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Chapter 94: Examples > Section 94.57: Descent data for schemes need not be effective, even for a projective morphism

Lemma 94.57.1. There is an etale covering $X\to S$ of schemes and a descent datum $(V/X,\varphi)$ relative to $X\to S$ such that $V\to X$ is projective, but the descent datum is not effective in the category of schemes.

Proof. We imitate Hironaka's example of a smooth separated complex algebraic space of dimension 3 which is not a scheme [H, Example B.3.4.2].

Consider the action of the group $G = \mathbf{Z}/2 = \{1, g\}$ on projective 3-space $\mathbf{P}^3$ over the complex numbers by $$ g[x,y,z,w] = [y,x,w,z]. $$ The action is free outside the two disjoint lines $L_1=\{ [x,x,z,z]\}$ and $L_2=\{ [x,-x,z,-z]\}$ in ${\mathbf P}^3$. Let $Y={\mathbf P}^3-(L_1\cup L_2)$. There is a smooth quasi-projective scheme $S=Y/G$ over ${\mathbf C}$ such that $Y\to S$ is a $G$-torsor (Groupoids, Definition 38.11.3). Explicitly, we can define $S$ as the image of the open subset $Y$ in ${\mathbf P}^3$ under the morphism \begin{align*} {\mathbf P}^3 & \to \text{Proj } {\mathbf C}[x,y,z,w]^G\\ & = \text{Proj } {\mathbf C}[u_0,u_1,v_0,v_1,v_2]/(v_0v_1=v_2^2), \end{align*} where $u_0=x+y$, $u_1=z+w$, $v_0=(x-y)^2$, $v_1=(z-w)^2$, and $v_2=(x-y)(z-w)$, and the ring is graded with $u_0,u_1$ in degree 1 and $v_0,v_1,v_2$ in degree 2.

Let $C=\{ [x,y,z,w]: xy=z^2, w=0\}$ and $D=\{ [x,y,z,w]: xy=w^2, z=0\}$. These are smooth conic curves in ${\mathbf P}^3$, contained in the $G$-invariant open subset $Y$, with $g(C)=D$. Also, $C\cap D$ consists of the two points $P:=[1,0,0,0]$ and $Q:=[0,1,0,0]$, and these two points are switched by the action of $G$.

Let $V_Y\to Y$ be the scheme which over $Y-P$ is defined by blowing up $D$ and then the strict transform of $C$, and over $Y-Q$ is defined by blowing up $C$ and then the strict transform of $D$. (This is the same construction as in the proof of Lemma 94.56.1, except that $Y$ here denotes an open subset of ${\mathbf P}^3$ rather than all of ${\mathbf P}^3$.) Then the action of $G$ on $Y$ lifts to an action of $G$ on $V_Y$, which switches the inverse images of $Y-P$ and $Y-Q$. This action of $G$ on $V_Y$ gives a descent datum $(V_Y/Y,\varphi_Y)$ on $V_Y$ relative to the $G$-torsor $Y\to S$. The morphism $V_Y\to Y$ is proper but not projective, as shown in the proof of Lemma 94.56.1.

Let $X$ be the disjoint union of the open subsets $Y-P$ and $Y-Q$; then we have surjective etale morphisms $X\to Y\to S$. Let $V$ be the pullback of $V_Y\to Y$ to $X$; then the morphism $V\to X$ is projective, since $V_Y\to Y$ is a blow-up over each of the open subsets $Y-P$ and $Y-Q$. Moreover, the descent datum $(V_Y/Y,\varphi_Y)$ pulls back to a descent datum $(V/X,\varphi)$ relative to the etale covering $X\to S$.

Suppose that this descent datum is effective in the category of schemes. That is, there is a scheme $U\to S$ which pulls back to the morphism $V\to X$ together with its descent datum. Then $U$ would be the quotient of $V_Y$ by its $G$-action. $$ \xymatrix{ V \ar[r]\ar[d]& X\ar[d] \\ V_Y \ar[r]\ar[d]& Y\ar[d] \\ U \ar[r]& S } $$

Let $E$ be the inverse image of $C\cup D\subset Y$ in $V_Y$; thus $E\rightarrow C\cup D$ is a proper morphism, with fibers isomorphic to ${\mathbf P}^1$ over $(C\cup D)-\{P,Q\}$. The inverse image of $P$ in $E$ is a union of two lines $L_0$ and $M_0$. It follows that the inverse image of $Q=g(P)$ in $E$ is the union of two lines $L_0'=g(M_0)$ and $M_0'=g(L_0)$. As shown in the proof of Lemma 94.56.1, we have a rational equivalence $L_0+M_0'=L_0+g(L_0)\sim 0$ on $E$.

By descent of closed subschemes, there is a curve $L_1\subset U$ (isomorphic to ${\mathbf P}^1$) whose inverse image in $V_Y$ is $L_0\cup g(L_0)$. (Use Descent, Lemma 34.34.1, noting that a closed immersion is an affine morphism.) Let $R$ be a complex point of $L_1$. Since we assumed that $U$ is a scheme, we can choose a function $f$ in the local ring $O_{U,R}$ that vanishes at $R$ but not on the whole curve $L_1$. Let $D_{\text{loc}}$ be an irreducible component of the closed subset $\{f = 0\}$ in $\mathop{\rm Spec} O_{U,R}$; then $D_{\text{loc}}$ has codimension 1. The closure of $D_{\text{loc}}$ in $U$ is an irreducible divisor $D_U$ in $U$ which contains the point $R$ but not the whole curve $L_1$. The inverse image of $D_U$ in $V_Y$ is an effective divisor $D$ which intersects $L_0\cup g(L_0)$ but does not contain either curve $L_0$ or $g(L_0)$.

Since the complex 3-fold $V_Y$ is smooth, $O(D)$ is a line bundle on $V_Y$. We use here that a regular local ring is factorial, or in other words is a UFD, see More on Algebra, Lemma 15.94.7. The restriction of $O(D)$ to the proper surface $E\subset V_Y$ is a line bundle which has positive degree on the 1-cycle $L_0+g(L_0)$, by our information on $D$. Since $L_0+g(L_0)\sim 0$ on $E$, this contradicts that the degree of a line bundle is well-defined on 1-cycles modulo rational equivalence on a proper scheme over a field (Chow Homology, Lemma 41.21.2 and Lemma 41.27.2). Therefore the descent datum $(V/X,\varphi)$ is in fact not effective; that is, $U$ does not exist as a scheme. $\square$

    The code snippet corresponding to this tag is a part of the file examples.tex and is located in lines 4229–4235 (see updates for more information).

    \begin{lemma}
    \label{lemma-non-effective-descent-projective}
    There is an etale covering $X\to S$ of schemes and a descent datum
    $(V/X,\varphi)$ relative to $X\to S$ such that 
    $V\to X$ is projective,
    but the descent datum is not effective in the category of schemes.
    \end{lemma}
    
    \begin{proof}
    We imitate Hironaka's example of a smooth separated complex
    algebraic space of dimension 3
    which is not a scheme \cite[Example B.3.4.2]{H}.
    
    \medskip\noindent
    Consider the action of the group $G = \mathbf{Z}/2 = \{1, g\}$
    on projective 3-space
    $\mathbf{P}^3$ over the complex numbers by
    $$
    g[x,y,z,w] = [y,x,w,z].
    $$
    The action is free outside the two disjoint lines
    $L_1=\{ [x,x,z,z]\}$ and $L_2=\{ [x,-x,z,-z]\}$ in
    ${\mathbf P}^3$. Let $Y={\mathbf P}^3-(L_1\cup L_2)$. There is a
    smooth quasi-projective scheme $S=Y/G$ over ${\mathbf C}$ such that
    $Y\to S$ is a $G$-torsor (Groupoids,
    Definition \ref{groupoids-definition-principal-homogeneous-space}).
    Explicitly, we can define $S$ as the image of the open subset $Y$
    in ${\mathbf P}^3$ under the morphism
    \begin{align*}
    {\mathbf P}^3 & \to \text{Proj } {\mathbf C}[x,y,z,w]^G\\
       & = \text{Proj } {\mathbf C}[u_0,u_1,v_0,v_1,v_2]/(v_0v_1=v_2^2),
    \end{align*}
    where $u_0=x+y$, $u_1=z+w$, $v_0=(x-y)^2$, $v_1=(z-w)^2$,
    and $v_2=(x-y)(z-w)$, and the ring is graded with $u_0,u_1$
    in degree 1 and $v_0,v_1,v_2$ in degree 2.
    
    \medskip\noindent
    Let $C=\{ [x,y,z,w]: xy=z^2, w=0\}$ and $D=\{ [x,y,z,w]:
    xy=w^2, z=0\}$. These are smooth conic curves in ${\mathbf P}^3$, contained
    in the $G$-invariant open subset $Y$, with $g(C)=D$. Also,
    $C\cap D$ consists of the two points $P:=[1,0,0,0]$
    and $Q:=[0,1,0,0]$, and these two points are switched by the action
    of $G$. 
    
    \medskip\noindent
    Let $V_Y\to Y$ be the scheme which over $Y-P$
    is defined by blowing up $D$ and then the strict transform
    of $C$, and over $Y-Q$ is defined by blowing up $C$ and then
    the strict transform of $D$. (This is the same construction
    as in the proof of Lemma \ref{lemma-non-descending-property-projective},
    except that $Y$ here denotes an open subset of ${\mathbf P}^3$
    rather than all of ${\mathbf P}^3$.)
    Then the action of $G$ on $Y$ lifts to an action of $G$ on $V_Y$,
    which switches the inverse images of $Y-P$ and $Y-Q$. This action
    of $G$ on $V_Y$ gives a descent datum $(V_Y/Y,\varphi_Y)$
    on $V_Y$ relative to the $G$-torsor
    $Y\to S$. The morphism $V_Y\to Y$
    is proper but not projective, as shown in the proof
    of Lemma \ref{lemma-non-descending-property-projective}.
    
    \medskip\noindent
    Let $X$ be the disjoint union of the open subsets $Y-P$ and $Y-Q$;
    then we have surjective etale morphisms $X\to Y\to S$.
    Let $V$ be the pullback of $V_Y\to Y$ to $X$; then the morphism
    $V\to X$ is projective, since $V_Y\to Y$ is a blow-up over each of the open
    subsets $Y-P$ and $Y-Q$. Moreover, the descent datum $(V_Y/Y,\varphi_Y)$
    pulls back to a descent datum $(V/X,\varphi)$ relative to the
    etale covering $X\to S$.
    
    \medskip\noindent
    Suppose that this descent datum is effective in the category
    of schemes. That is, there is a scheme $U\to S$
    which pulls back to the morphism $V\to X$ together
    with its descent datum. Then $U$ would be the quotient
    of $V_Y$ by its $G$-action.
    $$
    \xymatrix{
    V \ar[r]\ar[d]&  X\ar[d] \\
    V_Y \ar[r]\ar[d]& Y\ar[d] \\
    U \ar[r]& S
    }
    $$
    
    \medskip\noindent
    Let $E$ be the inverse image of $C\cup D\subset Y$ in $V_Y$;
    thus $E\rightarrow C\cup D$ is a proper morphism, with fibers
    isomorphic to ${\mathbf P}^1$ over $(C\cup D)-\{P,Q\}$.
    The inverse image of $P$ in $E$ is a union of two lines $L_0$
    and $M_0$. It follows that the inverse image of $Q=g(P)$ in $E$
    is the union of two lines $L_0'=g(M_0)$ and $M_0'=g(L_0)$.
    As shown in the proof
    of Lemma \ref{lemma-non-descending-property-projective},
    we have a rational equivalence $L_0+M_0'=L_0+g(L_0)\sim 0$ on $E$.
    
    \medskip\noindent
    By descent of closed subschemes, there is a curve $L_1\subset U$
    (isomorphic to ${\mathbf P}^1$)
    whose inverse image in $V_Y$ is $L_0\cup g(L_0)$. (Use Descent, Lemma
    \ref{descent-lemma-affine}, noting that a closed immersion is an affine
    morphism.)
    Let $R$ be a complex point of $L_1$. Since
    we assumed that $U$ is a scheme, we can choose a function
    $f$ in the local ring $O_{U,R}$ that vanishes at $R$ but not
    on the whole curve $L_1$. Let $D_{\text{loc}}$ be an irreducible component
    of the closed subset $\{f = 0\}$ in $\Spec O_{U,R}$; then
    $D_{\text{loc}}$ has codimension 1.
    The closure of $D_{\text{loc}}$ in $U$ is an irreducible divisor $D_U$
    in $U$ which contains the point $R$ but not the whole curve $L_1$.
    The inverse image of $D_U$ in $V_Y$ is an effective divisor $D$
    which intersects $L_0\cup g(L_0)$ but does not contain either
    curve $L_0$ or $g(L_0)$.
    
    \medskip\noindent
    Since the complex 3-fold $V_Y$ is smooth, $O(D)$ is a line
    bundle on $V_Y$. We use here that a regular local ring is factorial,
    or in other words is a UFD, see
    More on Algebra, Lemma \ref{more-algebra-lemma-regular-local-UFD}.
    The restriction of $O(D)$ to the proper surface
    $E\subset V_Y$ is a line bundle which has positive degree on the 1-cycle
    $L_0+g(L_0)$, by our information on $D$. Since
    $L_0+g(L_0)\sim 0$ on $E$, this contradicts 
    that the degree of a line bundle is well-defined
    on 1-cycles modulo rational equivalence on a proper scheme
    over a field (Chow Homology,
    Lemma \ref{chow-lemma-proper-pushforward-rational-equivalence}
    and Lemma \ref{chow-lemma-factors}). Therefore the descent datum
    $(V/X,\varphi)$ is in fact not effective; that is, $U$ does not exist
    as a scheme.
    \end{proof}

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