Universal functor

In particular we can identify the set Xtnl(fc) of fc-valued points of Xtn) with the set of closed zero-dimensional subschemes of length n of X which are defined over k. In the simplest case such a subscheme is just a set of n distinct points of X with the reduced induced structure. The length of a zero-dimensional subscheme Z C X is dimkH0(Z, Oz)- The fact that Hilbn(X/T) represents the functor 7iilbn(X/T) means that there is a universal subscheme... 

Is the same as to give an element c F(X). If defines an Isomorphism of functors it is called a universal element for F one then says that F is represented by the couple (X,t). 

The definition and construction of the Hilbert schemes, carried out in section 7, can be extended quite easily to yield more general "universal" objects which arise very naturally in the study of families of projective schemes. In this section we will discuss some of these objects. We will define the corresponding functors first, and then we will prove that they are representable. 

Theorem. For every r and fi(t) as above the flag-Hilbert functor FHrB(t) is represented by a projective scheme FHr (t), called the flao-Hilbert scheme relative to >(t), and by a "universal family"... 

The functors of Artin rings that ire considered in algebraic geometry are rarely prorepresentable but they often have a semiuniversal element. In the next section we will give a criterion, easy to verify in practise, for the existence of a semiuniversal or of a universal element. 

Though the variables A and B have been treated differently in the foregoing, their roles are essentially equivalent. Indeed, there is a universal property analogous to (the dual of) that in (2.4.4), characterizing the natural composite map of A-functors from K x K to D ... 

Remarks, (i) In fact, the A-functors RTfom and are defined only up to canonical isomorphism, by universal properties, as in (2.5.9). We leave it to the reader to verify that the map in (2.6.1) (to be constructed below) is compatible, in the obvious sense, with such canonical isomorphisms. 

Freedman, M.H., Larsen, M., Wang, Z. A modular Functor which is universal for quantum computation. Commun.Math.Phys. 227(3), 605-622 (2002)... 

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