I'm reading through and trying to understand the proof of Proposition 5.8 in Chapter II of Algebraic Geometry by Hartshorne that the pushforward of quasicoherent sheaf $\mathcal{F}$ by a morphism $\phi : X \rightarrow Y$ is a quasicoherent sheaf on $Y$ when $X$ is noetherian.

The step I'm actually struggling to understand is the following: we assume that $Y$ is affine, as this is a local quesion on $Y$. Since $X$ is noetherian, it may be covered by finitely many open affines $U_i$, and there are finitely many open affines $U_{ijk}$ covering $U_i \cap U_j$. Then if $V \subset Y$ is open, a section $s \in \phi_* \mathcal{F}(V) = \mathcal{F}( \phi^{-1}(V))$ is determined uniquely by sections $s_i$ in $\mathcal{F}( \phi^{-1}(V)\cap U_i)$ which agree on $\phi^{-1}(V)\cap U_{ijk}$.

Then we get an injection $\phi_* \mathcal{F}(V) \hookrightarrow \bigoplus_i \mathcal{F}|_{U_i}( \phi^{-1}(V))$ by sending $s \mapsto (s|_{\phi^{-1}(V)\cap U_i})_i$ by the identity axiom.

However, we also are supposed to have a morphism $\bigoplus_i \mathcal{F}|_{U_i}( \phi^{-1}(V)) \rightarrow \bigoplus_{i,j,k} \mathcal{F}|_{U_{ijk}}( \phi^{-1}(V))$ with kernel $\phi_* \mathcal{F}(V)$. In a naive way I can see where this comes from, but it doesn't make sense to me as a morphism of rings to me. I think I'm really confusing myself and would appreciate some help clarifying things.

  • $\begingroup$ But if it's a sheaf morphism doesn't it need to be give ring morphisms on the open subsets? That's what I don't get - the only reasonable ring morphism to my mind is the one which sends $(s_i)_i \mapsto (s_i|_{U_{ijk}})_{i,j,k}$, but that would just have as a kernel sections which are $0$ where the $U_i$ and $U_j$ overlap. $\endgroup$ – Harry Dec 19 '15 at 17:54
  • $\begingroup$ Backing up a bit: $\mathcal{F}$ here is not the structure sheaf and is just an $\mathcal{O}_X$-module for the purposes of this definition. So I don't see why you want a ring to come out of it. $\endgroup$ – Hoot Dec 19 '15 at 18:00
  • $\begingroup$ The map you want, I think, sends $s_i$ to $s_i|_{U_{ijk}}$ in the $i,j,k$ slot and $-s_i$ in the $j,i,k$ slot. Think about what you want the kernel to be. [Maybe I'm missing some of your $\phi$s and $V$s but that's really not the point. We're more or less restating the definition of a sheaf here.] $\endgroup$ – Hoot Dec 19 '15 at 18:02
  • $\begingroup$ Perhaps could we send $s_i$ to $s_i|_{U_{ijk}} - s_j|_{U_{ijk}}$. I think that would be a module homomorphism and has kernel the $(s_i)$-tuples which agree on overlaps? $\endgroup$ – Harry Dec 22 '15 at 1:17

Yes (well spotted btw) your last comment is exactly the case. This is also in the proof of the Gathmann's online algebraic geometry script (Prop. 7.2.9): the last map is $$(\dots,s_i,\dots)\mapsto (\dots,s_i|U_{ijk}-s_j|U_{ijk},\dots)$$

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