Characterisation of Gorenstein Curve

Question

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:

$$C \text{ regular} \Rightarrow C \text{ is local complete intersection} \Rightarrow C \text{ is Gorenstein, so } \omega_C \in Pic(C) \Rightarrow C \text{ has not embedded components}$$

Answer 1

I'll abbreviate local complete intersection as LCI.

1. 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.
2. 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.
3. 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.