duality-lemma-shriek-over-CM

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Let $x \in X$ with image $y \in Y$. Assume \begin{enumerate} \item $\mathcal{O}_{Y, y}$ is Cohen-Macaulay, and \item $\text{trdeg}_{\kappa(f(\xi))}(\kappa(\xi)) \leq r$ for any generic point $\xi$ of an irreducible component of $X$ containing $x$. \end{enumerate} Then $$ H^i(f^!\mathcal{O}_Y)_x \not = 0 \Rightarrow - r \leq i $$ and the stalk $H^{-r}(f^!\mathcal{O}_Y)_x$ is $(S_2)$ as an $\mathcal{O}_{X, x}$-module.


duality-lemma-flat-quasi-finite-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. If $f$ is flat and quasi-finite, then $$ f^!\mathcal{O}_Y = \omega_{X/Y}[0] $$ for some coherent $\mathcal{O}_X$-module $\omega_{X/Y}$ flat over $Y$.


duality-lemma-CM-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. If $f$ is Cohen-Macaulay (More on Morphisms, Definition \ref{more-morphisms-definition-CM}), then $$ f^!\mathcal{O}_Y = \omega_{X/Y}[d] $$ for some coherent $\mathcal{O}_X$-module $\omega_{X/Y}$ flat over $Y$ where $d$ is the locally constant function on $X$ which gives the relative dimension of $X$ over $Y$.


duality-lemma-dualizing-module

Let $X$ be a connected Noetherian scheme and let $\omega_X$ be a dualizing module on $X$. The support of $\omega_X$ is the union of the irreducible components of maximal dimension with respect to any dimension function and $\omega_X$ is a coherent $\mathcal{O}_X$-module having property $(S_2)$.


duality-lemma-vanishing-good-dualizing

Let $X/A$ with $\omega_X^\bullet$ and $\omega_X$ be as in Example \ref{example-proper-over-local}. Then \begin{enumerate} \item $H^i(\omega_X^\bullet) \not = 0 \Rightarrow i \in \{-\dim(X), \ldots, 0\}$, \item the dimension of the support of $H^i(\omega_X^\bullet)$ is at most $-i$, \item $\text{Supp}(\omega_X)$ is the union of the components of dimension $\dim(X)$, and \item $\omega_X$ has property $(S_2)$. \end{enumerate}


duality-lemma-dualizing-module-proper-over-A

Let $X/A$ with dualizing module $\omega_X$ be as in Example \ref{example-proper-over-local}. Let $d = \dim(X_s)$ be the dimension of the closed fibre. If $\dim(X) = d + \dim(A)$, then the dualizing module $\omega_X$ represents the functor $$ \mathcal{F} \longmapsto \Hom_A(H^d(X, \mathcal{F}), \omega_A) $$ on the category of coherent $\mathcal{O}_X$-modules.


duality-lemma-dualizing-module-CM-scheme

Let $X$ be a locally Noetherian scheme with dualizing complex $\omega_X^\bullet$. \begin{enumerate} \item $X$ is Cohen-Macaulay $\Leftrightarrow$ $\omega_X^\bullet$ locally has a unique nonzero cohomology sheaf, \item $\mathcal{O}_{X, x}$ is Cohen-Macaulay $\Leftrightarrow$ $\omega_{X, x}^\bullet$ has a unique nonzero cohomology, \item $U = \{x \in X \mid \mathcal{O}_{X, x}\text{ is Cohen-Macaulay}\}$ is open and Cohen-Macaulay. \end{enumerate} If $X$ is connected and Cohen-Macaulay, then there is an integer $n$ and a coherent Cohen-Macaulay $\mathcal{O}_X$-module $\omega_X$ such that $\omega_X^\bullet = \omega_X[-n]$.


duality-lemma-affine-flat-Noetherian-CM

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Let $x \in X$. If $f$ is flat, then the following are equivalent \begin{enumerate} \item $f$ is Cohen-Macaulay at $x$, \item $f^!\mathcal{O}_Y$ has a unique nonzero cohomology sheaf in a neighbourhood of $x$. \end{enumerate}


duality-lemma-gorenstein-CM

A Gorenstein scheme is Cohen-Macaulay.


duality-lemma-regular-gorenstein

A regular scheme is Gorenstein.


duality-lemma-gorenstein

Let $X$ be a locally Noetherian scheme. \begin{enumerate} \item If $X$ has a dualizing complex $\omega_X^\bullet$, then \begin{enumerate} \item $X$ is Gorenstein $\Leftrightarrow$ $\omega_X^\bullet$ is an invertible object of $D(\mathcal{O}_X)$, \item $\mathcal{O}_{X, x}$ is Gorenstein $\Leftrightarrow$ $\omega_{X, x}^\bullet$ is an invertible object of $D(\mathcal{O}_{X, x})$, \item $U = \{x \in X \mid \mathcal{O}_{X, x}\text{ is Gorenstein}\}$ is an open Gorenstein subscheme. \end{enumerate} \item If $X$ is Gorenstein, then $X$ has a dualizing complex if and only if $\mathcal{O}_X[0]$ is a dualizing complex. \end{enumerate}


duality-lemma-gorenstein-lci

If $f : Y \to X$ is a local complete intersection morphism with $X$ a Gorenstein scheme, then $Y$ is Gorenstein.


duality-lemma-gorenstein-local-syntomic

The property $\mathcal{P}(S) =$``$S$ is Gorenstein'' is local in the syntomic topology.


duality-lemma-gorenstein-base-change

Let $X$ be a locally Noetherian scheme over the field $k$. Let $k \subset k'$ be a finitely generated field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. Then we have $$ \mathcal{O}_{X, x}\text{ is Gorenstein} \Leftrightarrow \mathcal{O}_{X_{k'}, x'}\text{ is Gorenstein} $$ If $X$ is locally of finite type over $k$, the same holds for any field extension $k \subset k'$.


duality-lemma-gorenstein-morphism

Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is Gorenstein, and \item $f$ is flat and its fibres are Gorenstein schemes. \end{enumerate}


duality-lemma-gorenstein-CM-morphism

A Gorenstein morphism is Cohen-Macaulay.


duality-lemma-lci-gorenstein

A syntomic morphism is Gorenstein. Equivalently a flat local complete intersection morphism is Gorenstein.


duality-lemma-composition-gorenstein

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms. Assume that the fibres $X_y$, $Y_z$ and $X_z$ of $f$, $g$, and $g \circ f$ are locally Noetherian. \begin{enumerate} \item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$, then $g \circ f$ is Gorenstein at $x$. \item If $f$ and $g$ are Gorenstein, then $g \circ f$ is Gorenstein. \item If $g \circ f$ is Gorenstein at $x$ and $f$ is flat at $x$, then $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$. \item If $f \circ g$ is Gorenstein and $f$ is flat, then $f$ is Gorenstein and $g$ is Gorenstein at every point in the image of $f$. \end{enumerate}


duality-lemma-flat-morphism-from-gorenstein-scheme

\begin{slogan} Gorensteinnes of the total space of a flat fibration implies same for base and fibres \end{slogan} Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Gorenstein, then $f$ is Gorenstein and $\mathcal{O}_{Y, f(x)}$ is Gorenstein for all $x \in X$.


duality-lemma-base-change-gorenstein

Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in X'$ be a point with image $x \in X$. \begin{enumerate} \item If $f$ is Gorenstein at $x$, then $f' : X' \to Y'$ is Gorenstein at $x'$. \item If $f$ is flat and $x$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \item If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \end{enumerate}


duality-lemma-flat-lft-base-change-gorenstein

Let $f : X \to Y$ be a morphism of schemes which is flat and locally of finite type. Then formation of the set $\{x \in X \mid f\text{ is Gorenstein at }x\}$ commutes with arbitrary base change.


duality-lemma-affine-flat-Noetherian-gorenstein

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Let $x \in X$. If $f$ is flat, then the following are equivalent \begin{enumerate} \item $f$ is Gorenstein at $x$, \item $f^!\mathcal{O}_X$ is isomorphic to an invertible object in a neighbourhood of $x$. \end{enumerate} In particular, the set of points where $f$ is Gorenstein is open in $X$.


duality-lemma-flat-finite-presentation-characterize-gorenstein

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $x \in X$ with image $s \in S$. Set $d = \dim_x(X_s)$. The following are equivalent \begin{enumerate} \item $f$ is Gorenstein at $x$, \item there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite morphism $U \to \mathbf{A}^d_S$ over $S$ which is Gorenstein at $x$, \item there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite Gorenstein morphism $U \to \mathbf{A}^d_S$ over $S$, \item for any $S$-morphism $g : U \to \mathbf{A}^d_S$ of an open neighbourhood $U \subset X$ of $x$ we have: $g$ is quasi-finite at $x$ $\Rightarrow$ $g$ is Gorenstein at $x$. \end{enumerate} In particular, the set of points where $f$ is Gorenstein is open in $X$.


duality-lemma-descent-ascent

Let $f : X \to Y$ be a morphism of locally Noetherian schemes. Assume \begin{enumerate} \item $f$ is syntomic and surjective, or \item $f$ is a surjective flat local complete intersection morphism, or \item $f$ is a surjective Gorenstein morphism of finite type. \end{enumerate} Then $K \in D_\QCoh(\mathcal{O}_Y)$ is a dualizing complex on $Y$ if and only if $Lf^*K$ is a dualizing complex on $X$.


duality-lemma-duality-proper-over-field

Let $X$ be a proper scheme 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 $i \in [-\dim(X), 0]$, \item $\omega_X = H^{-\dim(X)}(\omega_X^\bullet)$ is a coherent $(S_2)$-module whose support is the irreducible components of dimension $d$, \item the dimension of the support of $H^i(\omega_X^\bullet)$ is at most $-i$, \item for $x \in X$ closed the module $H^i(\omega_{X, x}^\bullet) \oplus \ldots \oplus H^0(\omega_{X, x}^\bullet)$ is nonzero if and only if $\text{depth}(\mathcal{O}_{X, x}) \leq -i$, \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 $\Hom(\mathcal{F}, \omega_X) = \Hom_k(H^{\dim(X)}(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$, and \item if $X \to \Spec(k)$ is smooth of relative dimension $d$, then $\omega_X^\bullet \cong \wedge^d\Omega_{X/k}[d]$ and $\omega_X \cong \wedge^d\Omega_{X/k}$. \end{enumerate}


duality-lemma-duality-proper-over-field-perfect

Let $k$, $X$, and $\omega_X^\bullet$ be as in Lemma \ref{lemma-duality-proper-over-field}. Let $t : H^0(X, \omega_X^\bullet) \to k$ be as in Remark \ref{remark-duality-proper-over-field}. Let $E \in D(\mathcal{O}_X)$ be perfect. Then the pairings $$ H^i(X, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} E^\vee) \times H^{-i}(X, E) \longrightarrow k, \quad (\xi, \eta) \longmapsto t((1_{\omega_X^\bullet} \otimes \epsilon)(\xi \cup \eta)) $$ are perfect for all $i$. Here $\cup$ denotes the cupproduct of Cohomology, Section \ref{cohomology-section-cup-product} and $\epsilon : E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E \to \mathcal{O}_X$ is as in Cohomology, Example \ref{cohomology-example-dual-derived}.


duality-lemma-duality-proper-over-field-CM

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


duality-lemma-relative-dualizing-complex-algebra

Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $(K, \xi)$ be a relative dualizing complex. Then for any commutative diagram $$ \xymatrix{ \Spec(A) \ar[d] \ar[r] & X \ar[d] \\ \Spec(R) \ar[r] & S } $$ whose horizontal arrows are open immersions, the restriction of $K$ to $\Spec(A)$ corresponds via Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded} to a relative dualizing complex for $R \to A$ in the sense of Dualizing Complexes, Definition \ref{dualizing-definition-relative-dualizing-complex}.


duality-lemma-relative-dualizing-RHom

Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $(K, \xi)$ be a relative dualizing complex. Then $\mathcal{O}_X \to R\SheafHom_{\mathcal{O}_X}(K, K)$ is an isomorphism.


duality-lemma-uniqueness-relative-dualizing

Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. If $(K, \xi)$ and $(L, \eta)$ are two relative dualizing complexes on $X/S$, then there is a unique isomorphism $K \to L$ sending $\xi$ to $\eta$.


duality-lemma-existence-relative-dualizing

Let $X \to S$ be a morphism of schemes which is flat and locally of finite presentation. There exists a relative dualizing complex $(K, \xi)$.


duality-lemma-base-change-relative-dualizing

Consider a cartesian square $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ of schemes. Assume $X \to S$ is flat and locally of finite presentation. Let $(K, \xi)$ be a relative dualizing complex for $f$. Set $K' = L(g')^*K$. Let $\xi'$ be the derived base change of $\xi$ (see proof). Then $(K', \xi')$ is a relative dualizing complex for $f'$.


duality-lemma-flat-proper-relative-dualizing

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a proper, flat morphism of finite presentation. The relative dualizing complex $\omega_{X/S}^\bullet$ of Remark \ref{remark-relative-dualizing-complex} together with (\ref{equation-pre-rigid}) is a relative dualizing complex in the sense of Definition \ref{definition-relative-dualizing-complex}.


duality-lemma-determinant

Let $X$ be a locally ringed space. Let $$ \mathcal{E}_1 \xrightarrow{\alpha} \mathcal{E}_0 \to \mathcal{F} \to 0 $$ be a short exact sequence of $\mathcal{O}_X$-modules. Assume $\mathcal{E}_1$ and $\mathcal{E}_0$ are locally free of ranks $r_1, r_0$. Then there is a canonical map $$ \wedge^{r_0 - r_1}\mathcal{F} \longrightarrow \wedge^{r_1}(\mathcal{E}_1^\vee) \otimes \wedge^{r_0}\mathcal{E}_0 $$ which is an isomorphism on the stalk at $x \in X$ if and only if $\mathcal{F}$ is locally free of rank $r_0 - r_1$ in an open neighbourhood of $x$.


duality-lemma-fundamental-class-lci

Let $Y$ be a Noetherian scheme. Let $f : X \to Y$ be a local complete intersection morphism which factors as an immersion $X \to P$ followed by a proper smooth morphism $P \to Y$. Let $r$ be the locally constant function on $X$ such that $\omega_{Y/X} = H^{-r}(f^!\mathcal{O}_Y)$ is the unique nonzero cohomology sheaf of $f^!\mathcal{O}_Y$, see Lemma \ref{lemma-lci-shriek}. Then there is a map $$ \wedge^r\Omega_{X/Y} \longrightarrow \omega_{Y/X} $$ which is an isomorphism on the stalk at a point $x$ if and only if $f$ is smooth at $x$.


duality-lemma-fundamental-class-almost-lci

Let $f : X \to Y$ be a morphism of schemes. Let $r \geq 0$. Assume \begin{enumerate} \item $Y$ is Cohen-Macaulay (Properties, Definition \ref{properties-definition-Cohen-Macaulay}), \item $f$ factors as $X \to P \to Y$ where the first morphism is an immersion and the second is smooth and proper, \item if $x \in X$ and $\dim(\mathcal{O}_{X, x}) \leq 1$, then $f$ is Koszul at $x$ (More on Morphisms, Definition \ref{more-morphisms-definition-lci}), and \item if $\xi$ is a generic point of an irreducible component of $X$, then we have $\text{trdeg}_{\kappa(f(\xi))} \kappa(\xi) = r$. \end{enumerate} Then with $\omega_{Y/X} = H^{-r}(f^!\mathcal{O}_Y)$ there is a map $$ \wedge^r\Omega_{X/Y} \longrightarrow \omega_{Y/X} $$ which is an isomorphism on the locus where $f$ is smooth.


duality-lemma-lift-map

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $(K_n)$ be a Deligne system and denote $K \in D^b_{\textit{Coh}}(\mathcal{O}_U)$ the value of the constant system $(K_n|_U)$. Let $L$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$. Then $\colim \Hom_X(K_n, L) = \Hom_U(K, L|_U)$.


duality-lemma-lift-map-plus

The result of Lemma \ref{lemma-lift-map} holds even for $L \in D^+_{\textit{Coh}}(\mathcal{O}_X)$.


duality-lemma-extension-by-zero

Let $j : U \to X$ be an open immersion of Noetherian schemes. \begin{enumerate} \item Let $(K_n)$ and $(L_n)$ be Deligne systems. Let $K$ and $L$ be the values of the constant systems $(K_n|_U)$ and $(L_n|_U)$. Given a morphism $\alpha : K \to L$ of $D(\mathcal{O}_U)$ there is a unique morphism of pro-systems $(K_n) \to (L_n)$ of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ whose restriction to $U$ is $\alpha$. \item Given $K \in D^b_{\textit{Coh}}(\mathcal{O}_U)$ there exists a Deligne system $(K_n)$ such that $(K_n|_U)$ is constant with value $K$. \item The pro-object $(K_n)$ of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ of (2) is unique up to unique isomorphism (as a pro-object). \end{enumerate}


duality-lemma-extension-by-zero-triangle

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $$ K \to L \to M \to K[1] $$ be a distinguished triangle of $D^b_{\textit{Coh}}(\mathcal{O}_U)$. Then there exists an inverse system of distinguished triangles $$ K_n \to L_n \to M_n \to K_n[1] $$ in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ such that $(K_n)$, $(L_n)$, $(M_n)$ are Deligne systems and such that the restriction of these distinguished triangles to $U$ is isomorphic to the distinguished triangle we started out with.


duality-lemma-deligne-system-2-out-of-3

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $$ K_n \to L_n \to M_n \to K_n[1] $$ be an inverse system of distinguished triangles in $D^b_{\textit{Coh}}(\mathcal{O}_X)$. If $(K_n)$ and $(M_n)$ are pro-isomorphic to Deligne systems, then so is $(L_n)$.


duality-lemma-consequence-Artin-Rees-bis

Let $X$ be a Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}^\bullet$ be a complex of coherent $\mathcal{O}_X$-modules. Let $p \in \mathbf{Z}$. Set $\mathcal{H} = H^p(\mathcal{F}^\bullet)$ and $\mathcal{H}_n = H^p(\mathcal{I}^n\mathcal{F}^\bullet)$. Then there are canonical $\mathcal{O}_X$-module maps $$ \ldots \to \mathcal{H}_3 \to \mathcal{H}_2 \to \mathcal{H}_1 \to \mathcal{H} $$ There exists a $c > 0$ such that for $n \geq c$ the image of $\mathcal{H}_n \to \mathcal{H}$ is contained in $\mathcal{I}^{n - c}\mathcal{H}$ and there is a canonical $\mathcal{O}_X$-module map $\mathcal{I}^n\mathcal{H} \to \mathcal{H}_{n - c}$ such that the compositions $$ \mathcal{I}^n \mathcal{H} \to \mathcal{H}_{n - c} \to \mathcal{I}^{n - 2c}\mathcal{H} \quad\text{and}\quad \mathcal{H}_n \to \mathcal{I}^{n - c}\mathcal{H} \to \mathcal{H}_{n - 2c} $$ are the canonical ones. In particular, the inverse systems $(\mathcal{H}_n)$ and $(\mathcal{I}^n\mathcal{H})$ are isomorphic as pro-objects of $\textit{Mod}(\mathcal{O}_X)$.


duality-lemma-characterize-extension-by-zero-algebra

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $a \leq b$ be integers. Let $(K_n)$ be an inverse system of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ such that $H^i(K_n) = 0$ for $i \not \in [a, b]$. The following are equivalent \begin{enumerate} \item $(K_n)$ is pro-isomorphic to a Deligne system, \item for every $p \in \mathbf{Z}$ there exists a coherent $\mathcal{O}_X$-module $\mathcal{F}$ such that the pro-systems $(H^p(K_n))$ and $(\mathcal{I}^n\mathcal{F})$ are pro-isomorphic. \end{enumerate}


duality-lemma-characterize-extension-by-zero

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $(K_n)$ be an inverse system in $D^b_{\textit{Coh}}(\mathcal{O}_X)$. Let $X = W_1 \cup \ldots \cup W_r$ be an open covering. The following are equivalent \begin{enumerate} \item $(K_n)$ is pro-isomorphic to a Deligne system, \item for each $i$ the restriction $(K_n|_{W_i})$ is pro-isomorphic to a Deligne system with respect to the open immersion $U \cap W_i \to W_i$. \end{enumerate}


duality-lemma-tensoring-Deligne-system

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals with $V(\mathcal{I}) = X \setminus U$. Let $K$ be in $D^b_{\textit{Coh}}(\mathcal{O}_X)$. Then $$ K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{I}^n $$ is pro-isomorphic to a Deligne system with constant value $K|_U$ over $U$.


duality-lemma-well-defined-pre

Let $f : X \to Y$ be a proper morphism of Noetherian schemes. Let $V \subset Y$ be an open subscheme and set $U = f^{-1}(V)$. Picture $$ \xymatrix{ U \ar[r]_j \ar[d]_g & X \ar[d]^f \\ V \ar[r]^{j'} & Y } $$ Then we have a canonical isomorphism $Rj'_! \circ Rg_* \to Rf_* \circ Rj_!$ of functors $D^b_{\textit{Coh}}(\mathcal{O}_U) \to \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)$ where $Rj_!$ and $Rj'_!$ are as in Remark \ref{remark-extension-by-zero}.


duality-lemma-well-defined

Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $j' : U \to X'$ be a compactification of $U$ over $X$ (see proof) and denote $f : X' \to X$ the structure morphism. Then we have a canonical isomorphism $Rj_! \to Rf_* \circ R(j')_!$ of functors $D^b_{\textit{Coh}}(\mathcal{O}_U) \to \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)$ where $Rj_!$ and $Rj'_!$ are as in Remark \ref{remark-extension-by-zero}.


duality-lemma-composition-lower-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ and $g : Y \to Z$ be composable morphisms of $\textit{FTS}_S$. With notation as in Remark \ref{remark-composition-lower-shriek} we have $Rg_! \circ Rf_! = R(g \circ f)_!$.


duality-lemma-duality-compact-support

Let $p : U \to \Spec(k)$ be separated of finite type where $k$ is a field. Let $\omega_{U/k}^\bullet = p^!\mathcal{O}_{\Spec(k)}$. There are canonical isomorphisms $$ \Hom_k(H^i(U, K), k) = H^{-i}_c(U, R\SheafHom_{\mathcal{O}_U}(K, \omega_{U/k}^\bullet)) $$ of topological $k$-vector spaces functorial for $K$ in $D^b_{\textit{Coh}}(\mathcal{O}_U)$.


duality-lemma-h0-compactly-supported

Let $X$ be a proper scheme over a field $k$. Let $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ with $H^i(K) = 0$ for $i < 0$. Set $\mathcal{F} = H^0(K)$. Let $Z \subset X$ be closed with complement $U = X \setminus U$. Then $$ H^0_c(U, K|_U) \subset H^0(X, \mathcal{F}) $$ is given by those global sections of $\mathcal{F}$ which vanish in an open neighbourhood of $Z$.


duality-lemma-lichtenbaum

Let $U$ be a variety. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. If $H^d(U, \mathcal{F})$ is nonzero, then $\dim(U) \geq d$ and if equality holds, then $U$ is proper.


duality-proposition-duality-compactly-supported

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Then the functors $Rf_!$ and $f^!$ are adjoint in the following sense: for all $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ and $L \in D^+_{\textit{Coh}}(\mathcal{O}_Y)$ we have $$ \Hom_X(K, f^!L) = \Hom_{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_!K, L) $$ bifunctorially in $K$ and $L$.


more-morphisms-theorem-openness-flatness

\begin{reference} \cite[IV Theorem 11.3.1]{EGA} \end{reference} Let $S$ be a scheme. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module which is locally of finite presentation. Then $$ U = \{x \in X \mid \mathcal{F}\text{ is flat over }S\text{ at }x\} $$ is open in $X$.


more-morphisms-theorem-criterion-flatness-fibre-Noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in X$. Set $y = f(x)$ and $s \in S$ the image of $x$ in $S$. Assume $S$, $X$, $Y$ locally Noetherian, $\mathcal{F}$ coherent, and $\mathcal{F}_x \not = 0$. Then the following are equivalent: \begin{enumerate} \item $\mathcal{F}$ is flat over $S$ at $x$, and $\mathcal{F}_s$ is flat over $Y_s$ at $x$, and \item $Y$ is flat over $S$ at $y$ and $\mathcal{F}$ is flat over $Y$ at $x$. \end{enumerate}


more-morphisms-theorem-criterion-flatness-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $S$, \item $\mathcal{F}$ an $\mathcal{O}_X$-module of finite presentation, and \item $Y$ is locally of finite type over $S$. \end{enumerate} Let $x \in X$. Set $y = f(x)$ and let $s \in S$ be the image of $x$ in $S$. If $\mathcal{F}_x \not = 0$, then the following are equivalent: \begin{enumerate} \item $\mathcal{F}$ is flat over $S$ at $x$, and $\mathcal{F}_s$ is flat over $Y_s$ at $x$, and \item $Y$ is flat over $S$ at $y$ and $\mathcal{F}$ is flat over $Y$ at $x$. \end{enumerate} Moreover, the set of points $x$ where (1) and (2) hold is open in $\text{Supp}(\mathcal{F})$.


more-morphisms-theorem-of-the-cube

Let $S$ be a scheme. Let $X$, $Y$, and $Z$ be schemes over $S$. Let $x : S \to X$ and $y : S \to Y$ be sections of the structure morphisms. Let $\mathcal{L}$ be an invertible module on $X \times_S Y \times_S Z$. If \begin{enumerate} \item $X \to S$ and $Y \to S$ are flat, proper morphisms of finite presentation with geometrically integral fibres, \item the pullbacks of $\mathcal{L}$ by $x \times \text{id}_Y \times \text{id}_Z$ and $\text{id}_X \times y \times \text{id}_Z$ are trivial over $Y \times_S Z$ and $X \times_S Z$, \item there is a point $z \in Z$ such that $\mathcal{L}$ restricted to $X \times_S Y \times_S z$ is trivial, and \item $Z$ is connected, \end{enumerate} then $\mathcal{L}$ is trivial.


more-morphisms-theorem-stein-factorization-Noetherian

Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a proper morphism. There exists a factorization $$ \xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & S' \ar[dl]^\pi \\ & S & } $$ with the following properties: \begin{enumerate} \item the morphism $f'$ is proper with geometrically connected fibres, \item the morphism $\pi : S' \to S$ is finite, \item we have $f'_*\mathcal{O}_X = \mathcal{O}_{S'}$, \item we have $S' = \underline{\Spec}_S(f_*\mathcal{O}_X)$, and \item $S'$ is the normalization of $S$ in $X$, see Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}. \end{enumerate}


more-morphisms-theorem-stein-factorization-general

Let $S$ be a scheme. Let $f : X \to S$ be a proper morphism. There exists a factorization $$ \xymatrix{ X \ar[rr]_{f'} \ar[rd]_f & & S' \ar[dl]^\pi \\ & S & } $$ with the following properties: \begin{enumerate} \item the morphism $f'$ is proper with geometrically connected fibres, \item the morphism $\pi : S' \to S$ is integral, \item we have $f'_*\mathcal{O}_X = \mathcal{O}_{S'}$, \item we have $S' = \underline{\Spec}_S(f_*\mathcal{O}_X)$, and \item $S'$ is the normalization of $S$ in $X$, see Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}. \end{enumerate}


more-morphisms-theorem-normalized-base-change-with-reduced-fibre

Let $A$ be a Dedekind ring with fraction field $K$. Let $X$ be a scheme flat and of finite type over $A$. Assume $A$ is a Nagata ring. There exists a finite extension $K \subset L$ such that the normalized base change $Y$ is smooth over $\Spec(B)$ at all generic points of all fibres.


more-morphisms-lemma-first-order-thickening

Let $X$ be a scheme over a base $S$. Consider a short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{A} \to \mathcal{O}_X \to 0 $$ of sheaves on $X$ where $\mathcal{A}$ is a sheaf of $f^{-1}\mathcal{O}_S$-algebras, $\mathcal{A} \to \mathcal{O}_X$ is a surjection of sheaves of $f^{-1}\mathcal{O}_S$-algebras, and $\mathcal{I}$ is its kernel. If \begin{enumerate} \item $\mathcal{I}$ is an ideal of square zero in $\mathcal{A}$, and \item $\mathcal{I}$ is quasi-coherent as an $\mathcal{O}_X$-module \end{enumerate} then $X' = (X, \mathcal{A})$ is a scheme and $X \to X'$ is a first order thickening over $S$. Moreover, any first order thickening over $S$ is of this form.


more-morphisms-lemma-thickening-affine-scheme

\begin{slogan} Affineness is insensitive to thickenings \end{slogan} Any thickening of an affine scheme is affine.


more-morphisms-lemma-base-change-thickening

Let $S \subset S'$ be a thickening of schemes. Let $X' \to S'$ be a morphism and set $X = S \times_{S'} X'$. Then $(X \subset X') \to (S \subset S')$ is a morphism of thickenings. If $S \subset S'$ is a first (resp.\ finite order) thickening, then $X \subset X'$ is a first (resp.\ finite order) thickening.


more-morphisms-lemma-composition-thickening

\begin{slogan} Compositions of thickenings are thickenings \end{slogan} If $S \subset S'$ and $S' \subset S''$ are thickenings, then so is $S \subset S''$.


more-morphisms-lemma-thicken-property-morphisms

Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism of thickenings. Then \begin{enumerate} \item $f$ is an affine morphism if and only if $f'$ is an affine morphism, \item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism, \item $f$ is quasi-compact if and only if $f'$ quasi-compact, \item $f$ is universally closed if and only if $f'$ is universally closed, \item $f$ is integral if and only if $f'$ is integral, \item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \item $f$ is universally injective if and only if $f'$ is universally injective, \item $f$ is universally open if and only if $f'$ is universally open, \item $f$ is quasi-affine if and only if $f'$ is quasi-affine, and \item add more here. \end{enumerate}


more-morphisms-lemma-thicken-property-relatively-ample

Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism of thickenings. Let $\mathcal{L}'$ be an invertible sheaf on $X'$ and denote $\mathcal{L}$ the restriction to $X$. Then $\mathcal{L}'$ is $f'$-ample if and only if $\mathcal{L}$ is $f$-ample.


more-morphisms-lemma-thicken-property-morphisms-cartesian

Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism of thickenings such that $X = S \times_{S'} X'$. If $S \subset S'$ is a finite order thickening, then \begin{enumerate} \item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \item $f$ is locally of finite type if and only if $f'$ is locally of finite type, \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \item $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \item $\Omega_{X/S} = 0$ if and only if $\Omega_{X'/S'} = 0$, \item $f$ is unramified if and only if $f'$ is unramified, \item $f$ is proper if and only if $f'$ is proper, \item $f$ is finite if and only if $f'$ is finite, \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \item $f$ is an immersion if and only if $f'$ is an immersion, and \item add more here. \end{enumerate}


more-morphisms-lemma-properties-that-extend-over-thickenings

Let $(f, f') : (X \subset X') \to (Y \to Y')$ be a morphism of thickenings. Assume $f$ and $f'$ are locally of finite type and $X = Y \times_{Y'} X'$. Then \begin{enumerate} \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \item $f$ is finite if and only if $f'$ is finite, \item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \item $\Omega_{X/Y} = 0$ if and only if $\Omega_{X'/Y'} = 0$, \item $f$ is unramified if and only if $f'$ is unramified, \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \item $f$ is an immersion if and only if $f'$ is an immersion, \item $f$ is proper if and only if $f'$ is proper, and \item add more here. \end{enumerate}


more-morphisms-lemma-picard-group-first-order-thickening

Let $X \subset X'$ be a first order thickening with ideal sheaf $\mathcal{I}$. Then there is a canonical exact sequence $$ \xymatrix{ 0 \ar[r] & H^0(X, \mathcal{I}) \ar[r] & H^0(X', \mathcal{O}_{X'}^*) \ar[r] & H^0(X, \mathcal{O}^*_X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^1(X, \mathcal{I}) \ar[r] & \Pic(X') \ar[r] & \Pic(X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^2(X, \mathcal{I}) \ar[r] & \ldots \ar[r] & \ldots } $$ of abelian groups.


more-morphisms-lemma-torsion-pic-thickening

Let $X \subset X'$ be a thickening. Let $n$ be an integer invertible in $\mathcal{O}_X$. Then the map $\Pic(X')[n] \to \Pic(X)[n]$ is bijective.


more-morphisms-lemma-first-order-infinitesimal-neighbourhood

Let $i : Z \to X$ be an immersion of schemes. The first order infinitesimal neighbourhood $Z'$ of $Z$ in $X$ has the following universal property: Given any commutative diagram $$ \xymatrix{ Z \ar[d]_i & T \ar[l]^a \ar[d] \\ X & T' \ar[l]_b } $$ where $T \subset T'$ is a first order thickening over $X$, there exists a unique morphism $(a', a) : (T \subset T') \to (Z \subset Z')$ of thickenings over $X$.


more-morphisms-lemma-infinitesimal-neighbourhood-conormal

Let $i : Z \to X$ be an immersion of schemes. Let $Z \subset Z'$ be the first order infinitesimal neighbourhood of $Z$ in $X$. Then the diagram $$ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Z \ar[r] & X } $$ induces a map of conormal sheaves $\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Z'}$ by Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial}. This map is an isomorphism.


more-morphisms-lemma-formally-unramified-not-affine

If $f : X \to S$ is a formally unramified morphism, then given any solid commutative diagram $$ \xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] } $$ where $T \subset T'$ is a first order thickening of schemes over $S$ there exists at most one dotted arrow making the diagram commute. In other words, in Definition \ref{definition-formally-unramified} the condition that $T$ be affine may be dropped.


more-morphisms-lemma-composition-formally-unramified

A composition of formally unramified morphisms is formally unramified.


more-morphisms-lemma-base-change-formally-unramified

A base change of a formally unramified morphism is formally unramified.


more-morphisms-lemma-formally-unramified-on-opens

Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$ and $V \subset S$ be open such that $f(U) \subset V$. If $f$ is formally unramified, so is $f|_U : U \to V$.


more-morphisms-lemma-formally-unramified-differentials

Let $f : X \to S$ be a morphism of schemes. Then $f$ is formally unramified if and only if $\Omega_{X/S} = 0$.


more-morphisms-lemma-unramified-formally-unramified

\begin{slogan} Unramified morphisms are the same as formally unramified morphism that are locally of finite type. \end{slogan} Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is unramified (resp.\ G-unramified), and \item the morphism $f$ is locally of finite type (resp.\ locally of finite presentation) and formally unramified. \end{enumerate}


more-morphisms-lemma-universal-thickening

Let $h : Z \to X$ be a formally unramified morphism of schemes. There exists a universal first order thickening $Z \subset Z'$ of $Z$ over $X$.


more-morphisms-lemma-universal-thickening-unramified

Let $Z \to X$ be a formally unramified morphism of schemes. Then the universal first order thickening $Z'$ is formally unramified over $X$.


more-morphisms-lemma-universal-thickening-functorial

Consider a commutative diagram of schemes $$ \xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y } $$ with $h$ and $h'$ formally unramified. Let $Z \subset Z'$ be the universal first order thickening of $Z$ over $X$. Let $W \subset W'$ be the universal first order thickening of $W$ over $Y$. There exists a canonical morphism $(f, f') : (Z, Z') \to (W, W')$ of thickenings over $Y$ which fits into the following commutative diagram $$ \xymatrix{ & & & Z' \ar[ld] \ar[d]^{f'} \\ Z \ar[rr] \ar[d]_f \ar[rrru] & & X \ar[d] & W' \ar[ld] \\ W \ar[rrru]|!{[rr];[rruu]}\hole \ar[rr] & & Y } $$ In particular the morphism $(f, f')$ of thickenings induces a morphism of conormal sheaves $f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}$.


more-morphisms-lemma-universal-thickening-fibre-product

Let $$ \xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y } $$ be a fibre product diagram in the category of schemes with $h'$ formally unramified. Then $h$ is formally unramified and if $W \subset W'$ is the universal first order thickening of $W$ over $Y$, then $Z = X \times_Y W \subset X \times_Y W'$ is the universal first order thickening of $Z$ over $X$. In particular the canonical map $f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}$ of Lemma \ref{lemma-universal-thickening-functorial} is surjective.


more-morphisms-lemma-universal-thickening-fibre-product-flat

Let $$ \xymatrix{ Z \ar[r]_h \ar[d]_f & X \ar[d]^g \\ W \ar[r]^{h'} & Y } $$ be a fibre product diagram in the category of schemes with $h'$ formally unramified and $g$ flat. In this case the corresponding map $Z' \to W'$ of universal first order thickenings is flat, and $f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}$ is an isomorphism.


more-morphisms-lemma-universal-thickening-localize

Taking the universal first order thickenings commutes with taking opens. More precisely, let $h : Z \to X$ be a formally unramified morphism of schemes. Let $V \subset Z$, $U \subset X$ be opens such that $h(V) \subset U$. Let $Z'$ be the universal first order thickening of $Z$ over $X$. Then $h|_V : V \to U$ is formally unramified and the universal first order thickening of $V$ over $U$ is the open subscheme $V' \subset Z'$ such that $V = Z \cap V'$. In particular, $\mathcal{C}_{Z/X}|_V = \mathcal{C}_{V/U}$.


more-morphisms-lemma-differentials-universally-unramified

Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$. Let $Z \subset Z'$ be the universal first order thickening of $Z$ over $X$ with structure morphism $h' : Z' \to X$. The canonical map $$ c_{h'} : (h')^*\Omega_{X/S} \longrightarrow \Omega_{Z'/S} $$ induces an isomorphism $h^*\Omega_{X/S} \to \Omega_{Z'/S} \otimes \mathcal{O}_Z$.


more-morphisms-lemma-universally-unramified-differentials-sequence

Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$. There is a canonical exact sequence $$ \mathcal{C}_{Z/X} \to h^*\Omega_{X/S} \to \Omega_{Z/S} \to 0. $$ The first arrow is induced by $\text{d}_{Z'/S}$ where $Z'$ is the universal first order neighbourhood of $Z$ over $X$.


more-morphisms-lemma-two-unramified-morphisms

Let $$ \xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d] \\ & Y } $$ be a commutative diagram of schemes where $i$ and $j$ are formally unramified. Then there is a canonical exact sequence $$ \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0 $$ where the first arrow comes from Lemma \ref{lemma-universal-thickening-functorial} and the second from Lemma \ref{lemma-universally-unramified-differentials-sequence}.


more-morphisms-lemma-transitivity-conormal

Let $Z \to Y \to X$ be formally unramified morphisms of schemes. \begin{enumerate} \item If $Z \subset Z'$ is the universal first order thickening of $Z$ over $X$ and $Y \subset Y'$ is the universal first order thickening of $Y$ over $X$, then there is a morphism $Z' \to Y'$ and $Y \times_{Y'} Z'$ is the universal first order thickening of $Z$ over $Y$. \item There is a canonical exact sequence $$ i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ where the maps come from Lemma \ref{lemma-universal-thickening-functorial} and $i : Z \to Y$ is the first morphism. \end{enumerate}


more-morphisms-lemma-composition-formally-etale

A composition of formally \'etale morphisms is formally \'etale.


more-morphisms-lemma-base-change-formally-etale

A base change of a formally \'etale morphism is formally \'etale.


more-morphisms-lemma-formally-etale-on-opens

Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$ and $V \subset S$ be open subschemes such that $f(U) \subset V$. If $f$ is formally \'etale, so is $f|_U : U \to V$.


more-morphisms-lemma-characterize-formally-etale

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is formally \'etale, \item $f$ is formally unramified and the universal first order thickening of $X$ over $S$ is equal to $X$, \item $f$ is formally unramified and $\mathcal{C}_{X/S} = 0$, and \item $\Omega_{X/S} = 0$ and $\mathcal{C}_{X/S} = 0$. \end{enumerate}


more-morphisms-lemma-unramified-flat-formally-etale

An unramified flat morphism is formally \'etale.


more-morphisms-lemma-affine-formally-etale

Let $f : X \to S$ be a morphism of schemes. Assume $X$ and $S$ are affine. Then $f$ is formally \'etale if and only if $\mathcal{O}_S(S) \to \mathcal{O}_X(X)$ is a formally \'etale ring map.


more-morphisms-lemma-etale-formally-etale

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is \'etale, and \item the morphism $f$ is locally of finite presentation and formally \'etale. \end{enumerate}


more-morphisms-lemma-difference-derivation

Let $S$ be a scheme. Let $X \subset X'$ and $Y \subset Y'$ be two first order thickenings over $S$. Let $(a, a'), (b, b') : (X \subset X') \to (Y \subset Y')$ be two morphisms of thickenings over $S$. Assume that \begin{enumerate} \item $a = b$, and \item the two maps $a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ (Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial}) are equal. \end{enumerate} Then the map $(a')^\sharp - (b')^\sharp$ factors as $$ \mathcal{O}_{Y'} \to \mathcal{O}_Y \xrightarrow{D} a_*\mathcal{C}_{X/X'} \to a_*\mathcal{O}_{X'} $$ where $D$ is an $\mathcal{O}_S$-derivation.


more-morphisms-lemma-action-by-derivations

Let $S$ be a scheme. Let $(a, a') : (X \subset X') \to (Y \subset Y')$ be a morphism of first order thickenings over $S$. Let $$ \theta : a^*\Omega_{Y/S} \to \mathcal{C}_{X/X'} $$ be an $\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs $(b, b') : (X \subset X') \to (Y \subset Y')$ such that (1) and (2) of Lemma \ref{lemma-difference-derivation} hold and the derivation $D$ and $\theta$ are related by Equation (\ref{equation-D}).


more-morphisms-lemma-sheaf

Let $S$ be a scheme. Let $X \subset X'$ and $Y \subset Y'$ be first order thickenings over $S$. Assume given a morphism $a : X \to Y$ and a map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ of $\mathcal{O}_X$-modules. For an open subscheme $U' \subset X'$ consider morphisms $a' : U' \to Y'$ such that \begin{enumerate} \item $a'$ is a morphism over $S$, \item $a'|_U = a|_U$, and \item the induced map $a^*\mathcal{C}_{Y/Y'}|_U \to \mathcal{C}_{X/X'}|_U$ is the restriction of $A$ to $U$. \end{enumerate} Here $U = X \cap U'$. Then the rule \begin{equation} \label{equation-sheaf} U' \mapsto \{a' : U' \to Y'\text{ such that (1), (2), (3) hold.}\} \end{equation} defines a sheaf of sets on $X'$.


more-morphisms-lemma-action-sheaf

Same notation and assumptions as in Lemma \ref{lemma-sheaf}. There is an action of the sheaf $$ \SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/S}, \mathcal{C}_{X/X'}) $$ on the sheaf (\ref{equation-sheaf}). Moreover, the action is simply transitive for any open $U' \subset X'$ over which the sheaf (\ref{equation-sheaf}) has a section.


more-morphisms-lemma-omega-deformation

Let $S$ be a scheme. Let $X \subset X'$ be a first order thickening over $S$. Let $Y$ be a scheme over $S$. Let $a', b' : X' \to Y$ be two morphisms over $S$ with $a = a'|_X = b'|_X$. This gives rise to a commutative diagram $$ \xymatrix{ X \ar[r] \ar[d]_a & X' \ar[d]^{(b', a')} \\ Y \ar[r]^-{\Delta_{Y/S}} & Y \times_S Y } $$ Since the horizontal arrows are immersions with conormal sheaves $\mathcal{C}_{X/X'}$ and $\Omega_{Y/S}$, by Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial}, we obtain a map $\theta : a^*\Omega_{Y/S} \to \mathcal{C}_{X/X'}$. Then this $\theta$ and the derivation $D$ of Lemma \ref{lemma-difference-derivation} are related by Equation (\ref{equation-D}).


more-morphisms-lemma-sheaf-differentials-etale-localization

Let $$ \xymatrix{ X_1 \ar[d] & X_2 \ar[l]^f \ar[d] \\ S_1 & S_2 \ar[l] } $$ be a commutative diagram of schemes with $X_2 \to X_1$ and $S_2 \to S_1$ \'etale. Then the map $c_f : f^*\Omega_{X_1/S_1} \to \Omega_{X_2/S_2}$ of Morphisms, Lemma \ref{morphisms-lemma-functoriality-differentials} is an isomorphism.


more-morphisms-lemma-action-by-derivations-etale-localization

Consider a commutative diagram of first order thickenings $$ \vcenter{ \xymatrix{ (T_2 \subset T_2') \ar[d]_{(h, h')} \ar[rr]_{(a_2, a_2')} & & (X_2 \subset X_2') \ar[d]^{(f, f')} \\ (T_1 \subset T_1') \ar[rr]^{(a_1, a_1')} & & (X_1 \subset X_1') } } \quad \begin{matrix} \text{and a commutative} \\ \text{diagram of schemes} \end{matrix} \quad \vcenter{ \xymatrix{ X_2' \ar[r] \ar[d] & S_2 \ar[d] \\ X_1' \ar[r] & S_1 } } $$ with $X_2 \to X_1$ and $S_2 \to S_1$ \'etale. For any $\mathcal{O}_{T_1}$-linear map $\theta_1 : a_1^*\Omega_{X_1/S_1} \to \mathcal{C}_{T_1/T'_1}$ let $\theta_2$ be the composition $$ \xymatrix{ a_2^*\Omega_{X_2/S_2} \ar@{=}[r] & h^*a_1^*\Omega_{X_1/S_1} \ar[r]^-{h^*\theta_1} & h^*\mathcal{C}_{T_1/T'_1} \ar[r] & \mathcal{C}_{T_2/T'_2} } $$ (equality sign is explained in the proof). Then the diagram $$ \xymatrix{ T_2' \ar[rr]_{\theta_2 \cdot a_2'} \ar[d] & & X'_2 \ar[d] \\ T_1' \ar[rr]^{\theta_1 \cdot a_1'} & & X'_1 } $$ commutes where the actions $\theta_2 \cdot a_2'$ and $\theta_1 \cdot a_1'$ are as in Remark \ref{remark-action-by-derivations}.


more-morphisms-lemma-deform

Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism of first order thickenings. Assume that $f$ is flat. Then the following are equivalent \begin{enumerate} \item $f'$ is flat and $X = S \times_{S'} X'$, and \item the canonical map $f^*\mathcal{C}_{S/S'} \to \mathcal{C}_{X/X'}$ is an isomorphism. \end{enumerate}


more-morphisms-lemma-flatness-morphism-thickenings

Consider a commutative diagram $$ \xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') } $$ of thickenings. Assume \begin{enumerate} \item $X'$ is flat over $S'$, \item $f$ is flat, \item $S \subset S'$ is a finite order thickening, and \item $X = S \times_{S'} X'$ and $Y = S \times_{S'} Y'$. \end{enumerate} Then $f'$ is flat and $Y'$ is flat over $S'$ at all points in the image of $f'$.


more-morphisms-lemma-deform-property

Consider a commutative diagram $$ \xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') } $$ of thickenings. Assume $S \subset S'$ is a finite order thickening, $X'$ flat over $S'$, $X = S \times_{S'} X'$, and $Y = S \times_{S'} Y'$. Then \begin{enumerate} \item $f$ is flat if and only if $f'$ is flat, \label{item-flat} \item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \label{item-isomorphism} \item $f$ is an open immersion if and only if $f'$ is an open immersion, \label{item-open-immersion} \item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \label{item-quasi-compact} \item $f$ is universally closed if and only if $f'$ is universally closed, \label{item-universally-closed} \item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \label{item-separated} \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \label{item-monomorphism} \item $f$ is surjective if and only if $f'$ is surjective, \label{item-surjective} \item $f$ is universally injective if and only if $f'$ is universally injective, \label{item-universally-injective} \item $f$ is affine if and only if $f'$ is affine, \label{item-affine} \item \label{item-finite-type} $f$ is locally of finite type if and only if $f'$ is locally of finite type, \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \label{item-quasi-finite} \item \label{item-finite-presentation} $f$ is locally of finite presentation if and only if $f'$ is locally of finite presentation, \item \label{item-relative-dimension-d} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \item $f$ is universally open if and only if $f'$ is universally open, \label{item-universally-open} \item $f$ is syntomic if and only if $f'$ is syntomic, \label{item-syntomic} \item $f$ is smooth if and only if $f'$ is smooth, \label{item-smooth} \item $f$ is unramified if and only if $f'$ is unramified, \label{item-unramified} \item $f$ is \'etale if and only if $f'$ is \'etale, \label{item-etale} \item $f$ is proper if and only if $f'$ is proper, \label{item-proper} \item $f$ is integral if and only if $f'$ is integral, \label{item-integral} \item $f$ is finite if and only if $f'$ is finite, \label{item-finite} \item \label{item-finite-locally-free} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \item add more here. \end{enumerate}


more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft

Consider a commutative diagram $$ \xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') } $$ of thickenings. Assume \begin{enumerate} \item $Y' \to S'$ is locally of finite type, \item $X' \to S'$ is flat and locally of finite presentation, \item $f$ is flat, and \item $X = S \times_{S'} X'$ and $Y = S \times_{S'} Y'$. \end{enumerate} Then $f'$ is flat and for all $y' \in Y'$ in the image of $f'$ the local ring $\mathcal{O}_{Y', y'}$ is flat and essentially of finite presentation over $\mathcal{O}_{S', s'}$.


more-morphisms-lemma-deform-property-fp-over-ft

Consider a commutative diagram $$ \xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') } $$ of thickenings. Assume $Y' \to S'$ locally of finite type, $X' \to S'$ flat and locally of finite presentation, $X = S \times_{S'} X'$, and $Y = S \times_{S'} Y'$. Then \begin{enumerate} \item $f$ is flat if and only if $f'$ is flat, \label{item-flat-fp-over-ft} \item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \label{item-isomorphism-fp-over-ft} \item $f$ is an open immersion if and only if $f'$ is an open immersion, \label{item-open-immersion-fp-over-ft} \item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \label{item-quasi-compact-fp-over-ft} \item $f$ is universally closed if and only if $f'$ is universally closed, \label{item-universally-closed-fp-over-ft} \item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \label{item-separated-fp-over-ft} \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \label{item-monomorphism-fp-over-ft} \item $f$ is surjective if and only if $f'$ is surjective, \label{item-surjective-fp-over-ft} \item $f$ is universally injective if and only if $f'$ is universally injective, \label{item-universally-injective-fp-over-ft} \item $f$ is affine if and only if $f'$ is affine, \label{item-affine-fp-over-ft} \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \label{item-quasi-finite-fp-over-ft} \item \label{item-relative-dimension-d-fp-over-ft} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \item $f$ is universally open if and only if $f'$ is universally open, \label{item-universally-open-fp-over-ft} \item $f$ is syntomic if and only if $f'$ is syntomic, \label{item-syntomic-fp-over-ft} \item $f$ is smooth if and only if $f'$ is smooth, \label{item-smooth-fp-over-ft} \item $f$ is unramified if and only if $f'$ is unramified, \label{item-unramified-fp-over-ft} \item $f$ is \'etale if and only if $f'$ is \'etale, \label{item-etale-fp-over-ft} \item $f$ is proper if and only if $f'$ is proper, \label{item-proper-fp-over-ft} \item $f$ is finite if and only if $f'$ is finite, \label{item-finite-fp-over-ft} \item \label{item-finite-locally-free-fp-over-ft} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \item add more here. \end{enumerate}


more-morphisms-lemma-deform-projective

Let $f : X \to S$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $\mathcal{L}$ be $f$-ample. Assume $S$ is quasi-compact. There exists a $d_0 \geq 0$ such that for every cartesian diagram $$ \vcenter{ \xymatrix{ X \ar[r]_{i'} \ar[d]_f & X' \ar[d]^{f'} \\ S \ar[r]^i & S' } } \quad\text{and}\quad \begin{matrix} \text{invertible }\mathcal{O}_{X'}\text{-module}\\ \mathcal{L}'\text{ with }\mathcal{L} \cong (i')^*\mathcal{L}' \end{matrix} $$ where $S \subset S'$ is a thickening and $f'$ is proper, flat, of finite presentation we have \begin{enumerate} \item $R^p(f')_*(\mathcal{L}')^{\otimes d} = 0$ for all $p > 0$ and $d \geq d_0$, \item $\mathcal{A}'_d = (f')_*(\mathcal{L}')^{\otimes d}$ is finite locally free for $d \geq d_0$, \item $\mathcal{A}' = \mathcal{O}_{S'} \oplus \bigoplus_{d \geq d_0} \mathcal{A}'_d$ is a quasi-coherent $\mathcal{O}_{S'}$-algebra of finite presentation, \item there is a canonical isomorphism $r' : X' \to \underline{\text{Proj}}_{S'}(\mathcal{A}')$, and \item there is a canonical isomorphism $\theta' : (r')^*\mathcal{O}_{\underline{\text{Proj}}_{S'}(\mathcal{A}')}(1) \to \mathcal{L}'$. \end{enumerate} The construction of $\mathcal{A}'$, $r'$, $\theta'$ is functorial in the data $(X', S', i, i', f', \mathcal{L}')$.


more-morphisms-lemma-composition-formally-smooth

A composition of formally smooth morphisms is formally smooth.


more-morphisms-lemma-formally-etale-unramified-smooth

Let $f : X \to S$ be a morphism of schemes. Then $f$ is formally \'etale if and only if $f$ is formally smooth and formally unramified.


more-morphisms-lemma-formally-smooth-on-opens

Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$ and $V \subset S$ be open subschemes such that $f(U) \subset V$. If $f$ is formally smooth, so is $f|_U : U \to V$.


more-morphisms-lemma-affine-formally-smooth

Let $f : X \to S$ be a morphism of schemes. Assume $X$ and $S$ are affine. Then $f$ is formally smooth if and only if $\mathcal{O}_S(S) \to \mathcal{O}_X(X)$ is a formally smooth ring map.


more-morphisms-lemma-smooth-formally-smooth

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is smooth, and \item the morphism $f$ is locally of finite presentation and formally smooth. \end{enumerate}


more-morphisms-lemma-formally-smooth-sheaf-differentials

Let $f : X \to Y$ be a formally smooth morphism of schemes. Then $\Omega_{X/Y}$ is locally projective on $X$.


more-morphisms-lemma-h1-is-zero

Let $T$ be an affine scheme. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_T$-modules. Consider $\mathcal{H} = \SheafHom_{\mathcal{O}_T}(\mathcal{F}, \mathcal{G})$. If $\mathcal{F}$ is locally projective, then $H^1(T, \mathcal{H}) = 0$.


more-morphisms-lemma-formally-smooth

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is formally smooth, \item for every $x \in X$ there exist opens $x \in U \subset X$ and $f(x) \in V \subset Y$ with $f(U) \subset V$ such that $f|_U : U \to V$ is formally smooth, \item for every pair of affine opens $U \subset X$ and $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}_Y(V) \to \mathcal{O}_X(U)$ is formally smooth, and \item there exists an affine open covering $Y = \bigcup V_j$ and for each $j$ an affine open covering $f^{-1}(V_j) = \bigcup U_{ji}$ such that $\mathcal{O}_Y(V) \to \mathcal{O}_X(U)$ is a formally smooth ring map for all $j$ and $i$. \end{enumerate}


more-morphisms-lemma-triangle-differentials-formally-smooth

Let $f : X \to Y$, $g : Y \to S$ be morphisms of schemes. Assume $f$ is formally smooth. Then $$ 0 \to f^*\Omega_{Y/S} \to \Omega_{X/S} \to \Omega_{X/Y} \to 0 $$ (see Morphisms, Lemma \ref{morphisms-lemma-triangle-differentials}) is short exact.


more-morphisms-lemma-two-unramified-morphisms-formally-smooth

Let $$ \xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d]^f \\ & Y } $$ be a commutative diagram of schemes where $i$ and $j$ are formally unramified and $f$ is formally smooth. Then the canonical exact sequence $$ 0 \to \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0 $$ of Lemma \ref{lemma-two-unramified-morphisms} is exact and locally split.


more-morphisms-lemma-lifting-along-artinian-at-point

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Assume that $S$ is locally Noetherian and $f$ locally of finite type. The following are equivalent: \begin{enumerate} \item $f$ is smooth at $x$, \item for every solid commutative diagram $$ \xymatrix{ X \ar[d]_f & \Spec(B) \ar[d]^i \ar[l]^-\alpha \\ S & \Spec(B') \ar[l]_-{\beta} \ar@{-->}[lu] } $$ where $B' \to B$ is a surjection of local rings with $\Ker(B' \to B)$ of square zero, and $\alpha$ mapping the closed point of $\Spec(B)$ to $x$ there exists a dotted arrow making the diagram commute, \item same as in (2) but with $B' \to B$ ranging over small extensions (see Algebra, Definition \ref{algebra-definition-small-extension}), and \item same as in (2) but with $B' \to B$ ranging over small extensions such that $\alpha$ induces an isomorphism $\kappa(x) \to \kappa(\mathfrak m)$ where $\mathfrak m \subset B$ is the maximal ideal. \end{enumerate}


more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds

Let $f : X \to S$ be a finite type morphism of locally Noetherian schemes. Let $Z \subset S$ be a closed subscheme with $n$th infinitesimal neighbourhood $Z_n \subset S$. Set $X_n = Z_n \times_S X$. \begin{enumerate} \item If $X_n \to Z_n$ is smooth for all $n$, then $f$ is smooth at every point of $f^{-1}(Z)$. \item If $X_n \to Z_n$ is \'etale for all $n$, then $f$ is \'etale at every point of $f^{-1}(Z)$. \end{enumerate}


more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds

Let $f : X \to S$ be a morphism of locally Noetherian schemes. Let $Z \subset S$ be a closed subscheme with $n$th infinitesimal neighbourhood $Z_n \subset S$. Set $X_n = Z_n \times_S X$. If $X_n \to Z_n$ is flat for all $n$, then $f$ is flat at every point of $f^{-1}(Z)$.


more-morphisms-lemma-NL-affine

Let $f : X \to Y$ be a morphism of schemes. Let $\Spec(A) = U \subset X$ and $\Spec(R) = V \subset S$ be affine opens with $f(U) \subset V$. There is a canonical map $$ \widetilde{\NL_{A/R}} \longrightarrow \NL_{X/Y}|_U $$ of complexes which is an isomorphism in $D(\mathcal{O}_U)$.


more-morphisms-lemma-netherlander-quasi-coherent

Let $f : X \to Y$ be a morphism of schemes. The cohomology sheaves of the complex $\NL_{X/Y}$ are quasi-coherent, zero outside degrees $-1$, $0$ and equal to $\Omega_{X/Y}$ in degree $0$.


more-morphisms-lemma-netherlander-fp

Let $f : X \to Y$ be a morphism of schemes. If $f$ is locally of finite presentation, then $\NL_{X/Y}$ is locally on $X$ quasi-isomorphic to a complex $$ \ldots \to 0 \to \mathcal{F}^{-1} \to \mathcal{F}^0 \to 0 \to \ldots $$ of quasi-coherent $\mathcal{O}_X$-modules with $\mathcal{F}^0$ of finite presentation and $\mathcal{F}^{-1}$ of finite type.


more-morphisms-lemma-NL-formally-smooth

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is formally smooth, \item $H^{-1}(\NL_{X/Y}) = 0$ and $H^0(\NL_{X/Y}) = \Omega_{X/Y}$ is locally projective. \end{enumerate}


more-morphisms-lemma-NL-smooth

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is smooth, and \item $f$ is locally of finite presentation, $H^{-1}(\NL_{X/Y}) = 0$, and $H^0(\NL_{X/Y}) = \Omega_{X/Y}$ is finite locally free. \end{enumerate}


more-morphisms-lemma-NL-etale

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is \'etale, and \item $f$ is locally of finite presentation and $H^{-1}(\NL_{X/Y}) = H^0(\NL_{X/Y}) = 0$. \end{enumerate}


more-morphisms-lemma-NL-immersion

Let $i : Z \to X$ be an immersion of schemes. Then $\NL_{Z/X}$ is isomorphic to $\mathcal{C}_{Z/X}[1]$ in $D(\mathcal{O}_Z)$ where $\mathcal{C}_{Z/X}$ is the conormal sheaf of $Z$ in $X$.


more-morphisms-lemma-exact-sequence-NL

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. There is a canonical six term exact sequence $$ H^{-1}(f^*\NL_{Y/Z}) \to H^{-1}(\NL_{X/Z}) \to H^{-1}(\NL_{X/Y}) \to f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0 $$ of cohomology sheaves.


more-morphisms-lemma-get-triangle-NL

Let $f : X \to Y$ and $Y \to Z$ be morphisms of schemes. Assume $X \to Y$ is a complete intersection morphism. Then there is a canonical distinguished triangle $$ f^*\NL_{Y/Z} \to \NL_{X/Z} \to \NL_{X/Y} \to f^*\NL_{Y/Z}[1] $$ in $D(\mathcal{O}_X)$ which recovers the $6$-term exact sequence of Lemma \ref{lemma-exact-sequence-NL}.


more-morphisms-lemma-get-NL

Let $f : X \to Y$ be a morphism of schemes which factors as $f = g \circ i$ with $i$ an immersion and $g : P \to Y$ formally smooth (for example smooth). Then there is a canonical isomorphism $$ \NL_{X/Y} \cong \left(\mathcal{C}_{X/P} \to i^*\Omega_{P/Y}\right) $$ in $D(\mathcal{O}_X)$ where the conormal sheaf $\mathcal{C}_{X/P}$ is placed in degree $-1$.


more-morphisms-lemma-pushout-fpqc-local

Let $\mathcal{I} \to (\Sch/S)_{fppf}$, $i \mapsto X_i$ be a diagram of schemes. Let $(W, X_i \to W)$ be a cocone for the diagram in the category of schemes (Categories, Remark \ref{categories-remark-cones-and-cocones}). If there exists a fpqc covering $\{W_a \to W\}_{a \in A}$ of schemes such that \begin{enumerate} \item for all $a \in A$ we have $W_a = \colim X_i \times_W W_a$ in the category of schemes, and \item for all $a, b \in A$ we have $W_a \times_W W_b = \colim X_i \times_W W_a \times_W W_b$ in the category of schemes, \end{enumerate} then $W = \colim X_i$ in the category of schemes.


more-morphisms-lemma-pushout-along-thickening

Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an affine morphism of schemes. Then there exists a pushout $$ \xymatrix{ X \ar[r] \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r] & Y' } $$ in the category of schemes. Moreover, $Y \subset Y'$ is a thickening, $X = Y \times_{Y'} X'$, and $$ \mathcal{O}_{Y'} = \mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'} $$ as sheaves on $|Y| = |Y'|$.


more-morphisms-lemma-equivalence-categories-schemes-over-pushout

Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an affine morphism of schemes. Let $Y' = Y \amalg_X X'$ be the pushout (see Lemma \ref{lemma-pushout-along-thickening}). Base change gives a functor $$ F : (\Sch/Y') \longrightarrow (\Sch/Y) \times_{(\Sch/Y')} (\Sch/X') $$ given by $V' \longmapsto (V' \times_{Y'} Y, V' \times_{Y'} X', 1)$ which has a left adjoint $$ G : (\Sch/Y) \times_{(\Sch/Y')} (\Sch/X') \longrightarrow (\Sch/Y') $$ which sends the triple $(V, U', \varphi)$ to the pushout $V \amalg_{(V \times_Y X)} U'$. Finally, $F \circ G$ is isomorphic to the identity functor.


more-morphisms-lemma-scheme-over-pushout-flat-modules

Let $X \to X'$ be a thickening of schemes and let $X \to Y$ be an affine morphism of schemes. Let $Y' = Y \amalg_X X'$ be the pushout (see Lemma \ref{lemma-pushout-along-thickening}). Let $V' \to Y'$ be a morphism of schemes. Set $V = Y \times_{Y'} V'$, $U' = X' \times_{Y'} V'$, and $U = X \times_{Y'} V'$. There is an equivalence of categories between \begin{enumerate} \item quasi-coherent $\mathcal{O}_{V'}$-modules flat over $Y'$, and \item the category of triples $(\mathcal{G}, \mathcal{F}', \varphi)$ where \begin{enumerate} \item $\mathcal{G}$ is a quasi-coherent $\mathcal{O}_V$-module flat over $Y$, \item $\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{U'}$-module flat over $X'$, and \item $\varphi : (U \to V)^*\mathcal{G} \to (U \to U')^*\mathcal{F}'$ is an isomorphism of $\mathcal{O}_U$-modules. \end{enumerate} \end{enumerate} The equivalence maps $\mathcal{G}'$ to $((V \to V')^*\mathcal{G}', (U' \to V')^*\mathcal{G}', can)$. Suppose $\mathcal{G}'$ corresponds to the triple $(\mathcal{G}, \mathcal{F}', \varphi)$. Then \begin{enumerate} \item[(a)] $\mathcal{G}'$ is a finite type $\mathcal{O}_{V'}$-module if and only if $\mathcal{G}$ and $\mathcal{F}'$ are finite type $\mathcal{O}_Y$ and $\mathcal{O}_{U'}$-modules. \item[(b)] if $V' \to Y'$ is locally of finite presentation, then $\mathcal{G}'$ is an $\mathcal{O}_{V'}$-module of finite presentation if and only if $\mathcal{G}$ and $\mathcal{F}'$ are $\mathcal{O}_Y$ and $\mathcal{O}_{U'}$-modules of finite presentation. \end{enumerate}


more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat

In the situation of Lemma \ref{lemma-equivalence-categories-schemes-over-pushout}. If $V' = G(V, U', \varphi)$ for some triple $(V, U', \varphi)$, then \begin{enumerate} \item $V' \to Y'$ is locally of finite type if and only if $V \to Y$ and $U' \to X'$ are locally of finite type, \item $V' \to Y'$ is flat if and only if $V \to Y$ and $U' \to X'$ are flat, \item $V' \to Y'$ is flat and locally of finite presentation if and only if $V \to Y$ and $U' \to X'$ are flat and locally of finite presentation, \item $V' \to Y'$ is smooth if and only if $V \to Y$ and $U' \to X'$ are smooth, \item $V' \to Y'$ is \'etale if and only if $V \to Y$ and $U' \to X'$ are \'etale, and \item add more here as needed. \end{enumerate} If $W'$ is flat over $Y'$, then the adjunction mapping $G(F(W')) \to W'$ is an isomorphism. Hence $F$ and $G$ define mutually quasi-inverse functors between the category of schemes flat over $Y'$ and the category of triples $(V, U', \varphi)$ with $V \to Y$ and $U' \to X'$ flat.


more-morphisms-lemma-flat-locus-base-change

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x' \in X'$ with images $x = g'(x')$ and $s' = g'(x')$. \begin{enumerate} \item If $\mathcal{F}$ is flat over $S$ at $x$, then $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$. \item If $g$ is flat at $s'$ and $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$, then $\mathcal{F}$ is flat over $S$ at $x$. \end{enumerate} In particular, if $g$ is flat, $f$ is locally of finite presentation, and $\mathcal{F}$ is locally of finite presentation, then formation of the open subset of Theorem \ref{theorem-openness-flatness} commutes with base change.


more-morphisms-lemma-morphism-between-flat-Noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume \begin{enumerate} \item $S$, $X$, $Y$ are locally Noetherian, \item $X$ is flat over $S$, \item for every $s \in S$ the morphism $f_s : X_s \to Y_s$ is flat. \end{enumerate} Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$.


more-morphisms-lemma-morphism-between-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $S$, \item $X$ is flat over $S$, \item for every $s \in S$ the morphism $f_s : X_s \to Y_s$ is flat, and \item $Y$ is locally of finite type over $S$. \end{enumerate} Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$.


more-morphisms-lemma-base-change-criterion-flatness-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X$ is locally of finite presentation over $S$, \item $\mathcal{F}$ an $\mathcal{O}_X$-module of finite presentation, \item $\mathcal{F}$ is flat over $S$, and \item $Y$ is locally of finite type over $S$. \end{enumerate} Then the set $$ U = \{x \in X \mid \mathcal{F} \text{ flat at }x \text{ over }Y\}. $$ is open in $X$ and its formation commutes with arbitrary base change: If $S' \to S$ is a morphism of schemes, and $U'$ is the set of points of $X' = X \times_S S'$ where $\mathcal{F}' = \mathcal{F} \times_S S'$ is flat over $Y' = Y \times_S S'$, then $U' = U \times_S S'$.


more-morphisms-lemma-flat-and-free-at-point-fibre

Let $f : X \to S$ be a morphism of schemes of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Let $x \in X$ with image $s \in S$. If $\mathcal{F}$ is flat at $x$ over $S$ and $(\mathcal{F}_s)_x$ is a flat $\mathcal{O}_{X_s, x}$-module, then $\mathcal{F}$ is finite free in a neighbourhood of $x$.


more-morphisms-lemma-finite-free-open

Let $f : X \to S$ be a morphism of schemes which is locally of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module flat over $S$. Then the set $$ \{x \in X : \mathcal{F}\text{ free in a neighbourhood of }x\} $$ is open in $X$ and its formation commutes with arbitrary base change $S' \to S$.


more-morphisms-lemma-integral-closure-smooth-pullback

Let $f : Y \to X$ be a smooth morphism of schemes. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. The integral closure of $\mathcal{O}_Y$ in $f^*\mathcal{A}$ is equal to $f^*\mathcal{A}'$ where $\mathcal{A}' \subset \mathcal{A}$ is the integral closure of $\mathcal{O}_X$ in $\mathcal{A}$.


more-morphisms-lemma-normalization-smooth-localization

Let $$ \xymatrix{ Y_2 \ar[r] \ar[d]_{f_2} & Y_1 \ar[d]^{f_1} \\ X_2 \ar[r]^\varphi & X_1 } $$ be a fibre square in the category of schemes. Assume $f_1$ is quasi-compact and quasi-separated, and $\varphi$ is smooth. Let $Y_i \to X_i' \to X_i$ be the normalization of $X_i$ in $Y_i$. Then $X_2' \cong X_2 \times_{X_1} X_1'$.


more-morphisms-lemma-normalization-and-smooth

Let $X \to Y$ be a smooth morphism of schemes. Assume every quasi-compact open of $Y$ has finitely many irreducible components. Then the same is true for $X$ and there is a canonical isomorphism $X^\nu = X \times_Y Y^\nu$ where $X^\nu$, $Y^\nu$ are the normalizations of $X$, $Y$.


more-morphisms-lemma-normalization-and-henselization

Let $X$ be a locally Noetherian scheme. Let $\nu : X^\nu \to X$ be the normalization morphism. Then for any point $x \in X$ the base change $$ X^\nu \times_X \Spec(\mathcal{O}_{X, x}^h) \to \Spec(\mathcal{O}_{X, x}^h), \quad\text{resp.}\quad X^\nu \times_X \Spec(\mathcal{O}_{X, x}^{sh}) \to \Spec(\mathcal{O}_{X, x}^{sh}) $$ is the normalization of $\Spec(\mathcal{O}_{X, x}^h)$, resp.\ $\Spec(\mathcal{O}_{X, x}^{sh})$.


more-morphisms-lemma-normal

Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is normal, and \item $f$ is flat and its fibres are geometrically normal schemes. \end{enumerate}


more-morphisms-lemma-smooth-normal

A smooth morphism is normal.


more-morphisms-lemma-locally-Noetherian-fibres-fppf-local-source-and-target

The property $\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian'' is local in the fppf topology on the source and the target.


more-morphisms-lemma-regular

Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is regular, \item $f$ is flat and its fibres are geometrically regular schemes, \item for every pair of affine opens $U \subset X$, $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}(V) \to \mathcal{O}(U)$ is regular, \item there exists an open covering $Y = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$ is regular, and \item there exists an affine open covering $Y = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring maps $\mathcal{O}(V_j) \to \mathcal{O}(U_i)$ are regular. \end{enumerate}


more-morphisms-lemma-CM

Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is Cohen-Macaulay, and \item $f$ is flat and its fibres are Cohen-Macaulay schemes. \end{enumerate}


more-morphisms-lemma-composition-CM

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. Assume that the fibres of $f$, $g$, and $g \circ f$ are locally Noetherian. Let $x \in X$ with images $y \in Y$ and $z \in Z$. \begin{enumerate} \item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \circ f$ is Cohen-Macaulay at $x$. \item If $f$ and $g$ are Cohen-Macaulay, then $g \circ f$ is Cohen-Macaulay. \item If $g \circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$, then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$. \item If $g \circ f$ is Cohen-Macaulay and $f$ is flat, then $f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in the image of $f$. \end{enumerate}


more-morphisms-lemma-base-change-CM

Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in X'$ be a point with image $x \in X$. \begin{enumerate} \item If $f$ is Cohen-Macaulay at $x$, then $f' : X' \to Y'$ is Cohen-Macaulay at $x'$. \item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$. \item If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$. \end{enumerate}


more-morphisms-lemma-flat-finite-presentation-CM-open

\begin{reference} \cite[IV Corollary 12.1.7(iii)]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $$ W = \{x \in X \mid f\text{ is Cohen-Macaulay at }x\} $$ Then \begin{enumerate} \item $W = \{x \in X \mid \mathcal{O}_{X_{f(x)}, x}\text{ is Cohen-Macaulay}\}$, \item $W$ is open in $X$, \item $W$ dense in every fibre of $X \to S$, \item the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : S' \to S$, consider the base change $f' : X' \to S'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$. \end{enumerate}


more-morphisms-lemma-flat-finite-presentation-characterize-CM

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $x \in X$ with image $s \in S$. Set $d = \dim_x(X_s)$. The following are equivalent \begin{enumerate} \item $f$ is Cohen-Macaulay at $x$, \item there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite morphism $U \to \mathbf{A}^d_S$ over $S$ which is flat at $x$, \item there exists an open neighbourhood $U \subset X$ of $x$ and a locally quasi-finite flat morphism $U \to \mathbf{A}^d_S$ over $S$, \item for any $S$-morphism $g : U \to \mathbf{A}^d_S$ of an open neighbourhood $U \subset X$ of $x$ we have: $g$ is quasi-finite at $x$ $\Rightarrow$ $g$ is flat at $x$. \end{enumerate}


more-morphisms-lemma-flat-finite-presentation-CM-pieces

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. For $d \geq 0$ there exist opens $U_d \subset X$ with the following properties \begin{enumerate} \item $W = \bigcup_{d \geq 0} U_d$ is dense in every fibre of $f$, and \item $U_d \to S$ is of relative dimension $d$ (see Morphisms, Definition \ref{morphisms-definition-relative-dimension-d}). \end{enumerate}


more-morphisms-lemma-flat-finite-presentation-specialization-dimension

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Suppose $x' \leadsto x$ is a specialization of points of $X$ with image $s' \leadsto s$ in $S$. If $x$ is a generic point of an irreducible component of $X_s$ then $\dim_{x'}(X_{s'}) = \dim_x(X_s)$.


more-morphisms-lemma-CM-local-source-and-target

The property $\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is Cohen-Macaulay'' is local in the fppf topology on the target and local in the syntomic topology on the source.


more-morphisms-lemma-slice-given-element

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Let $h \in \mathfrak m_x \subset \mathcal{O}_{X, x}$. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $f$ is flat at $x$, and \item the image $\overline{h}$ of $h$ in $\mathcal{O}_{X_s, x} = \mathcal{O}_{X, x}/\mathfrak m_s\mathcal{O}_{X, x}$ is a nonzerodivisor. \end{enumerate} Then there exists an affine open neighbourhood $U \subset X$ of $x$ such that $h$ comes from $h \in \Gamma(U, \mathcal{O}_U)$ and such that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \in D$ and $D \to S$ flat and locally of finite presentation.


more-morphisms-lemma-slice-given-elements

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Let $h_1, \ldots, h_r \in \mathcal{O}_{X, x}$. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $f$ is flat at $x$, and \item the images of $h_1, \ldots, h_r$ in $\mathcal{O}_{X_s, x} = \mathcal{O}_{X, x}/\mathfrak m_s\mathcal{O}_{X, x}$ form a regular sequence. \end{enumerate} Then there exists an affine open neighbourhood $U \subset X$ of $x$ such that $h_1, \ldots, h_r$ come from $h_1, \ldots, h_r \in \Gamma(U, \mathcal{O}_U)$ and such that $Z = V(h_1, \ldots, h_r) \to U$ is a regular immersion with $x \in Z$ and $Z \to S$ flat and locally of finite presentation. Moreover, the base change $Z_{S'} \to U_{S'}$ is a regular immersion for any scheme $S'$ over $S$.


more-morphisms-lemma-slice-once

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $f$ is flat at $x$, and \item $\mathcal{O}_{X_s, x}$ has $\text{depth} \geq 1$. \end{enumerate} Then there exists an affine open neighbourhood $U \subset X$ of $x$ and an effective Cartier divisor $D \subset U$ containing $x$ such that $D \to S$ is flat and of finite presentation.


more-morphisms-lemma-slice-CM

\begin{reference} \cite[IV Proposition 17.16.1]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $f$ is Cohen-Macaulay at $x$, and \item $x$ is a closed point of $X_s$. \end{enumerate} Then there exists a regular immersion $Z \to X$ containing $x$ such that \begin{enumerate} \item[(a)] $Z \to S$ is flat and locally of finite presentation, \item[(b)] $Z \to S$ is locally quasi-finite, and \item[(c)] $Z_s = \{x\}$ set theoretically. \end{enumerate}


more-morphisms-lemma-qf-fp-flat-neighbourhood-dominates-fppf

Let $f : X \to S$ be a flat morphism of schemes which is locally of finite presentation. Let $s \in S$ be a point in the image of $f$. Then there exists a commutative diagram $$ \xymatrix{ S' \ar[rr] \ar[rd]_g & & X \ar[ld]^f \\ & S } $$ where $g : S' \to S$ is flat, locally of finite presentation, locally quasi-finite, and $s \in g(S')$.


more-morphisms-lemma-qf-fp-flat-dominates-fppf

Let $S$ be a scheme. Let $\mathcal{U} = \{S_i \to S\}_{i \in I}$ be an fppf covering of $S$, see Topologies, Definition \ref{topologies-definition-fppf-covering}. Then there exists an fppf covering $\mathcal{V} = \{T_j \to S\}_{j \in J}$ which refines (see Sites, Definition \ref{sites-definition-morphism-coverings}) $\mathcal{U}$ such that each $T_j \to S$ is locally quasi-finite.


more-morphisms-lemma-empty-generic-fibre

Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta$. If $X_\eta = \emptyset$ then there exists a nonempty open $V \subset Y$ such that $X_V = V \times_Y X = \emptyset$.


more-morphisms-lemma-nonempty-generic-fibre

Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta$. If $X_\eta \not = \emptyset$ then there exists a nonempty open $V \subset Y$ such that $X_V = V \times_Y X \to V$ is surjective.


more-morphisms-lemma-nowhere-dense-generic-fibre

Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta$. If $Z \subset X$ is a closed subset with $Z_\eta$ nowhere dense in $X_\eta$, then there exists a nonempty open $V \subset Y$ such that $Z_y$ is nowhere dense in $X_y$ for all $y \in V$.


more-morphisms-lemma-scheme-theoretically-dense-generic-fibre

Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta$. Let $U \subset X$ be an open subscheme such that $U_\eta$ is scheme theoretically dense in $X_\eta$. Then there exists a nonempty open $V \subset Y$ such that $U_y$ is scheme theoretically dense in $X_y$ for all $y \in V$.


more-morphisms-lemma-cover-generic-fibre-neighbourhood

Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta$. Let $X_\eta = Z_{1, \eta} \cup \ldots \cup Z_{n, \eta}$ be a covering of the generic fibre by closed subsets of $X_\eta$. Let $Z_i$ be the closure of $Z_{i, \eta}$ in $X$ (see discussion above). Then there exists a nonempty open $V \subset Y$ such that $X_y = Z_{1, y} \cup \ldots \cup Z_{n, y}$ for all $y \in V$.


more-morphisms-lemma-reduction-generic-fibre

Let $f : X \to Y$ be a morphism of schemes. Let $\eta \in Y$ be a generic point of an irreducible component of $Y$. Then $(X_\eta)_{red} = (X_{red})_\eta$.


more-morphisms-lemma-make-generic-fibre-geometrically-reduced

Let $f : X \to Y$ be a morphism of schemes. Assume that $Y$ is irreducible and $f$ is of finite type. There exists a diagram $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_V \ar[r] \ar[d] & X \ar[d]^f \\ Y' \ar[r]^g & V \ar[r] & Y } $$ where \begin{enumerate} \item $V$ is a nonempty open of $Y$, \item $X_V = V \times_Y X$, \item $g : Y' \to V$ is a finite universal homeomorphism, \item $X' = (Y' \times_Y X)_{red} = (Y' \times_V X_V)_{red}$, \item $g'$ is a finite universal homeomorphism, \item $Y'$ is an integral affine scheme, \item $f'$ is flat and of finite presentation, and \item the generic fibre of $f'$ is geometrically reduced. \end{enumerate}


more-morphisms-lemma-make-components-generic-fibre-geometrically-irreducible

Let $f : X \to Y$ be a morphism of schemes. Assume that $Y$ is irreducible and $f$ is of finite type. There exists a diagram $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_V \ar[r] \ar[d] & X \ar[d]^f \\ Y' \ar[r]^g & V \ar[r] & Y } $$ where \begin{enumerate} \item $V$ is a nonempty open of $Y$, \item $X_V = V \times_Y X$, \item $g : Y' \to V$ is surjective finite \'etale, \item $X' = Y' \times_Y X = Y' \times_V X_V$, \item $g'$ is surjective finite \'etale, \item $Y'$ is an irreducible affine scheme, and \item all irreducible components of the generic fibre of $f'$ are geometrically irreducible. \end{enumerate}


more-morphisms-lemma-relative-assassin-in-neighbourhood

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\xi \in \text{Ass}_{X/S}(\mathcal{F})$ and set $Z = \overline{\{\xi\}} \subset X$. If $f$ is locally of finite type and $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, then there exists a nonempty open $V \subset Z$ such that for every $s \in f(V)$ the generic points of $V_s$ are elements of $\text{Ass}_{X/S}(\mathcal{F})$.


more-morphisms-lemma-bad-case

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $U \subset X$ be an open subscheme. Assume \begin{enumerate} \item $f$ is of finite type, \item $\mathcal{F}$ is of finite type, \item $Y$ is irreducible with generic point $\eta$, and \item $\text{Ass}_{X_\eta}(\mathcal{F}_\eta)$ is not contained in $U_\eta$. \end{enumerate} Then there exists a nonempty open subscheme $V \subset Y$ such that for all $y \in V$ the set $\text{Ass}_{X_y}(\mathcal{F}_y)$ is not contained in $U_y$.


more-morphisms-lemma-good-case

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $U \subset X$ be an open subscheme. Assume \begin{enumerate} \item $f$ is of finite type, \item $\mathcal{F}$ is of finite type, \item $Y$ is irreducible with generic point $\eta$, and \item $\text{Ass}_{X_\eta}(\mathcal{F}_\eta) \subset U_\eta$. \end{enumerate} Then there exists a nonempty open subscheme $V \subset Y$ such that for all $y \in V$ we have $\text{Ass}_{X_y}(\mathcal{F}_y) \subset U_y$.


more-morphisms-lemma-base-change-assassin-in-U

Let $f : X \to S$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $U \subset X$ be an open subscheme. Let $g : S' \to S$ be a morphism of schemes, let $f' : X' = X_{S'} \to S'$ be the base change of $f$, let $g' : X' \to X$ be the projection, set $\mathcal{F}' = (g')^*\mathcal{F}$, and set $U' = (g')^{-1}(U)$. Finally, let $s' \in S'$ with image $s = g(s')$. In this case $$ \text{Ass}_{X_s}(\mathcal{F}_s) \subset U_s \Leftrightarrow \text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) \subset U'_{s'}. $$


more-morphisms-lemma-relative-assassin-constructible

Let $f : X \to Y$ be a morphism of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite presentation. Let $U \subset X$ be an open subscheme such that $U \to Y$ is quasi-compact. Then the set $$ E = \{y \in Y \mid \text{Ass}_{X_y}(\mathcal{F}_y) \subset U_y\} $$ is locally constructible in $Y$.


more-morphisms-lemma-nonreduced-in-neighbourhood

Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\eta$ and $f$ of finite type. If $X_\eta$ is nonreduced, then there exists a nonempty open $V \subset Y$ such that for all $y \in V$ the fibre $X_y$ is nonreduced.


more-morphisms-lemma-base-change-fibres-geometrically-reduced

Let $f : X \to Y$ be a morphism of schemes. Let $g : Y' \to Y$ be any morphism, and denote $f' : X' \to Y'$ the base change of $f$. Then \begin{align*} \{y' \in Y' \mid X'_{y'}\text{ is geometrically reduced}\} \\ = g^{-1}(\{y \in Y \mid X_y\text{ is geometrically reduced}\}). \end{align*}


more-morphisms-lemma-not-geometrically-reduced-in-neighbourhood

Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\eta$ and $f$ of finite type. If $X_\eta$ is not geometrically reduced, then there exists a nonempty open $V \subset Y$ such that for all $y \in V$ the fibre $X_y$ is not geometrically reduced.


more-morphisms-lemma-geometrically-reduced-generic-fibre

Let $f : X \to Y$ be a morphism of schemes. Assume \begin{enumerate} \item $Y$ is irreducible with generic point $\eta$, \item $X_\eta$ is geometrically reduced, and \item $f$ is of finite type. \end{enumerate} Then there exists a nonempty open subscheme $V \subset Y$ such that $X_V \to V$ has geometrically reduced fibres.


more-morphisms-lemma-geometrically-reduced-constructible

Let $f : X \to Y$ be a morphism which is quasi-compact and locally of finite presentation. Then the set $$ E = \{y \in Y \mid X_y\text{ is geometrically reduced}\} $$ is locally constructible in $Y$.


more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre

Let $X \to \Spec(R)$ be a proper flat morphism where $R$ is a discrete valuation ring. If the special fibre is reduced, then both $X$ and the generic fibre $X_\eta$ are reduced.


more-morphisms-lemma-geometrically-reduced-open

Let $f : X \to Y$ be a flat proper morphism of finite presentation. Then the set $\{y \in Y \mid X_y\text{ is geometrically reduced}\}$ is open in $Y$.


more-morphisms-lemma-irreducible-components-in-neighbourhood

Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\eta$ and $f$ of finite type. If $X_\eta$ has $n$ irreducible components, then there exists a nonempty open $V \subset Y$ such that for all $y \in V$ the fibre $X_y$ has at least $n$ irreducible components.


more-morphisms-lemma-base-change-fibres-geometrically-irreducible

Let $f : X \to Y$ be a morphism of schemes. Let $g : Y' \to Y$ be any morphism, and denote $f' : X' \to Y'$ the base change of $f$. Then \begin{align*} \{y' \in Y' \mid X'_{y'}\text{ is geometrically irreducible}\} \\ = g^{-1}(\{y \in Y \mid X_y\text{ is geometrically irreducible}\}). \end{align*}


more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components

Let $f : X \to Y$ be a morphism of schemes. Let $$ n_{X/Y} : Y \to \{0, 1, 2, 3, \ldots, \infty\} $$ be the function which associates to $y \in Y$ the number of irreducible components of $(X_y)_K$ where $K$ is a separably closed extension of $\kappa(y)$. This is well defined and if $g : Y' \to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \circ g $$ where $X' \to Y'$ is the base change of $f$.


more-morphisms-lemma-irreducible-polynomial-over-domain

Let $A$ be a domain with fraction field $K$. Let $P \in A[x_1, \ldots, x_n]$. Denote $\overline{K}$ the algebraic closure of $K$. Assume $P$ is irreducible in $\overline{K}[x_1, \ldots, x_n]$. Then there exists a $f \in A$ such that $P^\varphi \in \kappa[x_1, \ldots, x_n]$ is irreducible for all homomorphisms $\varphi : A_f \to \kappa$ into fields.


more-morphisms-lemma-geom-irreducible-generic-fibre

Let $f : X \to Y$ be a morphism of schemes. Assume \begin{enumerate} \item $Y$ is irreducible with generic point $\eta$, \item $X_\eta$ is geometrically irreducible, and \item $f$ is of finite type. \end{enumerate} Then there exists a nonempty open subscheme $V \subset Y$ such that $X_V \to V$ has geometrically irreducible fibres.


more-morphisms-lemma-nr-geom-irreducible-components-good

Let $f : X \to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically irreducible components of fibres of $f$ introduced in Lemma \ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}. Assume $f$ of finite type. Let $y \in Y$ be a point. Then there exists a nonempty open $V \subset \overline{\{y\}}$ such that $n_{X/Y}|_V$ is constant.


more-morphisms-lemma-nr-geom-irreducible-components-constructible

Let $f : X \to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically irreducible components of fibres of $f$ introduced in Lemma \ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}. Assume $f$ of finite presentation. Then the level sets $$ E_n = \{y \in Y \mid n_{X/Y}(y) = n\} $$ of $n_{X/Y}$ are locally constructible in $Y$.


more-morphisms-lemma-connected-components-in-neighbourhood

Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\eta$ and $f$ of finite type. If $X_\eta$ has $n$ connected components, then there exists a nonempty open $V \subset Y$ such that for all $y \in V$ the fibre $X_y$ has at least $n$ connected components.


more-morphisms-lemma-base-change-fibres-geometrically-connected

Let $f : X \to Y$ be a morphism of schemes. Let $g : Y' \to Y$ be any morphism, and denote $f' : X' \to Y'$ the base change of $f$. Then \begin{align*} \{y' \in Y' \mid X'_{y'}\text{ is geometrically connected}\} \\ = g^{-1}(\{y \in Y \mid X_y\text{ is geometrically connected}\}). \end{align*}


more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components

Let $f : X \to Y$ be a morphism of schemes. Let $$ n_{X/Y} : Y \to \{0, 1, 2, 3, \ldots, \infty\} $$ be the function which associates to $y \in Y$ the number of connected components of $(X_y)_K$ where $K$ is a separably closed extension of $\kappa(y)$. This is well defined and if $g : Y' \to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \circ g $$ where $X' \to Y'$ is the base change of $f$.


more-morphisms-lemma-geometrically-connected-generic-fibre

Let $f : X \to Y$ be a morphism of schemes. Assume \begin{enumerate} \item $Y$ is irreducible with generic point $\eta$, \item $X_\eta$ is geometrically connected, and \item $f$ is of finite type. \end{enumerate} Then there exists a nonempty open subscheme $V \subset Y$ such that $X_V \to V$ has geometrically connected fibres.


more-morphisms-lemma-nr-geom-connected-components-good

Let $f : X \to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically connected components of fibres of $f$ introduced in Lemma \ref{lemma-base-change-fibres-nr-geometrically-connected-components}. Assume $f$ of finite type. Let $y \in Y$ be a point. Then there exists a nonempty open $V \subset \overline{\{y\}}$ such that $n_{X/Y}|_V$ is constant.


more-morphisms-lemma-nr-geom-connected-components-constructible

Let $f : X \to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometric connected components of fibres of $f$ introduced in Lemma \ref{lemma-base-change-fibres-nr-geometrically-connected-components}. Assume $f$ of finite presentation. Then the level sets $$ E_n = \{y \in Y \mid n_{X/Y}(y) = n\} $$ of $n_{X/Y}$ are locally constructible in $Y$.


more-morphisms-lemma-connected-flat-over-dvr

\begin{slogan} A flat degeneration of a disconnected scheme is either disconnected or nonreduced. \end{slogan} Let $f : X \to S$ be a morphism of schemes. Assume that \begin{enumerate} \item $S$ is the spectrum of a discrete valuation ring, \item $f$ is flat, \item $X$ is connected, \item the closed fibre $X_s$ is reduced. \end{enumerate} Then the generic fibre $X_\eta$ is connected.


more-morphisms-lemma-connected-along-section-open

Let $f : X \to Y$, $s : Y \to X$ be as in Situation \ref{situation-connected-along-section}. Assume \begin{enumerate} \item $f$ is of finite presentation and flat, and \item all fibres of $f$ are geometrically reduced. \end{enumerate} Then $X^0$ is open in $X$.


more-morphisms-lemma-dimension-in-neighbourhood

Let $f : X \to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\eta$ and $f$ of finite type. If $X_\eta$ has dimension $n$, then there exists a nonempty open $V \subset Y$ such that for all $y \in V$ the fibre $X_y$ has dimension $n$.


more-morphisms-lemma-base-change-dimension-fibres

Let $f : X \to Y$ be a morphism of finite type. Let $$ n_{X/Y} : Y \to \{0, 1, 2, 3, \ldots, \infty\} $$ be the function which associates to $y \in Y$ the dimension of $X_y$. If $g : Y' \to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \circ g $$ where $X' \to Y'$ is the base change of $f$.


more-morphisms-lemma-dimension-fibres-constructible

Let $f : X \to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \ref{lemma-base-change-dimension-fibres}. Assume $f$ of finite presentation. Then the level sets $$ E_n = \{y \in Y \mid n_{X/Y}(y) = n\} $$ of $n_{X/Y}$ are locally constructible in $Y$.


more-morphisms-lemma-dimension-fibres-flat

Let $f : X \to Y$ be a flat morphism of schemes of finite presentation. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is lower semi-continuous.


more-morphisms-lemma-dimension-fibres-proper

Let $f : X \to Y$ be a proper morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is upper semi-continuous.


more-morphisms-lemma-dimension-fibres-proper-flat

Let $f : X \to Y$ be a proper, flat morphism of schemes of finite presentation. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is locally constant.


more-morphisms-lemma-amazing-bertini-lemma

\begin{reference} See pages 71 and 72 of \cite{Jou} \end{reference} Let $K/k$ be a geometrically irreducible and finitely generated field extension. Let $n \geq 1$. Let $g_1, \ldots, g_n \in K$ be elements such that there exist $c_1, \ldots, c_n \in k$ such that the elements $$ x_1, \ldots, x_n, \sum g_ix_i, \sum c_ig_i \in K(x_1, \ldots, x_n) $$ are algebraically independent over $k$. Then $K(x_1, \ldots, x_n)$ is geometrically irreducible over $k(x_1, \ldots, x_n, \sum g_ix_i)$.


more-morphisms-lemma-algebraically-independent

Let $A$ be a domain of finite type over a field $k$. Let $n \geq 2$. Let $g_1, \ldots, g_n \in A$ be elements such that $V(g_1, g_2)$ has an irreducible component of dimension $\dim(A) - 2$. Then there exist $c_1, \ldots, c_n \in k$ such that the elements $$ x_1, \ldots, x_n, \sum g_ix_i, \sum c_ig_i \in \text{Frac}(A)(x_1, \ldots, x_n) $$ are algebraically independent over $k$.


more-morphisms-lemma-bertini-irreducible

\begin{reference} \cite[Theorem 6.3 part 4)]{Jou} \end{reference} In Varieties, Situation \ref{varieties-situation-family-divisors} assume \begin{enumerate} \item $X$ is of finite type over $k$, \item $X$ is geometrically irreducible over $k$, \item there exist $v_1, v_2, v_3 \in V$ and an irreducible component $Z$ of $H_{v_2} \cap H_{v_3}$ such that $Z \not \subset H_{v_1}$ and $\text{codim}(Z, X) = 2$, and \item every irreducible component $Y$ of $\bigcap_{v \in V} H_v$ has $\text{codim}(Y, X) \geq 2$. \end{enumerate} Then for general $v \in V \otimes_k k'$ the scheme $H_v$ is geometrically irreducible over $k'$.


more-morphisms-lemma-diagonal-picard-flat-proper

Let $f : X \to S$ be a flat, proper morphism of finite presentation. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. For a morphism $g : T \to S$ consider the base change diagram $$ \xymatrix{ X_T \ar[d]_p \ar[r]_q & X \ar[d]^f \\ T \ar[r]^g & S } $$ Assume $\mathcal{O}_T \to p_*\mathcal{O}_{X_T}$ is an isomorphism for all $g : T \to S$. Then there exists an immersion $j : Z \to S$ of finite presentation such that a morphism $g : T \to S$ factors through $Z$ if and only if there exists a finite locally free $\mathcal{O}_T$-module $\mathcal{N}$ with $p^*\mathcal{N} \cong q^*\mathcal{E}$.


more-morphisms-lemma-triviality-generic-fibre-valuation-ring

Let $f : X \to S$ be a proper flat morphism of finite presentation. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $S$ is the spectrum of a valuation ring, \item $\mathcal{L}$ is trivial on the generic fibre $X_\eta$ of $f$, \item the closed fibre $X_0$ of $f$ is integral, \item $H^0(X_\eta, \mathcal{O}_{X_\eta})$ is equal to the function field of $S$. \end{enumerate} Then $\mathcal{L}$ is trivial.


more-morphisms-lemma-get-a-closed

Let $f : X \to S$ and $\mathcal{E}$ be as in Lemma \ref{lemma-diagonal-picard-flat-proper} and in addition assume $\mathcal{E}$ is an invertible $\mathcal{O}_X$-module. If moreover the geometric fibres of $f$ are integral, then $Z$ is closed in $S$.