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Let $X$ be a smooth projective algebraic variety over a field $k$, of dimension $n$, and let $Z$ be a smooth closed subvariety of dimension $m$, with $i: Z \hookrightarrow X$ the inclusion map.

For any locally free coherent sheaf $\mathcal{F}$ on $X$, there is a pullback map $$\imath^*: H^i(X, \mathcal{F}) \to H^i(Z, \iota^* \mathcal{F});$$ and via Serre duality we have isomorphisms $H^i(X, \mathcal{F})^\vee = H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)$ and $H^i(Z, \iota^* \mathcal{F})^\vee = H^{m-i}(Z, \iota^* \mathcal{F}^\vee \otimes \omega_Z)$, where $\omega_X$ and $\omega_Z$ are the dualising sheaves. Setting $j=m-i$ and $\mathcal{G} = \mathcal{F}^\vee$, we conclude that there is a pushforward map $$\imath_*: H^j(Z, \iota^* \mathcal{G} \otimes \omega_Z) \to H^{j + c}(X, \mathcal{G} \otimes \omega_X),$$ for any $j$ and any locally free coherent sheaf $\mathcal{G}$ on $X$, where $c = n-m$ is the codimension of $Z$ in $X$.

Does this map have an intrinsic description (not using Serre duality)? Can it be defined without assuming that $X$ be projective, or that $\mathcal{G}$ be locally free?

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    $\begingroup$ Looks a bit like a coherent version of the Gysin map. $\endgroup$
    – Martin Bright
    Commented Nov 21, 2017 at 10:26
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    $\begingroup$ Yes, that's exactly what I'm hoping for. $\endgroup$
    – David Loeffler
    Commented Nov 21, 2017 at 10:40
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    $\begingroup$ Nice question. One more typo: $\omega_Y$ should be $\omega_Z$. One comment: writing $H$ for $G \otimes \omega_X$, the sheaf on $Z$ then becomes (if I calculated correctly) $i^*H \otimes \wedge^{top} N$ where $N$ is the normal bundle of $Z$ in $X$. I don't know if that helps. $\endgroup$
    – Pooter
    Commented Nov 21, 2017 at 10:41
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    $\begingroup$ The suggestion by @Pooter is correct. The result is, roughly, Lemma III.7.4, p. 242, and Theorem III.7.11, p. 245, of Hartshorne's "Algebraic Geometry." For the pushforward sheaf $\iota_*\omega_Z$, there is a Yoneda-Ext class $a_\iota\in \text{Ext}^c_{\mathcal{O}_X}(\iota_*\omega_Z,\omega_X).$ Tensoring $\mathcal{G}$ with the associated $(c+1)$-term acyclic complex on $X$ again gives an acyclic complex (since $\mathcal{G}$ is flat). Chasing through connecting maps gives $\iota_*$ (up to a sign). $\endgroup$
    – Jason Starr
    Commented Nov 21, 2017 at 10:54
  • $\begingroup$ @JasonStarr Thanks for the reference! Hartshorne only seems to consider the case where $X = \mathbf{P}^N$ for some $N$; but I guess the general case probably reduces to this. $\endgroup$
    – David Loeffler
    Commented Nov 21, 2017 at 14:45

1 Answer 1

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The map is induced by the right adjoint $i^!$ of the pushforward functor and the adjunction morphism $i_*i^! \to \mathrm{id}$, in view of the formula $i^!(F) \cong i^*(F) \otimes \omega_{Z/X}[\dim Z/X]$. This works for any locally complete intersection closed embedding.

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    $\begingroup$ In other words, the map is induced by the residue along $Z$. $\endgroup$
    – Leo Alonso
    Commented Nov 25, 2017 at 10:30
  • $\begingroup$ @Sasha Defining the map as being "induced by the right adjoint of the pushforward functor" is essentially re-stating Serre duality. What I was hoping for was a direct description of $\iota_*$ (one not going via duality). $\endgroup$
    – David Loeffler
    Commented Nov 25, 2017 at 13:35
  • $\begingroup$ @DavidLoeffler Not at all! Serre duality, as you mentioned, works only in projective setting, while the adjoint functor for a closed embedding always exists. Of course, when Serre functors exist you can express the right adjoint functor in terms of the left adjoint and duality isomorphisms, but I would say that right adjoint functor is more fundamental than Serre duality. $\endgroup$
    – Sasha
    Commented Nov 26, 2017 at 11:29

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