Lemma 109.65.1. 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 [Example B.3.4.2, H].

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 39.11.3). 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 109.64.1, 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 109.64.1.

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 blowup 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 109.64.1, 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 35.37.1, 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 $\mathop{\mathrm{Spec}}O_{U,R}$; 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 15.121.2. 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 42.20.3 and Lemma 42.28.2). Therefore the descent datum $(V/X,\varphi )$ is in fact not effective; that is, $U$ does not exist as a scheme. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).