Forcing algebra/Closure operations for ideals/Examples/Section

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Let denote a commutative ring and let be an ideal. Let and let

be the corresponding forcing algebra and

the corresponding spectrum morphism. How are properties of (or of the -algebra ) related to certain ideal closure operations?

We start with some examples. The element belongs to the ideal if and only if we can write with . By the universal property of the forcing algebra this means that there exists an -algebra-homomorphism

hence holds if and only if admits a scheme section. This is also equivalent to

admitting an -module section or being a pure -algebra (so for forcing algebras properties might be equivalent which are not equivalent for arbitrary algebras).

The radical of an ideal

Now we look at the radical of the ideal ,

The importance of the radical comes mainly from Hilbert's Nullstellensatz, saying that for algebras of finite type over an algebraically closed field there is a natural bijection between radical ideals and closed algebraic zero-sets. So geometrically one can see from an ideal only its radical. As this is quite a coarse closure operation we should expect that this corresponds to a quite coarse property of the morphism as well. Indeed, it is true that if and only if is surjective. This is true since the radical of an ideal is the intersection of all prime ideals in which it is contained. Hence an element belongs to the radical if and only if for all residue class homomorphisms

where is sent to , also is sent to . But this means for the forcing equation that whenever the equation degenerates to , then also the inhomogeneous part becomes zero, and so there will always be a solution to the inhomogeneous equation.

Integral closure of an ideal

Another closure operation is integral closure. It is defined by

This notion is important for describing the normalization of the blow up of the ideal . Another characterization (assume that is noetherian) is that there exists a , not contained in any minimal prime ideal of , such that holds for all . Another equivalent property (the valuative criterion) is that for all ring homomorphisms

to a discrete valuation domain the containment holds.

The characterization of the integral closure in terms of forcing algebras requires some notions from topology. A continuous map

between topological spaces and is called a submersion, if it is surjective and if carries the image topology (quotient topology) under this map. This means that a subset is open if and only if its preimage is open. Since the spectrum of a ring endowed with the Zarisiki topology is a topological space, this notion can be applied to the spectrum morphism of a ring homomorphism. With this notion we can state that if and only if the forcing morphism

is a universal submersion (universal means here that for any ring change to a noetherian ring , the resulting homomorphism still has this property). The relation between these two notions stems from the fact that also for universal submersions there exists a criterion in terms of discrete valuation domains: A morphism of finite type between two affine noetherian schemes is a universal submersion if and only if the base change to any discrete valuation domain yields a submersion. For a morphism

( a discrete valuation domain) to be a submersion means that above the only chain of prime ideals in , namely , there exists a chain of prime ideals in lying over this chain. This pair-lifting property holds for a universal submersion

for any pair of prime ideals in . This property is stronger than lying over (which means surjective) but weaker than the going-down or the going-up property (in the presence of surjectivity).

If we are dealing only with algebras of finite type over the complex numbers , then we may also consider the corresponding complex spaces with their natural topology induced from the euclidean topology of . Then universal submersive with respect to the Zariski topology is the same as submersive in the complex topology (the target space needs to be normal).

Let be a field and consider . Since this is a principal ideal domain, the only interesting forcing algebras (if we are only interested in the local behavior around ) are of the form . For this -algebra admits a section (corresponding to the fact that ), and if there exists an affine line over the maximal ideal . So now assume . If then we have a hyperbola mapping to an affine line, with the fiber over being empty, corresponding to the fact that does not belong to the radical of for . So assume finally . Then belongs to the radical of , but not to its integral closure (which is the identical closure on a one-dimensional regular ring). We can write the forcing equation as . So the spectrum of the forcing algebra consists of a (thickened) line over and of a hyperbola. The forcing morphism is surjective, but it is not a submersion. For example, the preimage of is a connected component hence open, but this single point is not open.

Let be a field and let be the polynomial ring in two variables. We consider the ideal and the element . This element belongs to the radical of this ideal, hence the forcing morphism

is surjective. We claim that it is not a submersion. For this we look at the reduction modulo . In the ideal becomes which does not contain . Hence by the valuative criterion for integral closure, does not belong to the integral closure of the ideal. One can also say that the chain in the affine plane does not have a lift (as a chain) to the spectrum of the forcing algebra.

For the ideal

and the element the situation looks different. Let

be a ring homomorphism to a discrete valuation domain . If or is mapped to , then also is mapped to and hence belongs to the extended ideal. So assume that and , where is a local parameter of and and are units. Then and the exponent is at least the minimum of and , hence

So belongs to the integral closure of and the forcing morphism

is a universal submersion.

Continuous closure

Suppose now that . Then every polynomial can be considered as a continuous function

in the complex topology. If is an ideal and is an element, we say that belongs to the continuous closure of , if there exist continuous functions

such that

(as an identity of functions). The same definition works for -algebras of finite type.

It is not at all clear at once that there may exist polynomials but inside the continuous closure of . For it is easy to show that the continuous closure is (like the integral closure) just the ideal itself. We also remark that when we would only allow holomorphic functions then we could not get something larger. However, with continuous functions

we can for example write

Continuous closure is always inside the integral closure and hence also inside the radical. The element does not belong to the continuous closure of , though it belongs to the integral closure of . In terms of forcing algebras, an element belongs to the continuous closure if and only if the complex forcing mapping

(between the corresponding complex spaces) admits a continuous section.