(Isomorphic stalks, nonisomorphic sheaves)
Let \(X = Y = S^1 \subset \CC\) be the unit circle, and let \(f \colon X \to Y\) be the two-sheeted covering map \(f(z) = z^2\). Let \(\mcf = \ul{\ZZ}_X\) be the constant sheaf with value \(\ZZ\) on \(X\).
Every \(y \in Y\) has arbitrarily small connected open neighbourhoods \(V\) with \(f^{-1}(V) = U_1 \amalg U_2\), each \(U_i\) connected and mapped homeomorphically onto \(V\). So \((f_* \mcf)(V) = \mcf(U_1) \oplus \mcf(U_2) \cong \ZZ^2\), the restriction maps between such neighbourhoods are isomorphisms, and \((f_* \mcf)_y \cong \ZZ^2\).
Let \(\mcg = \ul{\ZZ^2}_Y\) be the constant sheaf. Its stalks are also \(\ZZ^2\) at every point.
\(\Gamma(Y, \mcg) \cong \ZZ^2\) because \(Y\) is connected, while \(\Gamma(Y, f_* \mcf) = \mcf(X) \cong \ZZ\) because \(X\) is connected. Since \(\ZZ \not\cong \ZZ^2\), the sheaves \(f_* \mcf\) and \(\mcg\) are not isomorphic, although all their stalks are isomorphic.
By step 1, \(f_* \mcf\) restricts to a constant sheaf on each such \(V\), so it is a locally constant sheaf. It is not constant: going once around \(Y\) exchanges the two sheets, and its global sections are the invariants \((\ZZ^2)^{\ZZ/2} \cong \ZZ\). So this example shows that stalks do not determine a sheaf; it does not show that pushforward destroys local constancy, because the pushforward of a locally constant sheaf along a finite covering map is locally constant.
A morphism of sheaves \(\phi \colon \mcf \to \mcg\) is an isomorphism exactly when every stalk map \(\phi_x\) is an isomorphism. The hypothesis is a single morphism \(\phi\) inducing the stalk isomorphisms; isomorphisms between stalks chosen separately at each point do not suffice.
(A pushforward of a locally constant sheaf that is not locally constant)
Let \(j \colon U = (0, \infty) \hookrightarrow \RR\) be the inclusion and \(\mcf = \ul{\ZZ}_U\). For an open interval \(I \subseteq \RR\), \((j_* \mcf)(I) = \mcf(I \cap (0, \infty))\), which is \(\ZZ\) if \(I\) meets \((0,\infty)\) and \(0\) otherwise. So \((j_* \mcf)_x = \ZZ\) for \(x \geq 0\) and \((j_* \mcf)_x = 0\) for \(x < 0\). Every neighbourhood of \(0\) contains points of both kinds, and a locally constant sheaf has isomorphic stalks at all points of a connected neighbourhood, so \(j_* \mcf\) is not locally constant.
Dependencies
D-VJFAPThe espace étalé, and sheafification as continuous sectionsuses
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D-SHFLOCSYSLocal systems and monodromyrelated-to