# “Locally of finite type” is a local property

Previously we showed that morphisms locally of finite type are preserved under base change. We can use this to show that

Given a morphism of schemes , the preimage of any affine can be covered by affines such that the corresponding ring maps are of finite type.

Alternatively, if we define a morphism locally of finite type to be one that satisfies , then what we are saying is that such a property can be checked on a cover; we can replace “any affine” with “an affine in a cover of affines”.

Let’s try to prove . First, we base change to . Since the morphism is also locally of finite type, we can cover by affines such that their preimages can be covered by the spectra of finitely-generated -algebras . However, we don’t know if these are finitely-generated -algebras! To fix this, we base change to even smaller affines. Cover by basic open sets . This gives us a cover of each by basic open sets of the form . Since is of finite type, is of finite type. Since is clearly of finite type, is of finite type, giving us the desired cover of . The following diagram may be illustrative (every square is a pullback)

# Group schemes and graded rings

In this post we will describe how an action of the multiplicative group scheme on defines a -grading of . A future post may describe how this relates to projective schemes. (I will do all of this using diagrams, but there may be some easier way using the functors of points). All this was taught to me by Mark Haiman in Math 256B (Algebraic Geometry) at UC Berkeley.

Fix a field ; we will work in the category of -schemes. Thus will be a -algebra, and we will establish a graded -algebra structure on . However, none of our arguments change if we just let be . A group scheme is a group object in the category of -schemes. A precise definition can be found here. Most importantly, group schemes can act on other schemes. The definition of a group scheme action can be found here. Note that all definitions are given by diagrams (or functor of points). For example, we specify the “identity element” of a group scheme by a map , rather than selecting some point in the underlying topological space.

is defined as . (for shorthand, we will write as ). As a variety, it can be thought of as , the “punctured affine line”. Its group operation is given by a map which corresponds to the -algebra map defined by . The identity is given by a map corresponding to defined by .

Suppose acts on . The action map corresponds to a -algebra map such that the following diagrams commute:

Associativity:

Identity:

For , write , where almost all the are zero. Then the first diagram implies that

if (i.e. the polynomial is just a single monomial), then .

This is because, along the top and right arrows, we have and along the left and bottom arrows we have . Furthermore, the second diagram says that

for all , .

Therefore, letting stand for the degree homogenous component of (so that it consists of multiples of ), let . Since all the are disjoint, their preimages are disjoint as well. Furthermore, for an arbitrary element , we have by , and by , we have that each . Thus as a direct sum.

It remains to show that . But this is easy: if , then , so as desired.