more-morphisms-lemma-H1-O-picard-flat-proper

Consider a commutative diagram of schemes $$ \xymatrix{ X' \ar[rr] \ar[dr]_{f'} & & X \ar[dl]^f \\ & S } $$ with $f' : X' \to S$ and $f : X \to S$ satisfying the hypotheses of Lemma \ref{lemma-diagonal-picard-flat-proper}. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module and let $\mathcal{L}'$ be the pullback to $X'$. Let $Z \subset S$, resp.\ $Z' \subset S$ be the locally closed subscheme constructed in Lemma \ref{lemma-diagonal-picard-flat-proper} for $(f, \mathcal{L})$, resp.\ $(f', \mathcal{L}')$ so that $Z \subset Z'$. If $s \in Z$ and $$ H^1(X_s, \mathcal{O}) \longrightarrow H^1(X'_s, \mathcal{O}) $$ is injective, then $Z \cap U = Z' \cap U$ for some open neighbourhood $U$ of $s$.


more-morphisms-lemma-H1-O-multiple-picard-flat-proper

Consider $n$ commutative diagrams of schemes $$ \xymatrix{ X_i \ar[rr] \ar[dr]_{f_i} & & X \ar[dl]^f \\ & S } $$ with $f_i : X_i \to S$ and $f : X \to S$ satisfying the hypotheses of Lemma \ref{lemma-diagonal-picard-flat-proper}. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module and let $\mathcal{L}_i$ be the pullback to $X_i$. Let $Z \subset S$, resp.\ $Z_i \subset S$ be the locally closed subscheme constructed in Lemma \ref{lemma-diagonal-picard-flat-proper} for $(f, \mathcal{L})$, resp.\ $(f_i, \mathcal{L}_i)$ so that $Z \subset \bigcap_{i = 1, \ldots, n} Z_i$. If $s \in Z$ and $$ H^1(X_s, \mathcal{O}) \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} H^1(X_{i, s}, \mathcal{O}) $$ is injective, then $Z \cap U = (\bigcap_{i = 1, \ldots, n} Z_i) \cap U$ (scheme theoretic intersection) for some open neighbourhood $U$ of $s$.


more-morphisms-lemma-pic-of-product

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes satisfying the hypotheses of Lemma \ref{lemma-diagonal-picard-flat-proper}. Let $\sigma : S \to X$ and $\tau : S \to Y$ be sections of $f$ and $g$. Let $s \in S$. Let $\mathcal{L}$ be an invertible sheaf on $X \times_S Y$. If $(1 \times \tau)^*\mathcal{L}$ on $X$, $(\sigma \times 1)^*\mathcal{L}$ on $Y$, and $\mathcal{L}|_{(X \times_S Y)_s}$ are trivial, then there is an open neighbourhood $U$ of $s$ such that $\mathcal{L}$ is trivial over $(X \times_S Y)_U$.


more-morphisms-lemma-Noetherian-approximation

Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Then there exists a cartesian diagram $$ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^g \ar[d]^f \\ S_0 & S \ar[l] } $$ such that \begin{enumerate} \item $X_0$, $S_0$ are affine schemes, \item $S_0$ of finite type over $\mathbf{Z}$, \item $f_0$ is finite of finite type. \end{enumerate}


more-morphisms-lemma-Noetherian-approximation-module

Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite presentation. Then there exists a diagram as in Lemma \ref{lemma-Noetherian-approximation} such that there exists a coherent $\mathcal{O}_{X_0}$-module $\mathcal{F}_0$ with $g^*\mathcal{F}_0 = \mathcal{F}$.


more-morphisms-lemma-Noetherian-approximation-flat-module

Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite presentation and flat over $S$. Then we may choose a diagram as in Lemma \ref{lemma-Noetherian-approximation-module} and sheaf $\mathcal{F}_0$ such that in addition $\mathcal{F}_0$ is flat over $S_0$.


more-morphisms-lemma-Noetherian-approximation-flat

Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation and flat. Then there exists a diagram as in Lemma \ref{lemma-Noetherian-approximation} such that in addition $f_0$ is flat.


more-morphisms-lemma-Noetherian-approximation-geometrically-reduced

Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation with geometrically reduced fibres. Then there exists a diagram as in Lemma \ref{lemma-Noetherian-approximation} such that in addition $f_0$ has geometrically reduced fibres.


more-morphisms-lemma-Noetherian-approximation-geometrically-irreducible

Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation with geometrically irreducible fibres. Then there exists a diagram as in Lemma \ref{lemma-Noetherian-approximation} such that in addition $f_0$ has geometrically irreducible fibres.


more-morphisms-lemma-Noetherian-approximation-dimension-d

Let $d \geq 0$ be an integer. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation all of whose fibres have dimension $d$. Then there exists a diagram as in Lemma \ref{lemma-Noetherian-approximation} such that in addition all fibres of $f_0$ have dimension $d$.


more-morphisms-lemma-Noetherian-approximation-combine

(Noetherian approximation and combining properties.) Let $P$, $Q$ be properties of morphisms of schemes which are stable under base change. Let $f : X \to S$ be a morphism of finite presentation of affine schemes. Assume we can find cartesian diagrams $$ \vcenter{ \xymatrix{ X_1 \ar[d]_{f_1} & X \ar[l] \ar[d]^f \\ S_1 & S \ar[l] } } \quad\text{and}\quad \vcenter{ \xymatrix{ X_2 \ar[d]_{f_2} & X \ar[l] \ar[d]^f \\ S_2 & S \ar[l] } } $$ of affine schemes, with $S_1$, $S_2$ of finite type over $\mathbf{Z}$ and $f_1$, $f_2$ of finite type such that $f_1$ has property $P$ and $f_2$ has property $Q$. Then we can find a cartesian diagram $$ \xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l] \ar[d]^f \\ S_0 & S \ar[l] } $$ of affine schemes with $S_0$ of finite type over $\mathbf{Z}$ and $f_0$ of finite type such that $f_0$ has both property $P$ and property $Q$.


more-morphisms-lemma-realize-prescribed-residue-field-extension-etale

Let $S$ be a scheme. Let $s \in S$. Let $\kappa(s) \subset k$ be a finite separable field extension. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ such that the field extension $\kappa(s) \subset \kappa(u)$ is isomorphic to $\kappa(s) \subset k$.


more-morphisms-lemma-etale-neighbourhoods-not-quite-filtered

Let $S$ be a scheme, and let $s$ be a point of $S$. The category of \'etale neighborhoods has the following properties: \begin{enumerate} \item Let $(U_i, u_i)_{i=1, 2}$ be two \'etale neighborhoods of $s$ in $S$. Then there exists a third \'etale neighborhood $(U, u)$ and morphisms $(U, u) \to (U_i, u_i)$, $i = 1, 2$. \item Let $h_1, h_2: (U, u) \to (U', u')$ be two morphisms between \'etale neighborhoods of $s$. Assume $h_1$, $h_2$ induce the same map $\kappa(u') \to \kappa(u)$ of residue fields. Then there exist an \'etale neighborhood $(U'', u'')$ and a morphism $h : (U'', u'') \to (U, u)$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \circ h = h_2 \circ h$. \end{enumerate}


more-morphisms-lemma-elementary-etale-neighbourhoods

Let $S$ be a scheme, and let $s$ be a point of $S$. The category of elementary \'etale neighborhoods of $(S, s)$ is cofiltered (see Categories, Definition \ref{categories-definition-codirected}).


more-morphisms-lemma-describe-henselization

Let $S$ be a scheme. Let $s \in S$. Then we have $$ \mathcal{O}_{S, s}^h = \colim_{(U, u)} \mathcal{O}(U) $$ where the colimit is over the filtered category which is opposite to the category of elementary \'etale neighbourhoods $(U, u)$ of $(S, s)$.


more-morphisms-lemma-lift-etale-neighbourhood-fibre

\begin{slogan} Lift \'etale neighbourhood of point on fibre to total space. \end{slogan} Let $X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $(V, v) \to (X_s, x)$ be an \'etale neighbourhood. Then there exists an \'etale neighbourhood $(U, u) \to (X, x)$ such that there exists a morphism $(U_s, u) \to (V, v)$ of \'etale neighbourhoods of $(X_s, x)$ which is an open immersion.


more-morphisms-lemma-nr-minimal-primes

Let $R = \colim R_i$ be colimit of a directed system of rings whose transition maps are faithfully flat. Then the number of minimal primes of $R$ taken as an element of $\{0, 1, 2, \ldots, \infty\}$ is the supremum of the numbers of minimal primes of the $R_i$.


more-morphisms-lemma-nr-branches

Let $X$ be a scheme and $x \in X$ a point. Then \begin{enumerate} \item the number of branches of $X$ at $x$ is equal to the supremum of the number of irreducible components of $U$ passing through $u$ taken over elementary \'etale neighbourhoods $(U, u) \to (X, x)$, \item the number of geometric branches of $X$ at $x$ is equal to the supremum of the number of irreducible components of $U$ passing through $u$ taken over \'etale neighbourhoods $(U, u) \to (X, x)$, \item $X$ is unibranch at $x$ if and only if for every elementary \'etale neighbourhood $(U, u) \to (X, x)$ there is exactly one irreducible component of $U$ passing through $u$, and \item $X$ is geometrically unibranch at $x$ if and only if for every \'etale neighbourhood $(U, u) \to (X, x)$ there is exactly one irreducible component of $U$ passing through $u$. \end{enumerate}


more-morphisms-lemma-number-of-branches-and-smooth

Let $X \to S$ be a smooth morphism of schemes. Let $x \in X$ with image $s \in S$. Then \begin{enumerate} \item The number of geometric branches of $X$ at $x$ is equal to the number of geometric branches of $S$ at $s$. \item If $\kappa(x)/\kappa(s)$ is a purely inseparable\footnote{In fact, it would suffice if $\kappa(x)$ is geometrically irreducible over $\kappa(s)$. If we ever need this we will add a detailed proof.} extension of fields, then number of branches of $X$ at $x$ is equal to the number of branches of $S$ at $s$. \end{enumerate}


more-morphisms-lemma-slice-smooth-given-element

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


more-morphisms-lemma-slice-smooth-once

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume \begin{enumerate} \item $f$ is smooth at $x$, and \item the map $$ \Omega_{X_s/s, x} \otimes_{\mathcal{O}_{X_s, x}} \kappa(x) \longrightarrow \Omega_{\kappa(x)/\kappa(s)} $$ has a nonzero kernel. \end{enumerate} Then there exists an affine open neighbourhood $U \subset X$ of $x$ and an effective Cartier divisor $D \subset U$ containing $x$ such that $D \to S$ is smooth.


more-morphisms-lemma-slice-smooth

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume \begin{enumerate} \item $f$ is smooth at $x$, and \item $x$ is a closed point of $X_s$ and $\kappa(s) \subset \kappa(x)$ is separable. \end{enumerate} Then there exists an immersion $Z \to X$ containing $x$ such that \begin{enumerate} \item $Z \to S$ is \'etale, and \item $Z_s = \{x\}$ set theoretically. \end{enumerate}


more-morphisms-lemma-etale-nbhd-dominates-smooth

Let $f : X \to S$ be a smooth morphism of schemes. Let $s \in S$ be a point in the image of $f$. Then there exists an \'etale neighbourhood $(S', s') \to (S, s)$ and a $S$-morphism $S' \to X$.


more-morphisms-lemma-etale-dominates-smooth

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


more-morphisms-lemma-cover-smooth-by-special

Let $f : X \to S$ be a smooth morphism of schemes. Then there exists an \'etale covering $\{U_i \to X\}_{i \in I}$ such that $U_i \to S$ factors as $U_i \to V_i \to S$ where $V_i \to S$ is \'etale and $U_i \to V_i$ is a smooth morphism of affine schemes, which has a section, and has geometrically connected fibres.


more-morphisms-lemma-etale-local-structure

Let $S$ be a scheme. Let $Y \to X$ be a closed immersion of schemes smooth over $S$. For every $y \in Y$ there exist integers $0 \leq m, n$ and a commutative diagram $$ \xymatrix{ Y \ar[d] & V \ar[l] \ar[d] \ar[r] & \mathbf{A}^m_S \ar[d]^{(a_1, \ldots, a_m) \mapsto (a_1, \ldots, a_m, 0 \ldots, 0)} \\ X & U \ar[l] \ar[r]^-\pi & \mathbf{A}^{m + n}_S } $$ where $U \subset X$ is open, $V = Y \cap U$, $\pi$ is \'etale, $V = \pi^{-1}(\mathbf{A}^m_S)$, and $y \in V$.


more-morphisms-lemma-map-approximation

Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $x \in X$ and $y \in Y$ be points lying over the same point $s \in S$. Assume $\mathcal{O}_{S, s}$ is a G-ring. Assume further we are given a local $\mathcal{O}_{S, s}$-algebra map $$ \varphi : \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x}^\wedge $$ For every $N \geq 1$ there exists an elementary \'etale neighbourhood $(U, u) \to (X, x)$ and an $S$-morphism $f : U \to Y$ mapping $u$ to $y$ such that the diagram $$ \xymatrix{ \mathcal{O}_{X, x}^\wedge \ar[r] & \mathcal{O}_{U, u}^\wedge \\ \mathcal{O}_{Y, y} \ar[r]^{f^\sharp_u} \ar[u]^\varphi & \mathcal{O}_{U, u} \ar[u] } $$ commutes modulo $\mathfrak m_u^N$.


more-morphisms-lemma-isomorphism-approximation

Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $x \in X$ and $y \in Y$ be points lying over the same point $s \in S$. Assume $\mathcal{O}_{S, s}$ is a G-ring. Assume we have an $\mathcal{O}_{S, s}$-algebra isomorphism $$ \varphi : \mathcal{O}_{Y, y}^\wedge \longrightarrow \mathcal{O}_{X, x}^\wedge $$ between the complete local rings. Then for every $N \geq 1$ there exists morphisms $$ (X, x) \leftarrow (U, u) \rightarrow (Y, y) $$ of pointed schemes over $S$ such that both arrows define elementary \'etale neighbourhoods and such that the diagram $$ \xymatrix{ & \mathcal{O}_{U, u}^\wedge \\ \mathcal{O}_{Y, y}^\wedge \ar[rr]^\varphi \ar[ru] & & \mathcal{O}_{X, x}^\wedge \ar[lu] } $$ commutes modulo $\mathfrak m_u^N$.


more-morphisms-lemma-relative-map-approximation-pre

Let $X \to S$, $Y \to S$, $x$, $s$, $y$, $t$, $\sigma$, $y_\sigma$, and $\varphi$ be given as follows: we have morphisms of schemes $$ \vcenter{ \xymatrix{ X \ar[d] & Y \ar[d] \\ S & T } } \quad\text{with points}\quad \vcenter{ \xymatrix{ x \ar[d] & y \ar[d] \\ s & t } } $$ Here $S$ is locally Noetherian and $T$ is of finite type over $\mathbf{Z}$. The morphisms $X \to S$ and $Y \to T$ are locally of finite type. The local ring $\mathcal{O}_{S, s}$ is a G-ring. The map $$ \sigma : \mathcal{O}_{T, t} \longrightarrow \mathcal{O}_{S, s}^\wedge $$ is a local homomorphism. Set $Y_\sigma = Y \times_{T, \sigma} \Spec(\mathcal{O}_{S, s}^\wedge)$. Next, $y_\sigma$ is a point of $Y_\sigma$ mapping to $y$ and the closed point of $\Spec(\mathcal{O}_{S, s}^\wedge)$. Finally $$ \varphi : \mathcal{O}_{X, x}^\wedge \longrightarrow \mathcal{O}_{Y_\sigma, y_\sigma}^\wedge $$ is an isomorphism of $\mathcal{O}_{S, s}^\wedge$-algebras. In this situation there exists a commutative diagram $$ \xymatrix{ X \ar[d] & W \ar[l] \ar[rd] \ar[rr] & & Y \times_{T, \tau} V \ar[r] \ar[ld] & Y \ar[d] \\ S & & V \ar[ll] \ar[rr]^\tau & & T } $$ of schemes and points $w \in W$, $v \in V$ such that \begin{enumerate} \item $(V, v) \to (S, s)$ is an elementary \'etale neighbourhood, \item $(W, w) \to (X, x)$ is an elementary \'etale neighbourhood, and \item $\tau(v) = t$. \end{enumerate} Let $y_\tau \in Y \times_T V$ correspond to $y_\sigma$ via the identification $(Y_\sigma)_s = (Y \times_T V)_v$. Then \begin{enumerate} \item[(4)] $(W, w) \to (Y \times_{T, \tau} V, y_\tau)$ is an elementary \'etale neighbourhood. \end{enumerate}


more-morphisms-lemma-relative-map-approximation

Consider a diagram $$ \vcenter{ \xymatrix{ X \ar[d] & Y \ar[d] \\ S & T \ar[l] } } \quad\text{with points}\quad \vcenter{ \xymatrix{ x \ar[d] & y \ar[d] \\ s & t \ar[l] } } $$ where $S$ be a locally Noetherian scheme and the morphisms are locally of finite type. Assume $\mathcal{O}_{S, s}$ is a G-ring. Assume further we are given a local $\mathcal{O}_{S, s}$-algebra map $$ \sigma : \mathcal{O}_{T, t} \longrightarrow \mathcal{O}_{S, s}^\wedge $$ and a local $\mathcal{O}_{S, s}$-algebra map $$ \varphi : \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{Y_\sigma, y_\sigma}^\wedge $$ where $Y_\sigma = Y \times_{T, \sigma} \Spec(\mathcal{O}_{S, s}^\wedge)$ and $y_\sigma$ is the unique point of $Y_\sigma$ lying over $y$. For every $N \geq 1$ there exists a commutative diagram $$ \xymatrix{ X \ar[d] & X \times_S V \ar[l] \ar[rd] & W \ar[l]^-f \ar[r] \ar[d] & Y \times_{T, \tau} V \ar[r] \ar[ld] & Y \ar[d] \\ S & & V \ar[ll] \ar[rr]^\tau & & T } $$ of schemes over $S$ and points $w \in W$, $v \in V$ such that \begin{enumerate} \item $v \mapsto s$, $\tau(v) = t$, $f(w) = (x, v)$, and $w \mapsto (y, v)$, \item $(V, v) \to (S, s)$ is an elementary \'etale neighbourhood, \item the diagram $$ \xymatrix{ \mathcal{O}_{S, s}^\wedge \ar[r] & \mathcal{O}_{V, v}^\wedge \\ \mathcal{O}_{T, t} \ar[r]^{\tau^\sharp_v} \ar[u]_\sigma & \mathcal{O}_{V, v} \ar[u] } $$ commutes module $\mathfrak m_v^N$, \item $(W, w) \to (Y \times_{T, \tau} V, (y, v))$ is an elementary \'etale neighbourhood, \item the diagram $$ \xymatrix{ \mathcal{O}_{X, x} \ar[r]_\varphi & \mathcal{O}_{Y_\sigma, y_\sigma}^\wedge \ar[r] & \mathcal{O}_{Y_\sigma, y_\sigma}/\mathfrak m_{y_\sigma}^N \ar@{=}[r] & \mathcal{O}_{Y \times_{T, \tau} V, (y, v)}/\mathfrak m_{(y, v)}^N \ar[d]_{\cong} \\ \mathcal{O}_{X, x} \ar[r] \ar@{=}[u] & \mathcal{O}_{X \times_S V, (x, v)} \ar[r]^{f^\sharp_w} & \mathcal{O}_{W, w} \ar[r] & \mathcal{O}_{W, w}/\mathfrak m_w^N } $$ commutes. The equality comes from the fact that $Y_\sigma$ and $Y \times_{T, \tau} V$ are canonically isomorphic over $\mathcal{O}_{V, v}/\mathfrak m_v^N = \mathcal{O}_{S, s}/\mathfrak m_s^N$ by parts (2) and (3). \end{enumerate}


more-morphisms-lemma-control-agreement

Let $T \to S$ be finite type morphisms of Noetherian schemes. Let $t \in T$ map to $s \in S$ and let $\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge$ be a local $\mathcal{O}_{S, s}$-algebra map. For every $N \geq 1$ there exists a finite type morphism $(T', t') \to (T, t)$ such that $\sigma$ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ and such that for every local $\mathcal{O}_{S, s}$-algebra map $\sigma' : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge$ which factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ the maps $\sigma$ and $\sigma'$ agree modulo $\mathfrak m_s^N$.


more-morphisms-lemma-control-graded

Let $Y \to T \to S$ be finite type morphisms of Noetherian schemes. Let $t \in T$ map to $s \in S$ and let $\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge$ be a local $\mathcal{O}_{S, s}$-algebra map. There exists a finite type morphism $(T', t') \to (T, t)$ such that $\sigma$ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ and such that for every local $\mathcal{O}_{S, s}$-algebra map $\sigma' : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge$ which factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ the closed immersions $$ Y \times_{T, \sigma} \Spec(\mathcal{O}_{S, s}^\wedge) = Y_\sigma \longleftarrow Y_t \longrightarrow Y_{\sigma'} = Y \times_{T, \sigma'} \Spec(\mathcal{O}_{S, s}^\wedge) $$ have isomorphic conormal algebras.


more-morphisms-lemma-dominate-etale-neighbourhood-finite-flat

Let $S$ be a scheme. Let $s \in S$. Let $f : (U, u) \to (S, s)$ be an \'etale neighbourhood. There exists an affine open neighbourhood $s \in V \subset S$ and a surjective, finite locally free morphism $\pi : T \to V$ such that for every $t \in \pi^{-1}(s)$ there exists an open neighbourhood $t \in W_t \subset T$ and a commutative diagram $$ \xymatrix{ T \ar[d]^\pi & W_t \ar[l] \ar[rr]_{h_t} \ar[rd] & & U \ar[dl] \\ V \ar[rr] & & S } $$ with $h_t(t) = u$.


more-morphisms-lemma-dominate-etale-affine-finite-flat

Let $f : U \to S$ be a surjective \'etale morphism of affine schemes. There exists a surjective, finite locally free morphism $\pi : T \to S$ and a finite open covering $T = T_1 \cup \ldots \cup T_n$ such that each $T_i \to S$ factors through $U \to S$. Diagram: $$ \xymatrix{ & \coprod T_i  \ar[rd] \ar[ld] & \\ T \ar[rd]^\pi & & U \ar[ld]_f \\ & S & } $$ where the south-west arrow is a Zariski-covering.


more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume that \begin{enumerate} \item $f$ is locally of finite type, and \item $x \in X_s$ is isolated\footnote{In the presence of (1) this means that $f$ is quasi-finite at $x$, see Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}.}. \end{enumerate} Then there exist \begin{enumerate} \item[(a)] an elementary \'etale neighbourhood $(U, u) \to (S, s)$, \item[(b)] an open subscheme $V \subset X_U$ (see \ref{equation-basic-diagram}) \end{enumerate} such that \begin{enumerate} \item[(\romannumeral1)] $V \to U$ is a finite morphism, \item[(\romannumeral2)] there is a unique point $v$ of $V$ mapping to $u$ in $U$, and \item[(\romannumeral3)] the point $v$ maps to $x$ under the morphism $X_U \to X$, inducing $\kappa(x) = \kappa(v)$. \end{enumerate} Moreover, for any elementary \'etale neighbourhood $(U', u') \to (U, u)$ setting $V' = U' \times_U V \subset X_{U'}$ the triple $(U', u', V')$ satisfies the properties (\romannumeral1), (\romannumeral2), and (\romannumeral3) as well.


more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points

Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume that \begin{enumerate} \item $f$ is locally of finite type, and \item $x_i \in X_s$ is isolated for $i = 1, \ldots, n$. \end{enumerate} Then there exist \begin{enumerate} \item[(a)] an elementary \'etale neighbourhood $(U, u) \to (S, s)$, \item[(b)] for each $i$ an open subscheme $V_i \subset X_U$, \end{enumerate} such that for each $i$ we have \begin{enumerate} \item[(\romannumeral1)] $V_i \to U$ is a finite morphism, \item[(\romannumeral2)] there is a unique point $v_i$ of $V_i$ mapping to $u$ in $U$, and \item[(\romannumeral3)] the point $v_i$ maps to $x_i$ in $X$ and $\kappa(x_i) = \kappa(v_i)$. \end{enumerate}


more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var

Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume that \begin{enumerate} \item $f$ is locally of finite type, and \item $x_i \in X_s$ is isolated for $i = 1, \ldots, n$. \end{enumerate} Then there exist \begin{enumerate} \item[(a)] an \'etale neighbourhood $(U, u) \to (S, s)$, \item[(b)] for each $i$ an integer $m_i$ and open subschemes $V_{i, j} \subset X_U$, $j = 1, \ldots, m_i$ \end{enumerate} such that we have \begin{enumerate} \item[(\romannumeral1)] each $V_{i, j} \to U$ is a finite morphism, \item[(\romannumeral2)] there is a unique point $v_{i, j}$ of $V_{i, j}$ mapping to $u$ in $U$ with $\kappa(u) \subset \kappa(v_{i, j})$ finite purely inseparable, \item[(\romannumeral4)] if $v_{i, j} = v_{i', j'}$, then $i = i'$ and $j = j'$, and \item[(\romannumeral3)] the points $v_{i, j}$ map to $x_i$ in $X$ and no other points of $(X_U)_u$ map to $x_i$. \end{enumerate}


more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that \begin{enumerate} \item $f$ is locally of finite type, \item $f$ is separated, and \item $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(U, u) \to (S, s)$ and a decomposition $$ U \times_S X = W \amalg V_1 \amalg \ldots \amalg V_n $$ into open and closed subschemes such that the morphisms $V_i \to U$ are finite, the fibres of $V_i \to U$ over $u$ are singletons $\{v_i\}$, each $v_i$ maps to $x_i$ with $\kappa(x_i) = \kappa(v_i)$, and the fibre of $W \to U$ over $u$ contains no points mapping to any of the $x_i$.


more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that \begin{enumerate} \item $f$ is locally of finite type, \item $f$ is separated, and \item $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$. \end{enumerate} Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a decomposition $$ U \times_S X = W \amalg \ \coprod\nolimits_{i = 1, \ldots, n} \ \coprod\nolimits_{j = 1, \ldots, m_i} V_{i, j} $$ into open and closed subschemes such that the morphisms $V_{i, j} \to U$ are finite, the fibres of $V_{i, j} \to U$ over $u$ are singletons $\{v_{i, j}\}$, each $v_{i, j}$ maps to $x_i$, $\kappa(u) \subset \kappa(v_{i, j})$ is purely inseparable, and the fibre of $W \to U$ over $u$ contains no points mapping to any of the $x_i$.


more-morphisms-lemma-etale-splits-off-quasi-finite-part

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that \begin{enumerate} \item $f$ is locally of finite type, \item $f$ is separated, and \item $X_s$ has at most finitely many isolated points. \end{enumerate} Then there exists an elementary \'etale neighbourhood $(U, u) \to (S, s)$ and a decomposition $$ U \times_S X = W \amalg V $$ into open and closed subschemes such that the morphism $V \to U$ is finite, and the fibre $W_u$ of the morphism $W \to U$ contains no isolated points. In particular, if $f^{-1}(s)$ is a finite set, then $W_u = \emptyset$.


more-morphisms-lemma-etale-makes-integral-split

Let $R \to S$ be an integral ring map. Let $\mathfrak p \subset R$ be a prime ideal. Assume \begin{enumerate} \item there are finitely many primes $\mathfrak q_1, \ldots, \mathfrak q_n$ lying over $\mathfrak p$, and \item for each $i$ the maximal separable subextension $\kappa(\mathfrak q)/\kappa(\mathfrak q_i)_{sep}/\kappa(\mathfrak p)$ (Fields, Lemma \ref{fields-lemma-separable-first}) is finite over $\kappa(\mathfrak p)$. \end{enumerate} Then there exists an \'etale ring map $R \to R'$ and a prime $\mathfrak p'$ lying over $\mathfrak p$ such that $$ S \otimes_R R' = A_1 \times \ldots \times A_m $$ with $R' \to A_j$ integral having a unique prime $\mathfrak r_j$ over $\mathfrak p'$ such that $\kappa(\mathfrak r_j)/\kappa(\mathfrak p')$ is purely inseparable.


more-morphisms-lemma-finite-type-separated

Let $f : X \to S$ be a morphism of schemes. Assume $f$ is of finite type and separated. Let $S'$ be the normalization of $S$ in $X$, see Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}. Picture: $$ \xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & S' \ar[ld]^\nu \\ & S & } $$ Then there exists an open subscheme $U' \subset S'$ such that \begin{enumerate} \item $(f')^{-1}(U') \to U'$ is an isomorphism, and \item $(f')^{-1}(U') \subset X$ is the set of points at which $f$ is quasi-finite. \end{enumerate}


more-morphisms-lemma-quasi-finite-separated-quasi-affine

\begin{slogan} Quasi-finite, separated morphisms are quasi-affine \end{slogan} Let $f : X \to S$ be a morphism of schemes. Assume $f$ is quasi-finite and separated. Let $S'$ be the normalization of $S$ in $X$, see Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}. Picture: $$ \xymatrix{ X \ar[rd]_f \ar[rr]_{f'} & & S' \ar[ld]^\nu \\ & S & } $$ Then $f'$ is a quasi-compact open immersion and $\nu$ is integral. In particular $f$ is quasi-affine.


more-morphisms-lemma-quasi-finite-separated-pass-through-finite

\begin{reference} \cite[IV Corollary 18.12.13]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. Assume $f$ is quasi-finite and separated and assume that $S$ is quasi-compact and quasi-separated. Then there exists a factorization $$ \xymatrix{ X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\ & S & } $$ where $j$ is a quasi-compact open immersion and $\pi$ is finite.


more-morphisms-lemma-characterize-finite

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is finite, \item $f$ is proper with finite fibres, \item $f$ is proper and locally quasi-finite, \item $f$ is universally closed, separated, locally of finite type and has finite fibres. \end{enumerate}


more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that $f$ is proper and $f^{-1}(\{s\})$ is a finite set. Then there exists an open neighbourhood $V \subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.


more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre

Consider a commutative diagram of schemes $$ \xymatrix{ X \ar[rr]_h \ar[rd]_f & & Y \ar[ld]^g \\ & S } $$ Let $s \in S$. Assume \begin{enumerate} \item $X \to S$ is a proper morphism, \item $Y \to S$ is separated and locally of finite type, and \item the image of $X_s \to Y_s$ is finite. \end{enumerate} Then there is an open subspace $U \subset S$ containing $s$ such that $X_U \to Y_U$ factors through a closed subscheme $Z \subset Y_U$ finite over $U$.


more-morphisms-lemma-separated-locally-quasi-finite-over-affine

Let $f : X \to Y$ be a separated, locally quasi-finite morphism with $Y$ affine. Then every finite set of points of $X$ is contained in an open affine of $X$.


more-morphisms-lemma-quasi-finite-finite-over-dense-open

Let $f : Y \to X$ be a quasi-finite morphism. There exists a dense open $U \subset X$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is finite.


more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded

Let $f : X \to S$ be flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. Then there exist closed subsets $$ \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_n = S $$ such that with $S_r = Z_r \setminus Z_{r - 1}$ the stratification $S = \coprod_{r = 0, \ldots, n} S_r$ is characterized by the following universal property: Given $g : T \to S$ the projection $X \times_S T \to T$ is finite locally free of degree $r$ if and only if $g(T) \subset S_r$ (set theoretically).


more-morphisms-lemma-stratify-flat-fp-qf

Let $f : X \to S$ be a morphism of schemes which is flat, locally of finite presentation, separated, and quasi-finite. Then there exist closed subsets $$ \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset S $$ such that with $S_r = Z_r \setminus Z_{r - 1}$ the stratification $S = \coprod S_r$ is characterized by the following universal property: Given a morphism $g : T \to S$ the projection $X \times_S T \to T$ is finite locally free of degree $r$ if and only if $g(T) \subset S_r$ (set theoretically). Moreover, the inclusion maps $S_r \to S$ are quasi-compact.


more-morphisms-lemma-stratify-flat-fp-lqf

Let $f : X \to S$ be a flat, locally of finite presentation, separated, and locally quasi-finite morphism of schemes. Then there exist open subschemes $$ S = U_0 \supset U_1 \supset U_2 \supset \ldots $$ such that a morphism $\Spec(k) \to S$ factors through $U_d$ if and only if $X \times_S \Spec(k)$ has degree $\geq d$ over $k$.


more-morphisms-lemma-go-down-with-annihilators

Let $f : X \to S$ be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Let $g \in \Gamma(X, \mathcal{O}_X)$ nonzero. Then there exist an open $V \subset X$ such that $g|_V \not = 0$, an open $U \subset S$ fitting into a commutative diagram $$ \xymatrix{ V \ar[r] \ar[d]_\pi & X \ar[d]^f \\ U \ar[r] & S, } $$ a quasi-coherent subsheaf $\mathcal{F} \subset \mathcal{O}_U$, an integer $r > 0$, and an injective $\mathcal{O}_U$-module map $\mathcal{F}^{\oplus r} \to \pi_*\mathcal{O}_V$ whose image contains $g|_V$.


more-morphisms-lemma-there-is-a-scheme-integral-over

Let $U \to X$ be a surjective \'etale morphism of schemes. Assume $X$ is quasi-compact and quasi-separated. Then there exists a surjective integral morphism $Y \to X$, such that for every $y \in Y$ there is an open neighbourhood $V \subset Y$ such that $V \to X$ factors through $U$. In fact, we may assume $Y \to X$ is finite and of finite presentation.


more-morphisms-lemma-descent-connected-fibres

Consider a diagram of morphisms of schemes $$ \xymatrix{ Z \ar[r]_{\sigma} \ar[rd] & X \ar[d] \\ & Y } $$ an a point $y \in Y$. Assume \begin{enumerate} \item $X \to Y$ is of finite presentation and flat, \item $Z \to Y$ is finite locally free, \item $Z_y \not = \emptyset$, \item all fibres of $X \to Y$ are geometrically reduced, and \item $X_y$ is geometrically connected over $\kappa(y)$. \end{enumerate} Then there exists a quasi-compact open $X^0 \subset X$ such that $X^0_y = X_y$ and such that all nonempty fibres of $X^0 \to Y$ are geometrically connected.


more-morphisms-lemma-fibre-geometrically-connected-reduced

Let $h : Y \to S$ be a morphism of schemes. Let $s \in S$ be a point. Let $T \subset Y_s$ be an open subscheme. Assume \begin{enumerate} \item $h$ is flat and of finite presentation, \item all fibres of $h$ are geometrically reduced, and \item $T$ is geometrically connected over $\kappa(s)$. \end{enumerate} Then we can find an affine elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a quasi-compact open $V \subset Y_{S'}$ such that \begin{enumerate} \item[(a)] all fibres of $V \to S'$ are geometrically connected, \item[(b)] $V_{s'} = T \times_s s'$. \end{enumerate}


more-morphisms-lemma-cover-by-geometrically-connected

Let $f : X \to S$ be a morphism of schemes which is locally of finite presentation and flat with geometrically reduced fibres. Then there exists an \'etale covering $\{X_i \to X\}_{i \in I}$ such that $X_i \to S$ factors as $X_i \to S_i \to S$ where $S_i \to S$ is \'etale and $X_i \to S_i$ is flat of finite presentation with geometrically connected and geometrically reduced fibres.


more-morphisms-lemma-normal-morphism-irreducible

Let $h : Y \to S$ be a morphism of schemes. Let $s \in S$ be a point. Let $T \subset Y_s$ be an open subscheme. Assume \begin{enumerate} \item $h$ is of finite presentation, \item $h$ is normal, and \item $T$ is geometrically irreducible over $\kappa(s)$. \end{enumerate} Then we can find an affine elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a quasi-compact open $V \subset Y_{S'}$ such that \begin{enumerate} \item[(a)] all fibres of $V \to S'$ are geometrically integral, \item[(b)] $V_{s'} = T \times_s s'$. \end{enumerate}


more-morphisms-lemma-local-structure-finite-type

Let $f : X \to S$ be a morphism. Let $x \in X$ and set $s = f(x)$. Assume that $f$ is locally of finite type and that $n = \dim_x(X_s)$. Then there exists a commutative diagram $$ \xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l]  \ar@{|->}[d] \\ & Y \ar[d]^h & & y \ar@{|->}[d] \\ S \ar@{=}[r] & S & s & s \ar@{=}[l] } $$ and a point $x' \in X'$ with $g(x') = x$ such that with $y = \pi(x')$ we have \begin{enumerate} \item $h : Y \to S$ is smooth of relative dimension $n$, \item $g : (X', x') \to (X, x)$ is an elementary \'etale neighbourhood, \item $\pi$ is finite, and $\pi^{-1}(\{y\}) = \{x'\}$, and \item $\kappa(y)$ is a purely transcendental extension of $\kappa(s)$. \end{enumerate} Moreover, if $f$ is locally of finite presentation then $\pi$ is of finite presentation.


more-morphisms-lemma-local-local-structure-finite-type

\begin{slogan} A morphism of finite type is, in \'etale neighbourhoods, finite over a smooth morphism. \end{slogan} Let $f : X \to S$ be a morphism. Let $x \in X$ and set $s = f(x)$. Assume that $f$ is locally of finite type and that $n = \dim_x(X_s)$. Then there exists a commutative diagram $$ \xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y' \ar[d]^h & & y' \ar@{|->}[d] \\ S & S' \ar[l]_e & s & s' \ar@{|->}[l] } $$ and a point $x' \in X'$ with $g(x') = x$ such that with $y' = \pi(x')$, $s' = h(y')$ we have \begin{enumerate} \item $h : Y' \to S'$ is smooth of relative dimension $n$, \item all fibres of $Y' \to S'$ are geometrically integral, \item $g : (X', x') \to (X, x)$ is an elementary \'etale neighbourhood, \item $\pi$ is finite, and $\pi^{-1}(\{y'\}) = \{x'\}$, \item $\kappa(y')$ is a purely transcendental extension of $\kappa(s')$, and \item $e : (S', s') \to (S, s)$ is an elementary \'etale neighbourhood. \end{enumerate} Moreover, if $f$ is locally of finite presentation, then $\pi$ is of finite presentation.


more-morphisms-lemma-local-local-structure-finite-type-affine

Assumption and notation as in Lemma \ref{lemma-local-local-structure-finite-type}. In addition to properties (1) -- (6) we may also arrange it so that \begin{enumerate} \item[(7)] $S'$, $Y'$, $X'$ are affine. \end{enumerate}


more-morphisms-lemma-finite-morphism-single-point-in-fibre

Let $\pi : X \to Y$ be a finite morphism. Let $x \in X$ with $y = \pi(x)$ such that $\pi^{-1}(\{y\}) = \{x\}$. Then \begin{enumerate} \item For every neighbourhood $U \subset X$ of $x$ in $X$, there exists a neighbourhood $V \subset Y$ of $y$ such that $\pi^{-1}(V) \subset U$. \item The ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is finite. \item If $\pi$ is of finite presentation, then $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is of finite presentation. \item For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $\mathcal{F}_x = \pi_*\mathcal{F}_y$ as $\mathcal{O}_{Y, y}$-modules. \end{enumerate}


more-morphisms-lemma-dominate-fppf-etale-locally

Let $S$ be a scheme. Let $\{S_i \to S\}_{i \in I}$ be an fppf covering. Then there exist \begin{enumerate} \item an \'etale covering $\{S'_a \to S\}$, \item surjective finite locally free morphisms $V_a \to S'_a$, \end{enumerate} such that the fppf covering $\{V_a \to S\}$ refines the given covering $\{S_i \to S\}$.


more-morphisms-lemma-dominate-fppf

Let $S$ be a scheme. Let $\{S_i \to S\}_{i \in I}$ be an fppf covering. Then there exist \begin{enumerate} \item a Zariski open covering $S = \bigcup U_j$, \item surjective finite locally free morphisms $W_j \to U_j$, \item Zariski open coverings $W_j = \bigcup_k W_{j, k}$, \item surjective finite locally free morphisms $T_{j, k} \to W_{j, k}$ \end{enumerate} such that the fppf covering $\{T_{j, k} \to S\}$ refines the given covering $\{S_i \to S\}$.


more-morphisms-lemma-extend-integral-surjective-morphisms

Let $S$ be a scheme. If $U \subset S$ is open and $V \to U$ is a surjective integral morphism, then there exists a surjective integral morphism $\overline{V} \to S$ with $\overline{V} \times_S U$ isomorphic to $V$ as schemes over $U$.


more-morphisms-lemma-extend-finite-surjective-morphisms

Let $S$ be a quasi-compact and quasi-separated scheme. If $U \subset S$ is a quasi-compact open and $V \to U$ is a surjective finite morphism, then there exists a surjective finite morphism $\overline{V} \to S$ with $\overline{V} \times_S U$ isomorphic to $V$ as schemes over $U$.


more-morphisms-lemma-dominate-fppf-integral

Let $S$ be a scheme. Let $\{S_i \to S\}_{i \in I}$ be an fppf covering. Then there exists a surjective integral morphism $S' \to S$ and an open covering $S' = \bigcup U'_\alpha$ such that for each $\alpha$ the morphism $U'_\alpha \to S$ factors through $S_i \to S$ for some $i$.


more-morphisms-lemma-dominate-fppf-finite

Let $S$ be a quasi-compact and quasi-separated scheme. Let $\{S_i \to S\}_{i \in I}$ be an fppf covering. Then there exists a surjective finite morphism $S' \to S$ of finite presentation and an open covering $S' = \bigcup U'_\alpha$ such that for each $\alpha$ the morphism $U'_\alpha \to S$ factors through $S_i \to S$ for some $i$.


more-morphisms-lemma-fppf-ph

An fppf covering of schemes is a ph covering.


more-morphisms-lemma-quasi-projective

Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $X \to S$ is quasi-projective, \item $X \to S$ is H-quasi-projective, \item there exists a quasi-compact open immersion $X \to X'$ of schemes over $S$ with $X' \to S$ projective, \item $X \to S$ is of finite type and $X$ has an ample invertible sheaf, and \item $X \to S$ is of finite type and there exists an $f$-very ample invertible sheaf. \end{enumerate}


more-morphisms-lemma-projective

Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $X \to S$ is projective, \item $X \to S$ is H-projective, \item $X \to S$ is quasi-projective and proper, \item $X \to S$ is H-quasi-projective and proper, \item $X \to S$ is proper and $X$ has an ample invertible sheaf, \item $X \to S$ is proper and there exists an $f$-ample invertible sheaf, \item $X \to S$ is proper and there exists an $f$-very ample invertible sheaf, \item there is a quasi-coherent graded $\mathcal{O}_S$-algebra $\mathcal{A}$ generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ with $\mathcal{A}_1$ a finite type $\mathcal{O}_S$-module such that $X = \underline{\text{Proj}}_S(\mathcal{A})$.  \end{enumerate}


more-morphisms-lemma-category-projective

Let $S$ be a scheme which has an ample invertible sheaf. Let $\text{P}_S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma \ref{lemma-projective}. \begin{enumerate} \item if $S' \to S$ is a morphism of schemes and $S'$ has an ample invertible sheaf, then base change determines a functor $\text{P}_S \to \text{P}_{S'}$, \item if $X \in \text{P}_S$ and $Y \in \text{P}_X$, then $Y \in \text{P}_S$, \item the category $\text{P}_S$ is closed under fibre products, \item the category $\text{P}_S$ is closed under finite disjoint unions, \item if $X \to S$ is finite, then $X$ is in $\text{P}_S$, \item add more here. \end{enumerate}


more-morphisms-lemma-ample-in-neighbourhood

\begin{reference} \cite[IV Corollary 9.6.4]{EGA} \end{reference} Let $f : X \to Y$ be a proper morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.


more-morphisms-lemma-apply-proj-spec

Let $R$ be a ring. Let $P$ be a proper scheme over $R$ and let $\mathcal{L}$ be an ample invertible $\mathcal{O}_P$-module. Set $A = \bigoplus_{m \geq 0} \Gamma(P, \mathcal{L}^{\otimes m})$. Then $P = \text{Proj}(A)$ and diagram (\ref{equation-proj-and-spec}) becomes the diagram $$ \xymatrix{ \underline{\Spec}_P \left( \bigoplus\nolimits_{m \in \mathbf{Z}} \mathcal{L}^{\otimes m} \right) \ar[r] \ar@{=}[d] & L = \underline{\Spec}_P \left( \bigoplus\nolimits_{m \geq 0} \mathcal{L}^{\otimes m} \right) \ar[d]^\sigma \ar[r]_-\pi & P \ar[d] \\ U \ar[r] & X \ar[r] & Z } $$ having the properties explained above.


more-morphisms-lemma-locally-principal-vertical

Let $f : X \to S$ be a morphism of schemes. Let $Z \subset X$ be a closed subscheme. Let $s \in S$. Assume \begin{enumerate} \item $S$ is irreducible with generic point $\eta$, \item $X$ is irreducible, \item $f$ is dominant, \item $f$ is locally of finite type, \item $\dim(X_s) \leq \dim(X_\eta)$, \item $Z$ is locally principal in $X$, and \item $Z_\eta = \emptyset$. \end{enumerate} Then the fibre $Z_s$ is (set theoretically) a union of irreducible components of $X_s$.


more-morphisms-lemma-closed-point-nearby-fibre

Let $f : X \to S$ be a morphism of schemes. Let $x \leadsto x'$ be a specialization of points in $X$. Set $s = f(x)$ and $s' = f(x')$. Assume \begin{enumerate} \item $x'$ is a closed point of $X_{s'}$, and \item $f$ is locally of finite type. \end{enumerate} Then the set $$ \{x_1 \in X \text{ such that } f(x_1) = s \text{ and } x_1\text{ is closed in }X_s \text{ and } x \leadsto x_1 \leadsto x' \} $$ is dense in the closure of $x$ in $X_s$.


more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open-X

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $U \subset X$ be an open subscheme. Assume $f$ locally of finite type, $S$ locally Noetherian, $x$ a closed point of $X_s$, and assume there exists a point $x' \in U$ with $x' \leadsto x$ and $f(x') \not = s$. Then there exists a closed subscheme $Z \subset X$ such that (a) $x \in Z$, (b) $f|_Z : Z \to S$ is quasi-finite at $x$, and (c) there exists a $z \in Z$, $z \in U$, $z \leadsto x$ and $f(z) \not = s$.


more-morphisms-lemma-stein-universally-closed

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


more-morphisms-lemma-stein-universally-closed-residue-fields

In Lemma \ref{lemma-stein-universally-closed} assume in addition that $f$ is locally of finite type. Then for $s \in S$ the fibre $\pi^{-1}(\{s\}) = \{s_1, \ldots, s_n\}$ is finite and the field extensions $\kappa(s_i)/\kappa(s)$ are finite.


more-morphisms-lemma-characterize-geometrically-connected-fibres

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Then $X_s$ is geometrically connected, if and only if for every \'etale neighbourhood $(U, u) \to (S, s)$ the base change $X_U \to U$ has connected fibre $X_u$.


more-morphisms-lemma-geometrically-connected-fibres-towards-normal

Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is proper, \item $S$ is integral with generic point $\xi$, \item $S$ is normal, \item $X$ is reduced, \item every generic point of an irreducible component of $X$ maps to $\xi$, \item we have $H^0(X_\xi, \mathcal{O}) = \kappa(\xi)$. \end{enumerate} Then $f_*\mathcal{O}_X = \mathcal{O}_S$ and $f$ has geometrically connected fibres.


more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous

Let $X \to S$ be a flat proper morphism of finite presentation. Let $n_{X/S}$ be the function on $S$ counting the numbers of geometric connected components of fibres of $f$ introduced in Lemma \ref{lemma-base-change-fibres-nr-geometrically-connected-components}. Then $n_{X/S}$ is lower semi-continuous.


more-morphisms-lemma-split-off-proper-part-henselian

\begin{reference} A reference for the case of an adic Noetherian base is \cite[III, Proposition 5.5.1]{EGA} \end{reference} Let $(A, I)$ be a henselian pair. Let $X \to \Spec(A)$ be separated and of finite type. Set $X_0 = X \times_{\Spec(A)} \Spec(A/I)$. Let $Y \subset X_0$ be an open and closed subscheme such that $Y \to \Spec(A/I)$ is proper. Then there exists an open and closed subscheme $W \subset X$ which is proper over $A$ with $W \times_{\Spec(A)} \Spec(A/I) = Y$.


more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend

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


more-morphisms-lemma-relatively-pseudo-coherent

Let $X \to S$ be a finite type morphism of affine schemes. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item for some closed immersion $i : X \to \mathbf{A}^n_S$ the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^n_S})$ is $m$-pseudo-coherent, and \item for all closed immersions $i : X \to \mathbf{A}^n_S$ the object $Ri_*E$ of $D(\mathcal{O}_{\mathbf{A}^n_S})$ is $m$-pseudo-coherent. \end{enumerate}


more-morphisms-lemma-localize-relative-pseudo-coherent

Let $S$ be an affine scheme. Let $V \subset S$ be a standard open. Let $X \to V$ be a finite type morphism of affine schemes. Let $U \subset X$ be an affine open. Let $E$ be an object of $D(\mathcal{O}_X)$. If the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(X \to V, E)$, then the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(U \to S, E|_U)$.


more-morphisms-lemma-glue-relative-pseudo-coherent

Let $X \to S$ be a finite type morphism of affine schemes. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$. Let $X = \bigcup U_i$ be a standard affine open covering. The following are equivalent \begin{enumerate} \item the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} hold for the pairs $(U_i \to S, E|_{U_i})$, \item the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} hold for the pair $(X \to S, E)$. \end{enumerate}


more-morphisms-lemma-relative-pseudo-coherence-characterize

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\mathcal{O}_X)$. Fix $m \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $E$ is $m$-pseudo-coherent relative to $S$, \item for every affine opens $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(U \to V, E|_U)$. \end{enumerate} Moreover, if this is true, then for every open subschemes $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the restriction $E|_U$ is $m$-pseudo-coherent relative to $V$.


more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Fix $m \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $E$ is $m$-pseudo-coherent relative to $S$, \item there exists an affine open covering $S = \bigcup V_i$ and for each $i$ an affine open covering $f^{-1}(V_i) = \bigcup U_{ij}$ such that the complex of $\mathcal{O}_X(U_{ij})$-modules $R\Gamma(U_{ij}, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_S(V_i)$, and \item for every affine opens $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the complex of $\mathcal{O}_X(U)$-modules $R\Gamma(U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_S(V)$. \end{enumerate}


more-morphisms-lemma-finite-morphism-relative-pseudo-coherence

Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $R\pi_*E$ is $m$-pseudo-coherent relative to $S$.


more-morphisms-lemma-cone-relatively-pseudo-coherent

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $(E, E', E'')$ be a distinguished triangle of $D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$. \begin{enumerate} \item If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is $m$-pseudo-coherent relative to $S$. \item If $E, E''$ are $m$-pseudo-coherent relative to $S$, then $E'$ is $m$-pseudo-coherent relative to $S$. \item If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E''$ is $m$-pseudo-coherent relative to $S$, then $E$ is $(m + 1)$-pseudo-coherent relative to $S$. \end{enumerate} Moreover, if two out of three of $E, E', E''$ are pseudo-coherent relative to $S$, the so is the third.


more-morphisms-lemma-summands-relative-pseudo-coherent

Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E, K$ be objects of $D(\mathcal{O}_X)$. If $E \oplus K$ is $m$-pseudo-coherent relative to $S$ so are $E$ and $K$.


more-morphisms-lemma-base-change-relative-pseudo-coherent

Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$ which is $m$-pseudo-coherent relative to $S$. Let $S' \to S$ be a morphism of schemes. Set $X' = X \times_S S'$ and denote $E'$ the derived pullback of $E$ to $X'$. If $S'$ and $X$ are Tor independent over $S$, then $E'$ is $m$-pseudo-coherent relative to $S'$.


more-morphisms-lemma-pull-relative-pseudo-coherent

Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_Y)$. Assume \begin{enumerate} \item $\mathcal{O}_X$ is pseudo-coherent relative to $Y$\footnote{This means $f$ is pseudo-coherent, see Definition \ref{definition-pseudo-coherent}.}, and \item $E$ is $m$-pseudo-coherent relative to $S$. \end{enumerate} Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$.


more-morphisms-lemma-composition-relative-pseudo-coherent

Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_X)$. Assume $\mathcal{O}_Y$ is pseudo-coherent relative to $S$\footnote{This means $Y \to S$ is pseudo-coherent, see Definition \ref{definition-pseudo-coherent}.}. Then the following are equivalent \begin{enumerate} \item $E$ is $m$-pseudo-coherent relative to $Y$, and \item $E$ is $m$-pseudo-coherent relative to $S$. \end{enumerate}


more-morphisms-lemma-check-relative-pseudo-coherence-on-charts

Let $$ \xymatrix{ X \ar[rd] \ar[rr]_i & & P \ar[ld] \\ & S } $$ be a commutative diagram of schemes. Assume $i$ is a closed immersion and $P \to S$ flat and locally of finite presentation. Let $E$ be an object of $D(\mathcal{O}_X)$. Then the following are equivalent \begin{enumerate} \item $E$ is $m$-pseudo-coherent relative to $S$, \item $Ri_*E$ is $m$-pseudo-coherent relative to $S$, and \item $Ri_*E$ is $m$-pseudo-coherent on $P$. \end{enumerate}


more-morphisms-lemma-pseudo-coherent

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item there exist an affine open covering $S = \bigcup V_j$ and for each $j$ an affine open covering $f^{-1}(V_j) = \bigcup U_{ji}$ such that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_{ij})$ is a pseudo-coherent ring map, \item for every pair of affine opens $U \subset X$, $V \subset S$ such that $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is pseudo-coherent, and \item $f$ is locally of finite type and $\mathcal{O}_X$ is pseudo-coherent relative to $S$. \end{enumerate}


more-morphisms-lemma-flat-base-change-pseudo-coherent

A flat base change of a pseudo-coherent morphism is pseudo-coherent.


more-morphisms-lemma-composition-pseudo-coherent

A composition of pseudo-coherent morphisms of schemes is pseudo-coherent.


more-morphisms-lemma-pseudo-coherent-finite-presentation

A pseudo-coherent morphism is locally of finite presentation.


more-morphisms-lemma-flat-finite-presentation-pseudo-coherent

A flat morphism which is locally of finite presentation is pseudo-coherent.


more-morphisms-lemma-permanence-pseudo-coherent

Let $f : X \to Y$ be a morphism of schemes pseudo-coherent over a base scheme $S$. Then $f$ is pseudo-coherent.


more-morphisms-lemma-finite-pseudo-coherent

Let $f : X \to S$ be a finite morphism of schemes. Then $f$ is pseudo-coherent if and only if $f_*\mathcal{O}_X$ is pseudo-coherent as an $\mathcal{O}_S$-module.


more-morphisms-lemma-Noetherian-pseudo-coherent

Let $f : X \to S$ be a morphism of schemes. If $S$ is locally Noetherian, then $f$ is pseudo-coherent if and only if $f$ is locally of finite type.


more-morphisms-lemma-quotient-of-flat-finitely-presented

Let $A \to B$ be a flat ring map of finite presentation. Let $I \subset B$ be an ideal. Then $A \to B/I$ is pseudo-coherent if and only if $I$ is pseudo-coherent as a $B$-module.


more-morphisms-lemma-pseudo-coherent-fppf-local-source

The property $\mathcal{P}(f) =$``$f$ is pseudo-coherent'' is fppf local on the source.


more-morphisms-lemma-perfect

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. The following are equivalent \begin{enumerate} \item there exist an affine open covering $S = \bigcup V_j$ and for each $j$ an affine open covering $f^{-1}(V_j) = \bigcup U_{ji}$ such that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_{ij})$ is a perfect ring map, and \item for every pair of affine opens $U \subset X$, $V \subset S$ such that $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is perfect. \end{enumerate}


more-morphisms-lemma-flat-base-change-perfect

A flat base change of a perfect morphism is perfect.


more-morphisms-lemma-composition-perfect

A composition of perfect morphisms of schemes is perfect.


more-morphisms-lemma-flat-finite-presentation-perfect

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


more-morphisms-lemma-regular-target-perfect

Let $f : X \to S$ be a morphism of schemes. Assume $S$ is regular and $f$ is locally of finite type. Then $f$ is perfect.


more-morphisms-lemma-regular-immersion-perfect

A regular immersion of schemes is perfect. A Koszul-regular immersion of schemes is perfect.


more-morphisms-lemma-perfect-permanence

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $Y \to S$ smooth and $X \to S$ perfect. Then $f : X \to Y$ is perfect.


more-morphisms-lemma-check-perfect-stalks

Let $f : X \to S$ be a pseudo-coherent morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is perfect, \item $\mathcal{O}_X$ locally has finite tor dimension as a sheaf of $f^{-1}\mathcal{O}_S$-modules, and \item for all $x \in X$ the ring $\mathcal{O}_{X, x}$ has finite tor dimension as an $\mathcal{O}_{S, f(x)}$-module. \end{enumerate}


more-morphisms-lemma-perfect-proper-perfect-direct-image

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a perfect proper morphism of schemes. Let $E \in D(\mathcal{O}_X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_S)$.


more-morphisms-lemma-lci

Let $f : X \to S$ be a local complete intersection morphism. Let $P$ be a scheme smooth over $S$. Let $U \subset X$ be an open subscheme and $i : U \to P$ an immersion of schemes over $S$. Then $i$ is a Koszul-regular immersion.


more-morphisms-lemma-lci-properties

Let $f : X \to S$ be a local complete intersection morphism. Then \begin{enumerate} \item $f$ is locally of finite presentation, \item $f$ is pseudo-coherent, and \item $f$ is perfect. \end{enumerate}


more-morphisms-lemma-affine-lci

Let $f : X = \Spec(B) \to S = \Spec(A)$ be a morphism of affine schemes. Then $f$ is a local complete intersection morphism if and only if $A \to B$ is a local complete intersection homomorphism, see More on Algebra, Definition \ref{more-algebra-definition-local-complete-intersection}.


more-morphisms-lemma-flat-base-change-lci

A flat base change of a local complete intersection morphism is a local complete intersection morphism.


more-morphisms-lemma-composition-lci

A composition of local complete intersection morphisms is a local complete intersection morphism.


more-morphisms-lemma-flat-lci

\begin{slogan} A morphism is flat and lci if and only if it is syntomic. \end{slogan} Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is flat and a local complete intersection morphism, and \item $f$ is syntomic. \end{enumerate}


more-morphisms-lemma-regular-immersion-lci

A regular immersion of schemes is a local complete intersection morphism. A Koszul-regular immersion of schemes is a local complete intersection morphism.


more-morphisms-lemma-lci-permanence

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $Y \to S$ smooth and $X \to S$ is a local complete intersection morphism. Then $f : X \to Y$ is a local complete intersection morphism.


more-morphisms-lemma-morphism-regular-schemes-is-lci

Let $f : X \to Y$ be a morphism of schemes. If $f$ is locally of finite type and $X$ and $Y$ are regular, then $f$ is a local complete intersection morphism.


more-morphisms-lemma-lci-avramov

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume \begin{enumerate} \item $S$ is locally Noetherian, \item $Y \to S$ is locally of finite type, \item $f : X \to Y$ is perfect, \item $X \to S$ is a local complete intersection morphism. \end{enumerate} Then $X \to Y$ is a local complete intersection morphism and $Y \to S$ is Koszul at $f(x)$ for all $x \in X$.


more-morphisms-lemma-flat-fp-NL-lci

Let $f : X \to Y$ be a flat morphism of finite presentation. The following are equivalent \begin{enumerate} \item $f$ is a local complete intersection morphism, \item $f$ is syntomic, \item $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and \item $\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$. \end{enumerate}


more-morphisms-lemma-smooth-diagonal-perfect

Let $f : X \to Y$ be a finite type morphism of locally Noetherian schemes. Denote $\Delta : X \to X \times_Y X$ the diagonal morphism. The following are equivalent \begin{enumerate} \item $f$ is smooth, \item $f$ is flat and $\Delta : X \to X \times_Y X$ is a regular immersion, \item $f$ is flat and $\Delta : X \to X \times_Y X$ is a local complete intersection morphism, \item $f$ is flat and $\Delta : X \to X \times_Y X$ is perfect. \end{enumerate}


more-morphisms-lemma-descending-property-lci

The property $\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is fpqc local on the base.


more-morphisms-lemma-lci-syntomic-local-source

The property $\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is syntomic local on the source.


more-morphisms-lemma-base-change-lci-fibres

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume both $X$ and $Y$ are flat and locally of finite presentation over $S$. Then the set $$ \{x \in X \mid f\text{ Koszul at }x\}. $$ is open in $X$ and its formation commutes with arbitrary base change $S' \to S$.


more-morphisms-lemma-unramified-lci

Let $f : X \to Y$ be a local complete intersection morphism of schemes. Then $f$ is unramified if and only if $f$ is formally unramified and in this case the conormal sheaf $\mathcal{C}_{X/Y}$ is finite locally free on $X$.


more-morphisms-lemma-transitivity-conormal-lci

Let $Z \to Y \to X$ be formally unramified morphisms of schemes. Assume that $Z \to Y$ is a local complete intersection morphism. The exact sequence $$ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 $$ of Lemma \ref{lemma-transitivity-conormal} is short exact.


more-morphisms-lemma-check-weakly-etale-stalks

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $X \to Y$ is weakly \'etale, and \item for every $x \in X$ the ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is weakly \'etale. \end{enumerate}


more-morphisms-lemma-key

Let $X \to Y$ be a morphism of schemes such that $X \to X \times_Y X$ is flat. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. If $\mathcal{F}$ is flat over $Y$, then $\mathcal{F}$ is flat over $X$.


more-morphisms-lemma-weakly-etale-characterize

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is weakly \'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 weakly \'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 weakly \'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 of weakly \'etale, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is weakly \'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 weakly-\'etale.


more-morphisms-lemma-composition-weakly-etale

Let $X \to Y \to Z$ be morphisms of schemes. \begin{enumerate} \item If $X \to X \times_Y X$ and $Y \to Y \times_Z Y$ are flat, then $X \to X \times_Z X$ is flat. \item If $X \to Y$ and $Y \to Z$ are weakly \'etale, then $X \to Z$ is weakly \'etale. \end{enumerate}


more-morphisms-lemma-base-change-weakly-etale

Let $X \to Y$ and $Y' \to Y$ be morphisms of schemes and let $X' = Y' \times_Y X$ be the base change of $X$. \begin{enumerate} \item If $X \to X \times_Y X$ is flat, then $X' \to X' \times_{Y'} X'$ is flat. \item If $X \to Y$ is weakly \'etale, then $X' \to Y'$ is weakly \'etale. \end{enumerate}


more-morphisms-lemma-when-weakly-etale

Let $f : X \to Y$ be a morphism of schemes. Then $X \to Y$ is weakly \'etale in each of the following cases \begin{enumerate} \item $X \to Y$ is a flat monomorphism, \item $X \to Y$ is an open immersion, \item $X \to Y$ is flat and unramified, \item $X \to Y$ is \'etale. \end{enumerate}


more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is weakly \'etale, and \item for $x \in X$ the local ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ induces an isomorphism on strict henselizations. \end{enumerate}


more-morphisms-lemma-weakly-etale-permanence

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. If $X$, $Y$ are weakly \'etale over $S$, then $f$ is weakly \'etale.


more-morphisms-lemma-weakly-etale-universal-homeomorphism

Let $f : X \to Y$ be a morphism of schemes. If $f$ is weakly \'etale and a universal homeomorphism, it is an isomorphism.


more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve

Let $f : X \to S$ be a flat, finite type morphism of schemes. Assume $S$ is Nagata, integral with function field $K$, and regular of dimension $1$. Then there exists a finite extension $L/K$ such that in the diagram $$ \xymatrix{ Y \ar[rd]_g \ar[r]_-\nu & X \times_S T \ar[d] \ar[r] & X \ar[d]_f \\ & T \ar[r] & S } $$ the morphism $g$ is smooth at all generic points of fibres. Here $T$ is the normalization of $S$ in $\Spec(L)$ and $\nu : Y \to X \times_S T$ is the normalization.


more-morphisms-lemma-normalized-base-change-with-reduced-fibre-separable

Let $A$ be a Dedekind ring with fraction field $K$. Let $X$ be a scheme flat and of finite type over $A$. Assume $A$ is a Nagata ring and that for every generic point $\eta$ of an irreducible component of $X$ the field extension $K \subset \kappa(\eta)$ is separable. Then there exists a finite separable extension $K \subset L$ such that the normalized base change $Y$ is smooth over $\Spec(B)$ at all generic points of all fibres.


more-morphisms-lemma-ind-quasi-affine-alternative-definition

For a morphism of schemes $f : X \to Y$, the following are equivalent: \begin{enumerate} \item $f$ is ind-quasi-affine, \item for every affine open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine.  \item for some cover $\{ V_j \}_{j \in J}$ of $Y$ by quasi-compact and quasi-separated open subschemes $V_j \subset Y$, every $j \in J$, and every quasi-compact open subscheme $U \subset f^{-1}(V_j)$, the induced morphism $U \to V_j$ is quasi-affine. \item for every quasi-compact and quasi-separated open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine. \end{enumerate} In particular, the property of being an ind-quasi-affine morphism is Zariski local on the base.


more-morphisms-lemma-ind-quasi-affine-composition

The property of being an ind-quasi-affine morphism is stable under composition.


more-morphisms-lemma-ind-quasi-affine-examples

Any quasi-affine morphism is ind-quasi-affine. Any immersion is ind-quasi-affine.


more-morphisms-lemma-ind-quasi-affine-permanence

If $f : X \to Y$ and $g : Y \to Z$ are morphisms of schemes such that $g \circ f$ is ind-quasi-affine, then $f$ is ind-quasi-affine.


more-morphisms-lemma-base-change-ind-quasi-affine

The property of being ind-quasi-affine is stable under base change.


more-morphisms-lemma-etale-separated-ind-quasi-affine

A separated locally quasi-finite morphism of schemes is ind-quasi-affine.


more-morphisms-lemma-prepare-pushout-along-closed-immersion-and-integral

In Situation \ref{situation-pushout-along-closed-immersion-and-integral} then for $y \in Y$ there exist affine opens $U \subset X$ and $V \subset Y$ with $i^{-1}(U) = j^{-1}(V)$ and $y \in V$.


more-morphisms-lemma-pushout-functor

In Situation \ref{situation-pushout-along-closed-immersion-and-integral} suppose given a commutative diagram $$ \xymatrix{ Y' \ar[d]^g & Z' \ar[l]^{j'} \ar[r]_{i'} \ar[d]^h & X' \ar[d]^f \\ Y & Z \ar[l] \ar[r] & X } $$ with cartesian squares and $f, g, h$ separated and locally quasi-finite. Then \begin{enumerate} \item the pushouts $Y \amalg_Z X$ and $Y' \amalg_{Z'} X'$ exist, \item $Y' \amalg_{Z'} X' \to Y \amalg_Z X$ is separated and locally quasi-finite, and \item the squares $$ \xymatrix{ Y' \ar[r] \ar[d] & Y' \amalg_{Z'} X' \ar[d] & X' \ar[l] \ar[d] \\ Y \ar[r] & Y \amalg_Z X & X \ar[l] } $$ are cartesian. \end{enumerate}


more-morphisms-lemma-pushout-functor-equivalence-flat

In Situation \ref{situation-pushout-along-closed-immersion-and-integral} the category of schemes flat, separated, and locally quasi-finite over the pushout $Y \amalg_Z X$ is equivalent to the category of $(X', Y', Z', i', j', f, g, h)$ as in Lemma \ref{lemma-pushout-functor} with $f, g, h$ flat. Similarly with ``flat'' replaced with ``\'etale''.


more-morphisms-lemma-pushout-along-closed-immersions

Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes. Then the pushout $Y \amalg_Z X$ exists in the category of schemes. Picture $$ \xymatrix{ Z \ar[r]_i \ar[d]_j & X \ar[d]^a \\ Y \ar[r]^-b & Y \amalg_Z X } $$ The diagram is a fibre square, the morphisms $a$ and $b$ are closed immersions, and there is a short exact sequence $$ 0 \to \mathcal{O}_{Y \amalg_Z X} \to a_*\mathcal{O}_X \oplus b_*\mathcal{O}_Y \to c_*\mathcal{O}_Z \to 0 $$ where $c = a \circ i = b \circ j$.


more-morphisms-lemma-pushout-along-closed-immersions-properties-above

Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes. Let $f : X' \to X$ and $g : Y' \to Y$ be morphisms of schemes and let $\varphi : X' \times_{X, i} Z \to Y' \times_{Y, j} Z$ be an isomorphism of schemes over $Z$. Consider the morphism $$ h : X' \amalg_{X' \times_{X, i} Z, \varphi} Y' \longrightarrow X \amalg_Z Y $$ Then we have \begin{enumerate} \item $h$ is locally of finite type if and only if $f$ and $g$ are locally of finite type, \item $h$ is flat if and only if $f$ and $g$ are flat, \item $h$ is flat and locally of finite presentation if and only if $f$ and $g$ are flat and locally of finite presentation, \item $h$ is smooth if and only if $f$ and $g$ are smooth, \item $h$ is \'etale if and only if $f$ and $g$ are \'etale, and \item add more here as needed. \end{enumerate}


more-morphisms-lemma-hom-from-finite-free-into-affine

Let $Z \to S$ and $X \to S$ be morphisms of affine schemes. Assume $\Gamma(Z, \mathcal{O}_Z)$ is a finite free $\Gamma(S, \mathcal{O}_S)$-module. Then $\mathit{Mor}_S(Z, X)$ is representable by an affine scheme over $S$.


more-morphisms-lemma-hom-from-finite-locally-free-into-affine

Let $Z \to S$ and $X \to S$ be morphisms of schemes. If $Z \to S$ is finite locally free and $X \to S$ is affine, then $\mathit{Mor}_S(Z, X)$ is representable by a scheme affine over $S$.


more-morphisms-lemma-hom-from-finite-locally-free-representable

Let $Z \to S$ and $X \to S$ be morphisms of schemes. Assume \begin{enumerate} \item $Z \to S$ is finite locally free, and \item for all $(s, x_1, \ldots, x_d)$ where $s \in S$ and $x_1, \ldots, x_d \in X_s$ there exists an affine open $U \subset X$ with $x_1, \ldots, x_d \in U$. \end{enumerate} Then $\mathit{Mor}_S(Z, X)$ is representable by a scheme.


more-morphisms-lemma-hom-from-finite-locally-free-separated-lqf

Let $Z \to S$ and $X \to S$ be morphisms of schemes. Assume $Z \to S$ is finite locally free and $X \to S$ is separated and locally quasi-finite. Then $\mathit{Mor}_S(Z, X)$ is representable by a scheme.


more-morphisms-lemma-case-of-tor-independence

Consider a commutative diagram of schemes $$ \xymatrix{ Z' \ar[d] \ar[r] & Y' \ar[d] \\ X' \ar[r] & S' } $$ Let $S \to S'$ be a morphism. Denote by $X$ and $Y$ the base changes of $X'$ and $Y'$ to $S$. Assume $Y' \to S'$ and $Z' \to X'$ are flat. Then $X \times_S Y$ and $Z'$ are Tor independent over $X' \times_{S'} Y'$.


more-morphisms-lemma-derived-chow

Let $A$ be a ring. Let $X$ be a separated scheme of finite presentation over $A$. Let $x \in X$. Then there exist an open neighbourhood $U \subset X$ of $x$, an $n \geq 0$, an open $V \subset \mathbf{P}^n_A$, a closed subscheme $Z \subset X \times_A \mathbf{P}^n_A$, a point $z \in Z$, and an object $E$ in $D(\mathcal{O}_{X \times_A \mathbf{P}^n_A})$ such that \begin{enumerate} \item $Z \to X \times_A \mathbf{P}^n_A$ is of finite presentation, \item $b : Z \to X$ is an isomorphism over $U$ and $b(z) = x$, \item $c : Z \to \mathbf{P}^n_A$ is a closed immersion over $V$, \item $b^{-1}(U) = c^{-1}(V)$, in particular $c(z) \in V$, \item $E|_{X \times_A V} \cong (b, c)_*\mathcal{O}_Z|_{X \times_A V}$, \item $E$ is pseudo-coherent and supported on $Z$. \end{enumerate}


more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow

Let $A$, $x \in X$, and $U, n, V, Z, z, E$ be as in Lemma \ref{lemma-derived-chow}. For any $K \in D_\QCoh(\mathcal{O}_X)$ we have $$ Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = R(U \to V)_*K|_U $$ where $p : X \times_A \mathbf{P}^n_A \to X$ and $q : X \times_A \mathbf{P}^n_A \to \mathbf{P}^n_A$ are the projections and where the morphism $U \to V$ is the finitely presented closed immersion $c \circ (b|_U)^{-1}$.


more-morphisms-lemma-characterize-pseudo-coherent

Let $A$ be a ring. Let $X$ be a scheme separated and of finite presentation over $A$. Let $K \in D_\QCoh(\mathcal{O}_X)$. If $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$, then $K$ is pseudo-coherent relative to $A$.


more-morphisms-lemma-characterize-pseudo-coh-improved

Let $A$ be a ring. Let $X$ be a scheme separated and of finite presentation over $A$. Let  $K \in D_\QCoh(\mathcal{O}_X).$ If  $R \Gamma (X, E \otimes ^{\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every perfect  $E \in D(\mathcal{O}_X)$, then $K$ is pseudo-coherent relative to $A$.


more-morphisms-lemma-characterize-relatively-perfect

Let $A$ be a ring. Let $X$ be a scheme separated, of finite presentation, and flat over $A$. Let  $K \in D_\QCoh(\mathcal{O}_X).$ If  $R \Gamma (X, E \otimes^\mathbf{L} K)$ is perfect in $D(A)$ for every perfect $E \in D(\mathcal{O}_X)$, then $K$ is $\Spec(A)$-perfect.


more-morphisms-lemma-relative-pseudo-coherent-descends-fppf

Let $X \to S$ be locally of finite type. Let $\{f_i : X_i \to X\}$ be an fppf covering of schemes. Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if each $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$.


more-morphisms-lemma-relative-pseudo-coherent-post-compose

Let $X \to T \to S$ be morphisms of schemes. Assume $T \to S$ is flat and locally of finite presentation and $X \to T$ locally of finite type. Let $E \in D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $E$ is $m$-pseudo-coherent relative to $T$.


more-morphisms-lemma-thickening-pseudo-coherent

Let $i : X \to X'$ be a finite order thickening of schemes. Let $K' \in D(\mathcal{O}_{X'})$ be an object such that $K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent.


more-morphisms-lemma-thickening-relatively-perfect

Consider a cartesian diagram $$ \xymatrix{ X \ar[r]_i \ar[d]_f & X' \ar[d]^{f'} \\ Y \ar[r]^j & Y' } $$ of schemes. Assume $X' \to Y'$ is flat and locally of finite presentation and $Y \to Y'$ is a finite order thickening. Let $E' \in D(\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect, then $E'$ is $Y'$-perfect.


more-morphisms-lemma-check-h1-fibre-zero

Let $f : X \to Y$ be a proper morphism of schemes. Let $y \in Y$ be a point with $\dim(X_y) \leq 1$. If \begin{enumerate} \item $R^1f_*\mathcal{O}_X = 0$, or more generally \item there is a morphism $g : Y' \to Y$ such that $y$ is in the image of $g$ and such that $R'f'_*\mathcal{O}_{X'} = 0$ where $f' : X' \to Y'$ is the base change of $f$ by $g$. \end{enumerate} Then $H^1(X_y, \mathcal{O}_{X_y}) = 0$.


more-morphisms-lemma-h1-fibre-zero

Let $f : X \to Y$ be a proper morphism of schemes. Let $y \in Y$ be a point with $\dim(X_y) \leq 1$ and $H^1(X_y, \mathcal{O}_{X_y}) = 0$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $R^1f_*\mathcal{O}_X|_V = 0$ and the same is true after base change by any $Y' \to V$.


more-morphisms-lemma-globally-generated-vanishing

Let $f : X \to Y$ be a proper morphism of schemes such that $\dim(X_y) \leq 1$ and $H^1(X_y, \mathcal{O}_{X_y}) = 0$ for all $y \in Y$. Let $\mathcal{F}$ be quasi-coherent on $X$. Then \begin{enumerate} \item $R^pf_*\mathcal{F} = 0$ for $p > 1$, and \item $R^1f_*\mathcal{F} = 0$ if there is a surjection $f^*\mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ quasi-coherent on $Y$. \end{enumerate} If $Y$ is affine, then we also have \begin{enumerate} \item[(3)] $H^p(X, \mathcal{F}) = 0$ for $p \not \in \{0, 1\}$, and \item[(4)] $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ is globally generated. \end{enumerate}


more-morphisms-lemma-h1-fibre-zero-check-h0-kappa

Let $f : X \to Y$ be a proper morphism of schemes. Assume \begin{enumerate} \item for all $y \in Y$ we have $\dim(X_y) \leq 1$ and $H^1(X_y, \mathcal{O}_{X_y}) = 0$, and \item $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is surjective. \end{enumerate} Then $\mathcal{O}_{Y'} \to f'_*\mathcal{O}_{X'}$ is surjective for any base change $f' : X' \to Y'$ of $f$.


more-morphisms-lemma-h1-fibre-zero-h0-kappa

Consider a commutative diagram $$ \xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S } $$ of morphisms of schemes. Let $s \in S$ be a point. Assume \begin{enumerate} \item $X \to S$ is locally of finite presentation and flat at points of $X_s$, \item $f$ is proper, \item the fibres of $f_s : X_s \to Y_s$ have dimension $\leq 1$ and $R^1f_{s, *}\mathcal{O}_{X_s} = 0$, \item $\mathcal{O}_{Y_s} \to f_{s, *}\mathcal{O}_{X_s}$ is surjective. \end{enumerate} Then there is an open $Y_s \subset V \subset Y$ such that (a) $f^{-1}(V)$ is flat over $S$, (b) $\dim(X_y) \leq 1$ for $y \in V$, (c) $R^1f_*\mathcal{O}_X|_V = 0$, (d) $\mathcal{O}_V \to f_*\mathcal{O}_X|_V$ is surjective, and (b), (c), and (d) remain true after base change by any $Y' \to V$.


more-morphisms-lemma-h1-fibre-zero-h0-flat

Consider a commutative diagram $$ \xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S } $$ of morphisms of schemes. Assume $X \to S$ is flat, $f$ is proper, $\dim(X_y) \leq 1$ for $y \in Y$, and $R^1f_*\mathcal{O}_X = 0$. Then $f_*\mathcal{O}_X$ is $S$-flat and formation of $f_*\mathcal{O}_X$ commutes with arbitrary base change $S' \to S$.


more-morphisms-lemma-h1-fibre-zero-isom

Consider a commutative diagram $$ \xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S } $$ of morphisms of schemes. Let $s \in S$ be a point. Assume \begin{enumerate} \item $X \to S$ is locally of finite presentation and flat at points of $X_s$, \item $Y \to S$ is locally of finite presentation, \item $f$ is proper, \item the fibres of $f_s : X_s \to Y_s$ have dimension $\leq 1$ and $R^1f_{s, *}\mathcal{O}_{X_s} = 0$, \item $\mathcal{O}_{Y_s} \to f_{s, *}\mathcal{O}_{X_s}$ is an isomorphism. \end{enumerate} Then there is an open $Y_s \subset V \subset Y$ such that (a) $V$ is flat over $S$, (b) $f^{-1}(V)$ is flat over $S$, (c) $\dim(X_y) \leq 1$ for $y \in V$, (d) $R^1f_*\mathcal{O}_X|_V = 0$, (e) $\mathcal{O}_V \to f_*\mathcal{O}_X|_V$ is an isomorphism, and (a) -- (e) remain true after base change of $f^{-1}(V) \to V$ by any $S' \to S$.


more-morphisms-lemma-bijection-on-Pic

Let $f : X \to Y$ be a proper morphism of Noetherian schemes such that $f_*\mathcal{O}_X = \mathcal{O}_Y$, such that the fibres of $f$ have dimension $\leq 1$, and such that $H^1(X_y, \mathcal{O}_{X_y}) = 0$ for $y \in Y$. Then $f^* : \Pic(Y) \to \Pic(X)$ is a bijection onto the subgroup of $\mathcal{L} \in \Pic(X)$ with $\mathcal{L}|_{X_y} \cong \mathcal{O}_{X_y}$ for all $y \in Y$.


more-morphisms-lemma-test-universally-open

Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is universally open, \item for every morphism $S' \to S$ which is locally of finite presentation the base change $X_{S'} \to S'$ is open, and \item for every $n$ the morphism $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is open. \end{enumerate}


more-morphisms-lemma-characterize-universally-open-finite

Let $A \to B$ be a ring map. Say $B$ is generated as an $A$-module by $b_1, \ldots, b_d \in B$. Set $h = \sum x_ib_i \in B[x_1, \ldots, x_d]$. Then $\Spec(B) \to \Spec(A)$ is universally open if and only if the image of $D(h)$ in $\Spec(A[x_1, \ldots, x_d])$ is open.


more-morphisms-lemma-count-geometric-fibres

Let $f : X \to Y$ be a locally quasi-finite morphism. Then \begin{enumerate} \item the functions $n_{X/Y}$ of Lemmas \ref{lemma-base-change-fibres-nr-geometrically-irreducible-components} and \ref{lemma-base-change-fibres-nr-geometrically-connected-components} agree, \item if $X$ is quasi-compact, then $n_{X/Y}$ attains a maximum $d < \infty$. \end{enumerate}


more-morphisms-lemma-count-geometric-fibres-universally-open

Let $f : X \to Y$ be a separated, locally quasi-finite, and universally open morphism of schemes. Let $n_{X/Y}$ be as in Lemma \ref{lemma-count-geometric-fibres}. If $n_{X/Y}(y) \geq d$ for some $y \in Y$ and $d \geq 0$, then $n_{X/Y} \geq d$ in an open neighbourhood of $y$.


more-morphisms-lemma-weighting-check-after-etale-base-change

Given a cartesian square $$ \xymatrix{ U \ar[d]_\pi & U' \ar[l]^h \ar[d]^{\pi'} \\ V & V' \ar[l]_g } $$ with $\pi$ locally quasi-finite with finite fibres and a function $w : U \to \mathbf{Z}$ we have $(\int_\pi w) \circ g = \int_{\pi'} (w \circ h)$.


more-morphisms-lemma-weighting-base-change

Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}$ be a weighting. Let $f' : X' \to Y'$ be the base change of $f$ by a morphism $Y' \to Y$. Then the composition $w' : X' \to \mathbf{Z}$ of $w$ and the projection $X' \to X$ is a weighting of $f'$.


more-morphisms-lemma-weighting-universally-open

Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}$ be a weighting. If $w(x) > 0$ for all $x \in X$, then $f$ is universally open.


more-morphisms-lemma-jumps-w

Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. Then the level sets of the function $w$ are locally constructible in $X$.


more-morphisms-lemma-jumps-int-w

Let $f : X \to Y$ be a locally quasi-finite morphism of finite presentation. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. Then the level sets of the function $\int_f w$ are locally constructible in $Y$.


more-morphisms-lemma-semicontinuous-int-w

Let $f : X \to Y$ be a separated, locally quasi-finite morphism with finite fibres. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $\int_f w$ is lower semi-continuous.


more-morphisms-lemma-max-int-w

Let $f : X \to Y$ be a locally quasi-finite morphism with $X$ quasi-compact. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. Then $\int_f w$ attains its maximum.


more-morphisms-lemma-max-int-finite

Let $f : X \to Y$ be a separated, locally quasi-finite morphism. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Assume $\int_w f$ attains its maximum $d$ and let $Y_d \subset Y$ be the open set of points $y$ with $(\int_f w)(y) = d$. Then the morphism $f^{-1}(Y_d) \to Y_d$ is finite.


more-morphisms-lemma-open-and-closed-in-finite

Let $A \to B$ be a ring map which is finite and of finite presentation. There exists a finitely presented ring map $A \to A_{univ}$ and an idempotent $e_{univ} \in B \otimes_A A_{univ}$ such that for any ring map $A \to A'$ and idempotent $e \in B \otimes_A A'$ there is a ring map $A_{univ} \to A'$ mapping $e_{univ}$ to $e$.


more-morphisms-lemma-open-and-closed-in-quasi-finite

Let $X \to Y$ be a morphism of affine schemes which is quasi-finite and of finite presentation. There exists a morphism $Y_{univ} \to Y$ of finite presentation and an open subscheme $U_{univ} \subset Y_{univ} \times_Y X$ such that $U_{univ} \to Y_{univ}$ is finite with the following property: given any morphism $Y' \to Y$ of affine schemes and an open subscheme $U' \subset Y' \times_Y X$ such that $U' \to Y'$ is finite, there exists a morphism $Y' \to Y_{univ}$ such that the inverse image of $U_{univ}$ is $U'$.


more-morphisms-lemma-descend-weighting

Let $Y = \lim Y_i$ be a directed limit of affine schemes. Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of affine schemes which is quasi-finite and of finite presentation. Let $f : X  \to Y$ and $f_i : X_i \to Y_i$ for $i \geq 0$ be the base changes of $f_0$. If $w : X \to \mathbf{Z}$ is a weighting of $f$, then for sufficiently large $i$ there exists a weighting $w_i : X_i \to \mathbf{Z}$ of $f_i$ whose pullback to $X$ is $w$.


more-morphisms-lemma-affineness-of-large-open

Let $f : X \to Y$ be a morphism of affine schemes which is quasi-finite and of finite presentation. Let $w : X \to \mathbf{Z}_{> 0}$ be a postive weighting of $f$. Let $d < \infty$ be the maximum value of $\int_f w$. The open $$ Y_d = \{y \in Y \mid (\textstyle{\int}_f w)(y) = d \} $$ of $Y$ is affine.


more-morphisms-proposition-pushout-along-closed-immersion-and-integral

\begin{reference} \cite[Theorem 7.1 part iii]{Ferrand-Conducteur} \end{reference} In Situation \ref{situation-pushout-along-closed-immersion-and-integral} the pushout $Y \amalg_Z X$ exists in the category of schemes. Picture $$ \xymatrix{ Z \ar[r]_i \ar[d]_j & X \ar[d]^a \\ Y \ar[r]^-b & Y \amalg_Z X } $$ The diagram is a fibre square, the morphism $a$ is integral, the morphism $b$ is a closed immersion, and $$ \mathcal{O}_{Y \amalg_Z X} = b_*\mathcal{O}_Y \times_{c_*\mathcal{O}_Z} a_*\mathcal{O}_X $$ as sheaves of rings where $c = a \circ i = b \circ j$.


sites-modules-lemma-limits-colimits-abelian-presheaves

Let $\mathcal{C}$ be a category. \begin{enumerate} \item All limits and colimits exist in $\textit{PAb}(\mathcal{C})$. \item All limits and colimits commute with taking sections over objects of $\mathcal{C}$. \end{enumerate}


sites-modules-lemma-abelian-abelian

Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of abelian sheaves on $\mathcal{C}$. \begin{enumerate} \item The category $\textit{Ab}(\mathcal{C})$ is an abelian category. \item The kernel $\Ker(\varphi)$ of $\varphi$ is the same as the kernel of $\varphi$ as a morphism of presheaves. \item The morphism $\varphi$ is injective (Homology, Definition \ref{homology-definition-injective-surjective}) if and only if $\varphi$ is injective as a map of presheaves (Sites, Definition \ref{sites-definition-presheaves-injective-surjective}), if and only if $\varphi$ is injective as a map of sheaves (Sites, Definition \ref{sites-definition-sheaves-injective-surjective}). \item The cokernel $\Coker(\varphi)$ of $\varphi$ is the sheafification of the cokernel of $\varphi$ as a morphism of presheaves. \item The morphism $\varphi$ is surjective (Homology, Definition \ref{homology-definition-injective-surjective}) if and only if $\varphi$ is surjective as a map of sheaves (Sites, Definition \ref{sites-definition-sheaves-injective-surjective}). \item A complex of abelian sheaves $$ \mathcal{F} \to \mathcal{G} \to \mathcal{H} $$ is exact at $\mathcal{G}$ if and only if for all $U \in \Ob(\mathcal{C})$ and all $s \in \mathcal{G}(U)$ mapping to zero in $\mathcal{H}(U)$ there exists a covering $\{U_i \to U\}_{i \in I}$ in $\mathcal{C}$ such that each $s|_{U_i}$ is in the image of $\mathcal{F}(U_i) \to \mathcal{G}(U_i)$. \end{enumerate}


sites-modules-lemma-limits-colimits-abelian-sheaves

Let $\mathcal{C}$ be a site. \begin{enumerate} \item All limits and colimits exist in $\textit{Ab}(\mathcal{C})$. \item Limits are the same as the corresponding limits of abelian presheaves over $\mathcal{C}$ (i.e., commute with taking sections over objects of $\mathcal{C}$). \item Finite direct sums are the same as the corresponding finite direct sums in the category of abelian pre-sheaves over $\mathcal{C}$. \item A colimit is the sheafification of the corresponding colimit in the category of abelian presheaves. \item Filtered colimits are exact. \end{enumerate}


sites-modules-lemma-obvious-adjointness

Let $\mathcal{C}$ be a category. Let $\mathcal{G}$, $\mathcal{F}$ be a presheaves of sets. Let $\mathcal{A}$ be an abelian presheaf. Let $U$ be an object of $\mathcal{C}$. Then we have \begin{align*} \Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) & = \mathcal{F}(U), \\ \Mor_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_\mathcal{G}, \mathcal{A}) & = \Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\ \Mor_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_U, \mathcal{A}) & = \mathcal{A}(U). \end{align*} All of these equalities are functorial.


sites-modules-lemma-coproduct-sum-free-abelian-presheaf

Let $\mathcal{C}$ be a category. Let $I$ be a set. For each $i \in I$ let $\mathcal{G}_i$ be a presheaf of sets. Then $$ \mathbf{Z}_{\coprod_i \mathcal{G}_i} = \bigoplus\nolimits_{i \in I} \mathbf{Z}_{\mathcal{G}_i} $$ in $\textit{PAb}(\mathcal{C})$.


sites-modules-lemma-obvious-adjointness-sheaves

Let $\mathcal{C}$ be a site. Let $\mathcal{G}$, $\mathcal{F}$ be a sheaves of sets. Let $\mathcal{A}$ be an abelian sheaf. Let $U$ be an object of $\mathcal{C}$. Then we have \begin{align*} \Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F}) & = \mathcal{F}(U), \\ \Mor_{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_\mathcal{G}^\#, \mathcal{A}) & = \Mor_{\Sh(\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\ \Mor_{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_U^\#, \mathcal{A}) & = \mathcal{A}(U). \end{align*} All of these equalities are functorial.


sites-modules-lemma-may-sheafify-before-abelianize

Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets. Then $\mathbf{Z}_\mathcal{G}^\# = (\mathbf{Z}_{\mathcal{G}^\#})^\#$.


sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. There exists a factorization $$ \xymatrix{ (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[rr]_{(f, f^\sharp)} \ar[d]_{(g, g^\sharp)} & & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \ar[d]^{(e, e^\sharp)} \\ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[rr]^{(h, h^\sharp)} & & (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) } $$ where \begin{enumerate} \item $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{C} \to \mathcal{C}'$ (see Sites, Definition \ref{sites-definition-special-cocontinuous-functor}), \item $e : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{D} \to \mathcal{D}'$ (see Sites, Definition \ref{sites-definition-special-cocontinuous-functor}), \item $\mathcal{O}_{\mathcal{C}'} = g_*\mathcal{O}_\mathcal{C}$ and $g^\sharp$ is the obvious map, \item $\mathcal{O}_{\mathcal{D}'} = e_*\mathcal{O}_\mathcal{D}$ and $e^\sharp$ is the obvious map, \item the sites $\mathcal{C}'$ and $\mathcal{D}'$ have final objects and fibre products (i.e., all finite limits), \item $h$ is a morphism of sites induced by a continuous functor $u : \mathcal{D}' \to \mathcal{C}'$ which commutes with all finite limits (i.e., it satisfies the assumptions of Sites, Proposition \ref{sites-proposition-get-morphism}), and \item given any set of sheaves $\mathcal{F}_i$ (resp.\ $\mathcal{G}_j$) on $\mathcal{C}$ (resp.\ $\mathcal{D}$) we may assume each of these is a representable sheaf on $\mathcal{C}'$ (resp.\ $\mathcal{D}'$). \end{enumerate} Moreover, if $(f, f^\sharp)$ is an equivalence of ringed topoi, then we can choose the diagram such that $\mathcal{C}' = \mathcal{D}'$, $\mathcal{O}_{\mathcal{C}'} = \mathcal{O}_{\mathcal{D}'}$ and $(h, h^\sharp)$ is the identity.


sites-modules-lemma-adjointness-tensor-restrict-presheaves

With $\mathcal{C}$, $\mathcal{O}_1 \to \mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection $$ \Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} ) $$ In other words, the restriction and change of rings functors defined above are adjoint to each other.


sites-modules-lemma-sheafification-presheaf-modules

Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $\mathcal{O}^\#$ be the sheafification of $\mathcal{O}$ as a presheaf of rings, see Sites, Section \ref{sites-section-sheaves-algebraic-structures}. Let $\mathcal{F}^\#$ be the sheafification of $\mathcal{F}$ as a presheaf of abelian groups. There exists a unique map of sheaves of sets $$ \mathcal{O}^\# \times \mathcal{F}^\# \longrightarrow \mathcal{F}^\# $$ which makes the diagram $$ \xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{O}^\# \times \mathcal{F}^\# \ar[r] & \mathcal{F}^\# } $$ commute and which makes $\mathcal{F}^\#$ into a sheaf of $\mathcal{O}^\#$-modules. In addition, if $\mathcal{G}$ is a sheaf of $\mathcal{O}^\#$-modules, then any morphism of presheaves of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$ (into the restriction of $\mathcal{G}$ to a $\mathcal{O}$-module) factors uniquely as $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ where $\mathcal{F}^\# \to \mathcal{G}$ is a morphism of $\mathcal{O}^\#$-modules.


sites-modules-lemma-sheafification-exact

Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$ The sheafification functor $$ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}^\#), \quad \mathcal{F} \longmapsto \mathcal{F}^\# $$ is exact.


sites-modules-lemma-adjointness-tensor-restrict

With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection $$ \Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}, \mathcal{F} ) $$ In other words, the restriction and change of rings functors are adjoint to each other.


sites-modules-lemma-epimorphism-modules

Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be an epimorphism of sheaves of rings. Let $\mathcal{G}_1, \mathcal{G}_2$ be $\mathcal{O}'$-modules. Then $$ \Hom_{\mathcal{O}'}(\mathcal{G}_1, \mathcal{G}_2) = \Hom_\mathcal{O}(\mathcal{G}_1, \mathcal{G}_2). $$ In other words, the restriction functor $\textit{Mod}(\mathcal{O}') \to \textit{Mod}(\mathcal{O})$ is fully faithful.


sites-modules-lemma-pushforward-module

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. There is a natural map of sheaves of sets $$ f_*\mathcal{O} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F} $$ which turns $f_*\mathcal{F}$ into a sheaf of $f_*\mathcal{O}$-modules. This construction is functorial in $\mathcal{F}$.


sites-modules-lemma-pullback-module

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. There is a natural map of sheaves of sets $$ f^{-1}\mathcal{O} \times f^{-1}\mathcal{G} \longrightarrow f^{-1}\mathcal{G} $$ which turns $f^{-1}\mathcal{G}$ into a sheaf of $f^{-1}\mathcal{O}$-modules. This construction is functorial in $\mathcal{G}$.


sites-modules-lemma-adjoint-push-pull-modules

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a sheaf of $f^{-1}\mathcal{O}$-modules. Then $$ \Mor_{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). $$ Here we use Lemmas \ref{lemma-pullback-module} and \ref{lemma-pushforward-module}, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module by restriction via $\mathcal{O} \to f_*f^{-1}\mathcal{O}$.


sites-modules-lemma-adjoint-pullback-pushforward-modules

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi or ringed sites. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{C}$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal{D}$-modules. There is a canonical bijection $$ \Hom_{\mathcal{O}_\mathcal{C}}(f^*\mathcal{G}, \mathcal{F}) = \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, f_*\mathcal{F}). $$ In other words: the functor $f^*$ is the left adjoint to $f_*$.


sites-modules-lemma-push-pull-composition-modules

$(f, f^\sharp) : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to (\Sh(\mathcal{C}_2), \mathcal{O}_2)$ and $(g, g^\sharp) : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to (\Sh(\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi. There are canonical isomorphisms of functors $(g \circ f)_* \cong g_* \circ f_*$ and $(g \circ f)^* \cong f^* \circ g^*$.


sites-modules-lemma-abelian

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. The category $\textit{Mod}(\mathcal{O})$ is an abelian category. The forgetful functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ is exact, hence kernels, cokernels and exactness of $\mathcal{O}$-modules, correspond to the corresponding notions for abelian sheaves.