Definition 61.26.1. Let $j : U \to X$ be a weakly étale morphism of schemes.

The restriction functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{pro\text{-}\acute{e}tale})$ has a left adjoint $j_!^{Sh} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{pro\text{-}\acute{e}tale})$.

The restriction functor $j^{-1} : \textit{Ab}(X_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(U_{pro\text{-}\acute{e}tale})$ has a left adjoint which is denoted $j_! : \textit{Ab}(U_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(X_{pro\text{-}\acute{e}tale})$ and called

*extension by zero*.Let $\Lambda $ be a ring. The functor $j^{-1} : \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda ) \to \textit{Mod}(U_{pro\text{-}\acute{e}tale}, \Lambda )$ has a left adjoint $j_! : \textit{Mod}(U_{pro\text{-}\acute{e}tale}, \Lambda ) \to \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ and called

*extension by zero*.

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