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Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open.

For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\Gamma(V,\mathcal{F}) \rightarrow \Gamma(V-Z,\mathcal{F}))$ be the sections of $\mathcal{F}$ with support in $Z$. Denoting by $\mathfrak{Ab}(Y)$ the category of abelian sheaves on $Y$, we have the functor $\underline{\Gamma_Z}:\mathfrak{Ab}(Y) \rightarrow \mathfrak{Ab}(Y)$, such that for $\mathcal{F} \in \mathfrak{Ab}(Y)$ and $U\subset Y$ open, we have $\underline{\Gamma_Z}(\mathcal{F})(U)=\Gamma_{Z \cap U}(U,\mathcal{F}_{|U})$.

Let $\phi:(X,\mathcal{O}_X) \rightarrow (Y,\mathcal{O}_Y)$ be a flat morphism of locally noetherian locally ringed spaces, which induces the pullback $\phi^*:\mathfrak{QCoh}(Y) \rightarrow \mathfrak{QCoh}(X)$, where $\mathfrak{QCoh}(X)$ resp. $\mathfrak{QCoh}(Y)$ is the category of quasi-coherent sheaves on $X$ resp. $Y$. In this case we also have that $\underline{\Gamma_Z}:\mathfrak{QCoh}(Y) \rightarrow \mathfrak{QCoh}(Y)$.

Do we then have a commutative diagram

$\require{AMScd}$ \begin{CD} \mathfrak{QCoh}(Y) @>\underline{\Gamma_Z}>> \mathfrak{QCoh}(Y)\\ @V \phi^* V V @VV \phi^* V\\ \mathfrak{QCoh}(X) @>>\underline{\Gamma_{\phi^{-1}(Z)}}> \mathfrak{QCoh}(X) \end{CD}

?

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  • $\begingroup$ Is there a counter example if $\phi$ is not flat? $\endgroup$ – Karl Schwede Feb 10 at 5:25
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Yes, for a flat morphism $\phi:(X,\mathcal{O}_X) \rightarrow (Y,\mathcal{O}_Y)$ of quasi-compact and quasi-separated schemes one has an exact sequence $$ 0 \longrightarrow \underline{\Gamma_Z} \mathcal{F} \longrightarrow \mathcal{F} \longrightarrow j_*j^*\mathcal{F} $$ Where $j \colon X \setminus Z \to X$ denote the canonical embedding of the open complement. Applying $\phi^*$, one gets$$ 0 \longrightarrow \phi^*\underline{\Gamma_Z} \mathcal{F} \longrightarrow \phi^*\mathcal{F} \longrightarrow \phi^*j_*j^*\mathcal{F} $$ Let $j' \colon Y \setminus Z' \to y$ denote the corresponding canonical embedding with $Z' := \phi^{-1}(Z)$. We have the isomorphism $$ \phi^*j_*j^*\mathcal{F} \cong j'_*j'^*\phi^* \mathcal{F} $$ Whence, as $\underline{\Gamma_Z'}\phi^* \mathcal{F}$ is the kernel of the map $$ \phi^*\mathcal{F} \longrightarrow j'_*j'^*\phi^* \mathcal{F} $$ you obtain the desired identification $$ \phi^*\underline{\Gamma_Z} \mathcal{F} \cong \underline{\Gamma_Z'}\phi^* \mathcal{F}. $$

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  • $\begingroup$ Just wondering, the isomorphism $\phi^*j_*j^*\mathcal{F}\cong j'_*j'^*\phi^*\mathcal{F}$ comes from flat base change, or? Does this isomorphisim also exists for the adic analytification $\phi: X^{ad} \rightarrow X$? $\endgroup$ – CJS Feb 13 at 10:54
  • $\begingroup$ It's flat base change plus pseudo-functoriality (transitivity) of $(-)_*$. You need some sort of base change for your $\phi$, to repeat the argument. $\endgroup$ – Leo Alonso Feb 13 at 13:30
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I'd like to thank Leo Alonso for the answer and I'd like to add the case for $Z$ being locally closed.

So $Z=V \cap U^c \subset Y$, for $U,V \subset Y$ open. Hence we have the commutative diagram

$\require{AMScd}$ \begin{CD} \phi^{-1}(V-Z) @>j'>> \phi^{-1}(V) @>i'>> X\\ @V \phi_{|\phi^{-1}(V-Z)} V V @V \phi_{|\phi^{-1}(V)} V V @V \phi V V\\ V-Z @>j>> V @>i>>Y \end{CD}

Now let $W \subset Y$ be open, then

$i_*(\underline{\Gamma_Z}(\mathcal{F}_{|V}))(W)=\underline{\Gamma_Z}(\mathcal{F}_{|V}))(W\cap V) =\Gamma_{Z\cap W}(W\cap V,\mathcal{F}_{|V\cap W}) \\ \hspace{31mm}\stackrel{(*)} {\cong}\Gamma_{Z \cap W}(W,\mathcal{F}_{|W})=\underline{\Gamma_{Z}}(\mathcal{F})(W) $

where $(*)$ comes from excision. Hence we have an isomorphism $i_*(\underline{\Gamma_Z}(\mathcal{F}_{|V})) \ \cong \underline{\Gamma_{Z}}(\mathcal{F})$.

Then by this isomorphism and the explanation of Leo Alonso we have

$\phi^*({\underline{\Gamma_{Z}}}(\mathcal{F}))=\phi^*(i_*(\underline{\Gamma_Z}(\mathcal{F}_{|V})) \cong i'_*(\phi^*_{|\phi^{-1}(V)}(\underline{\Gamma_Z}(\mathcal{F}_{|V})) \\ \hspace{21mm}\cong i'_*(\underline{\Gamma_{\phi^{-1}(Z)}}(\phi^*_{|\phi^{-1}(V)}(\mathcal{F}_{|V})))\cong i'_*(\underline{\Gamma_{\phi^{-1}(Z)}}(\phi^*(\mathcal{F})_{|\phi^{-1}(V)}) \cong \underline{\Gamma_{\phi^{-1}(Z)}}(\phi^*(\mathcal{F}))$

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