Functors

In this section we introduce structure-preserving maps between categories. 

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Definition 1.1.1. [Grothendieck (1)] Let HUb(X/T) be the contravariant functor from the category SchlnT of locally noetherian T-schemes to the category Ens of sets, which for locally noetherian T-schemes U,V and a morphism V — U is given by... 

For an open subscheme Y C X the functor Hilbp(Y/T) is represented by an open subscheme... 

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... 

Remark 1.1.4. It is easy to see from the definitions that Zn(X/T) represents the functor Zn(X/T) from the category of locally noetherian schemes to the category of sets which is given by... 

Now we can easily see from the definitions that D n X) represents the functor... 

Later we will see that D X) is reduced and even smooth. D2n(X) is called the variety of second order data of m-dimensional subvarieties of X. Analogously we define D n(X) as the closed subscheme of X x Xtm+1l that represents the functor given by... 

To read this book the reader only needs to know the basics of algebraic geometry. For instance the knowledge of [Hartshorne (1)], is certainly enough, but also that of [Eisenbud-Harris (1)] suffices for reading most parts of the book. At some points a certain familiarity with the functor of points (like in the last chapter of [Eisenbud-Harris (1)]) will be useful. Of course we expect the reader to accept some results without proof, like the existence of the Hilbert scheme and obviously the Weil conjectures. 

First, we recall the definition of the Hilbert scheme in general (not necessarily of points, nor on a surface). Let X be a projective scheme over an algebraically closed field k and Gx(l) an ample line bundle on X. We consider the contra,variant functor Hilbx from the... 

Theorem 1.1 (Grothendieck [34]). The functor Hilbx is representable by a projective scheme Hilb. ... 

The category A9ts endowed with the natural functor Agj Sch is a stack. 

It is clear that it makes S(g, p) into a stack in groupoids over Sch. The forgetful functor of S(g,p) into the stack of principally polarized abelian varieties Ag defines a 1-morphism of stacks... 

Let us agree to denote by a v the functor on sheaves of Oscrys-modules... 

Let us remark that all such systems M. (resp. M.) over a ring R (resp. a scheme S) form in a natural way a category (morphisms are isomorphisms). For any ring homomorphism R — R (resp. morphism of schemes f Sr S) we get a pullback functor Jlf. M. R (resp. 

If So is the spectrum of a perfect field k then classical Dieudonne theory gives us a con-travariant functor G M(G) of C(l)s0 into the category of finite dimensional fc-vectorspaces. Since there is a canonical isomorphism M(GM) = M(G) P the module M(G) is endowed with the action of Frobenius F and Verschiebung V. The functor G (Af(G),F,V) is an anti equivalence of C(l)s0 with the category of triples (Af, F, C ), where M is a finite dimensional fc-vectors pace and F — Af, resp. V M are linear maps such that F o V — 0... 

Hence our aim is to construct a functor Af C(l)s0 — Af(l)s0 which is compatible with a and u. We do not construct such a functor for arbitrary schemes So and neither do we know whether this is always possible. 

Suppose we are given a scheme S, flat over Spec(Z/p2Z), such that So = V (p) C S. In 3 we construct a functor Ms from C(l)s0 into the category of finitely generated Os0-modules using the scheme S. It is exact (Theorem 3.10) in 5 we show that it is compatible with the functors a and w and in 6 we show that there is a natural isomorphism... 

Both a and uj give contravariant functors of the category of finite flat group schemes over S into the category of quasi-coherent Os-modules. The sheaf ug IS finitely generated as an Os-module. 

Theorem 8.5. Suppose S is the spectrum of a Noetherian complete local ring with perfect residue field. If the functor... 

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