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?



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


