Dualizing sheaf of nodal curve

Question

Let $C$ be a connected, nodal (I'm working with definition from Alper's notes on Stacks & Moduli, see p 210), projective curve over an alg closed field $k$, beeing everywhere smooth except at a node $z \in C(k)$.
Let $\pi:N \to C$ be its normalization resolving singularity, set $\Sigma:=\{p,q\}=\pi^{-1}(z)$,
and define by $\omega_C \subset \pi_*\Omega_N(\Sigma)$ the subsheaf declaring its sections at open $V \subset C$ to consisting of sections $s$ in $\Omega_N(\Sigma)(\pi^{−1}(V)$ (i.e. with at worst simple poles along $\Sigma$) such that at preimages $p,q$ of the node $z$ (if $z \in U$) we have relation $\text{res}_p(s) + \text{res}_q(s) = 0$.

Question: Exercise 5.2.21(d) in Alper's notes (page 215) asks to show that $\omega_C$ defined that way is indeed the dualizing sheaf, ie that for every coherent sheaf $\mathcal{F}$, the natural pairing

$$\operatorname{Hom}_{O_C}(\mathcal{F},\omega_C)\times H^1(C,\mathcal{F}) \to H^1(C,\omega_C) \xrightarrow{\text{trace}} k$$

is perfect.
Note that the claim itself is known and there are several ways to do it (see e.g. in Q. Liu's AGeo 10.3.12 or "adelic" construction in P. Belmans' notes, but both approaches seemingly not explicitly follow the strategy proposed in the hint in the adressed exercise to reduce to the case of a smooth curve by considering the normalization and my concern in this question is to understand the mechanism of this "reduction to smooth case" approach Alper proposed.

Could somebody help to understand the way of reasoning Alper suggesting there in detail, ie how the reduction step to duality for normalizated curve is performed?

my ideas so far: It is well known that $\Omega_N$ is dualizing sheaf of $N$, which normalizes $C$, but how to exploit this hint to show that $\omega_C$ defined that way above is indeed the legitime dualizing sheaf. Naively I interpreted the given hint animating the reader to build following (or similar) lovely naturally commuting comparison diagram

$$ \require{AMScd} \begin{CD} \operatorname{Hom}_{O_C}(\mathcal{F},\omega_C)\times H^1(C,\mathcal{F}) @>>> H^1(C,\omega_C) @>{\text{trace}}>> k\\ @VfVgV @VhVV @VjVV\\ \operatorname{Hom}_{O_C}(\widetilde{\mathcal{F}},\Omega_N)\times H^1(N,\widetilde{\mathcal{F}}) @>>> H^1(N,\Omega_N) @>{\text{trace}}>> k \end{CD} $$

for arbitrary choosed coherent $\mathcal{F}$ by constructing appropriate vertical maps $f,g,h,j$ and $\widetilde{\mathcal{F}}$ in order to reduce to story to smooth $N$ normalizing $C$.
If that's the right approach Alper suggesting there, then the vertical maps should allow control of the upper pairing by bottom pairing, eg if vertical maps are injective. But I not see how to archieve this.
Let's to to construct them:

The construction of $h$ and $j$ should be easily follow from ses $ 0 \to \pi_*\Omega_N \to \omega_C \to k_z \to 0$
which by forming les on cohomology shows that induced $H^1(N, \Omega_N) \to H^1(C, \Omega_C)$ is an iso (Note, here we had to use alg closedness of $k$) Especially, we inherit trace map from bottom diagram.

The left part appears to be weird, some Ideas for $f,g$: My guess is to take $\widetilde{\mathcal{F}}:= \pi^* \mathcal{F}(-\Sigma)$.
For $f$ there is another ses $0 \to \omega_C \to \pi_*\Omega_N(\Sigma) \to k_z \xrightarrow{\text{res}} 0$ inducing together with $(\pi^*,\pi_*)$-adjunction & twist shift

$$\operatorname{Hom}(\mathcal{F}, \omega_C) \to \operatorname{Hom}(\mathcal{F},\pi_*\Omega_N(\Sigma)) \cong \\ \operatorname{Hom}(\pi^*\mathcal{F}, \Omega_N(\Sigma)) \cong \operatorname{Hom}(\pi^*\mathcal{F}(-\Sigma),\Omega_N)$$

For $g$ we can use the induced map on cohomology $H^1(C, \mathcal{F}) \to H^1(N,\pi^*\mathcal{F}) $from unit map $\mathcal{F} \to \pi_*\pi^* \mathcal{F}$.

Problems with this approach: How we can compose $g$ to land actually in $H^1(N, \pi^*\mathcal{F}(-\Sigma))$? Moreover, there is no reason why $f$ and $g$ (once managed to construct it) are actually injective in order to exploit to perfectness of pairing on the bottom. Eg, to unit map is a priori not injective.

Does anybody see if the presented apprach could be "repaired" to deduce finally the claim of the exercise - or if I'm on wrong track - what would be theright approach going along Alper's hint to solve this exercise? So, in which way this reduction step to $N$ concretely enters the argumention?

#Added later: Maybe, the approach Alper suggests goes also in another direction to establish a natural "comparison map" between $\omega_C$ and "the real" dualizing sheaf $\omega_C^r$ beeing an iso on smooth locus and then to show somehow that they actually coincide. But again, where actually the proposed hint to "reduce to smooth curve via normalization map" enters properly the game?

Answer 1

This is explained nicely in Olsson's book "Algebraic Spaces and Stacks", Prop 13.2.9, where many of the details are worked out. There are other approaches as you point out, but Olsson's argument uses the normalization. Given that the explicit description of the dualizing sheaf is given in terms of the sheaf of differentials on the normalization, this seems to be a natural approach. However, I wouldn't say any of the approaches are easy and the exercise in my notes deserves perhaps a more detailed hint.

Here is an outline of Olsson's argument: Let $C$ be a nodal curve over a field $k$ with $\Sigma \subseteq C$ denoting the nodes, $\pi \colon \tilde{C} \to C$ the normalization, and $\tilde{\Sigma} := \pi^{-1}(\Sigma)$. Define the subsheaf $$K_C = \ker\left(\pi_* \Omega_{\tilde{C}}(\tilde{\Sigma}) \to \bigoplus_{z_i \in \Sigma} k\right),$$ where the map is defined by taking a rational section of $\pi_* \Omega_{\tilde{C}}$ to the vector whose coordinate at $i$ is the sum of the residues of the rational section at the two preimages in $\tilde{C}$ of the node $z_i \in \Sigma$.

(1) Since $K_C$ is preserved under etale base change, the explicit description of $K_C$ in the case of a nodal singularity ${\rm Spec} k[x,y]/(xy)$ shows that $K_C$ is a line bundle. Note that the dualizing sheaf $\omega_C$ is also a line bundle (as $C$ is a local complete intersection).

(2) To show that $K_C \cong \omega_C$, it suffices by Yoneda's Lemma to exhibit an isomorphism of the functors $${\rm Hom}(-,K_C), H^1(C,-)^{\vee} \colon {\rm Pic}(C) \to {\rm Vect}_k.$$

(3) For any $L \in {\rm Pic}(C)$, tensoring the exact sequence defining $K_C$ by $L^{\vee}$ and taking cohomology gives identifications $${\rm Hom}_{\mathcal{O}_C}(L, K_C) \cong H^0(C, L^{\vee} \otimes K_C) \cong \ker \left( H^0(\tilde{C}, \pi^* L^{\vee} \otimes \Omega_{\tilde{C}}(\tilde{\Sigma})) \to \bigoplus_{z_i \in \Sigma} k \right).$$

(4) On the other hand, the short exact sequence $0 \to \pi_* \pi^* (L (\tilde{\Sigma})) \to L \to \bigoplus_{z_i \in \Sigma} k \to 0$ induces an identification $$H^1(C, L)^{\vee} = \ker \left( H^1(\tilde{C}, \pi^* L(\tilde{\Sigma}))^{\vee} \to \bigoplus_{z_i \in \Sigma} k \right).$$

(5) One checks that the isomorphism $H^0(\tilde{C}, \pi^* L^{\vee} \otimes \Omega_{\tilde{C}}(\tilde{\Sigma})) \cong H^1(\tilde{C}, \pi^* L(\tilde{\Sigma}))^{\vee}$ given by Serre Duality on $\tilde{C}$ commutes with the two maps to $\bigoplus_{z_i \in \Sigma} k$. This gives an isomorphism ${\rm Hom}_{\mathcal{O}_C}(L, K_C) \cong H^1(C, L)^{\vee}$, which one checks is functorial in $L$.