3 corrected Grothendieck's example

Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologiquie", sec. 3.8. Edit: The space is the complement of plane, and the sheaf is constructed by using a union of two irreducible curves in the plane, intersecting at two points. The sheaf is a constant sheaf.

Q2: Cech cohomology and derived functor cohomology coincide on a Hausdorff paracompact space (the proof is given in Godement's "Topologie algébrique et théorie des faisceaux"). Are you looking for I don't know of an example on a non paracompact space ?where they differ.

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Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologiquie", sec. 3.8. The space is a the complement of the union of two irreducible curves in the plane, intersecting at two points. The sheaf is a constant sheaf.

Q2: Cech cohomology and derived functor cohomology coincide on a Hausdorff paracompact space (the proof is given in Godement's "Topologie algébrique et théorie des faisceaux"). Are you seriously looking for an example on a non paracompact space?

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Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologiquie", sec. 3.8. The space is a union of two irreducible curves intersecting at two points.

Q2: Cech cohomology and derived functor cohomology coincide on a Hausdorff paracompact space (the proof is given in Godement's "Topologie algébrique et théorie des faisceaux"). Are you seriously looking for an example on a non paracompact space?