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I'm reading Hartshorne Chapter 5 (1.3). He uses the following fact:

Assume $C,D$ are effective divisors over a projective smooth surface $X$. Then there is a short exact sequence: $$0\rightarrow\mathcal L(-D)\otimes\mathcal O_C\rightarrow\mathcal O_C\rightarrow\mathcal O_{C\cap D}\rightarrow 0$$

I'm trying to prove it and here is my attempt:

First, I try to make the statement more precise. Let $i:C\rightarrow X$ and $j:D\rightarrow X$ and $l:C\cap D\rightarrow C$ be closed immersions. Then it actually states $0\rightarrow i^*\mathcal L(-D)\rightarrow\mathcal O_C\rightarrow l_*\mathcal O_{C\cap D}\rightarrow 0$.

Now pullback the short exact sequence $0\rightarrow \mathcal L(-D)\rightarrow \mathcal O_X\rightarrow j_*\mathcal O_D\rightarrow 0$. We obtain: $$i^*\mathcal L(-D)\rightarrow\mathcal O_C\rightarrow i^*j_*\mathcal O_D\rightarrow 0$$

Here I meet 2 questions:

  1. How to show it is exact at left?

  2. How to show $i^*j_*\mathcal O_D = \mathcal l_*O_{C\cap D}$? (I try to draw the Cartesian diagram of $C\cap D$ and do pullback and pushforward. But I fail to work it out)

Thank you in advance!

Hydrogen
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  • is just saying that the restriction of an ideal sheaf to a closed subscheme is again an ideal sheaf; is this a familiar statement to you?
    • – Tabes Bridges Sep 14 '21 at 01:19
  • Tabes, I don’t think that’s correct. 1) is saying the divisor $D$ intersects the curve transversely. – Jake Levinson Sep 14 '21 at 12:47
  • And in general $I \otimes R/J$ is $I/IJ$, which may not be an ideal of $R/J$. – Jake Levinson Sep 14 '21 at 12:49
  • @JakeLevinson I find Hartshorne also uses this short exact sequence for nontransverse case in chapter 5 (1.4). Is this fact still true in that case? – Hydrogen Sep 14 '21 at 17:11
  • @TabesBridges Sorry, I'm not really familiar with this fact. Could you give a precise statement of it? – Hydrogen Sep 14 '21 at 17:12
  • @JakeLevinson OK, I thought transversality was needed for that claim but second-guessed myself based on the given statement. Checking in Hartshorne now I see that he assumes transversality in the statement of the lemma. – Tabes Bridges Sep 14 '21 at 18:51
  • @Hydrogen as Jake points out, this only holds in the transverse case where it corresponds affine-locally to the ideal-theoretic statement that he mentioned. As for the non-transversality in 1.4, I'm really not sure what to make of it. It seems like a rather large leap in logic, but perhaps the relevant $\operatorname{Tor}$ module vanishes for reasons that I'm not seeing right now. – Tabes Bridges Sep 14 '21 at 18:54
  • In that Hartshorne lemma V.1.4, he assumes the curves have no common component, which is "transverse" enough here: the curves are reduced and the ambient surface is smooth (running assumptions in this section). So, the assumption implies that the local equation for D is a non-zerodivisor on C. – Jake Levinson Sep 14 '21 at 19:35
  • I thought this felt familiar... https://math.stackexchange.com/questions/1028798/why-do-we-retain-exactness-when-tensoring-by-mathcalo-c-in-hartshorne-lemm?rq=1 – Jake Levinson Sep 14 '21 at 19:39
  • @JakeLevinson Thank you for your answer. Is there any hint about question 2? – Hydrogen Sep 14 '21 at 23:16
  • For 2) note that you have a corresponding Cartesian square of sheaves on $X$, with $\mathcal O_X$ at the top left and $i_* l_* \mathcal O_{C\cap D}$ at the bottom right. Try fiddling around with this square and adjunctions to find a map between $i^* j_* \mathcal O_D$ and $l_* \mathcal O_{C\cap D}\cong l_* k^* \mathcal O_D$, the latter of which seems more natural [since we are now looking at the opposite of the configuration studied by the base change theorems]). Then it should be straightforward (well, at least in the transverse case) to check that this map is an isomorphism on stalks. – Tabes Bridges Sep 15 '21 at 01:35
  • Note that by $k$ I mean the inclusion $k:C \cap D \hookrightarrow D$, and that I could be wildly overcomplicating things. – Tabes Bridges Sep 15 '21 at 01:36
  • @TabesBridges Sorry, but I'm a little confused about how to show $i^j_\mathcal O_D\cong l_k^\mathcal O_D$. This seems to be nontrivial because a similar version of this need flatness. (Hartshorne Chapter 3 Proposition 9.3) – Hydrogen Sep 15 '21 at 03:34
  • That is one of the theorems on cohomology and base change, which as I mentioned go in the opposite direction. So while I don't have any immediate citations, my suspicion is that this map exists due to some kind of abstract nonsense and can be shown to be an isomorphism due to "a sheaf map which is an isomorphism on stalks is itself an isomorphism," although this may be (again) tricker in the case that the intersection is not strictly transverse. – Tabes Bridges Sep 15 '21 at 07:00