$\def\sO{\mathcal{O}}
\def\sK{\mathcal{K}}
\def\sC{\mathscr{C}}$I am trying to understand what the maps are on a certain s.e.s. of sheaves of modules on an algebraic curve. It is \eqref{ses} on Conrad's Grothendieck Duality and Base Change, p. 227. I'm unable to find any other similar-looking s.e.s. or any reference on books, the Stacks Project, Google or Stack Exchange.
Here's the context in which the s.e.s. is stated: let $k$ be an algebraically closed field and let $X$ be a reduced and proper $k$-scheme of dimension $1$. Denote $\pi:\tilde{X}\to X$ to the normalization of $X$. In particular, $\sO_X\subset\pi_*\sO_{\tilde{X}}$ and the latter is the integral closure of $\sO_X$ in $\sK_X$. We define the conductor ideal
$$
\sC=\operatorname{Ann}_{\sO_X}(\pi_*\sO_{\tilde{X}}/\sO_X).
$$
This ideal sheaf cuts the non-normal locus (equivalently, the non-smooth locus, since $X$ is a curve) of $X$ (Bourbaki, Commutative Algebra, Ch. V, §1, no. 5, Corollary 5). We have that $\sC$ is a $\pi_*\sO_{\tilde{X}}$-module and that it is $\sO_X$-coherent (so is the annihilator of a coherent module on a locally Noetherian scheme; use 0H2L). Hence, it is $\pi_*\sO_{\tilde{X}}$-quasi-coherent (see first lemma here). Therefore, there is an ideal sheaf $\tilde{\sC}$ on $\tilde{X}$ with $\pi_*\tilde{\sC}=\sC$ (ibid., second lemma); unique up to unique isomorphism. Now Conrad argues “since $\tilde{\sC}$ is a coherent ideal sheaf on the smooth curve $\tilde{X}$ and is generically non-zero, it is an invertible sheaf.”
My first question is:
(Q1). Why does it follow $\tilde{\sC}$ is invertible?
I get that if $S\subset X$ denotes the singular locus of $X$, then $\tilde{U}=\pi^{-1}(X\setminus S)\subset\tilde{X}$ is open dense and $\sC|_{\tilde{U}}\cong\sO_{\tilde{U}}$. But I don't see why does this imply that $\sC$ is globally invertible.
He continues: “Thus, there is an exact sequence
$$
\label{ses}\tag{5.2.2}
0\to\Omega^1_{\tilde{X}/k}\to\Omega^1_{\tilde{X}/k}(\tilde{C})\to\sO_{\tilde{C}}\to 0
$$
where $\sO_{\tilde{C}}=\sO_{\tilde{X}}/\tilde{\sC}$ is the structure sheaf of the finite closed subscheme $\tilde{C}\subset\tilde{X}$ defined by $\tilde{\sC}$ and $\Omega^1_{\tilde{X}/k}(\tilde{C})=\Omega^1_{\tilde{X}/k}\otimes\tilde{\sC}^{-1}$ is the sheaf of meromorphic differentials on $\tilde{X}$ with ‘poles no worse than $\tilde{C}$’ (where we view $\tilde{C}$ as an effective Weil divisor on $\tilde{X}$).”
My other questions are:
(Q2). What are the maps in \eqref{ses}? I have no clue for any of them.
(Q3). What does it exactly mean that “$\Omega^1_{\tilde{X}/k}(\tilde{C})=\Omega^1_{\tilde{X}/k}\otimes\tilde{\sC}^{-1}$ is the sheaf of meromorphic differentials on $\tilde{X}$ with ‘poles no worse than $\tilde{C}$’”?
Best Answer
A subtlety here is that I don't think the second arrow in your short exact sequence is not supposed to be canonical. (It's not canonical in general, and I don't think the data of $\tilde{X} \to X$ is enough to make it so.) It is if $\tilde{C}$ is a reduced divisor, by the residue map, but that's probably not relevant here.
So one has the short exact sequence
$$ 0 \to \tilde{\mathscr C} \to \mathcal O_{\tilde{X}} \to \mathcal O_{\tilde{C}}\to 0$$
by definition.
Question: What invertible sheaf can we tensor the first term of this exact sequence by to get the first term of the desired short exact sequence?
Question: What happens to the second term if we tensor by the same invertible sheaf?
Question: What happens to the third term if we tensor by the same invertible (and thus locally trivial) sheaf?
For Q1, it is as Mohan says in the comment — it is a local question, and you can use local rings.
For Q3 it just means that for $I$ the ideal sheaf of a divisor $D$ and $L$ any line bundle, a section of $L \otimes I^{-1}$ gives a section of $L$ over the open set $\tilde{X}-D$, i.e. a meromorphic section of $L$. If we choose a local generator for $L$ at a point of $D$ and express this section as a rational function times the generator, the rational function may have a pole at that point (since it's not a section over that point) but the order of the pole is bounded by the multiplicity of the point in $D$ (again this is a local calculation and can be done in local rings).