Generalized differentials and prolongation spaces
Introduction
The literature contains several definitions of prolongations spaces. Most of them generalize the tangent bundle. Recently, a rather general definition of prolongation spaces was introduced by Moosa and Scanlon (cf. [25], [26]). While these prolongation spaces are defined in terms of Weil restrictions, we provide a direct construction in the case of affine schemes and show that they generalize the prolongation spaces defined previously by Buium, Rosen and Vojta (cf. [5], [28] and [34]). As the tangent bundle can be constructed using the Kähler differentials, one can construct the prolongation spaces of Buium, Vojta and Rosen using algebras of (higher) differentials. We propose a definition of generalized differentials that unifies the symmetric algebra of the Kähler differentials as well as the divided differentials defined by Vojta and the higher differentials as defined by Rosen. The spectra of our generalized differentials realize the prolongation spaces due to Moosa and Scanlon in the case of affine schemes.
Let A be a commutative ring and be a commutative A-algebra. Recall that the Kähler differentials of B over A together with have the universal property that for every B-module M and every A-derivation there exists a unique morphism of B-modules such that , giving rise to a bijection where denotes the homomorphisms of B-modules from to M. If is a commutative B-algebra, then this bijection induces a bijection where denotes the homomorphisms of B-algebras from to R. If is a derivation on A, then the derivation induces a derivation , where is the ideal of generated by for all , and the bijection (0.1) restricts to a bijection
Note that an A-derivation from B to a commutative B-algebra can be defined equivalently as a homomorphism of A-algebras such that , where is the homomorphism of R-algebras defined by . In positive characteristic, higher derivations are often more suitable than classical derivations. A higher derivation of length m from B to R over A is traditionally defined as a family of morphisms of A-modules fulfilling generalized Leibniz rules. Equivalently one can define higher derivations as follows: A (unital) higher derivation from B to R over A can be defined as a homomorphism of A-algebras (such that ). Given a family of morphisms of A-modules as above, we obtain such a homomorphism defined by for . Vojta and Rosen introduce differentials for higher derivations and generalize the bijections (0.1) and (0.2), respectively: Vojta defines for every a commutative B-algebra such that for every commutative A-algebra R there is a bijection the set on the left hand side describing higher derivations of length m from B to R over A, cf. [34]. Rosen generalizes Vojta's construction. Given a higher derivation on A and a commutative A-algebra B, Rosen defines a commutative B-algebra such that for every commutative B-algebra R there is a bijection where the A-algebra structure on is induced by the homomorphism , cf. [28]. Rosen's approach is the most general one of the aforementioned, specializing to the others as indicated by the arrows in the diagram These differentials share the property of being universal objects for certain classes of (higher) derivations.
The definition of prolongation spaces by Moosa and Scanlon is in terms of commutative -rings, where is a finite free -algebra scheme over a commutative ring k. A commutative -ring is a commutative ring A together with a homomorphism of k-algebras . This generalizes commutative rings equipped with higher derivations, replacing . In this article we mainly use D-measurings, where D is a coalgebra, instead of -rings. This is equivalent to their framework as shown in [15]: Given a finite free -algebra scheme over k, we define . The -algebra structure on induces a k-coalgebra structure on D and commutative -rings are equivalent to D-measurings from A to itself. The k-algebra is isomorphic to the set of homomorphisms of k-modules from D to A, which becomes a k-algebra thanks to the k-coalgebra structure on D. A D-measuring is equivalent to a homomorphism of k-algebras and the isomorphism allows to pass from -ring structures on A to D-measurings and vice versa.
More generally, given a commutative k-algebra A, a D-measuring , and two commutative A-algebras B and R, we can consider homomorphisms of A-algebras , a notion generalizing higher derivations from B to R over A. Our generalized differentials consist of a commutative A-algebra together with a homomorphism of A-algebras such that for every homomorphism of A-algebras there is a unique homomorphism of A-algebras such that , giving rise to a bijection that generalizes (0.3) and (0.4), cf. Proposition 3.2.
From our generalized differentials we also recover universal prolongations of differential kernels (cf. [19], [20]) and difference kernels (cf. [35], [36]).
The forgetful functor from the category of commutative differential algebras over a given commutative differential ring to the category of commutative A-algebras has a left adjoint, cf. [8]. A similar result for commutative unital iterative differential algebras is due to Rosen: The forgetful functor from the category of commutative unital iterative differential algebras over a given commutative unital iterative differential ring to the category of commutative -algebras (i.e. the category of commutative A-algebras such that Ker f is a -ideal) has a left adjoint. Similarly, the forgetful functor from the category of commutative difference algebras over a given commutative difference field to the category of commutative A-algebras has a left adjoint, as shown by Wibmer, cf. [36]. If the above-mentioned k-coalgebra D is a k-bialgebra and A is a commutative D-module algebra, then our generalized differentials again unify and generalize these results (cf. Proposition 3.5).
The simplest form of a prolongation space is the tangent bundle. In the case of an affine scheme over , the tangent bundle of X over Y is . Vojta defines, given a morphism of schemes , the scheme of m-jet differentials of X over Y, such that for every Y-scheme Z there is an isomorphism Buium defines, given a derivation on a commutative ring A and a scheme X over , an A-scheme such that for every A-scheme Z there is a bijection where the A-scheme structure of is induced by the exponential map . Buium's definition specializes to the one of Vojta if the derivation is trivial. Rosen generalizes Buium's definition by using higher derivations. Given a finite free commutative -algebra scheme over k, a commutative -ring and a scheme X over , Moosa and Scanlon define the prolongation space of X with respect to as the Weil restriction of from to A. Therefore for every A-scheme Z there is a bijection By taking and to be the ring homomorphism induced by a (higher) derivation, one recovers the definitions of Buium's jet spaces and Rosen's prolongation spaces as well as the tangent bundle and Vojta's jet spaces if the (higher) derivation is trivial.
Our generalized differentials provide an alternative and more direct construction of the prolongation spaces defined by Moosa and Scanlon, at least in the case of affine schemes. Although it is not explicitly mentioned in their articles, it seems that they assume the finite free -algebra schemes to be commutative, which is equivalent to the coalgebra D associated to to be cocommutative. Here we try not to impose this condition when it is not necessary, since operators like skew-derivations are described as a D-measurings for coalgebras D that are not cocommutative. We note that commutative rings with iterative q-difference operators, as introduced by Hardouin in [11], can be described as D-module algebras for a cocommutative bialgebra D, as Masuoka and Yanagawa show, cf. [27].
Our interest in generalized differentials arose from the use of D-measurings and D-module algebras in Galois theories of functional equations, cf. [30], [1], [2], [13], [12], [14].
This article is organized as follows: In the first section we begin with a review of Kähler differentials and a version of Kähler differentials relative to a derivation on the base ring. We recall higher derivations, introduced by Hasse and Schmidt (cf. [17]), which generalize classical derivations. Then we review divided differentials as defined by Vojta [34] and higher differentials as defined by Rosen [28]. We also recall differential prolongations due to Gillet [8] and difference prolongations due to Hrushovski, Tomasic and Wibmer, cf. [16], [31] and [36]. Finally we also include the definition of differential kernels of Johnson [19], [20]) and of difference kernels due to Wibmer [35].
In section 2 we briefly recall the notion of a measuring and of module algebras, which we use in section 3 to define generalized differentials. We prove universal properties of these generalized differentials and show how they generalize and unify several of the objects introduced in the first section: Example 3.8 shows that the universal prolongation of differential kernels due to Johnson is a special case of our generalized differentials. In fact, we recover Proposition 1.35 from Remark 3.3. The Kähler differentials relative to a derivation introduced in subsection 1.2 as well as the symmetric algebra of the classical module of Kähler differentials are special cases of our generalized differentials. Proposition 1.22 due to Gillet becomes a corollary of our Proposition 3.5. Example 3.7 shows that the algebra defined by Rosen is a special case of our generalized differentials, that Proposition 1.17 becomes a corollary of Proposition 3.2 and that Proposition 1.20 becomes a corollary of Proposition 3.5. In the case where is the trivial higher derivation on A, the algebra specializes to Vojta's divided differentials and Proposition 1.9 becomes a corollary of Proposition 3.2. In the difference case Example 3.10 shows that Proposition 1.30 is a corollary of Proposition 3.5. We close this section with some remarks on functorial properties of our generalized differentials.
Section 4 recalls several definitions of prolongation spaces. The most well-known of them is the tangent bundle. We recall Vojta's scheme of jet differentials, Buium's jet spaces and Rosen's prolongation spaces and see how they generalize the tangent bundle. We also observe how these jet and prolongation spaces can be constructed.
Finally, in section 5 we recall the definition of prolongation spaces due to Moosa and Scanlon ([25], [26]) and provide a new construction of them in terms of our generalized differentials. We show how they specialize to the spaces introduced in section 4.
Notation and conventions We assume all rings and algebras to be unital and associative, but not necessarily to be commutative, and all coalgebras to be counital and coassociative, but not necessarily to be cocommutative. Homomorphisms of (co)algebras are assumed to respect the (co)units and modules over (unital) rings are assumed to be unitary. If A is an algebra, then we denote its multiplication and unit maps by and , respectively. Similarly, given a coalgebra D, we denote its comultiplication and counit maps by and , respectively. If there is no risk of confusion, we also write m, η, Δ and ε for the multiplication, unit, comultiplication and counit maps, respectively. If is a coalgebra and , then we use the Sweedler notation and denote by . Let R be a commutative ring. We denote the category of algebras over R by and the category of left R-modules by . The category of commutative R-algebras is denoted by . We denote the symmetric algebra of an R-module M by . The category of schemes over R is denoted by . An R-ring is a ring together with a ring homomorphism from R into it. If is a category and A and B are objects in , then we denote the class of morphisms from A to B in by . If is a morphism and A is an object in , then we define by for all , and similarly for a morphism and an object B of we define by . The category of sets is denoted by . If A and B are sets and , then we denote by the evaluation map, i.e. for all . For elements we denote by the Kronecker delta, i.e. and if . We denote by the set of natural numbers including 0. Let k be a commutative ring.
Section snippets
Review of differentials, differential- and difference kernels and of prolongations
This section is of introductory nature and we do not claim originality of most of its definitions and results. We first recall the classical Kähler differentials and a version of them relative to a derivation on the base ring. The latter is probably known, but we do not know any reference. Then we recall the definition of higher derivations. We show how Vojta's divided differentials and a similar object, introduced by Rosen, generalize the symmetric algebra of the Kähler differentials and the
Measurings and module algebras
In this section we recall, given a k-coalgebra D, the definition of D-measurings and, in the case where D is a k-bialgebra, of D-module algebras.
First, recall that for k-modules and D there is an isomorphism of (left) k-modules
Lemma 2.1 If is a k-coalgebra and is a k-algebra, then the k-module becomes a k-algebra with respect to the convolution product, defined by for , and unit element given by the
Generalized differentials
In this section we introduce generalized differentials and show how they specialize to objects introduced in section 1.
Notation Let D be a k-coalgebra, A be a commutative k-algebra, be a commutative A-algebra, be a D-measuring from A to itself, and be the homomorphism of k-algebras associated to via the isomorphism (2.1).
Review of prolongation spaces
In this section we review the tangent bundle as well as several constructions of jet and prolongation spaces due to Buium, Rosen and Vojta.
Notation Let A be a commutative ring and be a commutative A-algebra.
Generalized prolongation spaces
In section 4 of [25], Moosa and Scanlon define general prolongation spaces in terms of Weil restrictions and show their existence in important cases. Here we give a direct construction in the case of affine schemes, which seems to be more direct and is analogous to the constructions of Buium, Rosen and Vojta that we reviewed in section 4. We obtain the prolongation spaces as the spectra of the A-algebras defined in section 3.
For the convenience of the reader we first recall the
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The author acknowledges partial support by grant MTM2012-33830 from the Ministerio de Economía y Competitividad, Spain.