# Torsion free sheaves on reducible curves

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?

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

• No. For one thing, $\mathcal{F}\otimes \mathcal{G}$ may have torsion. – 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.
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.
• 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... – Mohsen Karkheiran Sep 4 '18 at 13:50
• The specific example I have in mind is genus 1 reducible curve. Specially $I_N$ – Mohsen Karkheiran Sep 4 '18 at 13:54
{"ft_lang_label":"__label__en","ft_lang_prob":0.6759858,"math_prob":0.9993937,"size":1215,"snap":"2019-13-2019-22","text_gpt3_token_len":434,"char_repetition_ratio":0.25020644,"word_repetition_ratio":0.0,"special_character_ratio":0.3037037,"punctuation_ratio":0.052401748,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99997985,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-23T07:20:31Z\",\"WARC-Record-ID\":\"<urn:uuid:1fd278a4-bcb1-4e2d-9e75-a0f1081e0add>\",\"Content-Length\":\"116204\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6d4f2d37-b7b6-4f17-8132-40cde15940bb>\",\"WARC-Concurrent-To\":\"<urn:uuid:0484f37d-3c28-4eeb-8d3b-35fba2e8f0a7>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/309803/torsion-free-sheaves-on-reducible-curves\",\"WARC-Payload-Digest\":\"sha1:VY6423SQ7HI7CYKCO7ZS5K5IHRVA5ZZU\",\"WARC-Block-Digest\":\"sha1:5DGRK4Y3EHWCFVL56EMI2ORW37M2RPAM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202728.21_warc_CC-MAIN-20190323060839-20190323082839-00524.warc.gz\"}"}