Ringed Spaces (1)

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:

r_{V, U} : C^∞(V) → C^∞(U), ƒ ↦ ƒ|_U

which is such that

i) r_{U, U} = id

ii) r_{W, U} = r_{V, U} ○ r_{W, V}

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

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 C_M^∞ and C_N^∞, 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^*: C_N^∞(V) →  C_M^∞(ƒ^-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,…., x_n]/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 ↦ O_X(U)

where O_X(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 ↦ O_X(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, O_X) is another example of a (locally) ringed space.

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Weyl and Automorphism of Nature. Drunken Risibility.

In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.

Superalgebras

Let k be an algebraically closed field. Given a superalgebra A we will denote with A₀ the even part, with A₁ the odd part and with I^A_odd the ideal generated by the odd part.

A superalgebra is said to be commutative (or supercommutative) if

xy = (−1)^p(x)p(y)yx, ∀ homogeneous x, y

where p denotes the parity of an homogeneous element (p(x) = 0 if x ∈ A₀, p(x) = 1 if x ∈ A₁).

Let’s denote with A the category of affine superalgebras that is commutative superalgebras such that, modulo the ideal generated by their odd part, they are affine algebras (an affine algebra is a finitely generated reduced commutative algebra).

Define affine algebraic supervariety over k a representable functor V from the category A of affine superalgebras to the category S of sets. Let’s call k[V] the commutative k-superalgebra representing the functor V,

V (A) = Hom_k−superalg(k[V], A), A ∈ A

We will call V (A) the A-points of the variety V. A morphism of affine supervarieties is identified with a morphism between the representing objects, that is a morphism of affine superalgebras.

We also define the functor V_red associated to V from the category A_c of affine k-algebras to the category of sets:

V_red(A_c)= Hom_k−alg(k[V]/I^k[V]_odd, A_c), A_c ∈ A_c

V_red is an affine algebraic variety and it is called the reduced variety associated to V. If the algebra k[V] representing the functor V has the additional structure of a commutative Hopf superalgebra, we say that V is an affine algebraic supergroup.

Let G be an affine algebraic supergroup. As in the classical setting, the condition k[G] being a commutative Hopf superalgebra makes the functor group valued, that is the product of two morphisms is still a morphism. In fact let A be a commutative superalgebra and let x, y ∈ Hom_k−superalg(k[G], A) be two points of G(A). The product of x and y is defined as:

x · y = _defm_A · x ⊗ y · ∆

where m_A is the multiplication in A and ∆ the comultiplication in k[G]. One can find that x · y ∈ Hom_k−superalg(k[G], A), that is:

(x · y)(ab) = (x · y)(a)(x · y)(b)

The non commutativity of the Hopf algebra in the quantum setting does not allow to multiply morphisms(=points). In fact in the quantum (super)group setting the product of two morphisms is not in general a morphism.

Let V be an affine algebraic supervariety. Let k₀ ⊂ k be a subfield of k. We say that V is a k₀-variety if there exists a k₀-superalgebra k₀[V] such that k[V] ≅ k₀[V] ⊗_k0 k and

V(A) = Hom_{k0 − superalg}(k₀[V], A) = Hom_k−superalg(k[V], A), A ∈ A.

We obtain a functor that we still denote by V from the category A_k0 of affine k₀-superalgebras to the category of sets:

V(A_k0) = Hom_{k0−superalg}(k₀[V], A_k0), A ∈ A_k0

thus opening up for consideration of rationality on supervariety.