I'm interested in extending Riemann-Roch and Serre duality to bundles or sheaves defined on nodal curves. The extension of Riemann-Roch I think is straightforward: simply replace the geometric genus by the arithmetic genus. Unless I'm mistaken, Serre duality can be extended to nodal curves by replacing the canonical bundle with the dualizing sheaf. I've never worked with a dualizing sheaf concretely, so I was hoping someone could help me in one very explicit example.
Let $C$ be the nodal curve given as the glueing of two elliptic curves at their origins. One can think of this as a particular kind of genus two curve. Obviously, the normalization is just the two disjoint elliptic curves. But what exactly is the dualizing sheaf $\omega_{C}$ and how does one carry out the computation?
First, simple nodal curves are locally planar and so the dualizing sheaf is a line bundle. So, if the elliptic curves are $E_1,E_2$, one has an exact sequence $0\to \mathcal{O}_C\to \mathcal{O}_{E_1}\oplus\mathcal{O}_{E_2}\to k(p)\to 0$, where $p$ is the origin. Dulaizing, you get, $0\to\mathcal{O}_{E_1}\oplus\mathcal{O}_{E_2}\to \omega_C\to k(p)\to 0$. Most calculations you need can be made using this sequence.