Ringed Spaces (1)

maxresdefault

A ringed space is a broad concept in which we can fit most of the interesting geometrical objects. It consists of a topological space together with a sheaf of functions on it.

Let M be a differentiable manifold, whose topological space is Hausdorff and second countable. For each open set U ⊂ M , let C(U) be the R-algebra of smooth functions on U .

The assignment

U ↦ C(U)

satisfies the following two properties:

(1) If U ⊂ V are two open sets in M, we can define the restriction map, which is an algebra morphism:

rV, U : C(V) → C(U), ƒ ↦ ƒ|U

which is such that

i) rU, U = id

ii) rW, U = rV, U ○ rW, V

(2) Let {Ui}i∈I be an open covering of U and let {ƒi}i∈I, ƒi ∈ C(Ui) be a family such that ƒi|Ui ∩ Uj = ƒj| Ui ∩ Uj ∀ i, j ∈ I. In other words the elements of the family {ƒi}i∈I agree on the intersection of any two open sets Ui ∩ Uj. Then there exists a unique ƒ ∈ C(U) such that ƒ|Ui = ƒi.

Such an assignment is called a sheaf. The pair (M, C), consisting of the topological space M, underlying the differentiable manifold, and the sheaf of the C functions on M is an example of locally ringed space (the word “locally” refers to a local property of the sheaf of C functions.

Given two manifolds M and N, and the respective sheaves of smooth functions CM and CN, a morphism ƒ from M to N, viewed as ringed spaces, is a morphism |ƒ|: M → N of the underlying topological spaces together with a morphism of algebras,

ƒ*: CN(V) →  CM-1(V)), ƒ*(φ)(x) = φ(|ƒ|(x))

compatible with the restriction morphisms.

Notice that, as soon as we give the continuous map |ƒ| between the topological spaces, the morphism ƒ* is automatically assigned. This is a peculiarity of the sheaf of smooth functions on a manifold. Such a property is no longer true for a generic ringed space and, in particular, it is not true for supermanifolds.

A morphism of differentiable manifolds gives rise to a unique (locally) ringed space morphism and vice versa.

Moreover, given two manifolds, they are isomorphic as manifolds iff they are isomorphic as (locally) ringed spaces. In the language of categories, we say we have a fully faithful functor from the category of manifolds to the category of locally ringed spaces.

The generalization of algebraic geometry to the super-setting comes somehow more naturally than the similar generalization of differentiable geometry. This is because the machinery of algebraic geometry was developed to take already into account the presence of (even) nilpotents and consequently, the language is more suitable to supergeometry.

Let X be an affine algebraic variety in the affine space An over an algebraically closed field k and let O(X) = k[x1,…., xn]/I be its coordinate ring, where the ideal I is prime. This corresponds topologically to the irreducibility of the variety X. We can think of the points of X as the zeros of the polynomials in the ideal I in An. X is a topological space with respect to the Zariski topology, whose closed sets are the zeros of the polynomials in the ideals of O(X). For each open U in X, consider the assignment

U ↦ OX(U)

where OX(U) is the k-algebra of regular functions on U. By definition, these are the functions ƒ X → k that can be expressed as a quotient of two polynomials at each point of U ⊂ X. The assignment U ↦ OX(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, OX) is another example of a (locally) ringed space.

Advertisement

How are Topological Equivalences of Structures Homeomorphic?

dxabC

Given a first-order vocabulary 𝜏, 𝐿𝜔𝜔(𝜏) is the set of first-order sentences of type 𝜏. The elementary topology on the class 𝑆𝑡𝜏 of first-order structures type 𝜏 is obtained by taking the family of elementary classes

𝑀𝑜𝑑(𝜑) = {𝑀:𝑀 |= 𝜑}, 𝜑 ∈ 𝐿𝜔𝜔(𝜏)

as an open basis. Due to the presence of classical negation, this family is also a closed basis and thus the closed classes of 𝑆𝑡𝜏 are the first-order axiomatizable classes 𝑀𝑜𝑑(𝑇), 𝑇 ⊆ 𝐿𝜔𝜔(𝜏). Possible foundational problems due to the fact that the topology is a class of classes may be settled observing that it is indexed by a set, namely the set of theories of type 𝜏.

The main facts of model theory are reflected by the topological properties of these spaces. Thus, the downward Löwenheim-Skolem theorem for sentences amounts to topological density of the subclass of countable structures. Łoś theorem on ultraproducts grants that U-limits exist for any ultrafilter 𝑈, condition well known to be equivalent to topological compactness, and to model theoretic compactness in this case.

These spaces are not Hausdorff or T1, but having a clopen basis they are regular; that is, closed classes and exterior points may be separated by disjoint open classes. All properties or regular compact spaces are then available: normality, complete regularity, uniformizability, the Baire property, etc.

Many model theoretic properties are related to the continuity of natural operations between classes of structures, where operations are seen to be continuous and play an important role in abstract model theory.

A topological space is regular if closed sets and exterior points may be separated by open sets. It is normal if disjoint closed sets may be separated by disjoint open sets. Thus, normality does not imply regularity here. However, a regular compact space is normal. Actually, a regular Lindelöf space is already normal

Consider the following equivalence relation in a space 𝑋: 𝑥 ≡ 𝑦 ⇔ 𝑐𝑙{𝑥} = 𝑐𝑙{𝑦}

where 𝑐𝑙 denotes topological adherence. Clearly, 𝑥 ≡ 𝑦 iff 𝑥 and 𝑦 belong to the same closed (open) subsets (of a given basis). Let 𝑋/≡ be the quotient space and 𝜂 : 𝑋 → 𝑋/≡ the natural projection. Then 𝑋/≡ is T0 by construction but not necessarily Hausdorff. The following claims thus follow:

a) 𝜂 : 𝑋 → 𝑋/≡ induces an isomorphism between the respective lattices of Borel subsets of 𝑋 and 𝑋/≡. In particular, it is open and closed, preserves disjointedness, preserves and reflects compactness and normality.

b) The assignment 𝑋 → 𝑋/≡ is functorial, because ≡ is preserved by continuous functions and thus any continuous map 𝑓 : 𝑋 → 𝑌 induces a continuous assignment 𝑓/≡ : 𝑋/≡ → 𝑌/≡ which commutes with composition.

c) 𝑋 → 𝑋/≡ preserves products; that is, (𝛱𝑖𝑋𝑖)/≡ is canonically homeomorphic to 𝛱𝑖(𝑋𝑖/≡) with the product topology (monomorphisms are not preserved).

d) If 𝑋 is regular, the equivalence class of 𝑥 is 𝑐𝑙{𝑥} (this may fail in the non-regular case).

e) If 𝑋 is regular, 𝑋/≡ is Hausdorff : if 𝑥 ≢ 𝑦 then 𝑥 ∉ 𝑐𝑙{𝑦} by (d); thus there are disjoint open sets 𝑈, 𝑉 in 𝑋 such that 𝑥 ∈ 𝑈, 𝑐𝑙{𝑦} ⊆ 𝑉, and their images under 𝜂 provide an open separation of 𝜂𝑥 and 𝜂𝑦 in 𝑋/≡ by (a).

f) If 𝐾1 and 𝐾2 are disjoint compact subsets of a regular topological space 𝑋 that cannot be separated by open sets there exist 𝑥𝑖 ∈ 𝐾𝑖, 𝑖 = 1, 2, such that 𝑥1 ≡ 𝑥2. Indeed, 𝜂𝐾1 and 𝜂𝐾2 are compact in 𝑋/≡ by continuity and thus closed because 𝑋/≡ is Hausdorff by (e). They can not be disjoint; otherwise, they would be separated by open sets whose inverse images would separate 𝐾1 and 𝐾2. Pick 𝜂𝑥 = 𝜂𝑦 ∈ 𝜂𝐾1 ∩ 𝜂𝐾2 with 𝑥 ∈ 𝐾1, 𝑦 ∈ 𝐾2.

Clearly then, for the elementary topology on 𝑆𝑡𝜏, the relation ≡ coincides with elementary equivalence of structures and 𝑆𝑡𝜏/≡ is homeomorphic to the Stone space of complete theories.

Conjuncted: Twistor Spaces

scan0003

The α-planes can be generalized to a suitable class of curved complex space-times. By a complex space-time (M,g) we mean a four-dimensional Hausdorff manifold M with holomorphic metric g. Thus, with respect to a holomorphic coordinate basis xa, g is a 4×4 matrix of holomorphic functions of xa, and its determinant is nowhere-vanishing. Remarkably, g determines a unique holomorphic connection ∇, and a holomorphic curvature tensor Rabcd. Moreover, the Ricci tensor Rab becomes complex-valued, and the Weyl tensor Cabcd may be split into independent holomorphic tensors, i.e. its self-dual and anti-self-dual parts, respectively. With our two-spinor notation, one has

Cabcd = ψABCD εA′B′ εC′D′ + ψ~A′B′C′D′ εAB εCD

where ψABCD = ψ(ABCD), ψA′B′C′D′ = ψ~(A′B′C′D′). The spinors ψ and ψ~ are the anti-self-dual and self-dual Weyl spinors, respectively.

ψ~A′B′C′D′ = 0, Rab = 0, are called right-flat or anti-self-dual, whereas complex vacuum space-times such that

ψABCD = 0, Rab = 0,

are called left-flat or self-dual. This definition only makes sense if space-time is complex or real Riemannian, since in this case no complex conjugation relates primed to unprimed spinors (i.e. the corresponding spin-spaces are no longer anti-isomorphic). Hence, for example, the self-dual Weyl spinor ψ~A′B′C′D′ may vanish without its anti-self-dual counterpart ψABCD having to vanish as well, or the converse may hold.

By definition, α-surfaces are complex two-surfaces S in a complex space-time (M, g) whose tangent vectors v have the two-spinor form, where λA is varying, and πA is a fixed primed spinor field on S. From this definition, the following properties can be derived:

(i) tangent vectors to α-surfaces are null;

(ii) any two null tangent vectors v and u to an α-surface are orthogonal to one another;

(iii) the holomorphic metric g vanishes on S in that g(v, u) = g(v, v) = 0, ∀ v, u, so that α-surfaces are totally null;

(iv) α-surfaces are self-dual, in that F ≡ v ⊗ u − u ⊗ v takes the two-spinor form;

(v) α-surfaces exist in (M,g) iff the self-dual Weyl spinor vanishes, so that (M, g) is anti-self-dual.

The properties (i)–(iv), are the same as in the flat-space-time case, provided we replace the flat metric η with the curved metric g. Condition (v), however, is a peculiarity of curved space-times.

We want to prove that, if (M,g) is anti-self-dual, it admits a three-complex- parameter family of self-dual α-surfaces. Indeed, given any point p ∈ M and a spinor μA at p, one can find a spinor field πA on M, satisfying the equation

πAAA πB = ξAπB,

and such that

πA(p) = μA(p)

Hence πA defines a holomorphic two-dimensional distribution, spanned by the vector fields of the form λAπA, which is integrable. Thus, in particular, there exists a self-dual α-surface through p, with tangent vectors of the form λAμA at p. Since p is arbitrary, this argument may be repeated ∀p ∈ M. The space P of all self-dual α-surfaces in (M,g) is three-complex-dimensional, and is called twistor space of (M, g).

I15-56-trouser

When is the Spacetime Temporally Orientable?

space-time

In both general relativity and Newtonian gravitation, forces are represented by vectors at a point. We assume that the total force acting on a particle at a point (computed by taking the vector sum of all of the individual forces acting at that point) must be proportional to the acceleration of the particle at that point, as in F = ma, which holds in both theories. We understand forces to give rise to acceleration, and so we expect the total force at a point to vanish just in case the acceleration vanishes. Since the acceleration of a curve at a point, as determined relative to some derivative operator, must satisfy certain properties, it follows that the vector representing total force must also satisfy certain properties. In particular, in relativity theory, the acceleration of a curve at a point is always orthogonal to the tangent vector of the curve at that point, and thus the total force on a particle at a point must always be orthogonal to the tangent vector of the particle’s worldline at that point.

More precisely, we take a model of relativity theory to be a relativistic spacetime, which is an ordered pair (M, gab), where M is a smooth, connected, paracompact, Hausdorff 4-manifold and gab is a smooth Lorentzian metric. A model of Newtonian gravitation, meanwhile, is a classical spacetime, which is an ordered quadruple (M, tab, hab, ∇), where M is again a smooth, connected, paracompact, Hausdorff 4-manifold, tab and hab are smooth fields with signatures (1, 0, 0, 0) and (0, 1, 1, 1), respectively, which together satisfy tabhbc = 0, and ∇ is a smooth derivative operator satisfying the compatibility conditions ∇atbc = 0 and ∇ahab = 0. The fields tab and hab may be interpreted as a (degenerate) “temporal metric” and a (degenerate) “spatial metric”, respectively. Note that the signature of tab guarantees that locally, we can always find a field ta such that tab = tatb. In the special case where this field can be smoothly extended to a global field with the stated property, we call the spacetime temporally orientable.

Irrationality. Note Quote.

owLvp

To mathematics it is unique, that two absolutely contrary opinions do not logically exclude each other but exist simultaneously while there seems to be no chance to pick out a false one and to establish a remaining truth. This case is realised by the philosophy and mathematics of the infinite. While transfinite set theory is impossible without different degrees of infinity, constructivists and intuitionists deny this notion without running into inconsistencies as is admitted by some of the foremost set theorists:

… the attitude of the (neo-)intuitionists that there do not exist altogether non-equivalent infinite sets is consistent, though almost suicidal for mathematics. [p. 62]

It would not be astonishing if in different axiomatic systems different results were obtained with respect to peculiarities of those systems. But set theorists on one side and constructivists and intuitionists on the other are certainly believing to address the same entities when speaking of “rational numbers” or of “irrational numbers”. In spite of that, the former are convinced that there are infinitely many more irrational numbers than rational numbers while the latter deny that:

Hence the continua of Weyl, Lebesgue, Lusin, etc. are denumerable … [p. 255]

This situation yields bewildering results:

Feferman and Levy showed that one cannot prove that there is any non-denumerable set of real numbers which can be well ordered. … Moreover, they also showed that the statement that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted. [p. 62]

Nevertheless, the great majority of mathematicians refuse to accept the thesis that Cantor’s ideas were but a pathological fancy. Though the foundations of set theory are still somewhat shaky. Most surprising and by no means to be expected of a pupil of Fraenkel’s is that Robinson states:

Infinite totalities do not exist in any sense of the word (i.e. either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. Nevertheless, we should act as if infinite totalities really existed. [3]

Does there exist a correct and an incorrect position? And, if so, who is right, who is wrong?

Following the advice of Fraenkel, namely to judge about the value and necessity of the basic axioms, in particular of the axiom of choice, by considering its consequences, in order to settle this question. These consequences will turn out to entail what, in an euphemistic way, by set theorists usually is called a “paradoxical result”, in order to avoid the term self-contradiction.

Apart from the well-ordering theorem some statements of quite different character – in particular geometrical statements – have been proved by means of the axiom of choice, which because of their paradoxical character induced some mathematicians to reject the axiom. Presumably the earliest statement of this kind is Hausdorff’s discovery that half of the sphere’s surface is congruent to a third of it. … It may surprise scholars working in the field … that even after more than half a century of utilising the axiom of choice and well-ordering theorem, a number of first-rate mathematicians (especially French) have not essentially changed their distrustful attitude.

Transfinite set theory arises from Cantor’s observation that the set of all irrational numbers has infinitely many more members than the set of all rational numbers. While the latter has the same cardinality χ0 as the set N of all natural numbers n, the cardinality χ of the set of all irrational numbers is larger, χ = 2χ0. It is proven to be uncountable, i.e., any bijection with N can be excluded.