morphisms-lemma-dimension-fibre-specialization

Let $f : X \to S$ be a morphism of schemes. Let $x \leadsto x'$ be a nontrivial specialization of points in $X$ lying over the same point $s \in S$. Assume $f$ is locally of finite type. Then \begin{enumerate} \item $\dim_x(X_s) \leq \dim_{x'}(X_s)$, \item $\dim(\mathcal{O}_{X_s, x}) < \dim(\mathcal{O}_{X_s, x'})$, and \item $\text{trdeg}_{\kappa(s)}(\kappa(x)) > \text{trdeg}_{\kappa(s)}(\kappa(x'))$. \end{enumerate}


morphisms-lemma-base-change-relative-dimension-d

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. If $f$ has relative dimension $d$, then so does any base change of $f$. Same for relative dimension $\leq d$.


morphisms-lemma-composition-relative-dimension-d

Let $f : X \to Y$, $g : Y \to Z$ be locally of finite type. If $f$ has relative dimension $\leq d$ and $g$ has relative dimension $\leq e$ then $g \circ f$ has relative dimension $\leq d + e$. If \begin{enumerate} \item $f$ has relative dimension $d$, \item $g$ has relative dimension $e$, and \item $f$ is flat, \end{enumerate} then $g \circ f$ has relative dimension $d + e$.


morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension

\begin{slogan} Cohen-Macaulay morphisms decompose into clopens of pure relative dimension \end{slogan} Let $f : X \to S$ be a morphism of schemes. Assume that \begin{enumerate} \item $f$ is flat, \item $f$ is locally of finite presentation, and \item for all $s \in S$ the fibre $X_s$ is Cohen-Macaulay (Properties, Definition \ref{properties-definition-Cohen-Macaulay}) \end{enumerate} Then there exist open and closed subschemes $X_d \subset X$ such that $X = \coprod_{d \geq 0} X_d$ and $f|_{X_d} : X_d \to S$ has relative dimension $d$.


morphisms-lemma-locally-quasi-finite-rel-dimension-0

Let $f : X \to S$ be a morphism of schemes. Assume $f$ is locally of finite type. Let $x \in X$ with $s = f(x)$. Then $f$ is quasi-finite at $x$ if and only if $\dim_x(X_s) = 0$. In particular, $f$ is locally quasi-finite if and only if $f$ has relative dimension $0$.


morphisms-lemma-rel-dimension-dimension

Let $f : X \to Y$ be a morphism of locally Noetherian schemes which is flat, locally of finite type and of relative dimension $d$. For every point $x$ in $X$ with image $y$ in $Y$ we have $\dim_x(X) = \dim_y(Y) + d$.


morphisms-lemma-syntomic-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is syntomic. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is syntomic. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is syntomic. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is syntomic, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is syntomic then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is syntomic.


morphisms-lemma-composition-syntomic

The composition of two morphisms which are syntomic is syntomic.


morphisms-lemma-base-change-syntomic

The base change of a morphism which is syntomic is syntomic.


morphisms-lemma-open-immersion-syntomic

Any open immersion is syntomic.


morphisms-lemma-syntomic-locally-finite-presentation

A syntomic morphism is locally of finite presentation.


morphisms-lemma-syntomic-flat

A syntomic morphism is flat.


morphisms-lemma-syntomic-open

A syntomic morphism is universally open.


morphisms-lemma-local-complete-intersection

Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. The following are equivalent: \begin{enumerate} \item $X$ is a local complete intersection over $k$, \item for every $x \in X$ there exists an affine open $U = \Spec(R) \subset X$ neighbourhood of $x$ such that $R \cong k[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ is a global complete intersection over $k$, and \item for every $x \in X$ the local ring $\mathcal{O}_{X, x}$ is a complete intersection over $k$. \end{enumerate}


morphisms-lemma-syntomic-locally-standard-syntomic

Let $f : X  \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s = f(x)$. Let $V \subset S$ be an affine open neighbourhood of $s$. The following are equivalent \begin{enumerate} \item The morphism $f$ is syntomic at $x$. \item There exist an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that $f|_U : U \to V$ is standard syntomic. \item The morphism $f$ is of finite presentation at $x$, the local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $\mathcal{O}_{X, x}/\mathfrak m_s \mathcal{O}_{X, x}$ is a complete intersection over $\kappa(s)$ (see Algebra, Definition \ref{algebra-definition-lci-local-ring}). \end{enumerate}


morphisms-lemma-syntomic-flat-fibres

Let $f : X \to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and all fibres $X_s$ are local complete intersections, then $f$ is syntomic.


morphisms-lemma-set-points-where-fibres-lci

Let $f : X \to S$ be a morphism of schemes. Assume $f$ locally of finite type. Formation of the set $$ T = \{x \in X \mid \mathcal{O}_{X_{f(x)}, x} \text{ is a complete intersection over }\kappa(f(x))\} $$ commutes with arbitrary base change: 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 $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$. In particular, if $f$ is assumed flat, and locally of finite presentation then the same holds for the open set of points where $f$ is syntomic.


morphisms-lemma-standard-syntomic-relative-dimension

Let $R$ be a ring. Let $R \to A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ be a relative global complete intersection. Set $S = \Spec(R)$ and $X = \Spec(A)$. Consider the morphism $f : X \to S$ associated to the ring map $R \to A$. The function $x \mapsto \dim_x(X_{f(x)})$ is constant with value $n - c$.


morphisms-lemma-syntomic-relative-dimension

Let $f : X \to S$ be a syntomic morphism. The function $x \mapsto \dim_x(X_{f(x)})$ is locally constant on $X$.


morphisms-lemma-syntomic-permanence

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \begin{enumerate} \item $f$ is surjective and syntomic, \item $p$ is syntomic, and \item $q$ is locally of finite presentation\footnote{In fact, if $f$ is surjective, flat, and of finite presentation and $p$ is syntomic, then both $q$ and $f$ are syntomic, see Descent, Lemma \ref{descent-lemma-syntomic-permanence}.}. \end{enumerate} Then $q$ is syntomic.


morphisms-lemma-affine-conormal

Let $i : Z \to X$ be an immersion. The conormal sheaf of $i$ has the following properties: \begin{enumerate} \item Let $U \subset X$ be any open subscheme such that $i$ factors as $Z \xrightarrow{i'} U \to X$ where $i'$ is a closed immersion. Let $\mathcal{I} = \Ker((i')^\sharp) \subset \mathcal{O}_U$. Then $$ \mathcal{C}_{Z/X} = (i')^*\mathcal{I}\quad\text{and}\quad i'_*\mathcal{C}_{Z/X} = \mathcal{I}/\mathcal{I}^2 $$ \item For any affine open $\Spec(R) = U \subset X$ such that $Z \cap U = \Spec(R/I)$ there is a canonical isomorphism $\Gamma(Z \cap U, \mathcal{C}_{Z/X}) = I/I^2$. \end{enumerate}


morphisms-lemma-conormal-functorial

Let $$ \xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' } $$ be a commutative diagram in the category of schemes. Assume $i$, $i'$ immersions. There is a canonical map of $\mathcal{O}_Z$-modules $$ f^*\mathcal{C}_{Z'/X'} \longrightarrow \mathcal{C}_{Z/X} $$ characterized by the following property: For every pair of affine opens $(\Spec(R) = U \subset X, \Spec(R') = U' \subset X')$ with $f(U) \subset U'$ such that $Z \cap U = \Spec(R/I)$ and $Z' \cap U' = \Spec(R'/I')$ the induced map $$ \Gamma(Z' \cap U', \mathcal{C}_{Z'/X'}) = I'/I'^2 \longrightarrow I/I^2 = \Gamma(Z \cap U, \mathcal{C}_{Z/X}) $$ is the one induced by the ring map $f^\sharp : R' \to R$ which has the property $f^\sharp(I') \subset I$.


morphisms-lemma-conormal-functorial-flat

Let $$ \xymatrix{ Z \ar[r]_i \ar[d]_f & X \ar[d]^g \\ Z' \ar[r]^{i'} & X' } $$ be a fibre product diagram in the category of schemes with $i$, $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ of Lemma \ref{lemma-conormal-functorial} is surjective. If $g$ is flat, then it is an isomorphism.


morphisms-lemma-transitivity-conormal

Let $Z \to Y \to X$ be immersions of schemes. Then 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-conormal-functorial} and $i : Z \to Y$ is the first morphism.


morphisms-lemma-universal-derivation-universal

Let $f : X \to S$ be a morphism of schemes. The map $$ \Hom_{\mathcal{O}_X}(\Omega_{X/S}, \mathcal{F}) \longrightarrow \text{Der}_S(\mathcal{O}_X, \mathcal{F}),\quad \alpha \longmapsto \alpha \circ \text{d}_{X/S} $$ is an isomorphism of functors $\textit{Mod}(\mathcal{O}_X) \to \textit{Sets}$.


morphisms-lemma-differentials-restrict-open

Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$, $V \subset S$ be open subschemes such that $f(U) \subset V$. Then there is a unique isomorphism $\Omega_{X/S}|_U = \Omega_{U/V}$ of $\mathcal{O}_U$-modules such that $\text{d}_{X/S}|_U = \text{d}_{U/V}$.


morphisms-lemma-affine-case-derivation

Let $R \to A$ be a ring map. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules on $X = \Spec(A)$. Set $S = \Spec(R)$. The rule which associates to an $S$-derivation on $\mathcal{F}$ its action on global sections defines a bijection between the set of $S$-derivations of $\mathcal{F}$ and the set of $R$-derivations on $M = \Gamma(X, \mathcal{F})$.


morphisms-lemma-differentials-affine

Let $f : X \to S$ be a morphism of schemes. For any pair of affine opens $\Spec(A) = U \subset X$, $\Spec(R) = V \subset S$ with $f(U) \subset V$ there is a unique isomorphism $$ \Gamma(U, \Omega_{X/S}) = \Omega_{A/R}. $$ compatible with $\text{d}_{X/S}$ and $\text{d} : A \to \Omega_{A/R}$.


morphisms-lemma-differentials-diagonal

Let $f : X \to S$ be a morphism of schemes. There is a canonical isomorphism between $\Omega_{X/S}$ and the conormal sheaf of the diagonal morphism $\Delta_{X/S} : X \longrightarrow X \times_S X$.


morphisms-lemma-functoriality-differentials

Let $$ \xymatrix{ X' \ar[d] \ar[r]_f & X \ar[d] \\ S' \ar[r] & S } $$ be a commutative diagram of schemes. The canonical map $\mathcal{O}_X \to f_*\mathcal{O}_{X'}$ composed with the map $f_*\text{d}_{X'/S'} : f_*\mathcal{O}_{X'} \to f_*\Omega_{X'/S'}$ is a $S$-derivation. Hence we obtain a canonical map of $\mathcal{O}_X$-modules $\Omega_{X/S} \to f_*\Omega_{X'/S'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\Omega_{X/S} \longrightarrow \Omega_{X'/S'}. $$ It is uniquely characterized by the property that $f^*\text{d}_{X/S}(h)$ maps to $\text{d}_{X'/S'}(f^* h)$ for any local section $h$ of $\mathcal{O}_X$.


morphisms-lemma-triangle-differentials

Let $f : X \to Y$, $g : Y \to S$ be morphisms of schemes. Then there is a canonical exact sequence $$ f^*\Omega_{Y/S} \to \Omega_{X/S} \to \Omega_{X/Y} \to 0 $$ where the maps come from applications of Lemma \ref{lemma-functoriality-differentials}.


morphisms-lemma-base-change-differentials

Let $X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Let $X' = X_{S'}$ be the base change of $X$. Denote $g' : X' \to X$ the projection. Then the map $$ (g')^*\Omega_{X/S} \to \Omega_{X'/S'} $$ of Lemma \ref{lemma-functoriality-differentials} is an isomorphism.


morphisms-lemma-differential-product

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $p : X \times_S Y \to X$ and $q : X \times_S Y \to Y$ be the projection morphisms. The maps from Lemma \ref{lemma-functoriality-differentials} $$ p^*\Omega_{X/S} \oplus q^*\Omega_{Y/S} \longrightarrow \Omega_{X \times_S Y/S} $$ give an isomorphism.


morphisms-lemma-finite-type-differentials

Let $f : X \to S$ be a morphism of schemes. If $f$ is locally of finite type, then $\Omega_{X/S}$ is a finite type $\mathcal{O}_X$-module.


morphisms-lemma-finite-presentation-differentials

Let $f : X \to S$ be a morphism of schemes. If $f$ is locally of finite presentation, then $\Omega_{X/S}$ is an $\mathcal{O}_X$-module of finite presentation.


morphisms-lemma-immersion-differentials

If $X \to S$ is an immersion, or more generally a monomorphism, then $\Omega_{X/S}$ is zero.


morphisms-lemma-differentials-relative-immersion

Let $i : Z \to X$ be an immersion of schemes over $S$. There is a canonical exact sequence $$ \mathcal{C}_{Z/X} \to i^*\Omega_{X/S} \to \Omega_{Z/S} \to 0 $$ where the first arrow is induced by $\text{d}_{X/S}$ and the second arrow comes from Lemma \ref{lemma-functoriality-differentials}.


morphisms-lemma-two-immersions

Let $$ \xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d] \\ & Y } $$ be a commutative diagram of schemes where $i$ and $j$ are immersions. 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-conormal-functorial} and the second from Lemma \ref{lemma-differentials-relative-immersion}.


morphisms-lemma-affine-case-differential-operators

Let $R \to A$ be a ring map. Denote $f : X \to S$ the corresponding morphism of affine schemes. Let $\mathcal{F}$ and $\mathcal{G}$ be $\mathcal{O}_X$-modules. If $\mathcal{F}$ is quasi-coherent then the map $$ \text{Diff}^k_{X/S}(\mathcal{F}, \mathcal{G}) \to \text{Diff}^k_{A/R}(\Gamma(X, \mathcal{F}), \Gamma(X, \mathcal{G})) $$ sending a differential operator to its action on global sections is bijective.


morphisms-lemma-base-change-differential-operators

Let $a : X \to S$ and $b : Y \to S$ be morphisms of schemes. Let $\mathcal{F}$ and $\mathcal{F}'$ be quasi-coherent $\mathcal{O}_X$-modules. Let $D : \mathcal{F} \to \mathcal{F}'$ be a differential operator of order $k$ on $X/S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Then there is a unique differential operator $$ D' : \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} \text{pr}_2^*\mathcal{G} \longrightarrow \text{pr}_1^*\mathcal{F}' \otimes_{\mathcal{O}_{X \times_S Y}} \text{pr}_2^*\mathcal{G} $$ of order $k$ on $X \times_S Y / Y$ such that $ D'(s \otimes t) = D(s) \otimes t $ for local sections $s$ of $\mathcal{F}$ and $t$ of $\mathcal{G}$.


morphisms-lemma-smooth-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is smooth. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is smooth. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is smooth. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is smooth, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is smooth then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is smooth.


morphisms-lemma-smooth-flat-smooth-fibres

Let $f : X \to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and all fibres $X_s$ are smooth, then $f$ is smooth.


morphisms-lemma-composition-smooth

The composition of two morphisms which are smooth is smooth.


morphisms-lemma-base-change-smooth

The base change of a morphism which is smooth is smooth.


morphisms-lemma-open-immersion-smooth

Any open immersion is smooth.


morphisms-lemma-smooth-syntomic

A smooth morphism is syntomic.


morphisms-lemma-smooth-locally-finite-presentation

A smooth morphism is locally of finite presentation.


morphisms-lemma-smooth-flat

A smooth morphism is flat.


morphisms-lemma-smooth-open

A smooth morphism is universally open.


morphisms-lemma-smooth-locally-standard-smooth

\begin{slogan} Smooth morphisms are locally standard smooth. \end{slogan} Let $f : X  \to S$ be a morphism of schemes. Let $x \in X$ be a point. Let $V \subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent \begin{enumerate} \item The morphism $f$ is smooth at $x$. \item There exist an affine open $U \subset X$, with $x \in U$ and $f(U) \subset V$ such that the induced morphism $f|_U : U \to V$ is standard smooth. \end{enumerate}


morphisms-lemma-smooth-omega-finite-locally-free

Let $f : X \to S$ be a morphism of schemes. Assume $f$ is smooth. Then the module of differentials $\Omega_{X/S}$ of $X$ over $S$ is finite locally free and $$ \text{rank}_x(\Omega_{X/S}) = \dim_x(X_{f(x)}) $$ for every $x \in X$.


morphisms-lemma-smooth-at-point

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume $f$ is locally of finite presentation. The following are equivalent: \begin{enumerate} \item The morphism $f$ is smooth at $x$. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $X_s \to \Spec(\kappa(s))$ is smooth at $x$. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\mathcal{O}_{X, x}$-module $\Omega_{X/S, x}$ can be generated by at most $\dim_x(X_{f(x)})$ elements. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\kappa(x)$-vector space $$ \Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) = \Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x) $$ can be generated by at most $\dim_x(X_{f(x)})$ elements. \item There exist affine opens $U \subset X$, and $V \subset S$ such that $x \in U$, $f(U) \subset V$ and the induced morphism $f|_U : U \to V$ is standard smooth. \item There exist affine opens $\Spec(A) = U \subset X$ and $\Spec(R) = V \subset S$ with $x \in U$ corresponding to $\mathfrak q \subset A$, and $f(U) \subset V$ such that there exists a presentation $$ A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c) $$ with $$ g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_c/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_c/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_c & \partial f_2/\partial x_c & \ldots & \partial f_c/\partial x_c \end{matrix} \right) $$ mapping to an element of $A$ not in $\mathfrak q$. \end{enumerate}


morphisms-lemma-set-points-where-fibres-smooth

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 $W \subset X$, resp.\ $W' \subset X'$ be the open subscheme of points where $f$, resp.\ $f'$ is smooth. Then $W' = (g')^{-1}(W)$ if \begin{enumerate} \item $f$ is flat and locally of finite presentation, or \item $f$ is locally of finite presentation and $g$ is flat. \end{enumerate}


morphisms-lemma-triangle-differentials-smooth

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


morphisms-lemma-smooth-permanence

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \begin{enumerate} \item $f$ is surjective, and smooth, \item $p$ is smooth, and \item $q$ is locally of finite presentation\footnote{In fact this is implied by (1) and (2), see Descent, Lemma \ref{descent-lemma-flat-finitely-presented-permanence}. Moreover, it suffices to assume $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma \ref{descent-lemma-smooth-permanence}.}. \end{enumerate} Then $q$ is smooth.


morphisms-lemma-section-smooth-morphism

Let $f : X \to S$ be a morphism of schemes. Let $\sigma : S \to X$ be a section of $f$. Let $s \in S$ be a point such that $f$ is smooth at $x = \sigma(s)$. Then there exist affine open neighbourhoods $\Spec(A) = U \subset S$ of $s$ and $\Spec(B) = V \subset X$ of $x$ such that \begin{enumerate} \item $f(V) \subset U$ and $\sigma(U) \subset V$, \item with $I = \Ker(\sigma^\# : B \to A)$ the module $I/I^2$ is a free $A$-module, and \item $B^\wedge \cong A[[x_1, \ldots, x_d]]$ as $A$-algebras where $B^\wedge$ denotes the completion of $B$ with respect to $I$. \end{enumerate}


morphisms-lemma-smoothness-dimension

Let $f : X \to Y$ be a smooth morphism of locally Noetherian schemes. For every point $x$ in $X$ with image $y$ in $Y$, $$ \dim_x(X) = \dim_y(Y) + \dim_x(X_y), $$ where $X_y$ denotes the fiber over $y$.


morphisms-lemma-unramified-omega-zero

Let $f : X \to S$ be a morphism of schemes. Then \begin{enumerate} \item $f$ is unramified if and only if $f$ is locally of finite type and $\Omega_{X/S} = 0$, and \item $f$ is G-unramified if and only if $f$ is locally of finite presentation and $\Omega_{X/S} = 0$. \end{enumerate}


morphisms-lemma-unramified-characterize

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). \item For every affine open $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is unramified (resp.\ G-unramified). \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is unramified (resp.\ G-unramified). \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is unramified (resp.\ G-unramified), for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is unramified (resp.\ G-unramified) then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is unramified (resp.\ G-unramified).


morphisms-lemma-composition-unramified

The composition of two morphisms which are unramified is unramified. The same holds for G-unramified morphisms.


morphisms-lemma-base-change-unramified

The base change of a morphism which is unramified is unramified. The same holds for G-unramified morphisms.


morphisms-lemma-open-immersion-unramified

Any open immersion is G-unramified.


morphisms-lemma-closed-immersion-unramified

A closed immersion $i : Z \to X$ is unramified. It is G-unramified if and only if the associated quasi-coherent sheaf of ideals $\mathcal{I} = \Ker(\mathcal{O}_X \to i_*\mathcal{O}_Z)$ is of finite type (as an $\mathcal{O}_X$-module).


morphisms-lemma-unramified-locally-finite-type

An unramified morphism is locally of finite type. A G-unramified morphism is locally of finite presentation.


morphisms-lemma-unramified-quasi-finite

Let $f : X \to S$ be a morphism of schemes. If $f$ is unramified at $x$ then $f$ is quasi-finite at $x$. In particular, an unramified morphism is locally quasi-finite.


morphisms-lemma-unramified-over-field

Fibres of unramified morphisms. \begin{enumerate} \item Let $X$ be a scheme over a field $k$. The structure morphism $X \to \Spec(k)$ is unramified if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$. \item If $f : X \to S$ is an unramified morphism then for every $s \in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\kappa(s)$. \end{enumerate}


morphisms-lemma-unramified-etale-fibres

Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item If $f$ is unramified then for any $x \in X$ the field extension $\kappa(f(x)) \subset \kappa(x)$ is finite separable. \item If $f$ is locally of finite type, and for every $s \in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\kappa(s)$ then $f$ is unramified. \item If $f$ is locally of finite presentation, and for every $s \in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\kappa(s)$ then $f$ is G-unramified. \end{enumerate}


morphisms-lemma-diagonal-unramified-morphism

Let $f : X \to S$ be a morphism. \begin{enumerate} \item If $f$ is unramified, then the diagonal morphism $\Delta : X \to X \times_S X$ is an open immersion. \item If $f$ is locally of finite type and $\Delta$ is an open immersion, then $f$ is unramified. \item If $f$ is locally of finite presentation and $\Delta$ is an open immersion, then $f$ is G-unramified. \end{enumerate}


morphisms-lemma-unramified-at-point

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume $f$ is locally of finite type (resp.\ locally of finite presentation). The following are equivalent: \begin{enumerate} \item The morphism $f$ is unramified (resp.\ G-unramified) at $x$. \item The fibre $X_s$ is unramified over $\kappa(s)$ at $x$. \item The $\mathcal{O}_{X, x}$-module $\Omega_{X/S, x}$ is zero. \item The $\mathcal{O}_{X_s, x}$-module $\Omega_{X_s/s, x}$ is zero. \item The $\kappa(x)$-vector space $$ \Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) = \Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x) $$ is zero. \item We have $\mathfrak m_s\mathcal{O}_{X, x} = \mathfrak m_x$ and the field extension $\kappa(s) \subset \kappa(x)$ is finite separable. \end{enumerate}


morphisms-lemma-set-points-where-fibres-unramified

Let $f : X \to S$ be a morphism of schemes. Assume $f$ locally of finite type. Formation of the open set \begin{align*} T & = \{x \in X \mid X_{f(x)}\text{ is unramified over }\kappa(f(x))\text{ at }x\} \\ & = \{x \in X \mid X\text{ is unramified over }S\text{ at }x\} \end{align*} commutes with arbitrary base change: 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 $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$. If $f$ is assumed locally of finite presentation then the same holds for the open set of points where $f$ is G-unramified.


morphisms-lemma-unramified-permanence

Let $f : X \to Y$ be a morphism of schemes over $S$. \begin{enumerate} \item If $X$ is unramified over $S$, then $f$ is unramified. \item If $X$ is G-unramified over $S$ and $Y$ of finite type over $S$, then $f$ is G-unramified. \end{enumerate}


morphisms-lemma-value-at-one-point

Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $f, g : X \to Y$ be morphisms over $S$. Let $x \in X$. Assume that \begin{enumerate} \item the structure morphism $Y \to S$ is unramified, \item $f(x) = g(x)$ in $Y$, say $y = f(x) = g(x)$, and \item the induced maps $f^\sharp, g^\sharp : \kappa(y) \to \kappa(x)$ are equal. \end{enumerate} Then there exists an open neighbourhood of $x$ in $X$ on which $f$ and $g$ are equal.


morphisms-lemma-etale-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is \'etale. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is \'etale. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is \'etale. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is \'etale, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is \'etale then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is \'etale.


morphisms-lemma-composition-etale

The composition of two morphisms which are \'etale is \'etale.


morphisms-lemma-base-change-etale

The base change of a morphism which is \'etale is \'etale.


morphisms-lemma-etale-smooth-unramified

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


morphisms-lemma-etale-locally-quasi-finite

An \'etale morphism is locally quasi-finite.


morphisms-lemma-etale-over-field

\begin{slogan} Description of the \'etale schemes over fields and fibres of \'etale morphisms. \end{slogan} Fibres of \'etale morphisms. \begin{enumerate} \item Let $X$ be a scheme over a field $k$. The structure morphism $X \to \Spec(k)$ is \'etale if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$. \item If $f : X \to S$ is an \'etale morphism, then for every $s \in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\kappa(s)$. \end{enumerate}


morphisms-lemma-etale-flat-etale-fibres

Let $f : X \to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and for every $s \in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\kappa(s)$, then $f$ is \'etale.


morphisms-lemma-open-immersion-etale

Any open immersion is \'etale.


morphisms-lemma-etale-syntomic

An \'etale morphism is syntomic.


morphisms-lemma-etale-locally-finite-presentation

An \'etale morphism is locally of finite presentation.


morphisms-lemma-etale-flat

An \'etale morphism is flat.


morphisms-lemma-etale-open

An \'etale morphism is open.


morphisms-lemma-etale-locally-standard-etale

Let $f : X  \to S$ be a morphism of schemes. Let $x \in X$ be a point. Let $V \subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent \begin{enumerate} \item The morphism $f$ is \'etale at $x$. \item There exist an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that the induced morphism $f|_U : U \to V$ is standard \'etale (see Definition \ref{definition-etale}). \end{enumerate}


morphisms-lemma-etale-at-point

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume $f$ is locally of finite presentation. The following are equivalent: \begin{enumerate} \item The morphism $f$ is \'etale at $x$. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $X_s \to \Spec(\kappa(s))$ is \'etale at $x$. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $X_s \to \Spec(\kappa(s))$ is unramified at $x$. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\mathcal{O}_{X, x}$-module $\Omega_{X/S, x}$ is zero. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\kappa(x)$-vector space $$ \Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) = \Omega_{X/S, x} \otimes_{\mathcal{O}_{X, x}} \kappa(x) $$ is zero. \item The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat, we have $\mathfrak m_s\mathcal{O}_{X, x} = \mathfrak m_x$ and the field extension $\kappa(s) \subset \kappa(x)$ is finite separable. \item There exist affine opens $U \subset X$, and $V \subset S$ such that $x \in U$, $f(U) \subset V$ and the induced morphism $f|_U : U \to V$ is standard smooth of relative dimension $0$. \item There exist affine opens $\Spec(A) = U \subset X$ and $\Spec(R) = V \subset S$ with $x \in U$ corresponding to $\mathfrak q \subset A$, and $f(U) \subset V$ such that there exists a presentation $$ A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_n) $$ with $$ g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_n/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_n/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_n & \partial f_2/\partial x_n & \ldots & \partial f_n/\partial x_n \end{matrix} \right) $$ mapping to an element of $A$ not in $\mathfrak q$. \item There exist affine opens $U \subset X$, and $V \subset S$ such that $x \in U$, $f(U) \subset V$ and the induced morphism $f|_U : U \to V$ is standard \'etale. \item There exist affine opens $\Spec(A) = U \subset X$ and $\Spec(R) = V \subset S$ with $x \in U$ corresponding to $\mathfrak q \subset A$, and $f(U) \subset V$ such that there exists a presentation $$ A = R[x]_Q/(P) = R[x, 1/Q]/(P) $$ with $P, Q \in R[x]$, $P$ monic and $P' = \text{d}P/\text{d}x$ mapping to an element of $A$ not in $\mathfrak q$. \end{enumerate}


morphisms-lemma-flat-unramified-etale

A morphism is \'etale at a point if and only if it is flat and G-unramified at that point. A morphism is \'etale if and only if it is flat and G-unramified.


morphisms-lemma-set-points-where-fibres-etale

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 $W \subset X$, resp.\ $W' \subset X'$ be the open subscheme of points where $f$, resp.\ $f'$ is \'etale. Then $W' = (g')^{-1}(W)$ if \begin{enumerate} \item $f$ is flat and locally of finite presentation, or \item $f$ is locally of finite presentation and $g$ is flat. \end{enumerate}


morphisms-lemma-etale-permanence

\begin{slogan} Cancellation law for \'etale morphisms \end{slogan} Let $f : X \to Y$ be a morphism of schemes over $S$. If $X$ and $Y$ are \'etale over $S$, then $f$ is \'etale.


morphisms-lemma-etale-permanence-two

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \begin{enumerate} \item $f$ is surjective, and \'etale, \item $p$ is \'etale, and \item $q$ is locally of finite presentation\footnote{In fact this is implied by (1) and (2), see Descent, Lemma \ref{descent-lemma-flat-finitely-presented-permanence}. Moreover, it suffices to assume that $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma \ref{descent-lemma-smooth-permanence}.}. \end{enumerate} Then $q$ is \'etale.


morphisms-lemma-smooth-etale-over-affine-space

\begin{slogan} Smooth schemes are \'etale-locally like affine spaces. \end{slogan} Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. Let $V \subset Y$ be an affine open neighbourhood of $\varphi(x)$. If $\varphi$ is smooth at $x$, then there exists an integer $d \geq 0$ and an affine open $U \subset X$ with $x \in U$ and $\varphi(U) \subset V$ such that there exists a commutative diagram $$ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^d_V \ar[ld] \\ Y & V \ar[l] } $$ where $\pi$ is \'etale.


morphisms-lemma-ample-power-ample

Let $X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $n \geq 1$. Then $\mathcal{L}$ is $f$-ample if and only if $\mathcal{L}^{\otimes n}$ is $f$-ample.


morphisms-lemma-relatively-ample-separated

Let $f : X \to S$ be a morphism of schemes. If there exists an $f$-ample invertible sheaf, then $f$ is separated.


morphisms-lemma-characterize-relatively-ample

\begin{reference} \cite[II, Proposition 4.6.3]{EGA} \end{reference} Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. The following are equivalent: \begin{enumerate} \item The invertible sheaf $\mathcal{L}$ is $f$-ample. \item There exists an open covering $S = \bigcup V_i$ such that each $\mathcal{L}|_{f^{-1}(V_i)}$ is ample relative to $f^{-1}(V_i) \to V_i$. \item There exists an affine open covering $S = \bigcup V_i$ such that each $\mathcal{L}|_{f^{-1}(V_i)}$ is ample. \item There exists a quasi-coherent graded $\mathcal{O}_S$-algebra $\mathcal{A}$ and a map of graded $\mathcal{O}_X$-algebras $\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}$ such that $U(\psi) = X$ and $$ r_{\mathcal{L}, \psi} : X \longrightarrow \underline{\text{Proj}}_S(\mathcal{A}) $$ is an open immersion (see Constructions, Lemma \ref{constructions-lemma-invertible-map-into-relative-proj} for notation). \item The morphism $f$ is quasi-separated and part (4) above holds with $\mathcal{A} = f_*(\bigoplus_{d \geq 0} \mathcal{L}^{\otimes d})$ and $\psi$ the adjunction mapping. \item Same as (4) but just requiring $r_{\mathcal{L}, \psi}$ to be an immersion. \end{enumerate}


morphisms-lemma-ample-over-affine

\begin{reference} \cite[II Corollary 4.6.6]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $S$ affine. Then $\mathcal{L}$ is $f$-relatively ample if and only if $\mathcal{L}$ is ample on $X$.


morphisms-lemma-quasi-affine-O-ample

\begin{reference} \cite[II Proposition 5.1.6]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. Then $f$ is quasi-affine if and only if $\mathcal{O}_X$ is $f$-relatively ample.


morphisms-lemma-pullback-ample-tensor-relatively-ample

Let $f : X \to Y$ be a morphism of schemes, $\mathcal{M}$ an invertible $\mathcal{O}_Y$-module, and $\mathcal{L}$ an invertible $\mathcal{O}_X$-module. \begin{enumerate} \item If $\mathcal{L}$ is $f$-ample and $\mathcal{M}$ is ample, then $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample for $a \gg 0$. \item If $\mathcal{M}$ is ample and $f$ quasi-affine, then $f^*\mathcal{M}$ is ample. \end{enumerate}


morphisms-lemma-ample-composition

Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. Let $\mathcal{M}$ be an invertible $\mathcal{O}_Y$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. If $S$ is quasi-compact, $\mathcal{M}$ is $g$-ample, and $\mathcal{L}$ is $f$-ample, then $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is $g \circ f$-ample for $a \gg 0$.


morphisms-lemma-ample-base-change

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $S' \to S$ be a morphism of schemes. Let $f' : X' \to S'$ be the base change of $f$ and denote $\mathcal{L}'$ the pullback of $\mathcal{L}$ to $X'$. If $\mathcal{L}$ is $f$-ample, then $\mathcal{L}'$ is $f'$-ample.


morphisms-lemma-ample-permanence

Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. If $\mathcal{L}$ is $g \circ f$-ample and $f$ is quasi-compact\footnote{This follows if $g$ is quasi-separated by Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence}.} then $\mathcal{L}$ is $f$-ample.


morphisms-lemma-ample-very-ample

\begin{reference} \cite[II, Proposition 4.6.2]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. If $f$ is quasi-compact and $\mathcal{L}$ is a relatively very ample invertible sheaf, then $\mathcal{L}$ is a relatively ample invertible sheaf.


morphisms-lemma-relatively-very-ample-separated

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. If $\mathcal{L}$ is relatively very ample on $X/S$ then $f$ is separated.


morphisms-lemma-relatively-very-ample

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is quasi-compact. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is relatively very ample on $X/S$, \item there exists an open covering $S = \bigcup V_j$ such that $\mathcal{L}|_{f^{-1}(V_j)}$ is relatively very ample on $f^{-1}(V_j)/V_j$ for all $j$, \item there exists a quasi-coherent sheaf of graded $\mathcal{O}_S$-algebras $\mathcal{A}$ generated in degree $1$ over $\mathcal{O}_S$ and a map of graded $\mathcal{O}_X$-algebras $\psi : f^*\mathcal{A} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}$ such that $f^*\mathcal{A}_1 \to \mathcal{L}$ is surjective and the associated morphism $r_{\mathcal{L}, \psi} : X \to \underline{\text{Proj}}_S(\mathcal{A})$ is an immersion, and \item $f$ is quasi-separated, the canonical map $\psi : f^*f_*\mathcal{L} \to \mathcal{L}$ is surjective, and the associated map $r_{\mathcal{L}, \psi} : X \to \mathbf{P}(f_*\mathcal{L})$ is an immersion. \end{enumerate}


morphisms-lemma-very-ample-base-change

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $S' \to S$ be a morphism of schemes. Let $f' : X' \to S'$ be the base change of $f$ and denote $\mathcal{L}'$ the pullback of $\mathcal{L}$ to $X'$. If $\mathcal{L}$ is $f$-very ample, then $\mathcal{L}'$ is $f'$-very ample.


morphisms-lemma-very-ample-finite-type-over-affine

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that \begin{enumerate} \item the invertible sheaf $\mathcal{L}$ is very ample on $X/S$, \item the morphism $X \to S$ is of finite type, and \item $S$ is affine. \end{enumerate} Then there exists an $n \geq 0$ and an immersion $i : X \to \mathbf{P}^n_S$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}^n_S}(1)$.


morphisms-lemma-quasi-affine-finite-type-over-S

Let $\pi : X \to S$ be a morphism of schemes. Assume that $X$ is quasi-affine and that $\pi$ is locally of finite type. Then there exist $n \geq 0$ and an immersion $i : X \to \mathbf{A}^n_S$ over $S$.


morphisms-lemma-quasi-projective-finite-type-over-S

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that \begin{enumerate} \item the invertible sheaf $\mathcal{L}$ is ample on $X$, and \item the morphism $X \to S$ is locally of finite type. \end{enumerate} Then there exists a $d_0 \geq 1$ such that for every $d \geq d_0$ there exists an $n \geq 0$ and an immersion $i : X \to \mathbf{P}^n_S$ over $S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^n_S}(1)$.


morphisms-lemma-finite-type-over-affine-ample-very-ample

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $S$ affine and $f$ of finite type. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X$, \item $\mathcal{L}$ is $f$-ample, \item $\mathcal{L}^{\otimes d}$ is $f$-very ample for some $d \geq 1$, \item $\mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$, \item for some $d \geq 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^n_S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^n_S}(1)$, and \item for all $d \gg 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^n_S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^n_S}(1)$. \end{enumerate}


morphisms-lemma-finite-type-ample-very-ample

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $S$ quasi-compact and $f$ of finite type. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is $f$-ample, \item $\mathcal{L}^{\otimes d}$ is $f$-very ample for some $d \geq 1$, \item $\mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$. \end{enumerate}


morphisms-lemma-characterize-very-ample-on-finite-type

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is of finite type. The following are equivalent: \begin{enumerate} \item $\mathcal{L}$ is $f$-relatively very ample, and \item there exist an open covering $S = \bigcup V_j$, for each $j$ an integer $n_j$, and immersions $$ i_j : X_j = f^{-1}(V_j) = V_j \times_S X \longrightarrow \mathbf{P}^{n_j}_{V_j} $$ over $V_j$ such that $\mathcal{L}|_{X_j} \cong i_j^*\mathcal{O}_{\mathbf{P}^{n_j}_{V_j}}(1)$. \end{enumerate}


morphisms-lemma-characterize-ample-on-finite-type

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is of finite type. The following are equivalent: \begin{enumerate} \item $\mathcal{L}$ is $f$-relatively ample, and \item there exist an open covering $S = \bigcup V_j$, for each $j$ an integers $d_j \geq 1$, $n_j \geq 0$, and immersions $$ i_j : X_j = f^{-1}(V_j) = V_j \times_S X \longrightarrow \mathbf{P}^{n_j}_{V_j} $$ over $V_j$ such that $\mathcal{L}^{\otimes d_j}|_{X_j} \cong i_j^*\mathcal{O}_{\mathbf{P}^{n_j}_{V_j}}(1)$. \end{enumerate}


morphisms-lemma-invertible-add-enough-ample-very-ample

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{N}$, $\mathcal{L}$ be invertible $\mathcal{O}_X$-modules. Assume $S$ is quasi-compact, $f$ is of finite type, and $\mathcal{L}$ is $f$-ample. Then $\mathcal{N} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$.


morphisms-lemma-base-change-quasi-projective

A base change of a quasi-projective morphism is quasi-projective.


morphisms-lemma-composition-quasi-projective

Let $f : X \to Y$ and $g : Y \to S$ be morphisms of schemes. If $S$ is quasi-compact and $f$ and $g$ are quasi-projective, then $g \circ f$ is quasi-projective.


morphisms-lemma-quasi-projective-properties

Let $f : X \to S$ be a morphism of schemes. If $f$ is quasi-projective, or H-quasi-projective or locally quasi-projective, then $f$ is separated of finite type.


morphisms-lemma-H-quasi-projective-quasi-projective

A H-quasi-projective morphism is quasi-projective.


morphisms-lemma-characterize-locally-quasi-projective

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is locally quasi-projective. \item There exists an open covering $S = \bigcup V_j$ such that each $f^{-1}(V_j) \to V_j$ is H-quasi-projective. \end{enumerate}


morphisms-lemma-quasi-affine-finite-type-quasi-projective

\begin{reference} \cite[II, Proposition 5.3.4 (i)]{EGA} \end{reference} A quasi-affine morphism of finite type is quasi-projective.


morphisms-lemma-universally-closed-local-on-the-base

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is universally closed. \item There exists an open covering $S = \bigcup V_j$ such that $f^{-1}(V_j) \to V_j$ is universally closed for all indices $j$. \end{enumerate}


morphisms-lemma-proper-local-on-the-base

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is proper. \item There exists an open covering $S = \bigcup V_j$ such that $f^{-1}(V_j) \to V_j$ is proper for all indices $j$. \end{enumerate}


morphisms-lemma-composition-proper

The composition of proper morphisms is proper. The same is true for universally closed morphisms.


morphisms-lemma-base-change-proper

The base change of a proper morphism is proper. The same is true for universally closed morphisms.


morphisms-lemma-closed-immersion-proper

A closed immersion is proper, hence a fortiori universally closed.


morphisms-lemma-image-proper-scheme-closed

Suppose given a commutative diagram of schemes $$ \xymatrix{ X \ar[rr] \ar[rd] & & Y \ar[ld] \\ & S & } $$ with $Y$ separated over $S$. \begin{enumerate} \item If $X \to S$ is universally closed, then the morphism $X \to Y$ is universally closed. \item If $X$ is proper over $S$, then the morphism $X \to Y$ is proper. \end{enumerate} In particular, in both cases the image of $X$ in $Y$ is closed.


morphisms-lemma-universally-closed-quasi-compact

\begin{reference} Due to Bjorn Poonen. \end{reference} A universally closed morphism of schemes is quasi-compact.


morphisms-lemma-image-proper-is-proper

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. If $X$ is universally closed over $S$ and $f$ is surjective then $Y$ is universally closed over $S$. In particular, if also $Y$ is separated and locally of finite type over $S$, then $Y$ is proper over $S$.


morphisms-lemma-scheme-theoretic-image-is-proper

Suppose given a commutative diagram of schemes $$ \xymatrix{ X \ar[rr]_h \ar[rd]_f & & Y \ar[ld]^g \\ & S } $$ Assume \begin{enumerate} \item $X \to S$ is a universally closed (for example proper) morphism, and \item $Y \to S$ is separated and locally of finite type. \end{enumerate} Then the scheme theoretic image $Z \subset Y$ of $h$ is proper over $S$ and $X \to Z$ is surjective.


morphisms-lemma-image-universally-closed-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a surjective universally closed morphism of schemes over $S$. \begin{enumerate} \item If $X$ is quasi-separated, then $Y$ is quasi-separated. \item If $X$ is separated, then $Y$ is separated. \item If $X$ is quasi-separated over $S$, then $Y$ is quasi-separated over $S$. \item If $X$ is separated over $S$, then $Y$ is separated over $S$. \end{enumerate}


morphisms-lemma-characterize-proper

\begin{reference} \cite[II Theorem 7.3.8]{EGA} \end{reference} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent \begin{enumerate} \item $f$ is proper, \item $f$ satisfies the valuative criterion (Schemes, Definition \ref{schemes-definition-valuative-criterion}), \item given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute. \end{enumerate}


morphisms-lemma-refined-valuative-criterion-universally-closed

Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $f$ and $h$ are quasi-compact and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & S } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is universally closed. If moreover $f$ is quasi-separated, then $f$ is separated.


morphisms-lemma-morphism-defined-local-ring

Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $s \in S$ and $x \in X$, $y \in Y$ points over $s$. \begin{enumerate} \item Let $f, g : X \to Y$ be morphisms over $S$ such that $f(x) = g(x) = y$ and $f^\sharp_x = g^\sharp_x : \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$. Then there is an open neighbourhood $U \subset X$ with $f|_U = g|_U$ in the following cases \begin{enumerate} \item $Y$ is locally of finite type over $S$, \item $X$ is integral, \item $X$ is locally Noetherian, or \item $X$ is reduced with finitely many irreducible components. \end{enumerate} \item Let $\varphi : \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ be a local $\mathcal{O}_{S, s}$-algebra map. Then there exists an open neighbourhood $U \subset X$ of $x$ and a morphism $f : U \to Y$ mapping $x$ to $y$ with $f^\sharp_x = \varphi$ in the following cases \begin{enumerate} \item $Y$ is locally of finite presentation over $S$, \item $Y$ is locally of finite type and $X$ is integral, \item $Y$ is locally of finite type and $X$ is locally Noetherian, or \item $Y$ is locally of finite type and $X$ is reduced with finitely many irreducible components. \end{enumerate} \end{enumerate}


morphisms-lemma-extend-across

Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $x \in X$. Let $U \subset X$ be an open and let $f : U \to Y$ be a morphism over $S$. Assume \begin{enumerate} \item $x$ is in the closure of $U$, \item $X$ is reduced with finitely many irreducible components or $X$ is Noetherian, \item $\mathcal{O}_{X, x}$ is a valuation ring, \item $Y \to S$ is proper \end{enumerate} Then there exists an open $U \subset U' \subset X$ containing $x$ and an $S$-morphism $f' : U' \to Y$ extending $f$.


morphisms-lemma-H-projective

An H-projective morphism is H-quasi-projective. An H-projective morphism is projective.


morphisms-lemma-characterize-locally-projective

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is locally projective. \item There exists an open covering $S = \bigcup U_i$ such that each $f^{-1}(U_i) \to U_i$ is H-projective. \end{enumerate}


morphisms-lemma-locally-projective-proper

A locally projective morphism is proper.


morphisms-lemma-proper-ample-locally-projective

Let $f : X \to S$ be a proper morphism of schemes. If there exists an $f$-ample invertible sheaf on $X$, then $f$ is locally projective.


morphisms-lemma-H-projective-composition

A composition of H-projective morphisms is H-projective.


morphisms-lemma-H-projective-base-change

A base change of a H-projective morphism is H-projective.


morphisms-lemma-base-change-projective

A base change of a (locally) projective morphism is (locally) projective.


morphisms-lemma-projective-quasi-projective

A projective morphism is quasi-projective.


morphisms-lemma-H-quasi-projective-open-H-projective

Let $f : X \to S$ be a H-quasi-projective morphism. Then $f$ factors as $X \to X' \to S$ where $X \to X'$ is an open immersion and $X' \to S$ is H-projective.


morphisms-lemma-quasi-projective-open-projective

Let $f : X \to S$ be a quasi-projective morphism with $S$ quasi-compact and quasi-separated. Then $f$ factors as $X \to X' \to S$ where $X \to X'$ is an open immersion and $X' \to S$ is projective.


morphisms-lemma-projective-is-quasi-projective-proper

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a morphism of schemes. Then \begin{enumerate} \item $f$ is projective if and only if $f$ is quasi-projective and proper, and \item $f$ is H-projective if and only if $f$ is H-quasi-projective and proper. \end{enumerate}


morphisms-lemma-composition-projective

Let $f : X \to Y$ and $g : Y \to S$ be morphisms of schemes. If $S$ is quasi-compact and quasi-separated and $f$ and $g$ are projective, then $g \circ f$ is projective.


morphisms-lemma-projective-permanence

Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. If $g \circ f$ is projective and $g$ is separated, then $f$ is projective.


morphisms-lemma-projective-over-quasi-projective-is-H-projective

Let $S$ be a scheme which admits an ample invertible sheaf. Then \begin{enumerate} \item any projective morphism $X \to S$ is H-projective, and \item any quasi-projective morphism $X \to S$ is H-quasi-projective. \end{enumerate}


morphisms-lemma-proper-ample-is-proj

Let $f : X \to S$ be a universally closed morphism. Let $\mathcal{L}$ be an $f$-ample invertible $\mathcal{O}_X$-module. Then the canonical morphism $$ r : X \longrightarrow \underline{\text{Proj}}_S \left( \bigoplus\nolimits_{d \geq 0} f_*\mathcal{L}^{\otimes d} \right) $$ of Lemma \ref{lemma-characterize-relatively-ample} is an isomorphism.


morphisms-lemma-proper-ample-delete-affine

Let $f : X \to S$ be a universally closed morphism. Let $\mathcal{L}$ be an $f$-ample invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Then $X_s \to S$ is an affine morphism.


morphisms-lemma-finite-local

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is finite. \item There exists an affine open covering $S = \bigcup U_i$ such that each $f^{-1}(U_i)$ is affine and $\mathcal{O}_S(U_i) \to \mathcal{O}_X(f^{-1}(U_i))$ is finite. \item There exists an open covering $S = \bigcup U_i$ such that each $f^{-1}(U_i) \to U_i$ is finite. \end{enumerate} Moreover, if $f$ is finite then for every open subscheme $U \subset S$ the morphism $f : f^{-1}(U) \to U$ is finite.


morphisms-lemma-finite-integral

A finite morphism is integral. An integral morphism which is locally of finite type is finite.


morphisms-lemma-composition-finite

A composition of finite morphisms is finite. Same is true for integral morphisms.


morphisms-lemma-base-change-finite

A base change of a finite morphism is finite. Same is true for integral morphisms.


morphisms-lemma-integral-universally-closed

\begin{slogan} integral $=$ affine $+$ universally closed \end{slogan} Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is integral, and \item $f$ is affine and universally closed. \end{enumerate}


morphisms-lemma-integral-fibres

Let $f : X \to S$ be an integral morphism. Then every point of $X$ is closed in its fibre.


morphisms-lemma-integral-dimension

Let $f : X \to Y$ be an integral morphism. Then $\dim(X) \leq \dim(Y)$. If $f$ is surjective then $\dim(X) = \dim(Y)$.


morphisms-lemma-finite-quasi-finite

A finite morphism is quasi-finite.


morphisms-lemma-finite-proper

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


morphisms-lemma-closed-immersion-finite

A closed immersion is finite (and a fortiori integral).


morphisms-lemma-finite-union-finite

Let $X_i \to Y$, $i = 1, \ldots, n$ be finite morphisms of schemes. Then $X_1 \amalg \ldots \amalg X_n \to Y$ is finite too.


morphisms-lemma-finite-permanence

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms. \begin{enumerate} \item If $g \circ f$ is finite and $g$ separated then $f$ is finite. \item If $g \circ f$ is integral and $g$ separated then $f$ is integral. \end{enumerate}


morphisms-lemma-finite-monomorphism-closed

Let $f : X \to Y$ be a morphism of schemes. If $f$ is finite and a monomorphism, then $f$ is a closed immersion.


morphisms-lemma-finite-projective

A finite morphism is projective.


morphisms-lemma-base-change-universal-homeomorphism

The base change of a universal homeomorphism of schemes by any morphism of schemes is a universal homeomorphism.


morphisms-lemma-composition-universal-homeomorphism

The composition of a pair of universal homeomorphisms of schemes is a universal homeomorphism.


morphisms-lemma-homeomorphism-affine

Let $f : X \to Y$ be a morphism of schemes. If $f$ is a homeomorphism onto a closed subset of $Y$ then $f$ is affine.


morphisms-lemma-universal-homeomorphism

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is a universal homeomorphism, and \item $f$ is integral, universally injective and surjective. \end{enumerate}


morphisms-lemma-reduction-universal-homeomorphism

Let $X$ be a scheme. The canonical closed immersion $X_{red} \to X$ (see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}) is a universal homeomorphism.


morphisms-lemma-check-closed-infinitesimally

Let $f : X \to S$ and $S' \to S$ be morphisms of schemes. Assume \begin{enumerate} \item $S' \to S$ is a closed immersion, \item $S' \to S$ is bijective on points, \item $X \times_S S' \to S'$ is a closed immersion, and \item $X \to S$ is of finite type or $S' \to S$ is of finite presentation. \end{enumerate} Then $f : X \to S$ is a closed immersion.


morphisms-lemma-subalgebra-inherits

Let $A \to B$ be a ring map such that the induced morphism of schemes $f : \Spec(B) \to \Spec(A)$ is a universal homeomorphism, resp.\ a universal homeomorphism inducing isomorphisms on residue fields, resp.\ universally closed, resp.\ universally closed and universally injective. Then for any $A$-subalgebra $B' \subset B$ the same thing is true for $f' : \Spec(B') \to \Spec(A)$.


morphisms-lemma-colimit-inherits

Let $A$ be a ring. Let $B = \colim B_\lambda$ be a filtered colimit of $A$-algebras. If each $f_\lambda : \Spec(B_\lambda) \to \Spec(A)$ is a universal homeomorphism, resp.\ a universal homeomorphism inducing isomorphisms on residue fields, resp.\ universally closed, resp.\ universally closed and universally injective, then the same thing is true for $f : \Spec(B) \to \Spec(A)$.


morphisms-lemma-special-elements-and-localization

Let $A \subset B$ be a ring extension. Let $S \subset A$ be a multiplicative subset. Let $n \geq 1$ and $b_i \in B$ for $1 \leq i \leq n$. If the set $$ \{x \in S^{-1}B \mid x \not \in S^{-1}A\text{ and } b_i x^i \in S^{-1}A\text{ for }i = 1, \ldots, n\} $$ is nonempty, then so is $$ \{x \in B \mid x \not \in A\text{ and } b_i x^i \in A\text{ for }i = 1, \ldots, n\} $$


morphisms-lemma-nth-and-nplusone-implies-square-and-cube

Let $A \subset B$ be a ring extension. If there exists $b \in B$, $b \not \in A$ and an integer $n \geq 2$ with $b^n \in A$ and $b^{n + 1} \in A$, then there exists a $b' \in B$, $b' \not \in A$ with $(b')^2 \in A$ and $(b')^3 \in A$.


morphisms-lemma-square-and-cube

Let $A \subset B$ be a ring extension such that $\Spec(B) \to \Spec(A)$ is a universal homeomorphism inducing isomorphisms on residue fields. If $A \not = B$, then there exists a $b \in B$, $b \not \in A$ with $b^2 \in A$ and $b^3 \in A$.


morphisms-lemma-pth-power-and-multiple

Let $A \subset B$ be a ring extension such that $\Spec(B) \to \Spec(A)$ is a universal homeomorphism. If $A \not = B$, then either there exists a $b \in B$, $b \not \in A$ with $b^2 \in A$ and $b^3 \in A$ or there exists a prime number $p$ and a $b \in B$, $b \not \in A$ with $pb \in A$ and $b^p \in A$.


morphisms-lemma-make-universal-homeo

Let $A$ be a ring. Let $x, y \in A$. \begin{enumerate} \item If $x^3 = y^2$ in $A$, then $A \to B = A[t]/(t^2 - x, t^3 - y)$ induces bijections on residue fields and a universal homeomorphism on spectra. \item If there is a prime number $p$ such that $p^px = y^p$ in $A$, then $A \to B = A[t]/(t^p - x, pt - y)$ induces a universal homeomorphism on spectra. \end{enumerate}


morphisms-lemma-universal-homeo-limit

Let $A \to B$ be a ring map. \begin{enumerate} \item If $A \to B$ induces a universal homeomorphism on spectra, then $B = \colim B_i$ is a filtered colimit of finitely presented $A$-algebras $B_i$ such that $A \to B_i$ induces a universal homeomorphism on spectra. \item If $A \to B$ induces isomorphisms on residue fields and a universal homeomorphism on spectra, then $B = \colim B_i$ is a filtered colimit of finitely presented $A$-algebras $B_i$ such that $A \to B_i$ induces isomorphisms on residue fields and a universal homeomorphism on spectra. \end{enumerate}


morphisms-lemma-seminormal-local-property

Being seminormal or being absolutely weakly normal is a local property of rings, see Properties, Definition \ref{properties-definition-property-local}.


morphisms-lemma-locally-seminormal

Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is seminormal. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is seminormal. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is seminormal. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is seminormal. \end{enumerate} Moreover, if $X$ is seminormal then every open subscheme is seminormal. The same statements are true with ``seminormal'' replaced by ``absolutely weakly normal''.


morphisms-lemma-seminormal-reduced

A seminormal scheme or ring is reduced. A fortiori the same is true for absolutely weakly normal schemes or rings.


morphisms-lemma-seminormalization-ring

Let $A$ be a ring. \begin{enumerate} \item The category of ring maps $A \to B$ inducing a universal homeomorphism on spectra has a final object $A \to A^{awn}$. \item Given $A \to B$ in the category of (1) the resulting map $B \to A^{awn}$ is an isomorphism if and only if $B$ is absolutely weakly normal. \item The category of ring maps $A \to B$ inducing isomorphisms on residue fields and a universal homeomorphism on spectra has a final object $A \to A^{sn}$. \item Given $A \to B$ in the category of (3) the resulting map $B \to A^{sn}$ is an isomorphism if and only if $B$ is seminormal. \end{enumerate} For any ring map $\varphi : A \to A'$ there are unique maps $\varphi^{awn} : A^{awn} \to (A')^{awn}$ and $\varphi^{sn} : A^{sn} \to (A')^{sn}$ compatible with $\varphi$.


morphisms-lemma-seminormalization

Let $X$ be a scheme. \begin{enumerate} \item The category of universal homeomorphisms $Y \to X$ has an initial object $X^{awn} \to X$. \item Given $Y \to X$ in the category of (1) the resulting morphism $X^{awn} \to Y$ is an isomorphism if and only if $Y$ is absolutely weakly normal. \item The category of universal homeomorphisms $Y \to X$ which induce ismomorphisms on residue fields has an initial object $X^{sn} \to X$. \item Given $Y \to X$ in the category of (3) the resulting morphism $X^{sn} \to Y$ is an isomorphism if and only if $Y$ is seminormal. \end{enumerate} For any morphism $h : X' \to X$ of schemes there are unique morphisms $h^{awn} : (X')^{awn} \to X^{awn}$ and $h^{sn} : (X')^{sn} \to X^{sn}$ compatible with $h$.


morphisms-lemma-finite-flat

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is finite locally free, \item $f$ is finite, flat, and locally of finite presentation. \end{enumerate} If $S$ is locally Noetherian these are also equivalent to \begin{enumerate} \item[(3)] $f$ is finite and flat. \end{enumerate}


morphisms-lemma-composition-finite-locally-free

A composition of finite locally free morphisms is finite locally free.


morphisms-lemma-base-change-finite-locally-free

A base change of a finite locally free morphism is finite locally free.


morphisms-lemma-finite-locally-free

Let $f : X \to S$ be a finite locally free morphism of schemes. There exists a disjoint union decomposition $S = \coprod_{d \geq 0} S_d$ by open and closed subschemes such that setting $X_d = f^{-1}(S_d)$ the restrictions $f|_{X_d}$ are finite locally free morphisms $X_d \to S_d$ of degree $d$.


morphisms-lemma-massage-finite

Let $f : Y \to X$ be a finite morphism with $X$ affine. There exists a diagram $$ \xymatrix{ Z' \ar[rd] & Y' \ar[l]^i \ar[d] \ar[r] & Y \ar[d] \\  & X' \ar[r] & X } $$ where \begin{enumerate} \item $Y' \to Y$ and $X' \to X$ are surjective finite locally free, \item $Y' = X' \times_X Y$, \item $i : Y' \to Z'$ is a closed immersion, \item $Z' \to X'$ is finite locally free, and \item $Z' = \bigcup_{j = 1, \ldots, m} Z'_j$ is a (set theoretic) finite union of closed subschemes, each of which maps isomorphically to $X'$. \end{enumerate}


morphisms-lemma-image-nowhere-dense-finite

Let $f : Y \to X$ be a finite morphism of schemes. Let $T \subset Y$ be a closed nowhere dense subset of $Y$. Then $f(T) \subset X$ is a closed nowhere dense subset of $X$.


morphisms-lemma-rational-map-finite-presentation

Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Assume $X$ has finitely many irreducible components with generic points $x_1, \ldots, x_n$. Let $s_i \in S$ be the image of $x_i$. Consider the map $$ \left\{ \begin{matrix} S\text{-rational maps} \\ \text{from }X\text{ to }Y \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} (y_1, \varphi_1, \ldots, y_n, \varphi_n)\text{ where } y_i \in Y\text{ lies over }s_i\text{ and}\\ \varphi_i : \mathcal{O}_{Y, y_i} \to \mathcal{O}_{X, x_i} \text{ is a local }\mathcal{O}_{S, s_i}\text{-algebra map} \end{matrix} \right\} $$ which sends $f : U \to Y$ to the $2n$-tuple with $y_i = f(x_i)$ and $\varphi_i = f^\sharp_{x_i}$. Then \begin{enumerate} \item If $Y \to S$ is locally of finite type, then the map is injective. \item If $Y \to S$ is locally of finite presentation, then the map is bijective. \item If $Y \to S$ is locally of finite type and $X$ reduced, then the map is bijective. \end{enumerate}


morphisms-lemma-integral-scheme-rational-functions

Let $X$ be a scheme with finitely many irreducible components $X_1, \ldots, X_n$. If $\eta_i \in X_i$ is the generic point, then $$ R(X) = \mathcal{O}_{X, \eta_1} \times \ldots \times \mathcal{O}_{X, \eta_n} $$ If $X$ is reduced this is equal to $\prod \kappa(\eta_i)$. If $X$ is integral then $R(X) = \mathcal{O}_{X, \eta} = \kappa(\eta)$ is a field.


morphisms-lemma-distinct-local-rings

Let $X$ be an integral separated scheme. Let $Z_1$, $Z_2$ be distinct irreducible closed subsets of $X$. Let $\eta_i$ be the generic point of $Z_i$. If $Z_1 \not\subset Z_2$, then $\mathcal{O}_{X, \eta_1} \not \subset \mathcal{O}_{X, \eta_2}$ as subrings of $R(X)$. In particular, if $Z_1 = \{x\}$ consists of one closed point $x$, there exists a function regular in a neighborhood of $x$ which is not in $\mathcal{O}_{X, \eta_{2}}$.


morphisms-lemma-rational-map-from-reduced-to-separated

Let $X$ and $Y$ be schemes. Assume $X$ reduced and $Y$ separated. Let $\varphi$ be a rational map from $X$ to $Y$ with domain of definition $U \subset X$. Then there exists a unique morphism $f : U \to Y$ representing $\varphi$. If $X$ and $Y$ are schemes over a separated scheme $S$ and if $\varphi$ is an $S$-rational map, then $f$ is a morphism over $S$.


morphisms-lemma-birational-integral

Let $X$ and $Y$ be irreducible schemes. \begin{enumerate} \item The schemes $X$ and $Y$ are birational if and only if they have isomorphic nonempty opens. \item Assume $X$ and $Y$ are schemes over a base scheme $S$. Then $X$ and $Y$ are $S$-birational if and only if there are nonempty opens $U \subset X$ and $V \subset Y$ which are $S$-isomorphic. \end{enumerate}


morphisms-lemma-birational-generic-fibres

Let $f : X \to Y$ be a birational morphism of schemes having finitely many irreducible components. If $y \in Y$ is the generic point of an irreducible component, then the base change $X \times_Y \Spec(\mathcal{O}_{Y, y}) \to \Spec(\mathcal{O}_{Y, y})$ is an isomorphism.


morphisms-lemma-birational-birational

Let $f : X \to Y$ be a birational morphism of schemes having finitely many irreducible components over a base scheme $S$. Assume one of the following conditions is satisfied \begin{enumerate} \item $f$ is locally of finite type and $Y$ reduced, \item $f$ is locally of finite presentation. \end{enumerate} Then there exist dense opens $U \subset X$ and $V \subset Y$ such that $f(U) \subset V$ and $f|_U : U \to V$ is an isomorphism. In particular if $X$ and $Y$ are irreducible, then $X$ and $Y$ are $S$-birational.


morphisms-lemma-criterion-birational-finite-presentation

Let $S$ be a scheme. Let $X$ and $Y$ be irreducible schemes locally of finite presentation over $S$. Let $x \in X$ and $y \in Y$ be the generic points. The following are equivalent \begin{enumerate} \item $X$ and $Y$ are $S$-birational, \item there exist nonempty opens of $X$ and $Y$ which are $S$-isomorphic, and \item $x$ and $y$ map to the same point $s$ of $S$ and $\mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ are isomorphic as $\mathcal{O}_{S, s}$-algebras. \end{enumerate}


morphisms-lemma-common-open

Let $S$ be a scheme. Let $X$ and $Y$ be integral schemes locally of finite type over $S$. Let $x \in X$ and $y \in Y$ be the generic points. The following are equivalent \begin{enumerate} \item $X$ and $Y$ are $S$-birational, \item there exist nonempty opens of $X$ and $Y$ which are $S$-isomorphic, and \item $x$ and $y$ map to the same point $s \in S$ and $\kappa(x) \cong \kappa(y)$ as $\kappa(s)$-extensions. \end{enumerate}


morphisms-lemma-generically-finite

Let $X$, $Y$ be schemes. Let $f : X \to Y$ be locally of finite type. Let $\eta \in Y$ be a generic point of an irreducible component of $Y$. The following are equivalent: \begin{enumerate} \item the set $f^{-1}(\{\eta\})$ is finite, \item there exist affine opens $U_i \subset X$, $i = 1, \ldots, n$ and $V \subset Y$ with $f(U_i) \subset V$, $\eta \in V$ and $f^{-1}(\{\eta\}) \subset \bigcup U_i$ such that each $f|_{U_i} : U_i \to V$ is finite. \end{enumerate} If $f$ is quasi-separated, then these are also equivalent to \begin{enumerate} \item[(3)] there exist affine opens $V \subset Y$, and $U \subset X$ with $f(U) \subset V$, $\eta \in V$ and $f^{-1}(\{\eta\}) \subset U$ such that $f|_U : U \to V$ is finite. \end{enumerate} If $f$ is quasi-compact and quasi-separated, then these are also equivalent to \begin{enumerate} \item[(4)] there exists an affine open $V \subset Y$, $\eta \in V$ such that $f^{-1}(V) \to V$ is finite. \end{enumerate}


morphisms-lemma-quasi-finiteness-over-generic-point

Let $X$, $Y$ be schemes. Let $f : X \to Y$ be locally of finite type. Let $X^0$, resp.\ $Y^0$ denote the set of generic points of irreducible components of $X$, resp.\ $Y$. Let $\eta \in Y^0$. The following are equivalent \begin{enumerate} \item $f^{-1}(\{\eta\}) \subset X^0$, \item $f$ is quasi-finite at all points lying over $\eta$, \item $f$ is quasi-finite at all $\xi \in X^0$ lying over $\eta$. \end{enumerate}


morphisms-lemma-finite-over-dense-open

Let $X$, $Y$ be schemes. Let $f : X \to Y$ be locally of finite type. Let $X^0$, resp.\ $Y^0$ denote the set of generic points of irreducible components of $X$, resp.\ $Y$. Assume \begin{enumerate} \item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$, \item either $f$ is quasi-compact or $f$ is separated. \end{enumerate} Then there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite.


morphisms-lemma-birational-isomorphism-over-dense-open

Let $X$, $Y$ be schemes. Let $f : X \to Y$ be a birational morphism between schemes which have finitely many irreducible components. Assume \begin{enumerate} \item either $f$ is quasi-compact or $f$ is separated, and \item either $f$ is locally of finite type and $Y$ is reduced or $f$ is locally of finite presentation. \end{enumerate} Then there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is an isomorphism.


morphisms-lemma-finite-degree

Let $X$, $Y$ be integral schemes. Let $f : X \to Y$ be locally of finite type. Assume $f$ is dominant. The following are equivalent: \begin{enumerate} \item the extension $R(Y) \subset R(X)$ has transcendence degree $0$, \item the extension $R(Y) \subset R(X)$ is finite, \item there exist nonempty affine opens $U \subset X$ and $V \subset Y$ such that $f(U) \subset V$ and $f|_U : U \to V$ is finite, and \item the generic point of $X$ is the only point of $X$ mapping to the generic point of $Y$. \end{enumerate} If $f$ is separated or if $f$ is quasi-compact, then these are also equivalent to \begin{enumerate} \item[(5)] there exists a nonempty affine open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite. \end{enumerate}


morphisms-lemma-degree-composition

Let $X$, $Y$, $Z$ be integral schemes. Let $f : X \to Y$ and $g : Y \to Z$ be dominant morphisms locally of finite type. Assume that $[R(X) : R(Y)] < \infty$ and $[R(Y) : R(Z)] < \infty$. Then $$ \deg(X/Z) = \deg(X/Y) \deg(Y/Z). $$


morphisms-lemma-dimension-formula

Let $S$ be a scheme. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$, and set $s = f(x)$. Assume \begin{enumerate} \item $S$ is locally Noetherian, \item $f$ is locally of finite type, \item $X$ and $S$ integral, and \item $f$ dominant. \end{enumerate} We have \begin{equation} \label{equation-dimension-formula} \dim(\mathcal{O}_{X, x}) \leq \dim(\mathcal{O}_{S, s}) + \text{trdeg}_{R(S)}R(X) - \text{trdeg}_{\kappa(s)} \kappa(x). \end{equation} Moreover, equality holds if $S$ is universally catenary.


morphisms-lemma-dimension-formula-general

Let $S$ be a scheme. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$, and set $s = f(x)$. Assume $S$ is locally Noetherian and $f$ is locally of finite type, We have \begin{equation} \label{equation-dimension-formula-general} \dim(\mathcal{O}_{X, x}) \leq \dim(\mathcal{O}_{S, s}) + E - \text{trdeg}_{\kappa(s)} \kappa(x). \end{equation} where $E$ is the maximum of $\text{trdeg}_{\kappa(f(\xi))}(\kappa(\xi))$ where $\xi$ runs over the generic points of irreducible components of $X$ containing $x$.


morphisms-lemma-dimension-function-propagates

Let $S$ be a locally Noetherian and universally catenary scheme. Let $\delta : S \to \mathbf{Z}$ be a dimension function. Let $f : X \to S$ be a morphism of schemes. Assume $f$ locally of finite type. Then the map \begin{align*} \delta = \delta_{X/S} : X & \longrightarrow \mathbf{Z} \\ x & \longmapsto \delta(f(x)) + \text{trdeg}_{\kappa(f(x))} \kappa(x) \end{align*} is a dimension function on $X$.


morphisms-lemma-alteration-dimension

Let $f : X \to Y$ be a morphism of schemes. Assume that \begin{enumerate} \item $Y$ is locally Noetherian, \item $X$ and $Y$ are integral schemes, \item $f$ is dominant, and \item $f$ is locally of finite type. \end{enumerate} Then we have $$ \dim(X) \leq \dim(Y) + \text{trdeg}_{R(Y)} R(X). $$ If $f$ is closed\footnote{For example if $f$ is proper, see Definition \ref{definition-proper}.} then equality holds.


morphisms-lemma-integral-closure

Let $X$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. The subsheaf $\mathcal{A}' \subset \mathcal{A}$ defined by the rule $$ U \longmapsto \{f \in \mathcal{A}(U) \mid f_x \in \mathcal{A}_x \text{ integral over } \mathcal{O}_{X, x} \text{ for all }x \in U\} $$ is a quasi-coherent $\mathcal{O}_X$-algebra, the stalk $\mathcal{A}'_x$ is the integral closure of $\mathcal{O}_{X, x}$ in $\mathcal{A}_x$, and for any affine open $U \subset X$ the ring $\mathcal{A}'(U) \subset \mathcal{A}(U)$ is the integral closure of $\mathcal{O}_X(U)$ in $\mathcal{A}(U)$.


morphisms-lemma-characterize-normalization

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. The factorization $f = \nu \circ f'$, where $\nu : X' \to X$ is the normalization of $X$ in $Y$ is characterized by the following two properties: \begin{enumerate} \item the morphism $\nu$ is integral, and \item for any factorization $f = \pi \circ g$, with $\pi : Z \to X$ integral, there exists a commutative diagram $$ \xymatrix{ Y \ar[d]_{f'} \ar[r]_g & Z \ar[d]^\pi \\ X' \ar[ru]^h \ar[r]^\nu & X } $$ for some unique morphism $h : X' \to Z$. \end{enumerate} Moreover, the morphism $f' : Y \to X'$ is dominant and in (2) the morphism $h : X' \to Z$ is the normalization of $Z$ in $Y$.


morphisms-lemma-functoriality-normalization

Let $$ \xymatrix{ Y_2 \ar[d]_{f_2} \ar[r] & Y_1 \ar[d]^{f_1} \\ X_2 \ar[r] & X_1 } $$ be a commutative diagram of morphisms of schemes. Assume $f_1$, $f_2$ quasi-compact and quasi-separated. Let $f_i = \nu_i \circ f_i'$, $i = 1, 2$ be the canonical factorizations, where $\nu_i : X_i' \to X_i$ is the normalization of $X_i$ in $Y_i$. Then there exists a unique arrow $X'_2 \to X'_1$ fitting into a commutative diagram $$ \xymatrix{ Y_2 \ar[d]_{f_2'} \ar[r] & Y_1 \ar[d]^{f_1'} \\ X_2' \ar[d]_{\nu_2} \ar[r] & X_1' \ar[d]^{\nu_1} \\ X_2 \ar[r] & X_1 } $$


morphisms-lemma-normalization-localization

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $U \subset X$ be an open subscheme and set $V = f^{-1}(U)$. Then the normalization of $U$ in $V$ is the inverse image of $U$ in the normalization of $X$ in $Y$.


morphisms-lemma-normalization-is-normalization

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X'$ be the normalization of $X$ in $Y$. Then the normalization of $X'$ in $Y$ is $X'$.


morphisms-lemma-normalization-in-reduced

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so is $X'$.


morphisms-lemma-normalization-generic

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \to X$ be the normalization of $X$ in $Y$. Every generic point of an irreducible component of $X'$ is the image of a generic point of an irreducible component of $Y$.


morphisms-lemma-normalization-in-disjoint-union

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Suppose that $Y = Y_1 \amalg Y_2$ is a disjoint union of two schemes. Write $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$. Then $X_1' \amalg X_2'$ is the normalization of $X$ in $Y$.


morphisms-lemma-normalization-in-universally-closed

Let $f : X \to S$ be a quasi-compact, quasi-separated and universally closed morphisms of schemes. Then $f_*\mathcal{O}_X$ is integral over $\mathcal{O}_S$. In other words, the normalization of $S$ in $X$ is equal to the factorization $$ X \longrightarrow \underline{\Spec}_S(f_*\mathcal{O}_X) \longrightarrow S $$ of Constructions, Lemma \ref{constructions-lemma-canonical-morphism}.


morphisms-lemma-normalization-in-integral

Let $f : Y \to X$ be an integral morphism. Then the normalization of $X$ in $Y$ is equal to $Y$.


morphisms-lemma-normal-normalization

Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X'$ be the normalization of $X$ in $Y$. Assume \begin{enumerate} \item $Y$ is a normal scheme, \item quasi-compact opens of $Y$ have finitely many irreducible components. \end{enumerate} Then $X'$ is a disjoint union of integral normal schemes. Moreover, the morphism $Y \to X'$ is dominant and induces a bijection of irreducible components.


morphisms-lemma-nagata-normalization-finite-general

Let $f : X \to S$ be a morphism. Assume that \begin{enumerate} \item $S$ is a Nagata scheme, \item $f$ is quasi-compact and quasi-separated, \item quasi-compact opens of $X$ have finitely many irreducible components, \item if $x \in X$ is a generic point of an irreducible component, then the field extension $\kappa(f(x)) \subset \kappa(x)$ is finitely generated, and \item $X$ is reduced. \end{enumerate} Then the normalization $\nu : S' \to S$ of $S$ in $X$ is finite.


morphisms-lemma-nagata-normalization-finite

Let $f : X \to S$ be a morphism. Assume that \begin{enumerate} \item $S$ is a Nagata scheme, \item $f$ is of finite type, \item $X$ is reduced. \end{enumerate} Then the normalization $\nu : S' \to S$ of $S$ in $X$ is finite.


morphisms-lemma-relative-normalization-normal-codim-1

Let $f : Y \to X$ be a finite type morphism of schemes with $Y$ reduced and $X$ Nagata. Let $X'$ be the normalization of $X$ in $Y$. Let $x' \in X'$ be a point such that \begin{enumerate} \item $\dim(\mathcal{O}_{X', x'}) = 1$, and \item the fibre of $Y \to X'$ over $x'$ is empty. \end{enumerate} Then $\mathcal{O}_{X', x'}$ is a discrete valuation ring.


morphisms-lemma-normalization-reduced

Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. The normalization morphism $\nu$ factors through the reduction $X_{red}$ and $X^\nu \to X_{red}$ is the normalization of $X_{red}$.


morphisms-lemma-description-normalization

Let $X$ be a reduced scheme such that every quasi-compact open has finitely many irreducible components. Let $\Spec(A) = U \subset X$ be an affine open. Then \begin{enumerate} \item $A$ has finitely many minimal primes $\mathfrak q_1, \ldots, \mathfrak q_t$, \item the total ring of fractions $Q(A)$ of $A$ is $Q(A/\mathfrak q_1) \times \ldots \times Q(A/\mathfrak q_t)$, \item the integral closure $A'$ of $A$ in $Q(A)$ is the product of the integral closures of the domains $A/\mathfrak q_i$ in the fields $Q(A/\mathfrak q_i)$, and \item $\nu^{-1}(U)$ is identified with the spectrum of $A'$ where $\nu : X^\nu \to X$ is the normalization morphism. \end{enumerate}


morphisms-lemma-stalk-normalization

Let $X$ be a scheme such that every quasi-compact open has a finite number of irreducible components. Let $\nu : X^\nu \to X$ be the normalization of $X$. Let $x \in X$. Then the following are canonically isomorphic as $\mathcal{O}_{X, x}$-algebras \begin{enumerate} \item the stalk $(\nu_*\mathcal{O}_{X^\nu})_x$, \item the integral closure of $\mathcal{O}_{X, x}$ in the total ring of fractions of $(\mathcal{O}_{X, x})_{red}$, \item the integral closure of $\mathcal{O}_{X, x}$ in the product of the residue fields of the minimal primes of $\mathcal{O}_{X, x}$ (and there are finitely many of these). \end{enumerate}


morphisms-lemma-normalization-normal

Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. \begin{enumerate} \item The normalization $X^\nu$ is a disjoint union of integral normal schemes. \item The morphism $\nu : X^\nu \to X$ is integral, surjective, and induces a bijection on irreducible components. \item For any integral morphism $\alpha : X' \to X$ such that for $U \subset X$ quasi-compact open the inverse image $\alpha^{-1}(U)$ has finitely many irreducible components and $\alpha|_{\alpha^{-1}(U)} : \alpha^{-1}(U) \to U$ is birational\footnote{This awkward formulation is necessary as we've only defined what it means for a morphism to be birational if the source and target have finitely many irreducible components. It suffices if $X'_{red} \to X_{red}$ satisfies the condition.} there exists a factorization $X^\nu \to X' \to X$ and $X^\nu \to X'$ is the normalization of $X'$. \item For any morphism $Z \to X$ with $Z$ a normal scheme such that each irreducible component of $Z$ dominates an irreducible component of $X$ there exists a unique factorization $Z \to X^\nu \to X$. \end{enumerate}


morphisms-lemma-normalization-in-terms-of-components

Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. Let $Z_i \subset X$, $i \in I$ be the irreducible components of $X$ endowed with the reduced induced structure. Let $Z_i^\nu \to Z_i$ be the normalization. Then $\coprod_{i \in I} Z_i^\nu \to X$ is the normalization of $X$.


morphisms-lemma-normalization-birational

Let $X$ be a reduced scheme with finitely many irreducible components. Then the normalization morphism $X^\nu \to X$ is birational.


morphisms-lemma-finite-birational-over-normal

A finite (or even integral) birational morphism $f : X \to Y$ of integral schemes with $Y$ normal is an isomorphism.


morphisms-lemma-Japanese-normalization

Let $X$ be an integral, Japanese scheme. The normalization $\nu : X^\nu \to X$ is a finite morphism.


morphisms-lemma-nagata-normalization

Let $X$ be a Nagata scheme. The normalization $\nu : X^\nu \to X$ is a finite morphism.


morphisms-lemma-quasi-finite-points-open

\begin{slogan} The locally quasi-finite locus of a morphism is open \end{slogan} Let $f : X \to S$ be a morphism of schemes. The set of points of $X$ where $f$ is quasi-finite is an open $U \subset X$. The induced morphism $U \to S$ is locally quasi-finite.


morphisms-lemma-quasi-finite-affine

Let $f : Y \to X$ be a morphism of schemes. Assume \begin{enumerate} \item $X$ and $Y$ are affine, and \item $f$ is quasi-finite. \end{enumerate} Then there exists a diagram $$ \xymatrix{ Y \ar[rd]_f \ar[rr]_j & & Z \ar[ld]^\pi \\ & X & } $$ with $Z$ affine, $\pi$ finite and $j$ an open immersion.


morphisms-lemma-characterize-universally-bounded

Let $f : X \to Y$ be a morphism of schemes. Let $n \geq 0$. The following are equivalent: \begin{enumerate} \item the integer $n$ bounds the degrees of the fibres of $f$, and \item for every morphism $\Spec(k) \to Y$, where $k$ is a field, the fibre product $X_k = \Spec(k) \times_Y X$ is finite over $k$ of degree $\leq n$. \end{enumerate} In this case the fibres of $f$ are universally bounded and the schemes $X_k$ have at most $n$ points. More precisely, if $X_k = \{x_1, \ldots, x_t\}$, then we have $$ n \geq \sum\nolimits_{i = 1, \ldots, t} [\kappa(x_i) : k] $$


morphisms-lemma-finite-locally-free-universally-bounded

If $f$ is a finite locally free morphism of degree $d$, then $d$ bounds the degree of the fibres of $f$.


morphisms-lemma-composition-universally-bounded

A composition of morphisms with universally bounded fibres is a morphism with universally bounded fibres. More precisely, assume that $n$ bounds the degrees of the fibres of $f : X \to Y$ and $m$ bounds the degrees of $g : Y \to Z$. Then $nm$ bounds the degrees of the fibres of $g \circ f : X \to Z$.


morphisms-lemma-base-change-universally-bounded

A base change of a morphism with universally bounded fibres is a morphism with universally bounded fibres. More precisely, if $n$ bounds the degrees of the fibres of $f : X \to Y$ and $Y' \to Y$ is any morphism, then the degrees of the fibres of the base change $f' : Y' \times_Y X \to Y'$ is also bounded by $n$.


morphisms-lemma-descent-universally-bounded

Let $f : X \to Y$ be a morphism of schemes. Let $Y' \to Y$ be a morphism of schemes, and let $f' : X' = X_{Y'} \to Y'$ be the base change of $f$. If $Y' \to Y$ is surjective and $f'$ has universally bounded fibres, then $f$ has universally bounded fibres. More precisely, if $n$ bounds the degree of the fibres of $f'$, then also $n$ bounds the degrees of the fibres of $f$.


morphisms-lemma-immersion-universally-bounded

An immersion has universally bounded fibres.


morphisms-lemma-etale-universally-bounded

Let $f : X \to Y$ be an \'etale morphism of schemes. Let $n \geq 0$. The following are equivalent \begin{enumerate} \item the integer $n$ bounds the degrees of the fibres, \item for every field $k$ and morphism $\Spec(k) \to Y$ the base change $X_k = \Spec(k) \times_Y X$ has at most $n$ points, and \item for every $y \in Y$ and every separable algebraic closure $\kappa(y) \subset \kappa(y)^{sep}$ the scheme $X_{\kappa(y)^{sep}}$ has at most $n$ points. \end{enumerate}


morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded

Let $f : X \to Y$ be a morphism of schemes. Assume that \begin{enumerate} \item $f$ is locally quasi-finite, and \item $X$ is quasi-compact. \end{enumerate} Then $f$ has universally bounded fibres.


morphisms-lemma-universally-bounded-permanence

Consider a commutative diagram of morphisms of schemes $$ \xymatrix{ X \ar[rd]_g \ar[rr]_f & & Y \ar[ld]^h \\ & Z & } $$ If $g$ has universally bounded fibres, and $f$ is surjective and flat, then also $h$ has universally bounded fibres. More precisely, if $n$ bounds the degree of the fibres of $g$, then also $n$ bounds the degree of the fibres of $h$.


morphisms-proposition-generic-flatness

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $S$ is integral, \item $f$ is of finite type, and \item $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module. \end{enumerate} Then there exists an open dense subscheme $U \subset S$ such that $X_U \to U$ is flat and of finite presentation and such that $\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation over $\mathcal{O}_{X_U}$.


morphisms-proposition-generic-flatness-reduced

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $S$ is reduced, \item $f$ is of finite type, and \item $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module. \end{enumerate} Then there exists an open dense subscheme $U \subset S$ such that $X_U \to U$ is flat and of finite presentation and such that $\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation over $\mathcal{O}_{X_U}$.


morphisms-proposition-universal-homeomorphism-equal-residue-fields

Let $A \subset B$ be a ring extension. The following are equivalent \begin{enumerate} \item $\Spec(B) \to \Spec(A)$ is a universal homeomorphism inducing isomorphisms on residue fields, and \item every finite subset $E \subset B$ is contained in an extension $$ A[b_1, \ldots, b_n] \subset B $$ such that $b_i^2, b_i^3 \in A[b_1, \ldots, b_{i - 1}]$ for $i = 1, \ldots, n$. \end{enumerate}


morphisms-proposition-universal-homeomorphism

Let $A \subset B$ be a ring extension. The following are equivalent \begin{enumerate} \item $\Spec(B) \to \Spec(A)$ is a universal homeomorphism, and \item every finite subset $E \subset B$ is contained in an extension $$ A[b_1, \ldots, b_n] \subset B $$ such that for $i = 1, \ldots, n$ we have \begin{enumerate} \item $b_i^2, b_i^3 \in A[b_1, \ldots, b_{i - 1}]$, or \item there exists a prime number $p$ with $pb_i, b_i^p \in A[b_1, \ldots, b_{i - 1}]$. \end{enumerate} \end{enumerate}


smoothing-theorem-popescu

Any regular homomorphism of Noetherian rings is a filtered colimit of smooth ring maps.


smoothing-theorem-approximation-property

Let $R$ be a Noetherian local ring. Let $f_1, \ldots, f_m \in R[x_1, \ldots, x_n]$. Suppose that $(a_1, \ldots, a_n) \in (R^\wedge)^n$ is a solution in $R^\wedge$. If $R$ is a henselian G-ring, then for every integer $N$ there exists a solution $(b_1, \ldots, b_n) \in R^n$ in $R$ such that $a_i - b_i \in \mathfrak m^NR^\wedge$.


smoothing-theorem-approximation-property-variant

Let $R$ be a Noetherian local ring. Let $f_1, \ldots, f_m \in R[x_1, \ldots, x_n]$. Suppose that $(a_1, \ldots, a_n) \in (R^\wedge)^n$ is a solution. If $R$ is a G-ring, then for every integer $N$ there exist \begin{enumerate} \item an \'etale ring map $R \to R'$, \item a maximal ideal $\mathfrak m' \subset R'$ lying over $\mathfrak m$ \item a solution $(b_1, \ldots, b_n) \in (R')^n$ in $R'$ \end{enumerate} such that $\kappa(\mathfrak m) = \kappa(\mathfrak m')$ and $a_i - b_i \in (\mathfrak m')^NR^\wedge$.


smoothing-lemma-find-strictly-standard

Let $R$ be a ring. Let $A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Let $\mathfrak q \subset A$ be a prime ideal. Assume $R \to A$ is smooth at $\mathfrak q$. Then there exists an $a \in A$, $a \not \in \mathfrak q$, an integer $c$, $0 \leq c \leq \min(n, m)$, subsets $U \subset \{1, \ldots, n\}$, $V \subset \{1, \ldots, m\}$ of cardinality $c$ such that $$ a = a' \det(\partial f_j/\partial x_i)_{j \in V, i \in U} $$ for some $a' \in A$ and $$ a f_\ell \in (f_j, j \in V) + (f_1, \ldots, f_m)^2 $$ for all $\ell \in \{1, \ldots, m\}$.


smoothing-lemma-elkik

Let $R \to A$ be a ring map of finite presentation. The singular ideal $H_{A/R}$ is the radical of the ideal generated by strictly standard elements in $A$ over $R$ and also the radical of the ideal generated by elementary standard elements in $A$ over $R$.


smoothing-lemma-strictly-standard-base-change

Let $R \to A$ be a ring map of finite presentation. Let $R \to R'$ be a ring map. If $a \in A$ is elementary, resp.\ strictly standard in $A$ over $R$, then $a \otimes 1$ is elementary, resp.\ strictly standard in $A \otimes_R R'$ over $R'$.


smoothing-lemma-final-solve

Let $R \to A \to \Lambda$ be ring maps with $A$ of finite presentation over $R$. Assume that $H_{A/R} \Lambda = \Lambda$. Then there exists a factorization $A \to B \to \Lambda$ with $B$ smooth over $R$.


smoothing-lemma-improve-presentation

Let $R$ be a ring and let $A$ be a finitely presented $R$-algebra. There exists finite type $R$-algebra map $A \to C$ which has a retraction with the following two properties \begin{enumerate} \item for each $a \in A$ such that $R \to A_a$ is a local complete intersection (More on Algebra, Definition \ref{more-algebra-definition-local-complete-intersection}) the ring $C_a$ is smooth over $A_a$ and has a presentation $C_a = R[y_1, \ldots, y_m]/J$ such that $J/J^2$ is free over $C_a$, and \item for each $a \in A$ such that $A_a$ is smooth over $R$ the module $\Omega_{C_a/R}$ is free over $C_a$. \end{enumerate}


smoothing-lemma-syntomic-complete-intersection

Let $R \to A$ be a syntomic ring map. Then there exists a smooth $R$-algebra map $A \to C$ with a retraction such that $C$ is a global relative complete intersection over $R$, i.e., $$ C \cong R[x_1, \ldots, x_n]/(f_1, \ldots, f_c) $$ flat over $R$ and all fibres of dimension $n - c$.


smoothing-lemma-smooth-standard-smooth

Let $R \to A$ be a smooth ring map. Then there exists a smooth $R$-algebra map $A \to B$ with a retraction such that $B$ is standard smooth over $R$, i.e., $$ B \cong R[x_1, \ldots, x_n]/(f_1, \ldots, f_c) $$ and $\det(\partial f_j/\partial x_i)_{i, j = 1, \ldots, c}$ is invertible in $B$.


smoothing-lemma-colimit-standard-smooth

Let $R \to \Lambda$ be a ring map. If $\Lambda$ is a filtered colimit of smooth $R$-algebras, then $\Lambda$ is a filtered colimit of standard smooth $R$-algebras.


smoothing-lemma-standard-smooth-include-generators

Let $R \to A$ be a standard smooth ring map. Let $E \subset A$ be a finite subset of order $|E| = n$. Then there exists a presentation $A = R[x_1, \ldots, x_{n + m}]/(f_1, \ldots, f_c)$ with $c \geq n$, with $\det(\partial f_j/\partial x_i)_{i, j = 1, \ldots, c}$ invertible in $A$, and such that $E$ is the set of congruence classes of $x_1, \ldots, x_n$.


smoothing-lemma-compare-standard

Let $R \to A$ be a ring map of finite presentation. Let $a \in A$. Consider the following conditions on $a$: \begin{enumerate} \item $A_a$ is smooth over $R$, \item $A_a$ is smooth over $R$ and $\Omega_{A_a/R}$ is stably free, \item $A_a$ is smooth over $R$ and $\Omega_{A_a/R}$ is free, \item $A_a$ is standard smooth over $R$, \item $a$ is strictly standard in $A$ over $R$, \item $a$ is elementary standard in $A$ over $R$. \end{enumerate} Then we have \begin{enumerate} \item[(a)] (4) $\Rightarrow$ (3) $\Rightarrow$ (2) $\Rightarrow$ (1), \item[(b)] (6) $\Rightarrow$ (5), \item[(c)] (6) $\Rightarrow$ (4), \item[(d)] (5) $\Rightarrow$ (2), \item[(e)] (2) $\Rightarrow$ the elements $a^e$, $e \geq e_0$ are strictly standard in $A$ over $R$, \item[(f)] (4) $\Rightarrow$ the elements $a^e$, $e \geq e_0$ are elementary standard in $A$ over $R$. \end{enumerate}


smoothing-lemma-neron-functorial

In Situation \ref{situation-neron} N\'eron's blowup is functorial in the following sense \begin{enumerate} \item if $a \in A$, $a \not \in \mathfrak p$, then N\'eron's blowup of $A_a$ is $A'_a$, and \item if $B \to A$ is a surjection of flat finite type $R$-algebras with kernel $I$, then $A'$ is the quotient of $B'/IB'$ by its $\pi$-power torsion. \end{enumerate}


smoothing-lemma-neron-blowup-smooth

In Situation \ref{situation-neron} assume that $R \to A$ is smooth at $\mathfrak p$ and that $R/\pi R \subset \Lambda/\pi \Lambda$ is a separable field extension. Then $R \to A'$ is smooth at $\mathfrak p'$ and there is a short exact sequence $$ 0 \to \Omega_{A/R} \otimes_A A'_{\mathfrak p'} \to \Omega_{A'/R, \mathfrak p'} \to (A'/\pi A')_{\mathfrak p'}^{\oplus c} \to 0 $$ where $c = \dim((A/\pi A)_\mathfrak p)$.


smoothing-lemma-neron-when-smooth

In Situation \ref{situation-neron} assume that $R \to A$ is smooth at $\mathfrak q$ and that we have a surjection of $R$-algebras $B \to A$ with kernel $I$. Assume $R \to B$ smooth at $\mathfrak p_B = (B \to A)^{-1}\mathfrak p$. If the cokernel of $$ I/I^2 \otimes_A \Lambda \to \Omega_{B/R} \otimes_B \Lambda $$ is a free $\Lambda$-module, then $R \to A$ is smooth at $\mathfrak p$.


smoothing-lemma-neron-desingularization

In Situation \ref{situation-neron} assume that $R \to A$ is smooth at $\mathfrak q$ and that $R/\pi R \subset \Lambda/\pi \Lambda$ is a separable extension of fields. Then after a finite number of affine N\'eron blowups the algebra $A$ becomes smooth over $R$ at $\mathfrak p$.


smoothing-lemma-lift-once

Let $R \to \Lambda$ be a ring map. Let $I \subset R$ be an ideal. Assume that \begin{enumerate} \item $I^2 = 0$, and \item $\Lambda/I\Lambda$ is a filtered colimit of smooth $R/I$-algebras. \end{enumerate} Let $\varphi : A \to \Lambda$ be an $R$-algebra map with $A$ of finite presentation over $R$. Then there exists a factorization $$ A \to B/J \to \Lambda $$ where $B$ is a smooth $R$-algebra and $J \subset IB$ is a finitely generated ideal.