Descent data for schemes over an fppf covering

Let $\{X_i\to X{\}}_{i\in I}$ be an fppf covering of an $S$-scheme $X$ (Fppf coverings and the fppf site, Schemes and morphisms over a base), and put $X_{ij}=X_i{\times }_X{X}_j,X_{ijk}=X_i{\times }_X{X}_j{\times }_X{X}_k$ (Fibre product of schemes). Write ${\mathrm{p}\mathrm{r}}_1,{\mathrm{p}\mathrm{r}}_2\text{\hspace{0.17em}⁣}:X_{ij}\to X_i,X_j$ and ${\mathrm{p}\mathrm{r}}_{12},{\mathrm{p}\mathrm{r}}_{13},{\mathrm{p}\mathrm{r}}_{23}\text{\hspace{0.17em}⁣}:X_{ijk}\to X_{ij},X_{ik},X_{jk}$ for the projections, and ${\mathrm{p}\mathrm{r}}_i$ for the projection of any of these fibre products to $X_i$.

A descent datum for schemes relative to this covering is a family of $X_i$-schemes $V_i\to X_i$ (Morphisms of schemes) together with isomorphisms ${\phi }_{ij}\text{\hspace{0.17em}⁣}:{\mathrm{p}\mathrm{r}}_1^{\ast }{V}_i=V_i{\times }_{X_i}{X}_{ij}⟶{\mathrm{p}\mathrm{r}}_2^{\ast }{V}_j=V_j{\times }_{X_j}{X}_{ij}$ over $X_{ij}$, one for each ordered pair $(i,j)$, satisfying the cocycle condition ${\mathrm{p}\mathrm{r}}_{13}^{\ast }{\phi }_{ik}={\mathrm{p}\mathrm{r}}_{23}^{\ast }{\phi }_{jk}\circ {\mathrm{p}\mathrm{r}}_{12}^{\ast }{\phi }_{ij}$ over $X_{ijk}$, where the pullbacks are taken along the displayed projections of $X_{ijk}$ and the composites are computed in the category of schemes over $X_{ijk}$. The condition is stated for all ordered triples and includes the case $i=j=k$; the identity structure of the fibre products identifies the pullbacks unambiguously.

A morphism of descent data $(V_i,{\phi }_{ij})\to (W_i,{\psi }_{ij})$ is a family of $X_i$-morphisms $f_i\text{\hspace{0.17em}⁣}:V_i\to W_i$ compatible with the isomorphisms, i.e. ${\psi }_{ij}\circ {\mathrm{p}\mathrm{r}}_1^{\ast }{f}_i={\mathrm{p}\mathrm{r}}_2^{\ast }{f}_j\circ {\phi }_{ij}$ over every $X_{ij}$.

The descent datum is effective when there is an $X$-scheme $V$ together with isomorphisms $V{\times }_X{X}_i\cong V_i$ over $X_i$ for all $i$ whose pullbacks to every $X_{ij}$ agree with the ${\phi }_{ij}$ through the canonical identifications $(V{\times }_X{X}_i){\times }_{X_i}{X}_{ij}\cong V{\times }_X{X}_{ij}\cong (V{\times }_X{X}_j){\times }_{X_j}{X}_{ij}$. Equivalently, the datum is effective precisely when it lies in the essential image of the base-change functor $V\mapsto (V{\times }_X{X}_i)$. Descent data and their morphisms form a category in the evident way, with composition componentwise.