# When/why is a point on a curve an effective Cartier divisor?

The definition of effective Cartier divisor that I'm using is: a closed immersion whose corresponding quasicoherent sheaf of ideals is an invertible sheaf.

Let $$X$$ be a scheme that is dimension 1 and locally of finite type over a field $$k$$ (but can be singular or have multiple components). Let $$p$$ be a closed point on $$X$$. Then why is the canonical morphism $$f : Spec(k(p)) \rightarrow X$$ is an effective Cartier divisor?

This is my attempt so far:

Since $$p$$ is a closed point, $$f : Spec(k(p)) \rightarrow X$$ is a homeomorphism onto its range and the range is closed. Also, the pullback morphism $$f^\# : O_X \rightarrow f_* O_{Spec(k(x))}$$ is surjective (as on an affine open neighbourhood $$U = Spec(A)$$ of $$p$$, we have $$A \rightarrow Quot(A/p) = A/p$$ since $$p$$ is a closed point so it's a maximal ideal, so pullback is surjective). Therefore it's a closed immersion.

Let $$\mathcal{I}$$ be the corresponding quasicoherent sheaf of ideals. For open sets U that don't contain $$p$$, $$\mathcal{I} \sim O_X|_{U}$$ so is an invertible sheaf on there. But now I'm stuck on the part where open sets $$U$$ that contain $$p$$.

Is the statement even true? If so, how do I prove $$\mathcal{I}$$ is invertible on some open set $$U$$ that contains $$p$$?

• the ideal sheaf is functions that vanish at $P$. picking a uniformizer at $P$, dividing by that uniformizer give an iso (in some neighborhood of $P$) from functions that vanish at $P$ to all functions. Commented Dec 20, 2020 at 7:12

This statement is true precisely when $$p$$ is a regular point of $$X$$. If $$p$$ is regular, then $$\mathcal{O}_{X,p}$$ is a regular local ring of dimension one and hence a DVR, so we can select a uniformizer which gives a generator on some neighborhood of $$p$$. Conversely, if the ideal sheaf of $$X$$ is locally principal then its stalk at $$p$$ must be a principal ideal of $$\mathcal{O}_{X,p}$$. This suffices to show that $$\mathcal{O}_{X,p}$$ is a DVR, which is a regular local ring.