology-theorem-henselian

Let $(R, \mathfrak m, \kappa)$ be a local ring. The following are equivalent: \begin{enumerate} \item $R$ is henselian, \item for any $f\in R[T]$ and any factorization $\bar f = g_0 h_0$ in $\kappa[T]$ with $\gcd(g_0, h_0)=1$, there exists a factorization $f = gh$ in $R[T]$ with $\bar g = g_0$ and $\bar h = h_0$, \item any finite $R$-algebra $S$ is isomorphic to a finite product of local rings finite over $R$, \item any finite type $R$-algebra $A$ is isomorphic to a product $A \cong A' \times C$ where $A' \cong A_1 \times \ldots \times A_r$ is a product of finite local $R$-algebras and all the irreducible components of $C \otimes_R \kappa$ have dimension at least 1, \item if $A$ is an \'etale $R$-algebra and $\mathfrak n$ is a maximal ideal of $A$ lying over $\mathfrak m$ such that $\kappa \cong A/\mathfrak n$, then there exists an isomorphism $\varphi : A \cong R \times A'$ such that $\varphi(\mathfrak n) = \mathfrak m \times A' \subset R \times A'$. \end{enumerate}


etale-cohomology-theorem-henselization

Let $(R, \mathfrak m, \kappa)$ be a local ring and $\kappa\subset\kappa^{sep}$ a separable algebraic closure. There exist canonical flat local ring maps $R \to R^h \to R^{sh}$ where \begin{enumerate} \item $R^h$, $R^{sh}$ are filtered colimits of \'etale $R$-algebras, \item $R^h$ is henselian, $R^{sh}$ is strictly henselian, \item $\mathfrak m R^h$ (resp.\ $\mathfrak m R^{sh}$) is the maximal ideal of $R^h$ (resp.\ $R^{sh}$), and \item $\kappa = R^h/\mathfrak m R^h$, and $\kappa^{sep} = R^{sh}/\mathfrak m R^{sh}$ as extensions of $\kappa$. \end{enumerate}


etale-cohomology-theorem-fully-faithful

Let $X$, $Y$ be schemes. Let $$ (g, g^\#) : (\Sh(X_\etale), \mathcal{O}_X) \longrightarrow (\Sh(Y_\etale), \mathcal{O}_Y) $$ be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes $f : X \to Y$ such that $(g, g^\#)$ is isomorphic to $(f_{small}, f_{small}^\sharp)$. In other words, the construction $$ \Sch \longrightarrow \textit{Locally ringed topoi}, \quad X \longrightarrow (X_\etale, \mathcal{O}_X) $$ is fully faithful (morphisms up to $2$-isomorphisms on the right hand side).


etale-cohomology-theorem-etale-topological

Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S' \to S$ be a universal homeomorphism. Denote $X'$ (resp.\ $Y'$) the base change to $S'$. If $X$ is \'etale over $S$, then the map $$ \Mor_S(Y, X) \longrightarrow \Mor_{S'}(Y', X') $$ is bijective.


etale-cohomology-theorem-topological-invariance

\begin{reference} \cite[IV Theorem 18.1.2]{EGA} \end{reference} Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is integral, universally injective and surjective (i.e., $f$ is a universal homeomorphism, see Morphisms, Lemma \ref{morphisms-lemma-universal-homeomorphism}). The functor $$ V \longmapsto V_X = X \times_Y V $$ defines an equivalence of categories $$ \{ \text{schemes }V\text{ \'etale over }Y \} \leftrightarrow \{ \text{schemes }U\text{ \'etale over }X \} $$


etale-cohomology-theorem-colimit

Let $X = \lim_{i \in I} X_i$ be a limit of a directed system of schemes with affine transition morphisms $f_{i'i} : X_{i'} \to X_i$. We assume that $X_i$ is quasi-compact and quasi-separated for all $i \in I$. Let $(\mathcal{F}_i, \varphi_{i'i})$ be a system of abelian sheaves on $(X_i, f_{i'i})$. Denote $f_i : X \to X_i$ the projection and set $\mathcal{F} = \colim f_i^{-1}\mathcal{F}_i$. Then $$ \colim_{i\in I} H_\etale^p(X_i, \mathcal{F}_i) = H_\etale^p(X, \mathcal{F}). $$ for all $p \geq 0$.


etale-cohomology-theorem-higher-direct-images

Let $f: X \to S$ be a quasi-compact and quasi-separated morphism of schemes, $\mathcal{F}$ an abelian sheaf on $X_\etale$, and $\overline{s}$ a geometric point of $S$ lying over $s \in S$. Then $$ \left(R^nf_* \mathcal{F}\right)_{\overline{s}} = H_\etale^n( X \times_S \Spec(\mathcal{O}_{S, s}^{sh}), p^{-1}\mathcal{F}) $$ where $p : X \times_S \Spec(\mathcal{O}_{S, s}^{sh}) \to X$ is the projection.


etale-cohomology-theorem-equivalence-sheaves-point

Let $S = \Spec(K)$ with $K$ a field. Let $\overline{s}$ be a geometric point of $S$. Let $G = \text{Gal}_{\kappa(s)}$ denote the absolute Galois group. Taking stalks induces an equivalence of categories $$ \Sh(S_\etale) \longrightarrow G\textit{-Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}. $$


etale-cohomology-theorem-central-simple-algebra

Let $K$ be a field. For a unital, associative (not necessarily commutative) $K$-algebra $A$ the following are equivalent \begin{enumerate} \item $A$ is finite central simple $K$-algebra, \item $A$ is a finite dimensional $K$-vector space, $K$ is the center of $A$, and $A$ has no nontrivial two-sided ideal, \item there exists $d \geq 1$ such that $A \otimes_K \bar K \cong \text{Mat}(d \times d, \bar K)$, \item there exists $d \geq 1$ such that $A \otimes_K K^{sep} \cong \text{Mat}(d \times d, K^{sep})$, \item there exist $d \geq 1$ and a finite Galois extension $K \subset K'$ such that $A \otimes_{K'} K' \cong \text{Mat}(d \times d, K')$, \item there exist $n \geq 1$ and a finite central skew field $D$ over $K$ such that $A \cong \text{Mat}(n \times n, D)$. \end{enumerate} The integer $d$ is called the {\it degree} of $A$.


etale-cohomology-theorem-brauer-delta

Let $K$ be a field with separable algebraic closure $K^{sep}$. The map $\delta : \text{Br}(K) \to H^2(\text{Gal}(K^{sep}/K), (K^{sep})^*)$ defined above is a group isomorphism.


etale-cohomology-theorem-C1-brauer-group-zero

Let $K$ be a $C_1$ field. Then $\text{Br}(K) = 0$.


etale-cohomology-theorem-tsen

The function field of a variety of dimension $r$ over an algebraically closed field $k$ is $C_r$.


etale-cohomology-theorem-vanishing-cohomology-Gm-curve

Let $X$ be a smooth curve over an algebraically closed field. Then $$ H_\etale^q(X, \mathbf{G}_m) = 0 \ \ \text{ for all } q \geq 2. $$


etale-cohomology-theorem-gabber

Let $(A, I)$ be a henselian pair. Set $X = \Spec(A)$ and $Z = \Spec(A/I)$. For any torsion abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $H^q_\etale(X, \mathcal{F}) = H^q_\etale(Z, \mathcal{F}|_Z)$.


etale-cohomology-theorem-vanishing-affine-curves

If $k$ is an algebraically closed field, $X$ is a separated, finite type scheme of dimension $\leq 1$ over $k$, and $\mathcal{F}$ is a torsion abelian sheaf on $X_\etale$, then \begin{enumerate} \item $H^q_\etale(X, \mathcal{F}) = 0$ for $q > 2$, \item $H^q_\etale(X, \mathcal{F}) = 0$ for $q > 1$ if $X$ is affine, \item $H^q_\etale(X, \mathcal{F}) = 0$ for $q > 1$ if $p = \text{char}(k) > 0$ and $\mathcal{F}$ is $p$-power torsion, \item $H^q_\etale(X, \mathcal{F})$ is finite if $\mathcal{F}$ is constructible and torsion prime to $\text{char}(k)$, \item $H^q_\etale(X, \mathcal{F})$ is finite if $X$ is proper and $\mathcal{F}$ constructible, \item $H^q_\etale(X, \mathcal{F}) \to H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k \subset k'$ of algebraically closed fields if $\mathcal{F}$ is torsion prime to $\text{char}(k)$, \item $H^q_\etale(X, \mathcal{F}) \to H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k \subset k'$ of algebraically closed fields if $X$ is proper, \item $H^2_\etale(X, \mathcal{F}) \to H^2_\etale(U, \mathcal{F})$ is surjective for all $U \subset X$ open. \end{enumerate}


etale-cohomology-theorem-vanishing-curves

Let $X$ be a finite type, dimension $1$ scheme over an algebraically closed field $k$. Let $\mathcal{F}$ be a torsion sheaf on $X_\etale$. Then $$ H_\etale^q(X, \mathcal{F}) = 0, \quad \forall q \geq 3. $$ If $X$ affine then also $H_\etale^2(X, \mathcal{F}) = 0$.


etale-cohomology-theorem-smooth-base-change

Consider a cartesian diagram of schemes $$ \xymatrix{ X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\ S & T \ar[l]_g } $$ where $f$ is smooth and $g$ quasi-compact and quasi-separated. Then $$ f^{-1}R^qg_*\mathcal{F} = R^qh_*e^{-1}\mathcal{F} $$ for any $q$ and any abelian sheaf $\mathcal{F}$ on $T_\etale$ all of whose stalks at geometric points are torsion of orders invertible on $S$.


etale-cohomology-theorem-proper-base-change

Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times_Y X$ and consider the cartesian diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$. Then the base change map $$ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} $$ is an isomorphism.


etale-cohomology-lemma-fpqc-sheaves

Let $\mathcal{F}$ be a presheaf on $\Sch/S$. Then $\mathcal{F}$ satisfies the sheaf property for the fpqc topology if and only if \begin{enumerate} \item $\mathcal{F}$ satisfies the sheaf property with respect to the Zariski topology, and \item for every faithfully flat morphism $\Spec(B) \to \Spec(A)$ of affine schemes over $S$, the sheaf axiom holds for the covering $\{\Spec(B) \to \Spec(A)\}$. Namely, this means that $$ \xymatrix{ \mathcal{F}(\Spec(A)) \ar[r] & \mathcal{F}(\Spec(B)) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(\Spec(B \otimes_A B)) } $$ is an equalizer diagram. \end{enumerate}


etale-cohomology-lemma-representable-sheaf-fpqc

Any representable presheaf on $\Sch/S$ satisfies the sheaf condition for the fpqc topology.


etale-cohomology-lemma-descent-modules

If $A \to B$ is faithfully flat and $M$ is an $A$-module, then the complex $(B/A)_\bullet \otimes_A M$ is exact in positive degrees, and $H^0((B/A)_\bullet \otimes_A M) = M$.


etale-cohomology-lemma-hom-injective

Let $\mathcal{C}$ be a category. If $\mathcal{I}$ is an injective object of $\textit{PAb}(\mathcal{C})$ and $\mathcal{U}$ is a family of morphisms with fixed target in $\mathcal{C}$, then $\check H^p(\mathcal{U}, \mathcal{I}) = 0$ for all $p > 0$.


etale-cohomology-lemma-forget-injectives

The forgetful functor $\textit{Ab}(\mathcal{C})\to \textit{PAb}(\mathcal{C})$ transforms injectives into injectives.


etale-cohomology-lemma-compare-cohomology-big-small

Let $\tau \in \{\etale, Zariski\}$. If $\mathcal{F}$ is an abelian sheaf defined on $(\Sch/S)_\tau$, then the cohomology groups of $\mathcal{F}$ over $S$ agree with the cohomology groups of $\mathcal{F}|_{S_\tau}$ over $S$.


etale-cohomology-lemma-alternative

Let $S$ be a scheme. Let $S_{affine, \etale}$ denote the full subcategory of $S_\etale$ whose objects are those $U/S \in \Ob(S_\etale)$ with $U$ affine. A covering of $S_{affine, \etale}$ will be a standard \'etale covering, see Topologies, Definition \ref{topologies-definition-standard-etale}. Then restriction $$ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}} $$ defines an equivalence of topoi $\Sh(S_\etale) \cong \Sh(S_{affine, \etale})$.


etale-cohomology-lemma-cech-complex

Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $(\Sch/S)_\tau$, or on $S_\tau$ in case $\tau = \etale$, and let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a standard $\tau$-covering of this site. Let $V = \coprod_{i \in I} U_i$. Then \begin{enumerate} \item $V$ is an affine scheme, \item $\mathcal{V} = \{V \to U\}$ is an fpqc covering and also a $\tau$-covering unless $\tau = Zariski$, \item the {\v C}ech complexes $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ and $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})$ agree. \end{enumerate}


etale-cohomology-lemma-locality-cohomology

Let $\mathcal{C}$ be a site, $\mathcal{F}$ an abelian sheaf on $\mathcal{C}$, $U$ an object of $\mathcal{C}$, $p > 0$ an integer and $\xi \in H^p(U, \mathcal{F})$. Then there exists a covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$ of $U$ in $\mathcal{C}$ such that $\xi |_{U_i} = 0$ for all $i \in I$.


etale-cohomology-lemma-kummer-sequence

If $n\in \mathcal{O}_S^*$ then $$ 0 \to \mu_{n, S} \to \mathbf{G}_{m, S} \xrightarrow{(\cdot)^n} \mathbf{G}_{m, S} \to 0 $$ is a short exact sequence of sheaves on both the small and big \'etale site of $S$.


etale-cohomology-lemma-kummer-sequence-syntomic

For any $n \in \mathbf{N}$ the sequence $$ 0 \to \mu_{n, S} \to \mathbf{G}_{m, S} \xrightarrow{(\cdot)^n} \mathbf{G}_{m, S} \to 0 $$ is a short exact sequence of sheaves on the site $(\Sch/S)_{fppf}$ and $(\Sch/S)_{syntomic}$.


etale-cohomology-lemma-cofinal-etale

Let $S$ be a scheme, and let $\overline{s}$ be a geometric point of $S$. The category of \'etale neighborhoods is cofiltered. More precisely: \begin{enumerate} \item Let $(U_i, \overline{u}_i)_{i = 1, 2}$ be two \'etale neighborhoods of $\overline{s}$ in $S$. Then there exists a third \'etale neighborhood $(U, \overline{u})$ and morphisms $(U, \overline{u}) \to (U_i, \overline{u}_i)$, $i = 1, 2$. \item Let $h_1, h_2: (U, \overline{u}) \to (U', \overline{u}')$ be two morphisms between \'etale neighborhoods of $\overline{s}$. Then there exist an \'etale neighborhood $(U'', \overline{u}'')$ and a morphism $h : (U'', \overline{u}'') \to (U, \overline{u})$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \circ h = h_2 \circ h$. \end{enumerate}


etale-cohomology-lemma-geometric-lift-to-cover

Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. Let $(U, \overline{u})$ be an \'etale neighborhood of $\overline{s}$. Let $\mathcal{U} = \{\varphi_i : U_i \to U \}_{i\in I}$ be an \'etale covering. Then there exist $i \in I$ and $\overline{u}_i : \overline{s} \to U_i$ such that $\varphi_i : (U_i, \overline{u}_i) \to (U, \overline{u})$ is a morphism of \'etale neighborhoods.


etale-cohomology-lemma-stalk-gives-point

Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. Consider the functor \begin{align*} u : S_\etale & \longrightarrow \textit{Sets}, \\ U & \longmapsto |U_{\overline{s}}| = \{\overline{u} \text{ such that }(U, \overline{u}) \text{ is an \'etale neighbourhood of }\overline{s}\}. \end{align*} Here $|U_{\overline{s}}|$ denotes the underlying set of the geometric fibre. Then $u$ defines a point $p$ of the site $S_\etale$ (Sites, Definition \ref{sites-definition-point}) and its associated stalk functor $\mathcal{F} \mapsto \mathcal{F}_p$ (Sites, Equation \ref{sites-equation-stalk}) is the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ defined above.


etale-cohomology-lemma-stalk-exact

Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. \begin{enumerate} \item The stalk functor $\textit{PAb}(S_\etale) \to \textit{Ab}$, $\mathcal{F}  \mapsto  \mathcal{F}_{\overline{s}}$ is exact. \item We have $(\mathcal{F}^\#)_{\overline{s}} = \mathcal{F}_{\overline{s}}$ for any presheaf of sets $\mathcal{F}$ on $S_\etale$. \item The functor $\textit{Ab}(S_\etale) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is exact. \item Similarly the functors $\textit{PSh}(S_\etale) \to \textit{Sets}$ and $\Sh(S_\etale) \to \textit{Sets}$ given by the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ are exact (see Categories, Definition \ref{categories-definition-exact}) and commute with arbitrary colimits. \end{enumerate}


etale-cohomology-lemma-points-small-etale-site

Let $S$ be a scheme. \begin{enumerate} \item Let $p$ be a point of the small \'etale site $S_\etale$ of $S$ given by a functor $u : S_\etale \to \textit{Sets}$. Then there exists a geometric point $\overline{s}$ of $S$ such that $p$ is isomorphic to the point of $S_\etale$ associated to $\overline{s}$ in Lemma \ref{lemma-stalk-gives-point}. \item Let $p : \Sh(pt) \to \Sh(S_\etale)$ be a point of the small \'etale topos of $S$. Then $p$ comes from a geometric point of $S$, i.e., the stalk functor $\mathcal{F} \mapsto \mathcal{F}_p$ is isomorphic to a stalk functor as defined in Definition \ref{definition-stalk}. \end{enumerate}


etale-cohomology-lemma-points-fppf

Let $S$ be a scheme. All of the following sites have enough points $S_{Zar}$, $S_\etale$, $(\Sch/S)_{Zar}$, $(\textit{Aff}/S)_{Zar}$, $(\Sch/S)_\etale$, $(\textit{Aff}/S)_\etale$, $(\Sch/S)_{smooth}$, $(\textit{Aff}/S)_{smooth}$, $(\Sch/S)_{syntomic}$, $(\textit{Aff}/S)_{syntomic}$, $(\Sch/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$.


etale-cohomology-lemma-support-subsheaf-final

Let $S$ be a scheme. Let $\mathcal{F}$ be a subsheaf of the final object of the \'etale topos of $S$ (see Sites, Example \ref{sites-example-singleton-sheaf}). Then there exists a unique open $W \subset S$ such that $\mathcal{F} = h_W$.


etale-cohomology-lemma-zero-over-image

Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_\etale$. Let $\sigma \in \mathcal{F}(U)$ be a local section. There exists an open subset $W \subset U$ such that \begin{enumerate} \item $W \subset U$ is the largest Zariski open subset of $U$ such that $\sigma|_W = 0$, \item for every $\varphi : V \to U$ in $S_\etale$ we have $$ \sigma|_V = 0 \Leftrightarrow \varphi(V) \subset W, $$ \item for every geometric point $\overline{u}$ of $U$ we have $$ (U, \overline{u}, \sigma) = 0\text{ in }\mathcal{F}_{\overline{s}} \Leftrightarrow \overline{u} \in W $$ where $\overline{s} = (U \to S) \circ \overline{u}$. \end{enumerate}


etale-cohomology-lemma-support-sheaf-rings-closed

The support of a sheaf of rings on $S_\etale$ is closed.


etale-cohomology-lemma-finite-over-henselian

If $R$ is henselian and $A$ is a finite $R$-algebra, then $A$ is a finite product of henselian local rings.


etale-cohomology-lemma-describe-etale-local-ring

\begin{slogan} The stalk of the structure sheaf of a scheme in the etale topology is the strict henselization. \end{slogan} Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$ lying over $s \in S$. Let $\kappa = \kappa(s)$ and let $\kappa \subset \kappa^{sep} \subset \kappa(\overline{s})$ denote the separable algebraic closure of $\kappa$ in $\kappa(\overline{s})$. Then there is a canonical identification $$ (\mathcal{O}_{S, s})^{sh} \cong \mathcal{O}_{S, \overline{s}} $$ where the left hand side is the strict henselization of the local ring $\mathcal{O}_{S, s}$ as described in Theorem \ref{theorem-henselization} and right hand side is the stalk of the structure sheaf $\mathcal{O}_S$ on $S_\etale$ at the geometric point $\overline{s}$.


etale-cohomology-lemma-etale-site-locally-ringed

Let $S$ be a scheme. The small \'etale site $S_\etale$ endowed with its structure sheaf $\mathcal{O}_S$ is a locally ringed site, see Modules on Sites, Definition \ref{sites-modules-definition-locally-ringed}.


etale-cohomology-lemma-stalk-pullback

Let $f : X \to Y$ be a morphism of schemes. \begin{enumerate} \item The functor $f^{-1} : \textit{Ab}(Y_\etale) \to \textit{Ab}(X_\etale)$ is exact. \item The functor $f^{-1} : \Sh(Y_\etale) \to \Sh(X_\etale)$ is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition \ref{categories-definition-exact}. \item Let $\overline{x} \to X$ be a geometric point. Let $\mathcal{G}$ be a sheaf on $Y_\etale$. Then there is a canonical identification $$ (f^{-1}\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}}. $$ where $\overline{y} = f \circ \overline{x}$. \item For any $V \to Y$ \'etale we have $f^{-1}h_V = h_{X \times_Y V}$. \end{enumerate}


etale-cohomology-lemma-where-sections-are-equal

Let $S$ be a scheme. Let $\mathcal{F}$ be a sheaf of sets on $S_\etale$. Let $s, t \in \mathcal{F}(S)$. Then there exists an open $W \subset S$ characterized by the following property: A morphism $f : T \to S$ factors through $W$ if and only if $s|_T = t|_T$ (restriction is pullback by $f_{small}$).


etale-cohomology-lemma-describe-pullback

Let $S$ be a scheme. Let $\tau \in \{Zariski, \etale\}$. Consider the morphism $$ \pi_S : (\Sch/S)_\tau \longrightarrow S_\tau $$ of Topologies, Lemma \ref{topologies-lemma-at-the-bottom} or \ref{topologies-lemma-at-the-bottom-etale}. Let $\mathcal{F}$ be a sheaf on $S_\tau$. Then $\pi_S^{-1}\mathcal{F}$ is given by the rule $$ (\pi_S^{-1}\mathcal{F})(T) = \Gamma(T_\tau, f_{small}^{-1}\mathcal{F}) $$ where $f : T \to S$. Moreover, $\pi_S^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to fpqc coverings.


etale-cohomology-lemma-sections-upstairs

Let $S$ be a scheme. Let $f : T \to S$ be a morphism such that \begin{enumerate} \item $f$ is flat and quasi-compact, and \item the geometric fibres of $f$ are connected. \end{enumerate} Let $\mathcal{F}$ be a sheaf on $S_\etale$. Then $\Gamma(S, \mathcal{F}) = \Gamma(T, f^{-1}_{small}\mathcal{F})$.


etale-cohomology-lemma-sections-base-field-extension

Let $k \subset K$ be an extension of fields with $k$ separably algebraically closed. Let $S$ be a scheme over $k$. Denote $p : S_K = S \times_{\Spec(k)} \Spec(K) \to S$ the projection. Let $\mathcal{F}$ be a sheaf on $S_\etale$. Then $\Gamma(S, \mathcal{F}) = \Gamma(S_K, p^{-1}_{small}\mathcal{F})$.