Zariski topology

The second chapter is devoted to computing the Betti numbers of Hilbert schemes of points. The main tool we want to use are the Weil conjectures. In section 2.1 we will study the structure of the closed subscheme of X which parametrizes subschemes of length nonl concentrated in a variable point of X. We will show that (X )rei is a locally trivial fibre bundle over X in the Zariski topology with fibre Hilbn( [[xi,... arj]]). We will then also globalize the stratification of Hilbn( [[xi,..., x ]]) from section 1.3 to a stratification of Some of the strata parametrize higher order data of smooth m-dimensional subvarieties Y C X for m < d. In chapter 3 we will study natural smooth compactifications of these strata. 

Lemma 2.1.4. n (X " j)red — X is a locally trivial fibre bundle in the Zariski topology with fibre Hilbn(J )red. 

F y = 0, therefore Y fl T G x = 0. Flere denotes the closure with respect to the ordinary topology (instead of the Zariski topology). Since Y is G-invariant, we have... 

Let us show that the arrow on the right is surjective. The question is local in the Zariski topology on S so we may assume that G is embedded into an abelian scheme A over So. The composition G - G A pn] is injective. Hence, by Proposition 3.5, we get that the composition Ms(A[pn]) —+ M G) Ms(Gr) is surjective. The result follows. 

By a theorem of Raynaud ([BBM] Theorem 3.1.1) any finite flat group scheme over S can locally in the Zariski topology on S be embedded into an abelian scheme. It follows that if S is the spectrum of a local ring then the category C(n)s is generated by the category BT(n)s-... 

Proof. Locally on 5b in the Zariski topology we can choose a resolution ... 

The second chapter is devoted to computing the Betti numbers of Hilbert schemes of points. The main tool we want to use are the Weil conjectures. In section 2.1 we will study the structure of the closed subscheme of which parametrizes subschemes of length n on X concentrated in a variable point of X. We will show that is a locally trivial fibre bundle over X in the Zariski topology... 

We now start afresh to consider the subject from a different viewpoint. Again we begin by looking at the solutions of sets of equations, but we consider only a fixed field fc. We call a subset S of fc" closed if it is the set of common zeros of some polynomials /, in k[Xu. .., X ]. Clearly an intersection of closed sets is closed. Also, if S is the zeros of / and T the zeros of gj, then S u T is the zeros of fipj, so finite unions of closed sets are closed. Thus we have a topology, the Zariski topology on fc". 

In fc1 the only closed sets—zero sets of polynomials—are k1 itself and the finite sets. The topology is thus quite coarse it will not be Hausdorff, and the integers for instance are dense in the real line. But this is actually just what we want we will only be considering polynomial functions, and a real polynomial is indeed determined by its values on integers. More generally, the only maps

closed sets are the polynomial maps, where the coordinates of [Pg.38]

Theorem. Let k L be fields. Then the Zariski topology on E induces that on If. 

If k is finite, the Zariski topology is discrete and contains no information. Consequently we assume k infinite in the rest of this chapter and in all subsequent references to closed sets in fc". We have then one simple fact to observe ... 

The basic fact we need is that on an algebraic matrix group S SL + t(/c) the functions xi bx, xh- x 1, and xi- x ibx for fixed b are continuous. This is clear, since they are given by polynomials, and polynomial maps are always continuous in the Zariski topology. It is worth mentioning only because multiplication is not jointly continuous (it is a continuous map S x S- S, but the topology on S x S is not the product topology). 

Show explicitly that a polynomial map

[Pg.44]

Proof of proposition (3. 2.9) (continued) Take G = P = our finite locally -free group scheme G, T = Sch/S with the Zariski topology (for example) on Sch/S. Take - fibered category of modules. Observe now that if we consider G c—> U(G), then sections of G induce commutative diagrams via translation ... 

The topology on Crys (X) is "that induced by the Zariski topology" it is defined by a pre-topology where... 

We now turn to the definition of crystals. Let J be a fibered category on (Sch) which is a stack with respect to the Zariski topology. This means that both morphisms and objects can be glued together. For a precise definition see [11, I 3.2],... 

It is easy to check that all morphisms are continuous in the Zariski topology. A basis for the open sets in the Zariski topology on E is given by the open sets ... 

One should notice that the Zariski topology is very weak. On k itself, for instance, it is just the topology of finite sets, the weakest T topology (since any ideal A in k[X] is principal - A = (/) - therefore V(A) is just the finite set of roots of /). It follows that any bijection a k -> k is continuous, so not all continuous maps are morphisms. In any case this is a very unclassical type of topological space. 

---