Characterisation of Gorenstein Curve
Let $C$ be a curve (so a $1$-dimensional, proper $k$-scheme) and $\omega_C$ it's dualizing sheaf.
A curve is called Gorenstein iff $\omega_C$ is an invertible sheaf on $C$, in other words $\omega_C \in Pic(C)$.
Futhermore the curve $C$ is called local complete intersection iff for each closed $a \in C$ one have
$$\widehat{\mathcal{O}_{C,a}} \cong \kappa(a)[[T_1,...,T_n]]/(f_1, ..., f_n)$$
where $f_1, ..., f_n$ is the regular sequence in the regular local ring $\kappa(a)[[T_1,...,T_n]]$.
Here $\widehat{\mathcal{O}_{C,a}}$ is the completion of the stalk $\mathcal{O}_{C,a}$ with respect of it's maximal ideal $m_a$ and $\kappa(a) = \mathcal{O}_{C,a} /m_a$.
My question is why are following implications true:
$$C \text{ regular} \Rightarrow C \text{ is local complete intersection} \Rightarrow C \text{ is Gorenstein, so } \omega_C \in Pic(C) \Rightarrow C \text{ has not embedded components}$$
I'll abbreviate local complete intersection as LCI.
- Regular $\Rightarrow$ LCI: Embedd any affine open subset into affine space over $k$. Since it is smooth of the right dimension, its conormal sheaf provides the correct choice of $f_i$ after some argument.
- LCI $\Rightarrow$ Gorenstein: Locally, one can define $\omega_C$ to be the top exterior power of the conormal sheaf, constructed above. See for instance Brian Conrad's Grothendieck Duality and Base Change, introduction section.
- This is really the unmixedness theorem. Gorenstein implies Cohen-Macaulay, because Cohen-Macaulay is equivalent to $\omega$ a sheaf, and then you may apply the theorem.