Equivariant sheaves What is the best introduction (textbook) to equivariant sheaves on algebraic varieties equipped with an action of an algebraic group? 
 A: If an algebraic group $G$ acts freely on a variety $X$, then  $G$-equivariant sheaves on $X$ are the same as sheaves on the quotient $X/G$.  (If the $G$-action is not free, the same statement is true if we instead talk about the quotient stack $[X/G]$, but it takes more work to give the statement meaning in this case.)
To see this, let $\pi: X \to X/G$ be the natural projection, let $g$ be an element of $G$, let $\alpha_g: X \to X$ be the automorphism of $X$ given by the $g$-action, and note that $\pi \circ g = \pi.$
Thus, if $\mathcal F$ is a sheaf on $X/G$, then there is anatural isomorphism 
$$\alpha_g^* \pi^* \mathcal F \cong (\pi \circ \alpha_g)^* \mathcal F =
\pi^* \mathcal F.$$
This is the equivariant structure on $\pi^* \mathcal F$.
To see that any equivariant sheaf on $X$ arises as $\pi^*\mathcal F$ for some $\mathcal F$ in this way, note that the equivariant structure gives descent data
on the equivariant sheaf for the map $\pi$, which allows us to descent the sheaf down to $X/G$.

In particular, if $R$ is a graded $\mathbb C$-algebra, say with $R_0 = \mathbb C$, so that $R$, and hence Spec $R$, is equipped with an action of $\mathbb C^{\times}$ (the action is being as follows: $z \in \mathbb C^{\times}$ acts on the $n$th graded piece of $R$ as mult. by $z^n$), then removing the point corresponding to the irrelevant ideal from Spec $R$, we get a free $\mathbb C^{\times}$-action, and equivariant sheaves for this action are the same as equivariant sheaves on the Proj.
As one example, if $R = \mathbb C[x_0,\ldots,x_N]$ and we consider the 
structure sheaf on Spec $R \setminus \{0\} = \mathbb A^{N+1} \setminus \{0\},$
this corresponds to the sheaf $\bigoplus_{n = 0}^{\infty} \mathcal O(n)$ 
on $\mathbb P^N$.
