Elsevier

Journal of Algebra

Volume 586, 15 November 2021, Pages 154-207
Journal of Algebra

Generalized differentials and prolongation spaces

https://doi.org/10.1016/j.jalgebra.2021.06.026Get rights and content

Abstract

Given a coalgebra D, a commutative algebra A, a commutative A-algebra B and a measuring ψ:DAB, we define an algebra [D](A,ψ)B that generalizes the symmetric algebra over the module of Kähler differentials SymB(ΩB/A1). We show that the spectrum of [D](A,ψ)B is isomorphic to a prolongation space as defined by Moosa and Scanlon, providing a direct construction of the latter. These prolongation spaces generalize those of Gillet, Rosen and Vojta. The universal prolongations of differential and difference kernels can also be recovered from our generalized differentials. When D is moreover a bialgebra, our generalized differentials provide a unified approach to the prolongations of commutative rings, unifying the well-known constructions in the differential and difference case.

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 f:AB be a commutative A-algebra. Recall that the Kähler differentials ΩB/A1 of B over A together with d:BΩB/A1 have the universal property that for every B-module M and every A-derivation :BM there exists a unique morphism of B-modules ϕ:ΩB/A1M such that =ϕd, giving rise to a bijectionDerA(B,M)BM(ΩB/A1,M), where MB(ΩB/A1,M) denotes the homomorphisms of B-modules from ΩB/A1 to M. If g:BR is a commutative B-algebra, then this bijection induces a bijectionDerA(B,R)AlgB(SymB(ΩB/A1),R), where AlgB(SymB(ΩB/A1),R) denotes the homomorphisms of B-algebras from SymB(ΩB/A1) to R. If δA:AA is a derivation on A, then the derivation d:BSymB(ΩB/Z1) induces a derivation d:BSymB(ΩB/Z1)/IδA, where IδA is the ideal of SymB(ΩB/Z1) generated by df(a)f(δA(a)) for all aA, and the bijection (0.1) restricts to a bijection{DerA(B,R)|f=gfδA}AlgB(SymB(ΩB/Z1)/IδA,R).

Note that an A-derivation from B to a commutative B-algebra g:BR can be defined equivalently as a homomorphism of A-algebras ρ:BR[t]/(t2) such that ev0ρ=g, where ev0:R[t]/(t2)R is the homomorphism of R-algebras defined by ev0(t)=0. 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 δ=(δ(i):BR)i=0,,m 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 ρ:BR[t]/(tm+1) (such that ev0ρ=g). Given a family of morphisms of A-modules (δ(i))i=0,,m as above, we obtain such a homomorphism ρ:BR[t]/(tm+1) defined by ρ(b):=i=0mδ(i)(b)ti for bB. Vojta and Rosen introduce differentials for higher derivations and generalize the bijections (0.1) and (0.2), respectively: Vojta defines for every mN a commutative B-algebra HSB/Am such that for every commutative A-algebra R there is a bijectionAlgA(B,R[t]/(tm+1))AlgA(HSB/Am,R), 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 δA=(δA(i))i=0,,m on A and a commutative A-algebra B, Rosen defines a commutative B-algebra HSB/(A,δA)m such that for every commutative B-algebra R there is a bijectionAlgA(B,R[t]/(tm+1))AlgA(HSB/(A,δA)m,R), where the A-algebra structure on R[t]/(tm+1) is induced by the homomorphism ρ:AA[t]/(tm+1),ai=0mδA(i)(a)ti, 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 E-rings, where E is a finite free S-algebra scheme over a commutative ring k. A commutative E-ring is a commutative ring A together with a homomorphism of k-algebras e:AE(A). This generalizes commutative rings equipped with higher derivations, E(A) replacing A[t]/(tm+1). In this article we mainly use D-measurings, where D is a coalgebra, instead of E-rings. This is equivalent to their framework as shown in [15]: Given a finite free S-algebra scheme E over k, we define D:=E(k). The S-algebra structure on E induces a k-coalgebra structure on D and commutative E-rings e:AE(A) are equivalent to D-measurings ψ:DkAA from A to itself. The k-algebra E(A) is isomorphic to the set Mk(D,A) 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 ψ:DkAA is equivalent to a homomorphism of k-algebras ρ:AkM(D,A) and the isomorphism E(A)kM(D,A) allows to pass from E-ring structures on A to D-measurings and vice versa.

More generally, given a commutative k-algebra A, a D-measuring ψA:DkAA, and two commutative A-algebras B and R, we can consider homomorphisms of A-algebras P:BkM(D,R), a notion generalizing higher derivations from B to R over A. Our generalized differentials consist of a commutative A-algebra [D](A,ψA)B together with a homomorphism of A-algebras ρu:BkM(D,[D](A,ψA)B) such that for every homomorphism of A-algebras P:BkM(D,R) there is a unique homomorphism of A-algebras ϕ:[D](A,ψA)BR such that P=kM(D,ϕ)ρu, giving rise to a bijectionAlgA(B,Mk(D,R))AlgA([D](A,ψA)B,R),Pϕ 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 (A,δA) 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 (A,δA) to the category of commutative (A,δA)-algebras (i.e. the category of commutative A-algebras f:AB such that Ker f is a δA-ideal) has a left adjoint. Similarly, the forgetful functor from the category of commutative difference algebras over a given commutative difference field (A,σA) 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 X=SpecB over Y=SpecA, the tangent bundle of X over Y is TX/Y=SpecSymB(ΩB/A1). Vojta defines, given a morphism of schemes XY, the scheme of m-jet differentials Jm(X/Y) of X over Y, such that for every Y-scheme Z there is an isomorphismSchY(Z×ZZ[t]/(tm+1),X)SchY(Z,Jm(X/Y)). Buium defines, given a derivation δA:AA on a commutative ring A and a scheme X over Y=SpecA, an A-scheme jetm(X/Y,δA) such that for every A-scheme Z there is a bijectionSchA(Z×AA[t]/(tm+1),X)SchA(Z,jetm(X/Y,δA)), where the A-scheme structure of Z×AA[t]/(tm+1) is induced by the exponential map e:AA[t]/(tm+1),ai=0mδAi(a)i!ti. Buium's definition specializes to the one of Vojta if the derivation δA is trivial. Rosen generalizes Buium's definition by using higher derivations. Given a finite free commutative S-algebra scheme E over k, a commutative E-ring e:AE(A) and a scheme X over Y=SpecA, Moosa and Scanlon define the prolongation space τ(X,E,e) of X with respect to e:AE(A) as the Weil restriction of X×AEe(A) from E(A) to A. Therefore for every A-scheme Z there is a bijectionSchA(Z×AE(A),X)SchA(Z,τ(X,E,e)). By taking E(A):=A[t]/(tm+1) and e:AE(A) 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 S-algebra schemes E to be commutative, which is equivalent to the coalgebra D associated to E 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 HSB/(A,δA)m 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 δA is the trivial higher derivation on A, the algebra HSB/(A,δA)m specializes to Vojta's divided differentials HSB/Am 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 mA and ηA, respectively. Similarly, given a coalgebra D, we denote its comultiplication and counit maps by ΔD and εD, 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 (D,Δ,ε) is a coalgebra and dD, then we use the Sweedler notation and denote Δ(d) by (d)d(1)d(2). Let R be a commutative ring. We denote the category of algebras over R by AlgR and the category of left R-modules by MR. The category of commutative R-algebras is denoted by CAlgR. We denote the symmetric algebra of an R-module M by SymR(M). The category of schemes over R is denoted by SchR. An R-ring is a ring together with a ring homomorphism from R into it.

If C is a category and A and B are objects in C, then we denote the class of morphisms from A to B in C by C(A,B). If f:BB is a morphism and A is an object in C, then we define C(A,f):C(A,B)C(A,B) by C(A,f)(g)=fg for all gC(A,B), and similarly for a morphism g:AA and an object B of C we define C(g,B):C(A,B)C(A,B) by C(g,B)(f):=fg.

The category of sets is denoted by Set. If A and B are sets and aA, then we denote by eva:Set(A,B)B the evaluation map, i.e. eva(f)=f(a) for all fSet(A,B). For elements a,bA we denote by δa,b the Kronecker delta, i.e. δa,a=1 and δa,b=0 if ab.

We denote by N 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 A,B and D there is an isomorphism of (left) k-modulesMk(DkA,B)kM(A,Mk(D,B)),ψ(a(dψ(da))).

Lemma 2.1

If (D,ΔD,εD) is a k-coalgebra and (B,mB,ηB) is a k-algebra, then the k-module Mk(D,B) becomes a k-algebra with respect to the convolution product, defined byfg:=mB(fg)ΔD for f,gkM(D,B), 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, f:AB be a commutative A-algebra, ψA:DkAA be a D-measuring from A to itself, and ρA:AkM(D,A) be the homomorphism of k-algebras associated to ψA 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 f:AB 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 [D](A,ψA)B 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.

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