My qualifying exam is fast approaching and so I’d like to use the next few months to refresh myself on the basics. In particular, I’d like to finally nail down a bunch of concepts that I somehow seem to forget about and then have to re-figure out every two months or so. Thus, I anticipate several very short blog posts in the coming weeks, which will hopefully serve me as a memory tool.
Today’s simple concept is how a group action is the same thing as a grading on . To be clear, this is a map of schemes such that:
- If is the multiplication map for , then .
- is the identity section, then is the identity map on .
We translate these conditions into maps of global sections. If takes to , then the condition we get is . This equality implies that if and .
The second condition implies that .
Thus, the map splits into infinitely many subsets corresponding to the images of the different . Since is a homomorphism of additive groups, is a group, and since is a homomorphism of algebras, . If for , then applying to both sides, we get that . Thus . Finally and so has the structure of a graded ring. Conversely projection maps from a graded ring to its components give maps satisfying the above properties.