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    Stalks and germs

    D-0QSI0

    (Stalk)

    For a presheaf \(\mcf\) on \(X\) and a point \(p\), \[ \mcf_p \da \colim_{U \ni p} \mcf(U) , \] the colimit over open neighbourhoods of \(p\) ordered by reverse inclusion. An element of \(\mcf_p\) is a germ of a section at \(p\): a pair \((U, s)\) with \(s \in \mcf(U)\), where two pairs are identified when the sections agree on some smaller neighbourhood.

    The open neighbourhoods of \(p\) form a directed set under reverse inclusion, so the colimit is filtered. Filtered colimits of abelian groups are exact, so \(\mcf\mapsto\mcf_p\) is an exact functor on sheaves of abelian groups. A sequence of sheaves of abelian groups on \(X\) is exact if and only if it is exact on the stalk at every point of \(X\); in particular, a morphism of sheaves is injective, surjective, or an isomorphism if and only if it is so on every stalk.

    For the sheaf \(C^\infty\) of smooth functions on \(\RR\), the map \(C^\infty(\RR)\to C^\infty_0\) is not injective: a nonzero smooth function supported in \([1,2]\) has germ \(0\) at \(0\). The germ of \(f\) at \(0\) determines every derivative \(f^{(k)}(0)\), but the derivatives do not determine the germ: the function equal to \(e^{-1/x^2}\) for \(x>0\) and to \(0\) for \(x\le0\) has Taylor series \(0\) at \(0\) and a nonzero germ at \(0\).

    What is the stalk of a sheaf?
    What is a germ?
    Show that the stalk at $p$ of the sheaf of holomorphic functions on an $n$-dimensional complex manifold is the ring $\CC\{z_1, \ldots, z_n\}$ of convergent power series.
    Show that for a real smooth manifold the Taylor map from the stalk of $C^\infty$ functions at $p$ to $\RR[[x_1, \ldots, x_n]]$ has a nonzero kernel.
    For an affine variety $X$ and $p \in X$, show that $\OO_{X,p} \cong A(X)_{\mfm_p}$.
    Give a morphism of sheaves that is surjective on stalks but not on global sections.

    Dependencies

    • D-RCCFYPresheaves, sheaves, and the sheaf axiomsuses

    Backlinks

    • D-COMMACATComma categories, and colimits as initial coconesrelated-to
    • D-UDIVHSupport of sections and sheavesuses
    • D-VJFAPThe espace étalé, and sheafification as continuous sectionsuses

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