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Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*}C$, where $C$ is a constant sheaf on a geometric (or Nisnevich) not necessarily closed (!!) point $x$ of $X$, $i_x:x\to X$ is the corresponding morphism. In this particular case the statement seems easy to prove; yet I wonder whether it follows from some general statement, and what are the 'standard' references for this. Are any additional restrictions needed here?

Also, I wonder whether sheaves of the type $i^*i_{x*}$ were studied somewhere in the literature?

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At least in topology the corresponding statement is true when $Y$ has a fundamental system of paracompact neighborhoods. See Godement, chapter II, \S 3.3. In particular, if $X$ is metrizable (which is the case in most situations) the restriction of a flabby sheaf to any subspace of $X$ is flabby (ibid, corollary 2). –  algori Oct 9 '11 at 21:21
    
The only thing I can say right now is that the result you want is not general nonsense, i.e. it is not a general property of topoi. In general, the functors that conserve flabby sheaves are $u_*$ ($u$ any morphism of topoi) and $i^!$ ($i$ a closed embedding), cf SGA 4 V 4.9 and 4.11. –  Alex Oct 10 '11 at 20:31

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