# VE and PE

A lot of things in algebraic geometry (if not all) can be thought of taking some commutative algebra construction and ‘schemifying’ it; that is making some construction about schemes that reduces to the original commutative algebra construction in the affine case. The functors and are no exceptions.

### Commutative Algebra

Let be R a ring and E an R module. Denote the category of R algebras as . Then define the functor

SETS

Then is represented by , the symmetric algebra on E. That is, for any map , there is a canonical inclusion and a unique map making the following diagram commute.

### Scheme

Now E is qcoh on a scheme X. For , let be the map to the base.

Define the functor

This functor is represented by (said relative spec; the construction of relative spec was one of the applications Ogus did of the all the open subfunctor business). That is, is a qcho sheaf of algebras, there is a sheaf map over X, and for any sheaf map there is a unique map etc. making the following diagram commute

And (by universal property of relative spec) such a diagram gives a unique scheme map .

Again, by adjunction I can look at sheaf maps into instead, then any , there is a sheaf map given by multiplication by a, and as is an module, multiplication by a also gives a sheaf map to itself; denote these sheaf maps as respectively. Then there is a commutative diagram

Or, in other words, is an module. Note, is represented by .

Exercise: As has a module structure, the scheme which represents it has a group action and a multiplication action of , this is encoded as saying there are morphisms

Describe the backwards map on structure sheaves.

### Commutative Algebra

The motivation is that should be ‘lines’ through . So consider

So to get scheme the notion of removing 0 and modding out by units need to be defined.

### Scheme

There is always the 0 map , and by universal property of relative spec, this gives a section , this is the ‘zero section.’ This zero section characterizes the set of points of X where all sections of E vanish ( or some generating set for E vanish). In particular, for a given map (recall this gives a map ), if there is some point of X such that all sections of E map to 0, then the map must hit the zero section.

Now consider the complement of the zero section; it represents a functor Ogus calls . Now is a subfunctor of and in particular, by the previous paragraph, consists of sheaf maps where for every point of X, some section of E doesn’t vanish. Now by adjunction, this becomes a sheaf map , and this has the property that for every point of T, there is some section of E that doesn’t vanish there (i.e. not in the maximal ideal of the local ring). In other words, every point has a small enough nbd s.t. some section of E maps to a unit in , i.e. the sheaf map is locally surjective, hence surjective. So

so in particular, consists of sheaf maps , or by adjunction .

The analogue to modding out by units requires the scheme . Then can be defined as taking the quotient of by some action of , but a more explicit definition comes from defining

a hyperplane of a qcoh sheaf of modules is a submodule such that is invertible, and an invertible quotient is a surjection where is invertible. Then is isomorphic to if . Then

Enough to show its representable when applied to affines. Two things need to be shown (using Thm1 from last post)

### PE is representable

- Show is a sheaf (this is true essentially because qcho sheafs are determined locally)
- has an open cover by representable open subfunctors.

For the second step, introduce , where , note defines an isomorphism . Now show is an open subfunctor, this amounts to looking at some fiber product diagrams an showing is represented by where is the restriction of to the stalk at .

Now check that is represented by , and if generate E, then cover .

It seems like you also have for a ring R and E and R module.

## About this entry

You’re currently reading “VE and PE,” an entry on Math Meandering

- Published:
- June 9, 2009 / 1:41 pm

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- Ogus Excerpts

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