Suppose $X=C_1\cup C_2\dots\cup C_N$, be a reduced but reducible curve, and $C_i$'s are $\mathbb{P}^1$. Then I think it is well known that every torsion free sheaf $\mathcal{F}$ with pure dimension one (which means that the support of all of the subsheaves of $\mathcal{F}$ has dimension 1), can be defined in a short exact sequence as,

$$0\rightarrow \mathcal{F}\rightarrow\mathcal{F}_{C_1}\oplus\mathcal{F}_{C_2}\oplus\dots\oplus\mathcal{F}_{C_N}\rightarrow T\rightarrow 0$$

where $T$ is a torsion sheaf supported on the intersection of $C_i$'s, and $\mathcal{F}_{C_i} = \mathcal{F}|_{C_i}/$torsion .

Then my question is if we have another pure dimension one sheaf $\mathcal{G}$, can I write the following short exact sequence?

$$0\rightarrow \mathcal{F}\otimes\mathcal{G} \rightarrow(\mathcal{F}\otimes\mathcal{G})_{C_1}\oplus(\mathcal{F}\otimes\mathcal{G})_{C_2}\oplus\dots\oplus(\mathcal{F}\otimes\mathcal{G})_{C_N}\rightarrow T\rightarrow 0$$

where $(\mathcal{F}\otimes\mathcal{G})_{C_i}$ is simply $\mathcal{F}_{C_i}\otimes\mathcal{G}_{C_i}$. In other words, is the following equality correct?

$$\mathcal{F}|_{C_i}/torsion \otimes \mathcal{G}|_{C_i}/torsion = (\mathcal{F}\otimes\mathcal{G})|_{C_i}/torsion$$

  • 2
    $\begingroup$ No. For one thing, $\mathcal{F}\otimes \mathcal{G}$ may have torsion. $\endgroup$ – abx Sep 4 '18 at 4:02

@Sasha I think that in your example (two lines meeting transversally at one point) we have $T=0$. I think that the first exact sequence is valid for every torsion free sheaf, if the components of $X$ meet transversally and there are only double points.

@Mohsen Karkheiran
Maybe you could first see what happens in intersection points (suppose only ordinary double points). Suppose $P\in C_1\cap C_2$. Then $\mathcal{F}_P$ is isomorphic to either $\mathcal{O}_{C_1,P}\oplus\mathcal{O}_{C_2,P}$ or $\mathcal{O}_{X,P}$, and the same for $\mathcal{G}$. You can then look what happens in each of the 4 cases.

  • $\begingroup$ I can refer to this paper arxiv.org/abs/math/0410393, right before Proposition 2.3, the author mentions that if X is projective, and $C_i$ are the irreducible components then (taken from a paper or book by Seshadri) the representation exists. Unfortunately, I don't have access to the reference by Seshadri to read the proof... $\endgroup$ – Mohsen Karkheiran Sep 4 '18 at 13:50
  • $\begingroup$ The specific example I have in mind is genus 1 reducible curve. Specially $I_N$ $\endgroup$ – Mohsen Karkheiran Sep 4 '18 at 13:54
  • $\begingroup$ @Hephaistos: You are right. $\endgroup$ – Sasha Sep 6 '18 at 8:06

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