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

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?

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.