h^0 of a sheaf supported at a point

Question

$C$ : a projective curve over an algebraically closed field, $I$ : the ideal sheaf of $\mathcal{O}_C$ defining a point $p\in C$. Why is the following hold?

$h^0(C,(\mathcal{O}_C/I^{r})\otimes\mathcal{O}_C(m))=dim(\mathcal{O}_{C,p}/m_{C,p}^r)\ge r$, where $\mathcal{O}_{C,p}$: the local ring at $p$ and $m_{C,p}$:the maximal ideal in $\mathcal{O}_{C,p}$.

the fact that $(\mathcal{O}_C/I^{r})\otimes\mathcal{O}_C(m)$ has support only at the point $p\in C$ implies "first equality". but I can't understand... and is "second inequality" from noetherian local ring?

I know my question is elementary..Please help me...

Answer 1

First of all you can get rid of $\mathcal{O}_C(m)$, since your sheaf is zero outside $p$ and $\mathcal{O}_C(m)\cong \mathcal{O}_C$ in a neighborhood of $p$. Then the fiber of $\mathcal{O}_C/I^r$ at $p$ is $\mathcal{O}_{C,p}/m_{C,p}^r$; since the sheaf is zero outside $p$, its sections are determined by their values at $p$, hence the isomorphism $H^0(C,\mathcal{O}_C/I^r)\cong \mathcal{O}_{C,p}/m_{C,p}^r$. Finally since $m_{C,p}^r\subsetneq m_{C,p}^{r-1}\subsetneq \ldots \subsetneq \mathcal{O}_{C,p}$, we have $\dim \mathcal{O}_{C,p}/m_{C,p}^r\geq r$.

Answer 2

The first equality just follows from the fact that the global sections of a sheaf supported at a point agree with the stalk of the sheaf at that point (think about the definition of a sheaf).

For the second equality. Here's one way consider $(R, m)$ a Noetherian local ring containing its residue field $k$ (if you want to remove the last requirement about the residue field, use length instead of dimension).

Of course, note that $m^i \neq m^{i+1}$ for any $i$ by Nakayama so that $\dim(m^i/m^{i+1}) \geq 1$. Then we certainly have $$ \dim(R/m^r) = \sum_{i = 0}^{r-1} \dim(m^{i}/m^{i+1}) \geq r $$ Also note that $\dim_k m^{r-1}/m^r$ is just the number of generators of $m^{r-1}$.

It turns out that these dimensions measure whether or not the ring is singular. They are always eventually polynomial. You may want to google/look up Hilbert-Samuel multiplicity for additional reading.