Chern class of torsion sheaf support on a point

Question

Let $X$ be a smooth projective surface. Let $p$ be a closed point of $X$. Let $k(p)$ be the corresponding skyscraper sheaf, then actually we could use Grothendieck-Riemann-Roch to calculate the Chern of the coherent sheaf $k(p)$.

However, now let $U\cong \mathrm{Spec}A$ be a open subset containing $p$, $m$ be the maximal idea of $A$ corresponding $p$. Then we have a closed embedding $$i: Y:=\mathrm{Spec}(A/m^2)\to X$$

Now, how do we calculate the Chern class of $i_{*}(\mathcal{O}_{Y})$.

Since $Y$ is not reduced, it seems we could not use GRR. Could you give me some ideas about it? Thanks a lot.

Answer 1

If you can figure out the Chern character of $i_*(\mathcal{O}_Z)$ where $Z = \text{Spec}(A/m)\hookrightarrow X$ by GRR then you use the following short exact sequence:

$$0 \rightarrow m/m^2 \rightarrow \mathcal{O}_Y\rightarrow \mathcal{O}_Z\rightarrow 0$$

Note the pushforward is also going to be exact. The kernel $m/m^2$ is a $2$-dimensional vector space over $A/m$ so its pushforward is going to be direct sum of two $i_*(\mathcal{O}_Z)$. Now you can calculate the Chern character of $i_*(\mathcal{O}_Y)$.