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    Exactness is checked on stalks, and global sections are only left exact

    PR-C9ZEK

    Sheaf exactness is stalkwise; global sections preserve kernels but can lose surjectivity.Suggest a slogan

    A sequence of sheaves \(\mcf \to \mcg \to \mch\) is exact exactly when \[ \mcf_p \to \mcg_p \to \mch_p \] is exact for every \(p \in X\). For \(0 \to \mcf \to \mcg \to \mch \to 0\) exact, the sequence of global sections \[ 0 \to \mcf(X) \to \mcg(X) \to \mch(X) \] is exact, but the last map need not be surjective.

    Surjectivity of \(\mcg \to \mch\) means every germ lifts, which is a statement about some neighbourhood of each point. A global section of \(\mch\) therefore lifts over each member of some cover, and the lifts need not agree on overlaps. The obstruction to correcting them is a Čech \(1\)-cocycle, and it is \(H^1(X, \mcf)\) that measures whether it can be corrected.

    The failure occurs on overlaps: local lifts differ by a Čech \(1\)-cocycle in the kernel, and its class in \(H^1(X,\mcf)\) is the obstruction to gluing them to a global lift.

    How do you check that a sequence of sheaves is exact?
    Is a surjection of sheaves surjective on sections?
    Where does the failure of right exactness go?

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    Backlinks

    • FE-Y12XBA surjection of sheaves that is not surjective on sectionsuses
    • PR-SHFPUSHEXExactness of pushforward, and pullback to a pointuses
    • D-PTIW0Čech cohomology and the comparison with sheaf cohomologyuses

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