The Stacks project

Remark 85.15.6 (Ringed variant). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. In this case, for any semi-representable object $K$ of $\mathcal{C}$ the site $\mathcal{C}/K$ is a ringed site with sheaf of rings $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$. The constructions above give

  1. a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C})$,

  2. a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\} $,

  3. a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ of ringed topoi,

  4. a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C})$.

Many of the results above hold in this setting. For example, the functor $j^*$ has an exact left adjoint

\[ j_! : \textit{Mod}(\mathcal{O}_ K) \to \textit{Mod}(\mathcal{O}_\mathcal {C}), \]

which in terms of the product decomposition given in (2) sends $(\mathcal{F}_ i)_{i \in I}$ to $\bigoplus j_{i, !}\mathcal{F}_ i$. Similarly, given $f : K \to L$ as above, the functor $f^*$ has an exact left adjoint $f_! : \textit{Mod}(\mathcal{O}_ K) \to \textit{Mod}(\mathcal{O}_ L)$. Thus the functors $j^*$ and $f^*$ are exact, i.e., $j$ and $f$ are flat morphisms of ringed topoi (also follows from the equalities $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$ and $\mathcal{O}_ K = f^{-1}\mathcal{O}_ L$).

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