# Chern class of torsion sheaf support on a point

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.

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 $$i_*(\mathcal{O}_Y)$$.