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39.15. Descending ind-quasi-affine morphisms

Ind-quasi-affine morphisms were defined in More on Morphisms, Section 36.55. This section is the analogue of Descent, Section 34.35 for ind-quasi-affine-morphisms.

Let $X$ be a quasi-separated scheme. Let $E \subset X$ be a subset which is an intersection of a nonempty family of quasi-compact opens of $X$. Say $E = \bigcap_{i \in I} U_i$ with $U_i \subset X$ quasi-compact open and $I$ nonempty. By adding finite intersections we may assume that for $i, j \in I$ there exists a $k \in I$ with $U_k \subset U_i \cap U_j$. In this situation we have \begin{equation} \tag{39.15.0.1} \Gamma(E, \mathcal{F}|_E) = \mathop{\rm colim}\nolimits \Gamma(U_i, \mathcal{F}|_{U_i}) \end{equation} for any sheaf $\mathcal{F}$ defined on $X$. Namely, fix $i_0 \in I$ and replace $X$ by $U_{i_0}$ and $I$ by $\{i \in I \mid U_i \subset U_{i_0}\}$. Then $X$ is quasi-compact and quasi-separated, hence a spectral space, see Properties, Lemma 27.2.4. Then we see the equality holds by Topology, Lemma 5.24.7 and Sheaves, Lemma 6.29.4. (In fact, the formula holds for higher cohomology groups as well if $\mathcal{F}$ is abelian, see Cohomology, Lemma 20.20.2.)

Lemma 39.15.1. Let $X$ be an ind-quasi-affine scheme. Let $E \subset X$ be an intersection of a nonempty family of quasi-compact opens of $X$. Set $A = \Gamma(E, \mathcal{O}_X|_E)$ and $Y = \mathop{\rm Spec}(A)$. Then the canonical morphism $$ j : (E, \mathcal{O}_X|_E) \longrightarrow (Y, \mathcal{O}_Y) $$ of Schemes, Lemma 25.6.4 determines an isomorphism $(E, \mathcal{O}_X|_E) \to (E', \mathcal{O}_Y|_{E'})$ where $E' \subset Y$ is an intersection of quasi-compact opens. If $W \subset E$ is open in $X$, then $j(W)$ is open in $Y$.

Proof. Note that $(E, \mathcal{O}_X|_E)$ is a locally ringed space so that Schemes, Lemma 25.6.4 applies to $A \to \Gamma(E, \mathcal{O}_X|_E)$. Write $E = \bigcap_{i \in I} U_i$ with $I \not = \emptyset$ and $U_i \subset X$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $U_k \subset U_i \cap U_j$. Set $A_i = \Gamma(U_i, \mathcal{O}_{U_i})$. We obtain commutative diagrams $$ \xymatrix{ (E, \mathcal{O}_X|_E) \ar[r] \ar[d] & (\mathop{\rm Spec}(A), \mathcal{O}_{\mathop{\rm Spec}(A)}) \ar[d] \\ (U_i, \mathcal{O}_{U_i}) \ar[r] & (\mathop{\rm Spec}(A_i), \mathcal{O}_{\mathop{\rm Spec}(A_i)}) } $$ Since $U_i$ is quasi-affine, we see that $U_i \to \mathop{\rm Spec}(A_i)$ is a quasi-compact open immersion. On the other hand $A = \mathop{\rm colim}\nolimits A_i$. Hence $\mathop{\rm Spec}(A) = \mathop{\rm lim}\nolimits \mathop{\rm Spec}(A_i)$ as topological spaces (Limits, Lemma 31.4.6). Since $E = \mathop{\rm lim}\nolimits U_i$ (by Topology, Lemma 5.24.7) we see that $E \to \mathop{\rm Spec}(A)$ is a homeomorphism onto its image $E'$ and that $E'$ is the intersection of the inverse images of the opens $U_i \subset \mathop{\rm Spec}(A_i)$ in $\mathop{\rm Spec}(A)$. For any $e \in E$ the local ring $\mathcal{O}_{X, e}$ is the value of $\mathcal{O}_{U_i, e}$ which is the same as the value on $\mathop{\rm Spec}(A)$.

To prove the final assertion of the lemma we argue as follows. Pick $i, j \in I$ with $U_i \subset U_j$. Consider the following commutative diagrams $$ \xymatrix{ U_i \ar[r] \ar[d] & \mathop{\rm Spec}(A_i) \ar[d] \\ U_i \ar[r] & \mathop{\rm Spec}(A_j) } \quad\quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\rm Spec}(A_i) \ar[d] \\ W \ar[r] & \mathop{\rm Spec}(A_j) } \quad\quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\rm Spec}(A) \ar[d] \\ W \ar[r] & \mathop{\rm Spec}(A_j) } $$ By Properties, Lemma 27.18.4 the first diagram is cartesian. Hence the second is cartesian as well. Passing to the limit we find that the third diagram is cartesian, so the top horizontal arrow of this diagram is an open immersion. $\square$

Lemma 39.15.2. Suppose given a cartesian diagram $$ \xymatrix{ X \ar[d]_f \ar[r] & \mathop{\rm Spec}(B) \ar[d] \\ Y \ar[r] & \mathop{\rm Spec}(A) } $$ of schemes. Let $E \subset Y$ be an intersection of a nonempty family of quasi-compact opens of $Y$. Then $$ \Gamma(f^{-1}(E), \mathcal{O}_X|_{f^{-1}(E)}) = \Gamma(E, \mathcal{O}_Y|_E) \otimes_A B $$ provided $Y$ is quasi-separated and $A \to B$ is flat.

Proof. Write $E = \bigcap_{i \in I} V_i$ with $V_i \subset Y$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $V_k \subset V_i \cap V_j$. Then we have similarly that $f^{-1}(E) = \bigcap_{i \in I} f^{-1}(V_i)$ in $X$. Thus the result follows from equation (39.15.0.1) and the corresponding result for $V_i$ and $f^{-1}(V_i)$ which is Cohomology of Schemes, Lemma 29.5.2. $\square$

Lemma 39.15.3 (Gabber). Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering. Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to S\}$, see Descent, Definition 34.31.3. If each morphism $V_i \to X_i$ is ind-quasi-affine, then the descent datum is effective.

Proof. Being ind-quasi-affine is a property of morphisms of schemes which is preserved under any base change, see More on Morphisms, Lemma 36.55.2. Hence Descent, Lemma 34.33.2 applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{X \to S\}$ flat surjective morphism of affines. Say $X = \mathop{\rm Spec}(A)$ and $S = \mathop{\rm Spec}(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$ and assume that $V \to X$ is ind-quasi-affine, in other words, $V$ is ind-quasi-affine.

Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with $U = X$ and $R = X \times_S X$ and $s$, $t$, $c$ as usual. By Groupoids, Lemma 38.21.3 the pair $(V, \varphi)$ corresponds to a cartesian morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes. Let $u' \in U'$ be any point. By Groupoids, Lemmas 38.19.2, 38.19.3, and 38.19.4 we can choose $u' \in W \subset E \subset U'$ where $W$ is open and $R'$-invariant, and $E$ is set-theoretically $R'$-invariant and an intersection of a nonempty family of quasi-compact opens.

Translating back to $(V, \varphi)$, for any $v \in V$ we can find $v \in W \subset E \subset V$ with the following properties: (a) $W$ is open and $\varphi(W \times_S X) = X \times_S W$ and (b) $E$ an intersection of quasi-compact opens and $\varphi(E \times_S X) = X \times_S E$ set-theoretically. Here we use the notation $E \times_S X$ to mean the inverse image of $E$ in $V \times_S X$ by the projection morphism and similarly for $X \times_S E$. By Lemma 39.15.2 this implies that $\varphi$ defines an isomorphism \begin{align*} \Gamma(E, \mathcal{O}_V|_E) \otimes_R A & = \Gamma(E \times_S X, \mathcal{O}_{V \times_S X}|_{E \times_S X}) \\ & \to \Gamma(X \times_S E, \mathcal{O}_{X \times_S V}|_{X \times_S E}) \\ & = A \otimes_R \Gamma(E, \mathcal{O}_V|_E) \end{align*} of $A \otimes_R A$-algebras which we will call $\psi$. The cocycle condition for $\varphi$ translates into the cocycle condition for $\psi$ as in Descent, Definition 34.3.1 (details omitted). By Descent, Proposition 34.3.9 we find an $R$-algebra $R'$ and an isomorphism $\chi : R' \otimes_R A \to \Gamma(E, \mathcal{O}_V|_E)$ of $A$-algebras, compatible with $\psi$ and the canonical descent datum on $R' \otimes_R A$.

By Lemma 39.15.1 we obtain a canonical ''embedding'' $$ j : (E, \mathcal{O}_V|_E) \longrightarrow \mathop{\rm Spec}(\Gamma(E, \mathcal{O}_V|_E)) = \mathop{\rm Spec}(R' \otimes_R A) $$ of locally ringed spaces. The construction of this map is canonical and we get a commutative diagram $$ \xymatrix{ & E \times_S X \ar[rr]_\varphi \ar[ld] \ar[rd]^{j'} & & X \times_S E \ar[rd] \ar[ld]_{j''} \\ E \ar[rd]^j & & \mathop{\rm Spec}(R' \otimes_R A \otimes_R A) \ar[ld] \ar[rd] & & E \ar[ld]_j \\ & \mathop{\rm Spec}(R' \otimes_R A) \ar[rd] && \mathop{\rm Spec}(R' \otimes_R A) \ar[ld] \\ & & \mathop{\rm Spec}(R') } $$ where $j'$ and $j''$ come from the same construction applied to $E \times_S X \subset V \times_S X$ and $X \times_S E \subset X \times_S V$ via $\chi$ and the identifications used to construct $\psi$. It follows that $j(W)$ is an open subscheme of $\mathop{\rm Spec}(R' \otimes_R A)$ whose inverse image under the two projections $\mathop{\rm Spec}(R' \otimes_R A \otimes_R A) \to \mathop{\rm Spec}(R' \otimes_R A)$ are equal. By Descent, Lemma 34.10.2 we find an open $W_0 \subset \mathop{\rm Spec}(R')$ whose base change to $\mathop{\rm Spec}(A)$ is $j(W)$. Contemplating the diagram above we see that the descent datum $(W, \varphi|_{W \times_S X})$ is effective. By Descent, Lemma 34.32.13 we see that our descent datum is effective. $\square$

    The code snippet corresponding to this tag is a part of the file more-groupoids.tex and is located in lines 2958–3199 (see updates for more information).

    \section{Descending ind-quasi-affine morphisms}
    \label{section-ind-quasi-affine}
    
    \noindent
    Ind-quasi-affine morphisms were defined in
    More on Morphisms, Section \ref{more-morphisms-section-ind-quasi-affine}.
    This section is the analogue of
    Descent, Section \ref{descent-section-quasi-affine}
    for ind-quasi-affine-morphisms.
    
    \medskip\noindent
    Let $X$ be a quasi-separated scheme. Let $E \subset X$ be a subset
    which is an intersection of a nonempty family of quasi-compact opens of $X$.
    Say $E = \bigcap_{i \in I} U_i$ with $U_i \subset X$ quasi-compact open
    and $I$ nonempty.
    By adding finite intersections we may assume that for $i, j \in I$
    there exists a $k \in I$ with $U_k \subset U_i \cap U_j$.
    In this situation we have
    \begin{equation}
    \label{equation-sections-of-intersection}
    \Gamma(E, \mathcal{F}|_E) = \colim \Gamma(U_i, \mathcal{F}|_{U_i})
    \end{equation}
    for any sheaf $\mathcal{F}$ defined on $X$. Namely, fix $i_0 \in I$
    and replace $X$ by $U_{i_0}$ and $I$ by
    $\{i \in I \mid U_i \subset U_{i_0}\}$. Then $X$ is quasi-compact
    and quasi-separated, hence a spectral space, see
    Properties, Lemma
    \ref{properties-lemma-quasi-compact-quasi-separated-spectral}.
    Then we see the equality holds by
    Topology, Lemma \ref{topology-lemma-make-spectral-space} and
    Sheaves, Lemma \ref{sheaves-lemma-descend-opens}.
    (In fact, the formula holds for higher cohomology groups
    as well if $\mathcal{F}$ is abelian, see
    Cohomology, Lemma \ref{cohomology-lemma-colimit}.)
    
    \begin{lemma}
    \label{lemma-sits-in-functions}
    Let $X$ be an ind-quasi-affine scheme. Let $E \subset X$ be an
    intersection of a nonempty family of quasi-compact opens of $X$.
    Set $A = \Gamma(E, \mathcal{O}_X|_E)$ and $Y = \Spec(A)$.
    Then the canonical morphism
    $$
    j : (E, \mathcal{O}_X|_E) \longrightarrow (Y, \mathcal{O}_Y)
    $$
    of Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}
    determines an isomorphism
    $(E, \mathcal{O}_X|_E) \to (E', \mathcal{O}_Y|_{E'})$
    where $E' \subset Y$ is an intersection of quasi-compact opens.
    If $W \subset E$ is open in $X$, then $j(W)$ is open in $Y$.
    \end{lemma}
    
    \begin{proof}
    Note that $(E, \mathcal{O}_X|_E)$ is a locally ringed space so that
    Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} applies
    to $A \to \Gamma(E, \mathcal{O}_X|_E)$. Write $E = \bigcap_{i \in I} U_i$
    with $I \not = \emptyset$ and $U_i \subset X$ quasi-compact open.
    We may and do assume that for $i, j \in I$ there exists a $k \in I$ with
    $U_k \subset U_i \cap U_j$. Set $A_i = \Gamma(U_i, \mathcal{O}_{U_i})$.
    We obtain commutative diagrams
    $$
    \xymatrix{
    (E, \mathcal{O}_X|_E) \ar[r] \ar[d] &
    (\Spec(A), \mathcal{O}_{\Spec(A)}) \ar[d] \\
    (U_i, \mathcal{O}_{U_i}) \ar[r] &
    (\Spec(A_i), \mathcal{O}_{\Spec(A_i)})
    }
    $$
    Since $U_i$ is quasi-affine, we see that $U_i \to \Spec(A_i)$
    is a quasi-compact open immersion. On the other hand
    $A = \colim A_i$. Hence $\Spec(A) = \lim \Spec(A_i)$ as topological
    spaces (Limits, Lemma \ref{limits-lemma-topology-limit}). Since
    $E = \lim U_i$ (by Topology, Lemma \ref{topology-lemma-make-spectral-space})
    we see that $E \to \Spec(A)$ is a homeomorphism onto its
    image $E'$ and that $E'$ is the intersection of the inverse images
    of the opens $U_i \subset \Spec(A_i)$ in $\Spec(A)$. For any
    $e \in E$ the local ring $\mathcal{O}_{X, e}$ is the value
    of $\mathcal{O}_{U_i, e}$ which is the same as the value on $\Spec(A)$.
    
    \medskip\noindent
    To prove the final assertion of the lemma we argue as follows.
    Pick $i, j \in I$ with $U_i \subset U_j$.
    Consider the following commutative diagrams
    $$
    \xymatrix{
    U_i \ar[r] \ar[d] & \Spec(A_i) \ar[d] \\
    U_i \ar[r] & \Spec(A_j)
    }
    \quad\quad
    \xymatrix{
    W \ar[r] \ar[d] & \Spec(A_i) \ar[d] \\
    W \ar[r] & \Spec(A_j)
    }
    \quad\quad
    \xymatrix{
    W \ar[r] \ar[d] & \Spec(A) \ar[d] \\
    W \ar[r] & \Spec(A_j)
    }
    $$
    By Properties, Lemma
    \ref{properties-lemma-cartesian-diagram-quasi-affine}
    the first diagram is cartesian. Hence the second is cartesian as well.
    Passing to the limit we find that the third diagram
    is cartesian, so the top horizontal arrow of this diagram is an open immersion.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-affine-base-change}
    Suppose given a cartesian diagram
    $$
    \xymatrix{
    X \ar[d]_f \ar[r] & \Spec(B) \ar[d] \\
    Y \ar[r] & \Spec(A)
    }
    $$
    of schemes. Let $E \subset Y$ be an intersection of a nonempty family
    of quasi-compact opens of $Y$. Then
    $$
    \Gamma(f^{-1}(E), \mathcal{O}_X|_{f^{-1}(E)}) =
    \Gamma(E, \mathcal{O}_Y|_E) \otimes_A B
    $$
    provided $Y$ is quasi-separated and $A \to B$ is flat.
    \end{lemma}
    
    \begin{proof}
    Write $E = \bigcap_{i \in I} V_i$ with $V_i \subset Y$ quasi-compact open.
    We may and do assume that for $i, j \in I$ there exists a $k \in I$ with
    $V_k \subset V_i \cap V_j$. Then we have similarly that
    $f^{-1}(E) = \bigcap_{i \in I} f^{-1}(V_i)$ in $X$.
    Thus the result follows from equation (\ref{equation-sections-of-intersection})
    and the corresponding result for $V_i$ and $f^{-1}(V_i)$ which is
    Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
    \end{proof}
    
    \begin{lemma}[Gabber]
    \label{lemma-ind-quasi-affine}
    Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering.
    Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to
    $\{X_i \to S\}$, see Descent, Definition
    \ref{descent-definition-descent-datum-for-family-of-morphisms}. 
    If each morphism $V_i \to X_i$ is ind-quasi-affine, then the descent datum
    is effective.
    \end{lemma}
    
    \begin{proof}
    Being ind-quasi-affine is a property of morphisms of schemes
    which is preserved under any base change, see
    More on Morphisms, Lemma
    \ref{more-morphisms-lemma-base-change-ind-quasi-affine}.
    Hence Descent, Lemma \ref{descent-lemma-descending-types-morphisms} applies
    and it suffices to prove the statement of the lemma
    in case the fpqc-covering is given by a single
    $\{X \to S\}$ flat surjective morphism of affines.
    Say $X = \Spec(A)$ and $S = \Spec(R)$ so
    that $R \to A$ is a faithfully flat ring map.
    Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$
    and assume that $V \to X$ is ind-quasi-affine, in other words,
    $V$ is ind-quasi-affine.
    
    \medskip\noindent
    Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with
    $U = X$ and $R = X \times_S X$ and $s$, $t$, $c$ as usual.
    By Groupoids, Lemma \ref{groupoids-lemma-cartesian-equivalent-descent-datum}
    the pair $(V, \varphi)$ corresponds to a cartesian morphism
    $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes.
    Let $u' \in U'$ be any point. By
    Groupoids, Lemmas \ref{groupoids-lemma-constructing-invariant-opens},
    \ref{groupoids-lemma-first-observation}, and
    \ref{groupoids-lemma-second-observation}
    we can choose $u' \in W \subset E \subset U'$
    where $W$ is open and $R'$-invariant, and
    $E$ is set-theoretically $R'$-invariant and
    an intersection of a nonempty family of quasi-compact opens.
    
    \medskip\noindent
    Translating back to $(V, \varphi)$, for any $v \in V$ we can find
    $v \in W \subset E \subset V$ with the following properties:
    (a) $W$ is open and $\varphi(W \times_S X) = X \times_S W$ and
    (b) $E$ an intersection of quasi-compact opens and
    $\varphi(E \times_S X) = X \times_S E$ set-theoretically.
    Here we use the notation $E \times_S X$ to mean the
    inverse image of $E$ in $V \times_S X$ by the projection morphism and
    similarly for $X \times_S E$. By Lemma \ref{lemma-affine-base-change}
    this implies that $\varphi$ defines an isomorphism
    \begin{align*}
    \Gamma(E, \mathcal{O}_V|_E) \otimes_R A
    & =
    \Gamma(E \times_S X, \mathcal{O}_{V \times_S X}|_{E \times_S X}) \\
    & \to
    \Gamma(X \times_S E, \mathcal{O}_{X \times_S V}|_{X \times_S E}) \\
    & =
    A \otimes_R \Gamma(E, \mathcal{O}_V|_E)
    \end{align*}
    of $A \otimes_R A$-algebras which we will call $\psi$.
    The cocycle condition for $\varphi$
    translates into the cocycle condition for $\psi$ as in
    Descent, Definition \ref{descent-definition-descent-datum-modules}
    (details omitted). By Descent, Proposition
    \ref{descent-proposition-descent-module}
    we find an $R$-algebra $R'$ and an isomorphism
    $\chi : R' \otimes_R A \to \Gamma(E, \mathcal{O}_V|_E)$
    of $A$-algebras, compatible with $\psi$ and the
    canonical descent datum on $R' \otimes_R A$.
    
    \medskip\noindent
    By Lemma \ref{lemma-sits-in-functions} we obtain a canonical ``embedding''
    $$
    j : (E, \mathcal{O}_V|_E) \longrightarrow
    \Spec(\Gamma(E, \mathcal{O}_V|_E)) = \Spec(R' \otimes_R A)
    $$
    of locally ringed spaces. The construction of this map is canonical
    and we get a commutative diagram
    $$
    \xymatrix{
    & E \times_S X \ar[rr]_\varphi \ar[ld] \ar[rd]^{j'} & &
    X \times_S E \ar[rd] \ar[ld]_{j''} \\
    E \ar[rd]^j  & &
    \Spec(R' \otimes_R A \otimes_R A) \ar[ld] \ar[rd] & &
    E \ar[ld]_j \\
    & \Spec(R' \otimes_R A) \ar[rd] && \Spec(R' \otimes_R A) \ar[ld] \\
    & & \Spec(R')
    }
    $$
    where $j'$ and $j''$ come from the same construction applied to
    $E \times_S X \subset V \times_S X$ and $X \times_S E \subset X \times_S V$
    via $\chi$ and the identifications used to construct $\psi$.
    It follows that $j(W)$ is an open subscheme of $\Spec(R' \otimes_R A)$
    whose inverse image under the two projections
    $\Spec(R' \otimes_R A \otimes_R A) \to \Spec(R' \otimes_R A)$
    are equal. By Descent, Lemma \ref{descent-lemma-open-fpqc-covering}
    we find an open $W_0 \subset \Spec(R')$ whose base change
    to $\Spec(A)$ is $j(W)$. Contemplating the diagram above
    we see that the descent datum $(W, \varphi|_{W \times_S X})$
    is effective. By Descent, Lemma
    \ref{descent-lemma-effective-for-fpqc-is-local-upstairs}
    we see that our descent datum is effective.
    \end{proof}
    
    
    
    
    
    
    \input{chapters}

    Comments (1)

    Comment #2763 by BCnrd on August 10, 2017 a 6:17 pm UTC

    To prove the instance of (34.15.0.1) that is relevant in this proof, namely $\{U_i\}$ an inverse system (under reverse inclusions) of quasi-compact open subspaces of a spectral space (rather than some more abstract kind of inverse system), one does not need the extreme generality of the links given above to prove it. Namely, one can given a less notationally intensive argument using basic "spreading out" properties involving quasi-compact open subsets, pro-constructible subsets, and open covers thereof in spectral spaces. I mention this because when trying to follow the links given for the proof of (34.15.0.1), one encounters a blizzard of notation which is unpleasant to unravel and too heavy for the actual situation relevant here. I found it easier to make my own proof of the relevant special case than to try to decipher whatever is going on in the given links. How about giving a Lemma here (or Corollary elsewhere) that focuses on the relevant special case of (34.15.0.1) with quasi-compact open subsets in a spectral space and gives a direct proof in that case, saving the ready from wading through the extra generality and heavier notation in the links presently given?

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