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I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\mathcal{O}_X$ module by $A=A[t,t^{-1}]/(t-1)$.

Let $\mathcal{F}$ be a coherent $\mathcal{O}_{ X}$ module on $X$. Does there exists an exact sequence $$0\to \mathcal{K} \to \mathcal{G} \to \mathcal{F} \to 0$$ in $Coh(X)$ such that $Tor^{\mathcal{O}_X}_1(\mathcal{G},\mathcal{O}_{Spec A}) =0$ ?

Thanks in advance.

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    $\begingroup$ In fact $\mathrm{Tor}_1^{\mathcal{O}_X}(G,\mathcal{O}_X) = 0$ for any quasicoherent sheaf $G$, because $\mathcal{O}_X$ is $\mathcal{O}_X$-flat. $\endgroup$
    – Sasha
    Commented Oct 17 at 5:22
  • $\begingroup$ @Sasha Sorry for the bad mistake. I have edited my question $\endgroup$
    – KAK
    Commented Oct 17 at 5:31
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    $\begingroup$ As you have already observed, the scheme $X=\operatorname{Spec}(A[t^\pm])$ is affine, so you can just formulate it as a problem of modules. However, I am not sure what assumptions you have on $A$. If $A$ is stably coherent (e.g. a Noetherian ring, or a valuation ring), then this simply follows from existence of flat resolutions. $\endgroup$
    – Z. M
    Commented Oct 17 at 9:52
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    $\begingroup$ When the ring $A[t^\pm]$ is coherent, you can pick any finite projective $A[t^\pm]$-module (which is coherent, by coherence of $A[t^\pm]$) along with a surjection $p$ to $\mathcal F(X)$, and the coherence of $A[t^\pm]$ guarantees that the $A[t^\pm]$-module $\ker p$, a priori of finite type, is also coherent. $\endgroup$
    – Z. M
    Commented Oct 17 at 10:18
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    $\begingroup$ I mean, any finitely projective projective module equipped with a surjection. This is how one produces projective resolutions, but since the module in question is coherent, you can pick finitely generated projective resolutions. $\endgroup$
    – Z. M
    Commented Oct 17 at 10:50

1 Answer 1

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This is to remove the question from unanswered list. This is little elaboration of the answer given in Z.M in comments.

Since $\mathcal{F}$ is coherent sheaf on affine scheme $X$, it corresponds to a finitely generated $A[t,t^{-1}]$ module $M$. Thus there exists a finitely generated free module $N$ and a surjection $q:N\to M$. Let $\mathcal{G}$ be the sheaf associated to the module $N$. Then Since $A$ is noetherian so is $A[t,t^{-1}]$. Thus $N$ is a noetherian module so the $\ker q$ is finitely generated. Let $\mathcal{K}$ be the sheaf associated to $\ker q$. Then we have the required exact sequence. $N$ being free, hence flat, the required Tor is 0.

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