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 \mathbb{V}E and \mathbb{P}E are no exceptions.

Commutative Algebra \mathbb{V}E

Let be R a ring and E an R module.  Denote the category of R algebras as \mathcal{A}_R .  Then define the functor  

\mathbb{V}E \colon \mathcal{A}_R \to SETS

A \mapsto \hom_{mod}(E,A)

Then \mathbb{V}E is represented by S^\bullet E , the symmetric algebra on E.  That is, for any map \phi \colon E \to A, there is a canonical inclusion E \to S^\bullet E and a unique map S^\bullet E \to A making the following diagram commute. 

\begin{array}{ccc} E & \to & A \\ \downarrow & \nearrow & \mbox{} \\ S^\bullet E & \mbox{} & \mbox{} \end{array}

Scheme \mathbb{V}E

Now E is qcoh on a scheme X. For T \in \mbox{Sch}_X, let p_T \colon T \to X be the map to the base. 

Define the functor

\begin{array}{ccc} \mathbb{V}E \colon \mbox{Sch}_X & \longrightarrow & \mbox{SETS} \\ T & \mapsto & \hom_{\mathcal{O}_X}(E, p_{T*}\mathcal{O}_T) \\ \mbox{} & \mbox{} & =\hom_{\mathcal{O}_T}(p^*_TE,\mathcal{O}_T) \end{array}

This functor is represented by \mbox{Spec}_X S^\bullet E (said relative spec; the construction of relative spec was one of the applications Ogus did of the all the open subfunctor business).  That is, S^\bullet E is a qcho sheaf of \mathcal{O}_X algebras, there is a sheaf map E \to S^\bullet E over X, and for any sheaf map E \to p_{T*}\mathcal{O}_T there is a unique map etc. making the following diagram commute

\begin{array}{ccc} E & \to & p_{T*}\mathcal{O}_T \\ \downarrow & \nearrow & \mbox{} \\ S^\bullet E & \mbox{} & \mbox{} \end{array}

And (by universal property of relative spec) such a diagram gives a unique scheme map T \to \mbox{Spec}_X S^\bullet E.

Again, by adjunction I can look at sheaf maps into \mathcal{O}_T instead, then any a \in \Gamma(T, \mathcal{O}_T) =: \mathbb{A}^1(T), there is a sheaf map \mathcal{O}_T \to \mathcal{O}_T given by multiplication by a, and as p_T^* E is an \mathcal{O}_T module, multiplication by a also gives a sheaf map to itself; denote these sheaf maps as a_T, a_E respectively.  Then there is a commutative diagram

\begin{array}{ccc} p_T^*E & \xrightarrow{a_E} & p_T^*E \\ \downarrow & \mbox{} & \downarrow \\ \mathcal{O}_T & \xrightarrow{a_T} & \mathcal{O}_T \end{array}

Or, in other words, \mathbb{V}E is an \mathbb{A}^1 module. Note, \mathbb{A}^1 is represented by \mbox{Spec}_X \mathcal{O}_X[t].

Exercise: As \mathbb{V}E has a module structure, the scheme which represents it \mbox{Spec}_X S^\bullet E has a group action and a multiplication action of \mathbb{A}^1, this is encoded as saying there are morphisms 

\mbox{Spec}_X S^\bullet E \times \mbox{Spec}_X S^\bullet E \to \mbox{Spec}_X S^\bullet E 

\mbox{Spec}_X S^\bullet E \times \mbox{Spec}_X \mathcal{O}_X[t] \to \mbox{Spec}_X S^\bullet E 

Describe the backwards map on structure sheaves.

Commutative Algebra \mathbb{P}E

The motivation is that \mathbb{P}E should be ‘lines’ through \mathbb{V}E.  So consider

\begin{array}{cc} \mathbb{P}E(k) & = \mbox{lines through 0 in }\hom(E,k) \\ \mbox{} & = \frac{\hom(E,k) \backslash 0}{k^*} \\ \mbox{} & = \mbox{hyperplanes in E} \end{array}

So to get scheme \mathbb{P}E the notion of removing 0 and modding out by units need to be defined.

Scheme \mathbb{P}E

There is always the 0 map E \to \mathcal{O}_X, and by universal property of relative spec, this gives a section X \to \mbox{Spec}_X S^\bullet E, 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 m \colon E \to p_{T*} \mathcal{O}_T (recall this gives a map T \to \mbox{Spec}_X S^\bullet E), if there is some point of X such that all sections of E map to 0, then the map T \to \mbox{Spec}_X S^\bullet E must hit the zero section.

Now consider the complement of the zero section; it represents a functor Ogus calls \mathbb{U}E.  Now \mathbb{U}E is a subfunctor of \mathbb{V}E and in particular, by the previous paragraph, \mathbb{U}E(T) consists of sheaf maps E \to p_{T*}\mathcal{O}_T where for every point of X, some section of E doesn’t vanish.  Now by adjunction, this becomes a sheaf map p_T^*E \to \mathcal{O}_T, 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 \mathcal{O}_T, i.e. the sheaf map is locally surjective, hence surjective.  So

\mathbb{U}E(T) = \{m\colon p_T^*E \to \mathcal{O}_T | m \mbox{ is surjective } \}

 

so in particular, \mathbb{U}E(T) consists of sheaf maps E \to p_{T*}\mathcal{O}_T, or by adjunction p_T^*E \to \mathcal{O}_T.    

The analogue to modding out by units requires the scheme \mathbb{G}_m := \mbox{Spec}k[x,1/x].  Then \mathbb{P}E can be defined as taking the quotient of \mathbb{U}E by some action of \mathbb{G}_m, but a more explicit definition comes from defining

a hyperplane H of a qcoh sheaf of modules G is a submodule such that G/H is invertible, and an invertible quotient is a surjection G \to L where L is invertible.  Then G \to L is isomorphic to G \to L' if L \cong L'.  Then

\begin{array}{rl} \mathbb{P}E(T) & =\{\mbox{hyperplanes in } p_T^*E \} \\ \mbox{} & =\{\mbox{isom. classes of inv. quot.}\} \end{array}

 

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

PE is representable

  1. Show \mathbb{P}E is a sheaf (this is true essentially because qcho sheafs are determined locally)
  2. \mathbb{P}E has an open cover by representable open subfunctors.

For the second step, introduce \mathbb{P}_e E(T) := \{H \subset p_T^* E =: E_T| \overline{e} \in E_T/H \mbox{ is nowhere zero} \}, where e \in \Gamma(X,E), note \bar e defines an isomorphism E_T/H \cong \mathcal{O}_T.  Now show \mathbb{P}_e E is an open subfunctor, this amounts to looking at some fiber product diagrams an showing \mathbb{P}_eE(T) is represented by \{ t \in T | 0 \ne \bar e (t) \in (E_T/H)_t \} where \bar e(t) is the restriction of \bar e to the stalk at t \in T.

Now check that \mathbb{P}_e E is represented by \mbox{Spec}_X S^\bullet E/(e - 1) , and if \{e_i\} generate E, then \{\mathbb{P}_{e_i} E\} cover \mathbb{P}E.

 

It seems like you also have \mathbb{P}E \cong \mbox{Proj} S^\bullet E for a ring R and E and R module.

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