Pushforward is left exact, closed-embedding pushforward is exact, and pulling a sheaf back to a point gives its stalk.Suggest a slogan
Let \(f \colon X \to Y\) be continuous and \(0 \to \mcf' \to \mcf \to \mcf''\) an exact sequence of sheaves of abelian groups on \(X\).
\(0 \to f_* \mcf' \to f_* \mcf \to f_* \mcf''\) is exact.
If \(f\) is a homeomorphism onto a closed subset of \(Y\), then \(f_*\) is exact.
For \(i \colon \{x\} \hookrightarrow X\) the inclusion of a point, \(i^{-1} \mcf \cong \mcf_x\).
Without the closed-image hypothesis part 2 fails: for \(j \colon \CC^* \hookrightarrow \CC\) the exponential sequence \(0 \to 2\pi i \ZZ \to \OO \to \OO^* \to 0\) on \(\CC^*\) is exact, but \(j_* \OO \to j_* \OO^*\) is not surjective on stalks at \(0\), since \(z\) has no logarithm on any punctured disc.
Show that pushforward is left exact, and exact along a closed embedding.
For $i \colon \{x\} \hookrightarrow X$, show that $i^{-1} \mathcal{F} \cong \mathcal{F}_x$.
Dependencies
PR-C9ZEKExactness is checked on stalks, and global sections are only left exactuses