Short exact sequence of equivariant line bundles on $\mathbb P^1$

Question

I have a two-dimensional vector space ${\mathbb C}^2$ with basis $e_m, f_1$ and action of ${\mathbb C}^*$ by $t \cdot e_m = t^m e_m$ and $t \cdot f_1 = f_1$ and I have the projective line ${\mathbb P}^1$ which is the set of lines through the origin in this ${\mathbb C}^2$. I have the tautological line bundle $\Lambda$ on this ${\mathbb P}^1$ whose fiber over the line $L$ in ${\mathbb C}^2$ is the vector space $L$. I also have the trivial line bundle ${\mathcal O}$ on ${\mathbb P}^1$. Finally, I have the skyscraper sheaf ${\mathbb C}_{e_m}$ on ${\mathbb P}^1$ supported on the point of ${\mathbb P}^1$ which corresponds to the line spanned by $e_m$ in ${\mathbb C}^2$.

Note that all of the sheaves $\Lambda$, $\mathcal O$ and ${\mathbb C}_{e_m}$ are ${\mathbb C}^*$-equivariant.

Question: how to write a short exact sequence of ${\mathbb C}^*$-equivariant sheaves on ${\mathbb P}^1$ which involves $\Lambda$, ${\mathcal O}$ and ${\mathbb C}_{e_m}$ possibly with twists of grading?

Namely, non-equivariantly we have a short exact sequence $0 \to \Lambda \to {\mathcal O} \to {\mathbb C}_{e_m} \to 0$, is this an exact sequence of equivariant sheaves, or do we need to add twists of grading somewhere to make this exact sequence equivariant?

Answer 1

This is equivariant. The map $\Lambda \to \mathcal O$ explicitly on sections sends a section $(e_m, f_1) $ valued in $ L \subseteq \mathbb C^2$ to the section $f_1$ of the trivial line bundle. This map is an isomorphism everywhere but the point corresponding to the line spanned by $e_m$, and vanishes to order $1$ there. So the cokernel is the map to $\mathbb C_{e_m}$ given by evaluating at that point.

Both the projection and evaluation maps are manifestly $\mathbb G_m$-equivariant.