Rings of differential operators as enveloping algebras of Hasse–Schmidt derivations
Abstract
Let k be a commutative ring and A a commutative k-algebra. In this paper we introduce the notion of enveloping algebra of Hasse–Schmidt derivations of A over k and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators DA/k is an isomorphism. This result generalizes the characteristic 0 case in which the ring DA/k appears as the enveloping algebra of the Lie-Rinehart algebra of the usual k-derivations of A provided that A is smooth over k.
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Journal of Pure and Applied Algebra 224 (2020) 320–361 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa Rings of differential operators as enveloping algebras of Hasse–Schmidt derivations Luis Narváez Macarro 1 Departamento de Álgebra & Instituto de Matemáticas (IMUS), Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia s/n, 41012 Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 3 December 2018 Received in revised form 13 April 2019 Available online 7 May 2019 Communicated by S. Iyengar MSC: 14F10; 13N10; 13N15 Keywords: Hasse–Schmidt derivation Integrable derivation Differential operator Substitution map Power divided algebra Let kbe a commutative ring and Aa commutative k-algebra. In this paper we introduce the notion of enveloping algebra of Hasse–Schmidt derivations of Aover kand we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators DA/k is an isomorphism. This result generalizes the characteristic 0 case in which the ring DA/k appears as the enveloping algebra of the Lie-Rinehart algebra of the usual k-derivations of Aprovided that Ais smooth over k. © 2019 Elsevier B.V. All rights reserved. Il semble donc (et c’est le point de vue de H. Hasse, F.K. Schmidt et O. Teichmül ler) que l’on ne puisse étudier les opérateurs Δkisolement, mais uniquement le système qu’ils forment avec les relations qui les relient. [Jean Dieudonné [3]] 0. Introduction In classical D-module theory, left DX-modules on a smooth space X(e.g. a smooth algebraic variety over a field of characteristic 0, or a complex smooth analytic manifold, or a smooth rigid analytic space over a complete ultrametric field of characteristic 0, etc.) are the same as modules over the structure sheaf OXendowed with an integrable connection, which is equivalent to an OX-linear action of the module of derivations Derk(OX) satisfying Leibniz rule and compatible with Lie brackets. A similar result holds for E-mail address: narv[email protected]. 1Partially supported by MTM2016-75027-P, P12-FQM-2696 and FEDER. https://doi.org/10.1016/j.jpaa.2019.05.009 0022-4049/© 2019 Elsevier B.V. All rights reserved.
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 321 right DX-modules. This fact plays a basic role in classical D-module theory, for instance in the definition of various operations or in the canonical right DX-module structure on top differential forms on X. It can be conceptually stated as saying that the sheaf DXis the enveloping algebra of the Lie algebroid Derk(OX) and it is strongly related with the canonical isomorphism of graded OX-algebras: SymOXDerk(OX)∼ −→ gr DX/k .(1) The main motivation of this paper is the existence of a canonical isomorphism: ΓADerk(A)∼ −→ gr DA/k (2) for any commutative ring k(of arbitrary characteristic) and any HS-smooth k-algebra A(see Definition 2.3.11), where ΓAdenotes the power divided algebra functor (remember that ΓA=Sym Aif Q ⊂A). The proof of (2)in [11] depends on the fact that for a HS-smooth k-algebra A, any k-derivation δ:A →A is integrable in the sense of Hasse–Schmidt (see Definition 2.3.1). This result suggests that, under these hypotheses, the ring of differential operators DA/k should be recovered in some canonical way from Hasse– Schmidt derivations. This paper is devoted to answering this question. The main difficulty is that Hasse–Schmidt derivations have a much less transparent algebraic structure than usual derivations. The module of usual derivations Derk(A) carries an A-module structure and a k-Lie algebra structure, and both are mixed on a Lie-Rinehart algebra structure, enough to recover the ring of differential operators as its enveloping algebra provided that Q ⊂kand Ais smooth over k(see [15]), although Hasse–Schmidt derivations were only known to carry a (non-commutative) group structure. In our previous paper [13], we introduced and studied the action of substitution maps (between power series rings) on Hasse–Schmidt derivations, to be thought as a substitute of the A-module structure on usual derivations. In this paper we prove that both the group structure and the action of substitution maps allow us to define the enveloping algebra of Hasse–Schmidt derivations and to prove that, under smoothness hypotheses, this enveloping algebra is canonically isomorphic to the ring of differential operators without any assumption on the characteristic of k. A key step in the proof is the existence of a canonical map of graded algebras from the power divided algebra of the module of integrable derivations (in the sense of Hasse–Schmidt) to the graded ring of the enveloping algebra of Hasse–Schmidt derivations. Let us now comment on the content of this paper. In section 1we recall and adapt, for the ease of the reader, the material in [13, §1, §2, §3]. We will concentrate ourselves in the case of power series rings and modules in a finite number of variables, which will be enough for our main results in section 3. In the last sub-section we recall the notions of exponential type series and power divided algebras. In section 2first we recall the notion of Hasse–Schmidt derivation and its basic properties. As we already did in [13, §4], we need to study, not only uni-variate Hasse–Schmidt derivations, but also multivariate ones: a (p, Δ)-variate Hasse–Schmidt derivation of our k-algebra Ais a family D=(Dα)α∈Δof k-linear endomorphisms of Asuch that D0is the identity map and Dα(xy)= β+γ=α Dβ(x)Dγ(y),∀α∈Δ,∀x, y ∈A, where Δ ⊂Npis a non-empty co-ideal, i.e. a subset of Npsuch that everytime α∈Δand α≤αwe have α∈Δ. An important idea is to think of Hasse–Schmidt derivations as series D=α∈ΔDαsαin the quotient ring R[[s]]Δof the power series ring R[[s]] =R[[s1, ..., sp]], R=End k(A), by the two-sided monomial ideal generated by all sαwith α∈Np\Δ. In the second sub-section we recall [13, §5] on the action of substitution maps on Hasse–Schmidt derivations. The starting point is simple: given a substitution
322 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 map ϕ :A[[s1, ..., sp]]Δ→A[[t1, ..., tq]]∇and a (p, Δ)-variate Hasse–Schmidt derivation D=α∈ΔDαsα we may consider a new (q, ∇)-variate Hasse–Schmidt derivation given by: ϕ•D:= α∈Δ ϕ(sα)Dα. In the last sub-section, we first recall the notion of integrable derivation: a k-derivation δ:A →Ais said to be m-integrable if there is a uni-variate Hasse–Schmidt derivation D=(Di)m i=0 such that D1=δ, and second we recall the main results in [11]. Section 3contains the original results of this paper. First, we introduce the notion of HS-module, as a generalization of the classical notion of module with an integrable connection. Roughly speaking, a left HS-module is a module Mover our k-algebra Aon which Hasse–Schmidt derivations act “globally”, in a compatible way with the group structure and the action of substitution maps, and satisfying a Leibniz rule. More precisely, for each (p, Δ)-variate Hasse–Schmidt derivation D=α∈ΔDαsαof A, Mis endowed with a k[[s]]Δ-linear automorphism Ψp Δ(D) :M[[s]]Δ→M[[s]]Δcongruent to the identity modulo s, in such a way that: -) The Ψp Δ(−)are group homomorphism. -) For each substitution map ϕ :A[[s]]Δ→A[[t]]∇we have Ψq ∇(ϕ•D) =ϕ•Ψp Δ(D). -) (Leibniz rule) For each a ∈Awe have Ψp Δ(D)a =D(a)Ψp Δ(D). Any DA/k-module is obviously a HS-module, since Hasse–Schmidt derivations act through their components, which are differential operators. Namely, if Mis a left DA/k-module, for each (p, Δ)-variate Hasse–Schmidt derivation D=α∈ΔDαsαof Awe define Ψp Δ(D)as: Ψp Δ(D)(m)= α∈Δ (Dαm)sα,∀m∈M. The basic question is whether a HS-module structure can be lifted to a DA/k-module structure or not. To illustrate the notion of HS-module, or more precisely, the notion of pre-HS-module structure (i.e. the compatibility with substitution maps only holds for substitution maps with constant coefficients), we give natural actions of Hasse–Schmidt derivations on ΩA/k and on Derk(A) generalizing, respectively, the classical Lie derivative and the adjoint representation of classical derivations. In the second sub-section we generalize the well known ⊗and Hom operations on modules with an integrable connection to the setting of HS-modules. In the last two sub-sections we define the enveloping algebra of Hasse–Schmidt derivations of a commutative algebra, and we prove, by imitating [11], that there is a canonical map of graded algebras from the power divided algebra of the module of integrable derivations to the graded ring of the enveloping algebra of Hasse–Schmidt derivations. We finally prove that, under the HS-smoothness hypothesis, the former map is an isomorphism and we deduce that the canonical map from the enveloping algebra of Hasse–Schmidt derivations to the ring of differential operators is an isomorphism. As a corollary, HS-modules coincide with D-modules for HS-smooth algebras. I would like to thank the referee for the careful reading of the paper. 1. Notations and preliminaries 1.1. Notations Throughout the paper we will use the following notations:
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 323 -) kis a commutative ring and Aa commutative k-algebra. -) DA/k is the ring of k-linear differential operators of A(see [4]). -) s ={s1, ..., sp}, t ={t1, ..., tq}, ...are sets of variables. -) k-algebra over A: see Definition 1.2.1. -) nβ:= {α∈Np| α≤β}) for β∈Np. -) tm:= {α∈Np| |α| ≤m}with m ≥0. -) CI (Np)is the set of all non-empty co-ideals of Np: see Notation 1.2.3. -) τΔΔis a truncation map: see (4). -) Up(R; Δ), Up fil(R; Δ), Up gr(R; Δ): see Notation 1.2.4. -) rr: see Definition 1.2.5. -) r→ r: see (7); g→ ge: see (8). -) Hom ◦ k(−, −), Aut ◦ k[[s]]Δ(−): see Notation 1.2.11. -) SA(p, q; Δ, ∇)is the set of substitution maps: see Definition 1.3.1. -) Ce(ϕ, α): see (13). -) ϕM, Mϕ: see 1.3.6; ϕ•r, r•ϕ: see 1.3.7. -) ϕ∗, ϕ∗: see (16)and (17). -) Em(B)is the set of exponential type series: see Definition 1.4.1. -) SymAMis the symmetric algebra of the A-module M. -) ΓAMis the power divided algebra of the A-module M: see Definition 1.4.3. -) HSp k(A; Δ) is the set of (p, Δ)-variate Hasse–Schmidt derivations: see Definition 2.1.1. -) a•D: see Definition 2.1.3. -) ϕD, for ϕa substitution map and Da Hasse–Schmidt derivation: see Proposition 2.2.3. -) UA/k =TA/k/Iis the enveloping algebra of the Hasse–Schmidt derivations of Aover k: see Definition 3.3.7. 1.2. Rings and modules of power series Throughout this section, kwill be a commutative ring, Aa commutative k-algebra and Ra ring, notnecessarily commutative. Let p ≥0be an integer and let us call s ={s1, ..., sp}a set of pvariables. The support of each α∈Np is defined as supp α:= {i | αi=0}. The monoid Npis endowed with a natural partial ordering. Namely, for α, β∈Np, we define α≤βdef. ⇐⇒ ∃ γ∈Npsuch that β=α+γ⇐⇒ αi≤βi∀i=1...,p. We denote |α| := α1+···+αp. If α≤βthen |α| ≤|β|. Moreover, if α≤βand |α| =|β|, then α=β. Let Mbe an abelian group and M[[s]] the abelian group of power series with coefficients in M. The support of a series m =αmαsα∈M[[s]] is supp(m) := {α∈Np| mα=0} ⊂Np. It is clear that m =0 ⇔supp(m) =∅. The order of a non-zero series m =αmαsα∈M[[s]] is ord(m):=min{|α||α∈supp(m)}∈N. If m =0we define ord(0) := ∞. If Mis an A-module, then M[[s]] is naturally an A[[s]]-module and for a ∈ A[[s]] and m, m∈M[[s]] we have supp(m +m) ⊂supp(m) ∪supp(m), supp(am), supp(ma) ⊂supp(m) +
324 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 supp(a), ord(m +m) ≥min{ord(m), ord(m)}and ord(am), ord(ma) ≥ord(a) +ord(m). Moreover, if ord(m) >ord(m), then ord(m +m) =ord(m). The abelian group M[[s]] is the completion of the abelian group M[s]of polynomials with coefficients in swith respect to the s-adic topology, and its natural topology is also the s-adic topology. When M=Ris a ring, R[[s]] is a topological ring. If Mis an A-module, there is a natural A[[s]]-linear bicontinuous isomorphism: A[[s]] ⊗AM∼ −→ M[[s]],(3) where ⊗Aindicates the completed tensor product with respect to the natural topology on A[[s]]. Definition 1.2.1. A k-algebra over Ais a (not-necessarily commutative) k-algebra Rendowed with a map of k-algebras ι :A →R. A map between two k-algebras ι :A →Rand ι:A →Rover Ais a map g:R→R of k-algebras such that ι=g◦ι. A filtered k-algebra over Ais a k-algebra (R, ι)over A, endowed with a ring filtration (Rk)k≥0such that ι(A) ⊂R0. A k-algebra over Ais obviously an (A; A)-bimodule. If Ris a k-algebra over A, then the power series ring R[[s]] is a k[[s]]-algebra over A[[s]]. Definition 1.2.2. We say that a subset Δ ⊂Npis an ideal (resp. a co-ideal) of Npif everytime α∈Δand α≤α(resp. α≤α), then α∈Δ. It is clear that Δis an ideal if and only if its complement Δcis a co-ideal, and that the union and the intersection of any family of ideals (resp. of co-ideals) of Npis again an ideal (resp. a co-ideal) of Np. Examples of ideals (resp. of co-ideals) of Npare the β+Np(resp. the nβ:= {α∈Np| α≤β}) with β∈Np. The tmdefined as tm:= {α∈Np| |α| ≤m}with m ≥0are also co-ideals. Notice that a co-ideal Δ ⊂Npis non-empty if and only if (t0=n0=){0} ⊂Δ. Notation 1.2.3. The set of all non-empty co-ideals of Npwill be denoted by CI (Np). For a co-ideal Δ ⊂NPand an integer m ≥0, we denote Δm:= Δ ∩tm. If Δ ⊂NPis a finite non-empty co-ideal, we define its height as ht(Δ) := min{m ∈N| Δ ⊂tm} =max{|α| | α∈Δ}. Let Mbe an (A; A)-bimodule central over k. For each co-ideal Δ ⊂Np, we denote by ΔMthe closed sub-(A[[s]; A[[s]])-bimodule of M[[s]] whose elements are the formal power series α∈Npmαsαsuch that mα= 0 whenever α∈Δ, i.e. ΔM={m∈M[[s]],supp(m)⊂Δc}=⎧ ⎨ ⎩m∈M[[s]],supp(m)⊂ β∈Δ nc β⎫ ⎬ ⎭= β∈Δm∈M[[s]],supp(m)⊂nc β= β∈Δ (nβ)M. For m ∈Nwe have (tm)M=sm+1M[[s]]. Let us denote by M[[s]]Δ:= M[[s]]/ΔMendowed with the quotient topology (it coincides with the s-adic topology regarded as a k[[s]]-module), for which it is a topological bimodule over (A[[s]]Δ; A[[s]]Δ). When Δ =nα, for some α∈Np, we will simply denote M[[s]]α:= M[[s]]nα. Similarly, when Δ =tm, for some m ≥0, we will simply denote M[[s]]m:= M[[s]]tm. The elements in M[[s]]Δare power series of the form α∈Δ mαsα,m α∈M.
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 325 The additive isomorphism α∈Δ mαsα∈M[[s]]Δ→{mα}α∈Δ∈MΔ is a homeomorphism, where MΔis endowed with the product of discrete topologies on each copy of M. For Δ ⊂Δco-ideals of Np, we have natural (A[[s]]Δ; A[[s]]Δ)-linear projections τΔΔ:M[[s]]Δ−→ M[[s]]Δ, that we call truncations: τΔΔ: α∈Δ mαsα∈M[[s]]Δ−→ α∈Δ mαsα∈M[[s]]Δ.(4) When Δ =tm, Δ=tm, m ≤m, we will simply denote τmm:= τtmtm. We have (A; A)-linear scissions: α∈Δ mαsα∈M[[s]]Δ−→ α∈Δ mαsα∈M[[s]]Δ which are topological immersions. In particular we have natural (A; A)-linear topological embeddings M[[s]]Δ→M[[s]] and we define the support (resp. the order) of any element in M[[s]]Δas its support (resp. its order) as element of M[[s]]. We have a bicontinuous isomorphism of (A[[s]]Δ; A[[s]]Δ)-bimodules M[[s]]Δ= lim ←− m∈N M[[s]]Δm, where transition maps in the inverse system are given by truncations. For a ring R, the ΔRare closed two-sided ideals of R[[s]] and we have a bicontinuous ring isomorphism R[[s]]Δ= lim ←− m∈N R[[s]]Δm. As in (3), for A[[s]]Δ⊗AM(resp. M⊗AA[[s]]Δ) endowed with the natural topology, we have that the natural map A[[s]]Δ⊗AM→M[[s]]Δ(resp. M⊗AA[[s]]Δ→M[[s]]Δ) is continuous and gives rise to a (A[[s]]Δ; A)-linear (resp. to a (A; A[[s]]Δ)-linear) isomorphism A[[s]]Δ ⊗AM∼ −→ M[[s]]Δ(resp. M ⊗AA[[s]]Δ ∼ −→ M[[s]]Δ). Each (A; A)-linear map h :M→Mbetween two bimodules induces a linear map (over ((A[[s]]Δ; A[[s]]Δ)) h: α∈Δ mαsα∈M[[s]Δ−→ α∈Δ h(mα)sα∈M[[s]Δ.(5) We have a commutative diagram A[[s]]Δ ⊗AMM[[s]]ΔM ⊗AA[[s]]Δ A[[s]]Δ ⊗AMM[[s]]ΔM ⊗AA[[s]]Δ. ≃ Id ⊗hh ≃ h ⊗Id ≃ ≃ Clearly, if Ris a k-algebra over A, then R[[s]]Δis a k[[s]]Δ-algebra over A[[s]]Δ. Notation 1.2.4. Let Rbe a ring, p ≥1and Δ ⊂Npa non-empty co-ideal. We denote by Up(R; Δ) the multiplicative sub-group of the units of R[[s]]Δwhose 0-degree coefficient is 1. The multiplicative inverse
326 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 of a unit r∈R[[s]]Δwill be denoted by r∗. Clearly, Up(R; Δ)opp =Up(Ropp; Δ). For Δ ⊂Δco-ideals we have τΔΔ(Up(R;Δ )) ⊂Up(R; Δ) and the truncation map τΔΔ:Up(R; Δ) →Up(R; Δ) is a group homomorphism. Clearly, we have: Up(R; Δ) = lim ←− m∈N Up(R;Δ m) = lim ←− Δ⊂Δ Δ<∞ Up(R;Δ ).(6) If p =1and Δ =tm={i ∈N| i ≤m}we will simply denote U(R; m) := U1(R; tm). If R=∪d≥0Rdis a filtered ring, we denote: Up fil(R;Δ):= α∈Δ rαsα∈Up(R;Δ) rα∈R|α|∀α∈Δ. It is clear that Up fil(R; Δ) is a subgroup of Up(R; Δ). If R=d∈NRdis a graded ring, we denote: Up gr(R;Δ):= α∈Δ rαsα∈Up(R;Δ) rα∈R|α|∀α∈Δ. It is clear that Up gr(R; Δ) is a subgroup of Up(R; Δ). If Rbe a filtered ring, we will denote by σ:Up fil(R; Δ) −→ Up gr(gr R; Δ) the total symbol map defined as: σ α∈Δ rαsα:= α∈Δ σ|α|(rα)sα. It is clear that σis a group homomorphism compatible with truncations. For any ring homomorphism f:R→R, the induced ring homomorphism f:R[[s]]Δ→R[[s]]Δsends Up(R; Δ) into Up(R; Δ) and so it induces natural group homomorphisms Up(R; Δ) →Up(R; Δ). Similar results hold for the filtered or graded cases. Definition 1.2.5. Let Rbe a ring, p, q≥0, s ={s1, ..., sp}, t ={t1, ..., tq}disjoint sets of variables and ∇ ⊂Np, Δ ⊂Nqnon-empty co-ideals. For each r∈R[[s]]∇, r∈R[[t]]Δ, the external product rr∈ R[[s t]]∇×Δ(notice that ∇ ×Δ ⊂Np+qis a non-empty co-ideal) is defined as rr:= (α,β)∈∇×Δ rαr βsαtβ. The above definition is consistent with the existence of natural isomorphism of (R; R)-bimodules R[[s]]∇ ⊗RR[[t]]Δ≃R[[s t]]∇×Δ≃R[[t s]]Δ×∇ ≃R[[t]]Δ ⊗RR[[s]]∇. Let us also notice that 1 1 =1 and rr=(r1)(1 r). Moreover, if r∈Up(R; ∇), r∈Uq(R; Δ), then rr∈Up+q(R; ∇ ×Δ) and (rr)∗=r∗r∗. Let E, Fbe two A-modules and Δ ⊂Npa non-empty co-ideal. The proof of the following proposition is straightforward. Proposition 1.2.6. Under the above hypotheses, any k[[s]]Δ-linear map f:E[[s]]Δ→F[[s]]Δis continuous for the natural topologies, and for any co-ideal Δ⊂Npwith Δ⊂Δwe have f(Δ E/ΔE)⊂Δ F/ΔFand so there is a unique k[[s]]Δ-linear map f:E[[s]]Δ→F[[s]]Δsuch that the following diagram is commutative:
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 327 E[[s]]ΔF[[s]]Δ E[[s]]ΔF[[s]]Δ. f trunc. trunc. f 1.2.7. For each r=βrβsβ∈Homk(E, F)[[s]]Δwe define r:E[[s]]Δ→F[[s]]Δby r α∈Δ eαsα:= α∈Δ β+γ=α rβ(eγ)sα, which is obviously a k[[s]]Δ-linear map. Let us notice that r=βsβrβ. It is clear that the map r∈Homk(E,F)[[s]]Δ−→ r∈Homk[[s]]Δ(E[[s]]Δ,F[[s]]Δ)(7) is (A[[s]]Δ; A[[s]]Δ)-linear. If f:E[[s]]Δ→F[[s]]Δis a k[[s]]Δ-linear map, let us denote by fα:E→F, α∈Δ, the k-linear maps defined by f(e)= α∈Δ fα(e)sα,∀e∈E. If g:E→F[[s]]Δis a k-linear map, we denote by ge:E[[s]]Δ→F[[s]]Δthe unique k[[s]]Δ-linear map extending gto E[[s]]Δ=k[[s]]Δ ⊗kE. It is given by ge α eαsα:= α g(eα)sα.(8) We have a k[[s]]Δ-bilinear and A[[s]]Δ-balanced map −,− :(r, e)∈Homk(E,F)[[s]]Δ×E[[s]]Δ−→ r, e:= r(e)∈F[[s]]Δ. Lemma 1.2.8. With the above hypotheses, the following properties hold: 1) The map (7)is an isomorphism of (A[[s]]Δ; A[[s]]Δ)-bimodules. When E=Fit is an isomorphism of k[[s]]Δ-algebras over A[[s]]Δ. 2) The restriction map f∈Homk[[s]]Δ(E[[s]]Δ,F[[s]]Δ)→ f|E∈Homk(E,F[[s]]Δ) is an isomorphism of (A[[s]]Δ; A)-bimodules. 3) For r∈Homk(A, F)[[s]]Δ, we have r∈Derk(A, F)[[s]]Δ⇐⇒ r∈Derk[[s]]Δ(A[[s]]Δ,F[[s]]Δ), and so the map (7)for E=Ainduces an isomorphism of A[[s]]Δ-modules Derk(A, F)[[s]]Δ ∼ −→ Derk[[s]]Δ(A[[s]]Δ,F[[s]]Δ).
328 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 Proof. Parts 1) and 2) are proven in [13, Lemma 3]. For part 3), let us write r=βrβsβ. (⇒) For all a =α, b =α∈A[[s]]Δwe have: r(ab)=···= α∈Δ β+γ+δ=α rβ(aγbδ)sα= α∈Δ β+γ+δ=α (bδrβ(aγ)+aγrβ(bδ))sα=···=br(a)+ar(b). (⇐) For all a, b ∈Awe have: β∈Δ rβ(ab)sβ=r(ab)=br(a)+ar(b)=···= β∈Δ (br β(a)+ar β(b))sβ and so rβ∈Derk(A, F)for all β∈Δ. 2 Let us call R=End k(E). As a consequence of the above lemma, the composition of the maps R[[s]]Δ r→r −−−→Endk[[s]]Δ(E[[s]]Δ)f→f|E −−−−−→Homk(E,E[[s]]Δ)(9) is an isomorphism of (A[[s]]Δ; A)-bimodules, and so Homk(E, E[[s]]Δ) inherits a natural structure of k[[s]]Δ-algebra over A[[s]]Δ. Namely, if g, h :E→E[[s]]Δare k-linear maps with g(e)= α∈Δ gα(e)sα,h(e)= α∈Δ hα(e)sα,∀e∈E, gα,h α∈Homk(E,E), then the product hg ∈Homk(E, E[[s]]Δ)is given by (hg)(e)= α∈Δ β+γ=α (hβ◦gγ)(e)sα.(10) Definition 1.2.9. Let p, q≥0, s ={s1, ..., sp}, t ={t1, ..., tq}disjoint sets of variables and Δ ⊂Np, ∇ ⊂Nq non-empty co-ideals. For each f∈Endk[[s]]Δ(E[[s]]Δ)and each g∈Endk[[t]]∇(E[[t]]∇), with f(e)= α∈Δ fα(e)sα,g(e)= β∈∇ gβ(e)tβ∀e∈E, we define fg∈Endk[[st]]Δ×∇ (E[[s t]]Δ×∇)as fg:= he, with: h(x):= (α,β)∈Δ×∇ (fα◦gβ)(x)sαtβ∀x∈E. The proof of the following lemma is clear and it is left to the reader. Lemma 1.2.10. With the above hypotheses, for each r∈R[[s]]Δ, r∈R[[t]]∇, we have rr=r r(see Definition 1.2.5). Notation 1.2.11. We denote: Hom ◦ k(E,E[[s]]Δ):={f∈Homk(E,E[[s]]Δ),f(e)≡emod (n0)E∀e∈E},
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 335 Definition 1.4.1. An element r=r(t) =m i=0 ritiin B[[t]]mis said to be of exponential type if r0=1and r(t +t) =r(t)r(t), i.e. ϕ•r=(ι•r)(ι•r), or equivalently, if i+j iri+j=rirj,whenever i+j<m+1. The set of elements in B[[t]]mof exponential type will be denoted by Em(B). The set E∞(B) will be simply denoted by E(B). The set Em(B)is a subgroup U(B; m)and the external operation a, m i=0 riti∈B×Em(B)→ m i=0 ri(at)i= m i=0 riaiti∈Em(B) (23) defines a natural B-module structure on Em(B). It is clear that E1(B)is canonically isomorphic to B(as B-module). Let Cbe another commutative A-algebra. For each m ≥1, any A-algebra map h :B→Cinduces obvious A-linear maps Em(h) :Em(B) →Em(C). In this way we obtain functors Emfrom the category of commutative A-algebras to the category of A-modules. For 1 ≤m ≤q≤∞, the projections B[[t]]q→B[[t]]m induce natural truncation maps Eq→Emand we have (see (6)): E(B) = lim ←− m∈N Em(B). When Q ⊂B, any r=m i=0 riti∈Em(B)is determined by r1, since ri=ri i!for all i =0 ..., m, and so all truncation maps Eq(B) →Em(B), 1 ≤m ≤q≤∞, are isomorphisms and B≃E1(B) ≃Em(B) ≃E(B). The following result is proven in [16, Chap. III] in the case m =∞. The proof for any integer m ≥1is completely similar. Proposition 1.4.2. For each A-module Mand each m ≥1there is an universal pair (ΓA,mM, γA,m), where ΓA,mMis a commutative A-algebra and γA,m :M→Em(ΓA,mM)is an A-linear map, satisfying the following universal property: for any commutative A-algebra Band any A-linear map H:M→Em(B) there is a unique homomorphism of A-algebras h :Γ A,mM→Bsuch that H=Em(h)◦γA,m, or equivalently, the map h∈HomA−alg(ΓA,mM,B)→ Em(h)◦γA,m ∈HomA(M,Em(B)) is bijective. The pair (ΓA,mM, γA,m)is unique up to a unique isomorphism. For m =1, we have a canonical isomorphism SymAM∼ −→ ΓA,1M. Definition 1.4.3. The A-algebra ΓA,mMis called the algebra of m-divided powers of Mand it is canonically N-graded with Γ0 A,mM=A, Γ1 A,mM=M. In the case m =∞, (ΓA,∞M, γA,∞)is simply denoted by (ΓAM, γA)and it is called the algebra of divided powers of M. In this way ΓA,m becomes a functor from the category of A-modules to the category of (N-graded) commutative A-algebras, which is left adjoint to Em. For 1 ≤m ≤q≤∞the truncations Eq→Eminduce natural transformations ΓA,m →ΓA,q and ΓA= lim −→ m∈N ΓA,m.
336 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 When Q ⊂A, we have SymA ∼ −→ ΓA,1 ∼ −→ ΓA,m ∼ −→ ΓAfor all m ≥1. For instance, for A =Zand M=Zxa free abelian group of rank 1, the algebra ΓZ,mMis the Z-subalgebra Z xi/i!,1≤i≤m!⊂Q[x] and γA,m :nx ∈Zx−→ m i=0 nixi i!ti∈Em"Z xi/i!,1≤i≤m!#. 2. Hasse–Schmidt derivations 2.1. Definitions and first results In this section we recall some notions and results of the theory of Hasse–Schmidt derivations [5]as developed in [13]. See also [6]. From now on kwill be a commutative ring, Aa commutative k-algebra, s ={s1, ..., sp}a set of variables and Δ ⊂Npa non-empty co-ideal. Definition 2.1.1. A (p, Δ)-variate Hasse–Schmidt derivation, or a (p, Δ)-variate HS-derivation for short, of Aover kis a family D=(Dα)α∈Δof k-linear maps Dα:A −→ A, with D0=Id Aand satisfying the following Leibniz type identities: Dα(xy)= β+γ=α Dβ(x)Dγ(y) for all x, y∈Aand for all α∈Δ. We denote by HSp k(A; Δ) the set of all (p, Δ)-variate HS-derivations of A over kand HSp k(A)for Δ =Np. When Δ =tmwe will simply denote HSp k(A; m) := HSp k(A; tm). For p =1, a 1-variate HS-derivation will be simply called a Hasse–Schmidt derivation (a HS-derivation for short), or a higher derivation,3and we will simply write HSk(A; m) := HS1 k(A; Δ) for Δ =tm={q∈N| q≤m}4 and HSk(A) := HS1 k(A). Any (p, Δ)-variate HS-derivation Dof Aover kcan be understood as a power series α∈Δ Dαsα∈R[[s]]Δ,R=End k(A), and so we consider HSp k(A; Δ) ⊂R[[s]]Δ. Actually HSp k(A; Δ) is a (multiplicative) sub-group of Up(R; Δ). The group operation in HSp k(A; Δ) is explicitly given by: (D,E)∈HSp k(A;Δ)×HSp k(A;Δ)−→ D◦E∈HSp k(A;Δ) with (D◦E)α= β+γ=α Dβ◦Eγ, and the identity element of HSp k(A; Δ) is Iwith I0=Idand Iα=0for all α=0. The inverse of a D∈HSp k(A; Δ) will be denoted by D∗. 3This terminology is used for instance in [9]. 4These HS-derivations are called of length min [12].
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 337 For Δ⊂Δ ⊂Npnon-empty co-ideals, we have truncations τΔΔ:HS p k(A;Δ)−→ HSp k(A;Δ ), which obviously are group homomorphisms. For m ≥nwe will denote τmn :HS p k(A; m) →HSp k(A; n)the truncation map. Since any D∈HSp k(A; Δ) is determined by its finite truncations, we have a natural group isomorphism HSp k(A) = lim ←− Δ⊂Δ Δ<∞ HSp k(A;Δ ).(24) The proof of the following proposition is clear and is left to the reader. Proposition 2.1.2. Let t ={t1, ..., tq}be another set of variables, ∇ ⊂Nqa non-empty co-ideal, and D∈HSp k(A; Δ), E∈HSq k(A; ∇)HS-derivations. Then its external product DE(see Definition 1.2.5) is a (p +q, ∇ ×Δ)-variate HS-derivation. Definition 2.1.3. For each a ∈Apand for each D∈HSp k(A; Δ), we define a•Das (a•D)α:= aαDα,∀α∈Δ. It is clear that a•D∈HSp k(A; Δ), a•(a•D) =(aa)•D, 1•D=Dand 0•D=I. If Δ⊂Δ ⊂Npare non-empty co-ideals, we have τΔΔ(a•D) =a•τΔΔ(D). In particular, the image of τm1:HS k(A; m) →HSk(A; 1) ≡Derk(A)is an A-submodule. Notation 2.1.4. Let us denote: Hom ◦ k−alg(A, A[[s]]Δ):={f∈Homk−alg(A, A[[s]]Δ),f(a)≡amod (n0)A∀a∈A}, Aut ◦ k[[s]]Δ−alg(A[[s]]Δ):=f∈Autk[[s]]Δ−alg(A[[s]]Δ)|f(a)≡a0mod (n0)A∀a∈A[[s]]Δ. It is clear that Hom ◦ k−alg(A, A[[s]]Δ) ⊂Hom ◦ k(A, A[[s]]Δ)and Aut ◦ k[[s]]Δ−alg(A[[s]]Δ)⊂Aut ◦ k[[s]]Δ(A[[s]]Δ) (see Notation 1.2.11) are subgroups and we have group isomorphisms (see (12)and (11)): HSp k(A;Δ) Aut◦ k[[s]]Δ−alg(A[[s]]Δ)Hom ◦ k−alg(A, A[[s]]Δ). D→ D ≃ restr. ≃(25) The composition of the above isomorphisms is given by: D∈HSp k(A;Δ) ∼ −→ ΦD:= $a∈A→ α∈Δ Dα(a)sα%∈Hom ◦ k−alg(A, A[[s]]Δ).(26) Notice that the identity D0= Id corresponds to the fact that ΦD(a) ≡amodulo (n0)Afor all a ∈A, Leibniz identities in Definition 2.1.1 correspond to the fact that ΦDis a ring homomorphism, and k-linearity of the Dαcorrespond to k-linearity of ΦD.
338 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 For each HS-derivation D∈HSp k(A; Δ) we have D=(Φ D)e, i.e.: D α∈Δ aαsα= α∈Δ ΦD(aα)sα for all αaαsα∈A[[s]]Δ, and for any E∈HSp k(A; Δ) we have ΦD◦E= D◦ΦE. If Δ⊂Δis another non-empty co-ideal and we denote by πΔΔ:A[[s]]Δ→A[[s]]Δthe projection (or truncation), one has ΦτΔΔ(D)=πΔΔ◦ΦD. Definition 2.1.5. For each HS-derivation E∈HSp k(A; Δ), we denote5 (E):=min{r≥1|∃α∈Δ,|α|=r, Eα=0}≥1 if E=Iand (E) =∞if E=I. In other words, (E) =ord(E−I). We obviously have (E◦E) ≥min{(E), (E)}and (E∗) =(E). Moreover, if (E) >(E), then (E◦E) =(E). The next two results are proven in Propositions 7 and 8 of [13]. Proposition 2.1.6. For each D∈HSp k(A; Δ) we have that Dαis a k-linear differential operator of order ≤ |α| (D)for all α∈Δ. As a consequence of the above proposition we have HSp k(A; Δ) ⊂Up fil(DA/k; Δ). Lemma 2.1.7. For any D, E∈HSs k(A; Δ) we have ([D, E]) ≥(D) +(E). Proof. It is a consequence of the identity [D, E] −I=[(D−I),(E−I)] D∗E∗.2 Proposition 2.1.6 can be improved in the following way. Definition 2.1.8. For each HS-derivation E∈HSp k(A; Δ) and each α∈Δ, we denote α(E) := (τΔ,nα(E)), i.e. α(E):=min{r≥1|∃β≤α, |β|=r, Eβ=0}≥1 if τΔ,nα(E) =Iand α(E) =∞if τΔ,nα(E) =I. It is clear that (E) ≤α(E)for all α∈Δ. Replacing Dwith τΔ,nα(D)makes obvious the following proposition. Proposition 2.1.9. For each D∈HSp k(A; Δ) we have that Dαis a k-linear differential operator or order ≤ |α| α(D)for all α∈Δ. 2.2. The action of substitution maps on HS-derivations In this section, kwill be a commutative ring, Aa commutative k-algebra, R=End k(A), s ={s1, ..., sp}, t ={t1, ..., tp}sets of variables and Δ ⊂Np, ∇ ⊂Nqnon-empty co-ideals. Let us recall Proposition 10 in [13]. 5This definition changes slightly with respect to Definition (1.2.7) in [12].
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 339 Proposition 2.2.1. For any substitution map ϕ :A[[s]]Δ→A[[t]]∇, we have: 1) ϕ∗"Hom ◦ k−alg(A, A[[s]]Δ)#⊂Hom ◦ k−alg(A, A[[t]]∇), 2) ϕ•HSp k(A; Δ) ⊂HSq k(A; ∇), 3) ϕ•Aut ◦ k[[s]]Δ−alg(A[[s]]Δ) ⊂Aut ◦ k[[t]]∇−alg(A[[t]]∇). We have then a commutative diagram: Hom ◦ k−alg(A, A[[s]]Δ)HS p k(A;Δ) Aut◦ k[[s]]Δ−alg(A[[s]]Δ) Hom ◦ k−alg(A, A[[t]]∇)HS q k(A;∇)Aut ◦ k[[t]]∇−alg(A[[t]]∇). ϕ∗ ∼ ΦD←[D ∼ ϕ•(−)ϕ•(−) ∼ ΦD←[D ∼ (27) In particular, for any HS-derivation D∈HSp k(A; Δ) we have ϕ•D∈HSq k(A; ∇)(see 1.3.7). Moreover Φϕ•D=ϕ◦ΦD. It is clear that for any co-ideals Δ⊂Δand ∇⊂∇with ϕ (Δ A/ΔA)⊂∇ A/∇Awe have τ∇∇(ϕ•D)=ϕ•τΔΔ(D),(28) where ϕ:A[[s]]Δ→A[[t]]∇is the substitution map induced by ϕ. Let us notice that any a ∈Apgives rise to a substitution map ϕ :A[[s]]Δ→A[[s]]Δgiven by ϕ(si) =aisi for all i =1, ..., p, and one has a•D=ϕ•D. 2.2.2. Let u={u1, ..., ur}be another set of variables, Ω ⊂Nra non-empty co-ideal, ϕ ∈SA(p, q; Δ, ∇), ψ∈SA(q, r; ∇, Ω) substitution maps and D, D∈HSp k(A; Δ) HS-derivations. From 1.3.7 we deduce the following properties: -) If we denote E:= ϕ•D∈HSq k(A; ∇), we have E0=Id,E e= α∈Δ |α|≤|e| Ce(ϕ, α)Dα,∀e∈∇.(29) -) If ϕ =0is the trivial substitution map or if D=I, then ϕ•D=I. -) If ϕhas constant coefficients, then (ϕ•D)∗=ϕ•D∗and ϕ•(D◦D) =(ϕ•D)◦(ϕ•D). The general case is treated in Proposition 2.2.3. -) ψ•(ϕ•D) =(ψ◦ϕ)•D. -) (ϕ•D) ≥ord(ϕ)(D). The following result is proven in Propositions 11 and 12 of [13]. Proposition 2.2.3. Let ϕ :A[[s]]Δ→A[[t]]∇be a substitution map. Then, the following assertions hold: (i) For each D∈HSp k(A; Δ) there is a unique substitution map ϕD:A[[s]]Δ→A[[t]]∇such that ϕ•D◦ϕD=ϕ◦ D. Moreover, (ϕ•D)∗=ϕD•D∗, ϕI=ϕand: Ce(ϕ, f +ν)= β+γ=e |f+g|≤|β|,|ν|≤|γ| Cβ(ϕ, f +g)Dg(Cγ(ϕD,ν)) for all e ∈Δand for all f, ν∈∇with |f+ν| ≤|e|.
340 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 (ii) For each D, E∈HSp k(A; Δ), we have ϕ•(D◦E) =(ϕ•D)◦(ϕD•E)and "ϕD#E=ϕD◦E. In particular, "ϕD#D∗ =ϕ. (iii) If ψis another composable substitution map, then (ϕ◦ψ)D=ϕψ•D◦ψD. (iv) If ϕhas constant coefficients then ϕD=ϕ. Definition 2.2.4. Let Sbe a k-algebra over A, D∈HSp k(A; Δ) and r∈Up(S; Δ). We say that ris a D-element if ra = D(a)rfor all a ∈A[[s]]Δ. Given D∈Up(Endk(A); Δ), it is clear that: D∈HSp k(A;Δ)⇐⇒ Dis a D-element. For D=Ithe identity HS-derivation, a r∈Up(S; Δ) is an I-element if and only if rcommutes with all a ∈A[[s]]Δ. If E∈HSp k(A; Δ) is another HS-derivation, r∈Up(S; Δ) is a D-element and s ∈Up(S; Δ) is an E-element, then rs is a (D◦E)-element. The proof of the following lemma is easy and it is left to the reader. Lemma 2.2.5. With the above notations, for each r=αrαsα∈Up(S; Δ) the following properties are equivalent: -) ris a D-element. -) br =r D∗(b)for all b ∈A[[s]]Δ. -) r∗is a D∗-element. -) If r=αrαsα, we have rαa =β+γ=αDβ(a)rγfor all a ∈Aand for all α∈Δ. -) ra = D(a)rfor all a ∈A. The following proposition generalizes Proposition 2.2.3. Proposition 2.2.6. Let Sbe a k-algebra over A, D∈HSp k(A; Δ), ϕ :A[[s]]Δ→A[t]]∇a substitution map and r∈Up(S; Δ) a D-element. Then the following properties hold: (a) ϕ•ris a (ϕ•D)-element. (b) ϕ•(rr) =(ϕ•r)(ϕD•r)for all r∈S[[s]]Δ. In particular, (ϕ•r)∗=ϕD•r∗. Moreover, if Eis an A-module and S=End k(E), then the following identity holds: (c) ϕ•r, ϕD E(e) =ϕE(r, e)for all e ∈E[[s]]Δ, i.e. (ϕ•r)◦ϕD E=ϕE◦r. Proof. (a) By Lemma 2.2.5 we need to prove that ϕR(r)b = ϕ•D(b) ϕR(r)for all b ∈A, but we know that rb = D(b)rand so, from Lemma 1.3.8 and (18), we deduce that (ϕ•r)b=ϕR(r)b=ϕR(rb)=ϕR D(b)r= ϕ D(b)ϕR(r)= ϕ•D(b)ϕR(r)= ϕ•D(b)(ϕ•r). (b) Since all the involved maps are k-linear and continuous, it is enough to prove the identity in the case where r=r αsαwith r α∈Rand α∈Δ. But, on one hand we have
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 341 ϕ•(rr)=ϕR(rr αsα)=ϕR(sαrr α)=ϕ(sα)ϕR(rr α)=ϕ(sα)ϕR(r)r α=ϕ(sα)(ϕ•r)r α, and on the other hand, by using (a), we have (ϕ•r)(ϕD•r)=(ϕ•r)ϕD R(r αsα)=(ϕ•r)ϕD(sα)r α= ϕ•D(ϕD(sα))(ϕ•r)r α= ϕ•D◦ϕD(sα)(ϕ•r)r α=ϕ◦ D(sα)(ϕ•r)r α=ϕ(sα)(ϕ•r)r α and we are done. For the last part, 1 =ϕR(1) =ϕR(rr∗) =ϕR(r)ϕD R(r∗). (c) As in part (b), it is enough to prove the identity for e =eαsα, with α∈Δand eα∈E. By using the fact that ∈Endk(E)[[s]]Δ−→ ∈Endk[[s]]Δ(E[[s]]Δ) is an (A[[s]]Δ;A[[s]]Δ)-linear isomorphism compatible with the ϕ•(−) operation (see Lemma 1.2.8 and (19)), we deduce from part (a) that (&ϕ•r)b = ϕ•D(b) (&ϕ•r)for all b ∈A[t]]∇, and from Proposition 2.2.3, (i) and (20)we obtain: ϕ•r, ϕD E(e)=(&ϕ•r)"ϕD E(eαsα)#=(&ϕ•r)"ϕD(sα)eα#= ϕ•D"ϕD(sα)#(&ϕ•r)(eα)= ϕ( D(sα))ϕE(r(eα)) = ϕ(sα)ϕE(r(eα)) = ϕE(sαr(eα)) = ϕE(r(sαeα)) = ϕE(r, e).2 2.3. Integrable derivations and HS-smooth algebras In this section we recall some notions and results of [11,12]. Let kbe a commutative ring and Aa commutative k-algebra. The following definition slightly changes with respect to Definition (2.1.1) in [12]. Definition 2.3.1. (Cf. [2,8]) Let m ≥1be an integer or m =∞, and δ:A →Aa k-derivation. We say that δis m-integrable (over k) if there is a HS-derivation D∈HSk(A; m)such that D1=δ. Any such Dwill be called an m-integral of δ. The set of m-integrable k-derivations of Ais denoted by IDerk(A; m). We simply say that δis integrable if it is ∞-integrable and denote IDerk(A) := IDerk(A; ∞). We say that δis f-integrable (finite integrable) if it is m-integrable for any integer m ≥1. The set of f-integrable k-derivations of Ais denoted by IDerf k(A). It is clear (see Definition 2.1.3) that the IDerk(A; m)and IDerf k(A)are A-submodules of Derk(A)and that we have exact sequences of groups: 1→ker τm1−→ HSk(A;m)→IDerk(A;m)→0,m≥1,(30) and Derk(A)=IDer k(A;1)⊃IDerk(A;2)⊃IDerk(A;3)⊃··· , IDerk(A;∞)⊂IDerf k(A)= m∈N m≥1 IDerk(A;m).(31) Example 2.3.2. Let m ≥1be an integer. If m!is invertible in A, then any k-derivation δof Ais m-integrable: we can take D∈HSk(A; m) defined by Di=δi i!for i =0, ..., m. If Q ⊂k, one proves in a similar way that any k-derivation of Ais ∞-integrable.
342 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 Let us recall the following result ([9, Theorem 27.1]): Proposition 2.3.3. Let us assume that Ais a 0-smooth k-algebra. Then any k-derivation of Ais integrable. Proposition 2.3.4. The following properties are equivalent: (a) Derk(A) =IDer k(A; ∞). (b) Derk(A) =IDer k(A; m)for all integers m ≥1(⇔Derk(A) =IDer f k(A)). Proof. The implication (a) ⇒(b) is clear. (b) ⇒(a) Let δbe a k-derivation of A. By hypothesis, there is a 2-integral D(2) =(Id, D1, D2) ∈HSk(A; 2) of δ, and by applying [13, Corollary 4] repeatedly we find a sequence D(m)∈HSk(A; m), m ≥2, such that τm,m−1D(m)=D(m−1) for each m ≥2. We can take D= lim ←− m D(m)∈HSk(A), that obviously is an ∞-integral of δ.2 Remark 2.3.5. In general, we know that IDerk(A;∞)⊂IDerf k(A)= m∈N+ IDerk(A;m)⊂Derk(A). Proposition 2.3.4 tells us that the above inclusion is an equality whenever all the k-derivations of Aare m-integrable for each m ∈N+. Otherwise, we do not know whether it is strict or not, or in other words, whether a derivation which is m-integrable for each integer m ≥1is ∞-integrable or not. Definition 2.3.6. Let mbe a non-negative integer or m =∞. For any HS-derivation D∈HSk(A; m)we define its total symbol by (see Notation 1.2.4): Σm(D):=σ(D)= m i=0 σi(Di)ti∈Ugr(gr DA/k;m). The total symbol map Σm:HS k(A; m) −→ Ugr(gr DA/k; m)is a group homomorphism. The following proposition is proven in [11, Proposition 2.5, Corollary 2.7]. Proposition 2.3.7. With the hypotheses above, the following properties hold: (1) The image of Σmis contained in Em"gr DA/k#. (2) For any D∈HSk(A; m)and any a ∈Awe have Σm(a •D) =aΣm(D). (3) The map Σminduces an A-linear map χm:IDer k(A; m) →Em(gr DA/k). It is clear that, for 1 ≤m ≤q≤∞, the following diagram is commutative: IDerk(A;q)Eq(gr DA/k) IDerk(A;m)Em(gr DA/k). χq inc. trunc. χm By taking the inverse limit of the χmfor 1 ≤m <∞we obtain an A-linear map χf:IDer f k(A) → E(gr DA/k). Explicitly, if δ∈IDerf k(A), then:
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 343 χf(δ)= ∞ m=0 σm(Dm m)tm where Dm="Dm j#0≤j≤m≡m j=0 Dm jtj∈HSk(A; m)is any m-integral of δfor each integer m ≥1 (D0=I). From the universal property of power divided algebras (see Proposition 1.4.2), we obtain a canonical homomorphism of graded A-algebras: ϑf A/k :ΓIDer f k(A)→gr DA/k .(32) It is clear that for each integer m ≥1, the following diagram is commutative: ΓIDer k(A;∞)ΓIDer f k(A)ΓIDer k(A;m) gr DA/k, nat. ϑA/k,∞ nat. ϑf A/k ϑA/k,m where the ϑA/k,m and ϑA/k,∞have been defined in [11, (2.6)]. The following two theorems are proven in [11], Theorem (2.8) and Theorem (2.14), for IDerk(A; ∞), ϑA/k,∞instead of IDerf k(A), ϑf A/k, but the proofs remain essentially the same. Theorem 2.3.8. With the above notations, there are canonical maps θA/k and φsuch that the following diagram of graded A-algebras is commutative: gr DA/k "SymAΩA/k#∗ gr ΓIDer f k(A)ΓDer k(A). θA/k ϑf A/k nat. φ Theorem 2.3.9. Assume that Derk(A)is a projective A-module of finite rank. The following properties are equivalent: (a) The homomorphism of graded A-algebras θA/k :grDA/k −→ "SymAΩA/k#∗ gr is an isomorphism. (b) The homomorphism of graded A-algebras ϑf A/k :Γ IDerf k(A) −→ gr DA/k is an isomorphism. (c) IDerf k(A) =Der k(A). Remark 2.3.10. After Theorem (2.14) in [11]or Proposition 2.3.4, the equivalent properties in Theorem 2.3.9 are also equivalent to: (b’) The homomorphism of graded A-algebras ϑA/k,∞:ΓIDer k(A;∞)−→ gr DA/k is an isomorphism. (c’) IDerk(A; ∞) =Der k(A). Definition 2.3.11. We say that a k-algebra Ais HS-smooth if Derk(A)is a projective A-module of finite rank and the equivalent properties (a), (b), (c) of Theorem 2.3.9 hold.
344 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 Let us recall the following result ([11, Corollary (2.16)]). Corollary 2.3.12. Assume that ΩA/k is a projective A-module of finite rank and that Ais differentially smooth over k(in the sense of [4, 16.10]). Then, Ais a HS-smooth k-algebra. In particular, after [4, Proposition 17.12.4], if Ais a smooth finitely presented k-algebra, then Ais a HS-smooth k-algebra. 3. Main results 3.1. Hasse–Schmidt modules Definition 3.1.1. Let Rbe a k-algebra over A. A pre-HS-structure on Rover A/k is a system of maps Ψ=Ψp Δ:HS p k(A;Δ)−→ Up(R;Δ),p∈N,Δ∈CI (Np) such that6: (i) The Ψp Δare group homomorphisms. (ii) (Leibniz rule) For any D∈HSp k(A; Δ), Ψp Δ(D)is a D-element, i.e. Ψp Δ(D) a = D(a)Ψp Δ(D)for all a ∈A (see Lemma 2.2.5). (iii) For any substitution map ϕ ∈Sk(p, q; Δ, ∇)and for any D∈HSp k(A; Δ) we have Ψq ∇(ϕ•D) = ϕ•Ψp Δ(D). We say that a pre-HS-structure Ψon Rover A/k is a HS-structure if property (iii) above holds for any substitution map ϕ ∈SA(p, q; Δ, ∇). If Ris another k-algebra over Aand f:R→Ris a map of k-algebras over A, then any (pre-)HS-structure Ψon Rover A/k gives rise to a (pre-)HS-structure f◦Ψon Rover A/k defined as (f◦Ψ)p Δ:= f◦Ψp Δ,p∈N,Δ∈CI (Np). If Ris filtered, we will say that a (pre-)HS-structure Ψon Rover A/k is filtered if Ψp Δ(HSp k(A;Δ))⊂Up fil(R;Δ) for all p ∈Nand all Δ ∈CI (Np). Let us notice that if Ψis a pre-HS-structure on Rover A/k, then the system of maps Γ={Γp Δ: HSp k(A; Δ) −→ Up(Ropp; Δ), p ∈N, Δ ∈CI (Np)}defined as Γp Δ(D) =Ψp Δ(D∗)for D∈HSp k(A; Δ) is a pre-structure on Ropp over A/k. However, if Ψis a HS-structure on Rover A/k, the system Γdefined above is not in general HS-structure on Ropp. More precisely, we have the following proposition. Proposition 3.1.2. Let Ψbe a pre-HS-structure on Rover A/k and let us consider the system of maps Γ= {Γp Δ:HS p k(A; Δ) −→ Up(Ropp; Δ), p ∈N, Δ ∈CI (Np)}defined as Γp Δ(D) =Ψp Δ(D∗)for D∈HSp k(A; Δ). The following properties are equivalent: (1) Γis a HS-structure on Ropp over A/k. 6Actually, from (6)and (24)we could restrict ourselves to non-empty finite co-ideals.
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 351 for all h∈HomA[[s]]Δ(E[[s]]Δ, F[[s]]Δ)and for all a ∈A[[s]]Δ. To finish, let us prove property (iii). Let us write M=Hom A(E, F). It is enough to prove that Ψq ∇(ϕ•D)|M="ϕ•Ψp Δ(D)#|Mfor all p, q∈N, for all Δ ⊂Np, ∇ ∈CI (Nq), for all substitution map ϕ ∈SA(p, q; Δ, ∇)(resp. ϕ ∈Sk(p, q; Δ, ∇)) and for all HS-derivation D∈HSp k(A; Δ). For each h ∈M we have ν(h) =h=hwith hβeαtβ=βh(eβ)tβfor each βeβtβ∈E[[t]]∇. So: (ν◦Ψq ∇(ϕ•D)) (h)|E='Ψq ∇(ϕ•D)◦ν(h)◦Ψq ∇((ϕ•D)∗)(|E (1) = ϕ•Ψp Δ(D)◦h◦'Ψq ∇(ϕD•D∗)|E(=ϕ•Ψp Δ(D)◦h◦'ϕD•Ψp Δ(D∗)|E((2) = ϕ•Ψp Δ(D)◦h◦'"ϕD#E◦Ψp Δ(D∗)|E(= ϕ•Ψp Δ(D)◦"ϕD#F◦h◦Ψp Δ(D∗)|E(3) =ϕF◦Ψp Δ(D)◦ν(h)◦Ψp Δ(D∗)|E= ϕF◦ "ν◦Ψp Δ(D)#(h)|E!=ϕF◦ ν"Ψp Δ(D)(h)#|E!(4) =ν"ϕM"Ψp Δ(D)(h)##|E= ν""ϕM◦Ψp Δ(D)#(h)#|E=ν""ϕ•Ψp Δ(D)#(h)#|E="ν◦"ϕ•Ψp Δ(D)##(h)|E, where equality (1) comes from Proposition 2.2.3, equality (2) comes from (20), equality (3) comes from Proposition 2.2.6, (c), and equality (4) comes from (18). We first deduce that (ν◦Ψq ∇(ϕ•D)) (h) = "ν◦"ϕ•Ψp Δ(D)##(h)for all h ∈M, i.e. ν◦(Ψq ∇(ϕ•D)|M)=ν◦""ϕ•Ψp Δ(D)#|M#, second, from the injectivity of ν, that Ψq ∇(ϕ•D)|M="ϕ•Ψp Δ(D)#|M, and we conclude that Ψq ∇(ϕ•D) = ϕ•Ψp Δ(D). 2 The proofs of the following three propositions are completely similar to the proofs of Propositions 3.2.2 and 3.2.1. Proposition 3.2.3. Under the above hypotheses, the following properties hold: (1) For any p ∈N, for any Δ ∈CI (Np)and for any D∈HSp k(A; Δ) there is a unique Γp Δ(D) ∈ Aut◦ k[[s]]Δ((P⊗AE)[[s]]Δ)such that the following diagram is commutative: P[[s]]Δ⊗k[[s]]ΔE[[s]]Δ(P⊗AE)[[s]]Δ P[[s]]Δ⊗k[[s]]ΔE[[s]]Δ(P⊗AE)[[s]]Δ, μ Γp Δ(D)⊗Ψp Δ(D∗)Γp Δ(D) μ where μis the natural (A[[s]]Δ; A[[s]]Δ)-linear map μ α pαsα⊗ α eαsα= α α+α=α pα⊗eα sα. (2) The system Γ={Γp Δ, p ∈N, Δ ∈CI (Np)}defines a right (pre-)HS-module structure over A/k on P⊗AE. Proposition 3.2.4. Under the above hypotheses, the following properties hold:
352 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 (1) For any p ∈N, for any Δ ∈CI (Np)and for any D∈HSp k(A; Δ) there is a unique Ψp Δ(D) ∈ Aut◦ k[[s]]Δ(HomA(P,Q)[[s]]Δ)such that the following diagram is commutative: HomA(P,Q)[[s]]ΔHomk[[s]]Δ(P[[s]]Δ,Q[[s]]Δ) HomA(P,Q)[[s]]ΔHomk[[s]]Δ(P[[s]]Δ,Q[[s]]Δ), ν Ψp Δ(D)Γp Δ(D∗)◦Γp Δ(D) ν where νis the natural (A[[s]]Δ; A[[s]]Δ)-linear map defined as ν(h) =h(see (7)). (2) The system Ψ ={Ψp Δ, p ∈N, Δ ∈CI (Np)}defines a left (pre-)HS-module structure over A/k on HomA(P, Q). Proposition 3.2.5. Under the above hypotheses, the following properties hold: (1) For any p ∈N, for any Δ ∈CI (Np)and for any D∈HSp k(A; Δ) there is a unique Γp Δ(D) ∈ Aut◦ k[[s]]Δ(HomA(E,P)[[s]]Δ)such that the following diagram is commutative: HomA(E,P)[[s]]ΔHomk[[s]]Δ(E[[s]]Δ,P[[s]]Δ) HomA(E,P)[[s]]ΔHomk[[s]]Δ(E[[s]]Δ,P[[s]]Δ), ν Γp Δ(D)Γp Δ(D)∗◦Ψp Δ(D)=Ψp Δ(D)◦Γp Δ(D)∗ ν where νis the natural (A[[s]]Δ; A[[s]]Δ)-linear map defined as ν(h) =h(see (7)). (2) The system Γ={Γp Δ, p ∈N, Δ ∈CI (Np)}defines a right (pre-)HS-module structure over A/k on HomA(E, P). The following proposition easily follows from Proposition 3.2.1 and its proof is left to the reader. Proposition 3.2.6. Under the above hypotheses, the left (pre-)HS-module structure over A/k on E⊗d= E⊗AE⊗A···⊗ AEdefined in Proposition 3.2.1 induces: 1) A unique (pre-)HS-module structure over A/k on Symd AEsuch that the natural map E⊗d→Symd AE is a HS-map. 2) A unique (pre-)HS-module structure over A/k on )d AEsuch that the natural map E⊗d→)d AEis a HS-map. 3.3. The enveloping algebra of Hasse–Schmidt derivations Let TA/k be the free k-algebra TA/k := kSa,T p,Δ,D,α;a∈A, p ∈N,Δ∈CI (Np),α∈Δ,D∈HSp k(A;Δ) and let us consider the two-sided ideal I ⊂TA/k with generators: (0) Sc1−c, Sa+a−Sa−Sa, Saa−SaSa, (i) Tp,{0},I,0−1,
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 353 (ii) Tp,Δ,I,α for |α| >0,7 (iii) Tp,Δ,D ◦E,α − β+γ=α Tp,Δ,D,β Tp,Δ,E,γ, (iv) Tp,Δ,D,α Sa− β+γ=α SDβ(a)Tp,Δ,D,γ, (v) Tq,∇,ϕ•D,β − α∈Δ |α|≤|β| SCβ(ϕ,α)Tp,Δ,D,α, for c ∈k, a, a∈A, p, q∈N, Δ ⊂Np, ∇ ∈CI (Nq), α∈Δ, β∈∇, D, E∈HSp k(A; Δ) and ϕ ∈ SA(p, q; Δ, ∇). We consider the N-grading in TA/k given by (see Definition 2.1.8): deg(k)=0,deg(Sa)=0,deg (Tp,Δ,D,α)=|α| α(D) for a ∈A, p ∈N, Δ ∈CI (Np), α∈Δand D∈HSp k(A; Δ). This grading is motivated by Proposition 2.1.9. Let us notice that deg (Tp,Δ,D,α)=deg"Tp,nα,τΔ,nα(D),α#. We will denote Td A/k the homogeneous component of degree dand T≤d A/k := e≤dTe A/k. Let us call UA/k := TA/k/Iand write Sa:= Sa+I, Tp,Δ,D,α := Tp,Δ,D,α +Ifor the generators of the k-algebra UA/k. The grading in TA/k induces a filtration on UA/k and let us also call deg : UA/k →Nthe corresponding map: deg(P):=min{deg(p)|p∈TA/k,P =p+I}for P∈UA/k,P =0, and deg(0) =−∞, with Ud A/k ={P∈UA/k | deg(P) ≤d} =T≤d A/k/ I∩T≤d A/k. The generators of type (0) of Igive rise to a natural k-algebra map a ∈A → Sa∈UA/k and so UA/k is a k-algebra over A. 3.3.1. We first collect some direct consequences of the above definitions. For p ∈N, s ={s1, ..., sp}, Δ ∈CI (Np), α∈Δand D∈HSp k(A; Δ) we have: (a) Since the quotient map π:A[[s]]Δ→A[[s]]nαis a substitution map (actually, a truncation map) and the action π•(−):HS p k(A;Δ)−→ HSp k(A;nα) coincides with the truncation τΔ,nα(see Lemma 1.3.2), by using the generators of type (v) and the fact that Cβ(π, α) =δαβ, we obtain Tp,Δ,D,α =Tp,nα,τΔ,nα(D),α (remember that deg (Tp,Δ,D,α)= deg "Tp,nα,τΔ,nα(D),α#). (b) From (a) and the generators of type (i) of Iwe deduce: Tp,Δ,D,0=Tp,{0},τΔ,{0}(D),0=1. (c) If 0 <|α| < α(D), then τΔ,nα(D) =Iand so from (a) and the generators of type (ii) of Iwe have Tp,Δ,D,α =Tp,nα,I,α =0. Lemma 3.3.2. The term U0 A/k is the k-module generated by the Sa, a ∈A, and coincides with the image of the natural map A →UA/k. 7Actually, generators (ii) can be avoided since they are deduced from generators (i) and (iii).
354 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 Proof. By definition, U0 A/k is the k-module generated by the monomials in the Sa, a ∈A, and the Tp,Δ,D,α with deg (Tp,Δ,D,α)=|α| α(D)=0, i.e. |α| < α(D). So, by (b) and (c) and the generators of type (0) of Iwe deduce that U0 A/k is the k-module generated by the Saand coincides with the image of A →UA/k.2 The proof of the following proposition is clear (see Proposition 2.1.6). Proposition 3.3.3. There is a unique k-algebra map υ:UA/k −→ DA/k sending Sa−→ a, Tp,Δ,D,α −→ Dα. Moreover, it is filtered. Corollary 3.3.4. The natural map A →UA/k is injective and A ≃U0 A/k. Proposition 3.3.5. The k-algebra UA/k over Ais endowed with a natural HS-structure Υover A/k. Moreover, the pair (UA/k, Υ)is universal among HS-structures, i.e. for any k-algebra Rover Aand any HS-structure Ψon Rover A/k, there is a unique map f:UA/k →Rof k-algebras over Asuch that f◦Υ =Ψ. Proof. We consider the system of maps Υgiven by: Υp Δ:D∈HSp k(A;Δ)−→ α∈Δ Tp,Δ,D,αsα∈Up(UA/k;Δ) for p ∈N, Δ ∈CI (Np). It is straightforward to see that properties in Definition 3.1.1 hold for Υ. Namely, property 1) follows from the generators of type (i), (ii) and (iii) of I, property 2) follows from the generators of type (iv) of I, and finally the generators of type (v) of Iguarantee property 3). For the universal property, let f0:TA/k →Rbe the k-algebra map determined by f0(Sa)=a1,f 0(Tp,Δ,D,α)=Ψp Δ(D)α for all a ∈A, for all p ∈N, for all Δ ∈CI (Np), for all α∈Δand for all D∈HSp k(A; Δ). It is clear that f0vanishes on Iand gives rise to our wanted map f:UA/k →Rof k-algebras over A. The uniqueness of f is clear. 2 Let us notice that the HS-structure Υin the above proposition is filtered. Corollary 3.3.6. The abelian category of left (resp. right) HS-modules over A/k is isomorphic to the category of left (resp. right) UA/k-modules. Definition 3.3.7. The enveloping algebra of the Hasse–Schmidt derivations of Aover kis the k-algebra UA/k =TA/k/Idefined above. It is a filtered k-algebra over A. Theorem 3.3.8. The graded ring gr UA/k is commutative. Proof. We need to prove that the degree of the bracket of the classes in UA/k of any two variables generating TA/k is strictly less than the sum of the degrees of these variables.
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 355 -) For the variables Sathe result is clear since SaSa−SaSa=Saa−Saa=0. -) Let us see the case of one variable Saand one variable Tp,Δ,D,α, with a ∈A, p ∈N, Δ ∈CI (Np), α∈Δ and D∈HSp k(A; Δ), and set =α(D). We know from (b) that Tp,Δ,D,0=1, and from (c) that whenever 0 <|α| <, then Tp,Δ,D,α =0, and of course Dα=0. So, if |α| <then Tp,Δ,D,αSa−SaTp,Δ,D,α =0. Otherwise |α| ≥and, by using the generators of type (iv) of I, we have: Tp,Δ,D,α Sa−SaTp,Δ,D,α = β+γ=α |β|>0 SDβ(a)Tp,Δ,D,γ = β+γ=α |β|≥ SDβ(a)Tp,Δ,D,γ. We conclude that: deg (Tp,Δ,D,α Sa−SaTp,Δ,D,α)≤max {deg (Tp,Δ,D,γ)|β+γ=α, |β|≥}≤ max *|γ| γ(D)|γ≤α, |γ|≤|α|−+≤max *|γ| α(D)|γ≤α, |γ|≤|α|−+< |α| =deg(Tp,Δ,D,α)=deg(Tp,Δ,D,α)+deg(Sa). -) It remains to treat the case of two variables Tp,Δ,D,α and Tq,∇,E,β. We need to prove that: deg (Tp,Δ,D,α Tq,∇,E,β −Tq,∇,E,β Tp,Δ,D,α)<deg (Tp,Δ,D,α)+deg(Tq,∇,E,β).(34) From (b), we may assume α, β=0; by taking into account generators of Iof type (ii), we may assume D, E=I; from (c), we may assume α(D) ≤|α|and β(E) ≤|β|; and finally, from (a), we may assume that Δ =nαand ∇ =nβ. Let us denote s ={s1, ..., sp}, t ={t1, ..., tq}, ι:A[[s]]nα→A[[st]]nα×nβ=A[[st]]n(α,β),κ:A[[t]]∇→A[[st]]n(α,β) the combinatorial substitution maps given by the inclusions s, t →s t, F:= ι•D, G := κ•E, 1:= (D) = α(D), 2:= (E) =β(E). From Proposition 2.2.3 we have F∗=ι•D∗and G∗=κ•E∗. We will proceed in several steps. First, by using the generators of type (v) of Iand the fact that: C(γ,σ)(ι, α)=1ifγ=αand σ=0 0otherwise, C(γ,σ)(κ, β)=1ifγ=0 and σ=β 0otherwise, we deduce that: (1) Tp+q,n(α,β),F,(α,0) =Tp,nα,D,α, Tp+q,n(α,β),G,(0,β)=Tq,nβ,E,β. (2) Tp+q,n(α,β),F,(α,β)=0for β=0and Tp+q,n(α,β),G,(α,β)=0for α=0. (3) (α,0)(F) =α(D), (0,β)(G) =β(E)(in particular, (F) =(α,0)(F) =α(D) =(D) =1, (G) = (0,β)(G) =β(E) =(E) =2) and deg Tp+q,n(α,β),F,(α,0)=|(α,0)| (α,0)(F)=|α| α(D)=deg(Tp,nα,D,α), deg Tp+q,n(α,β),G,(0,β)=|(0,β)| (0,β)(G)=|β| β(E)=deg"Tq,nβ,E,β#.
356 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 (4) From 1.3.9 and the generators of type (iii) and (v) of Iwe have: Tp+q,n(α,β),D E,(α,β)=Tp+q,n(α,β),F ◦G,(α,β)=Tp,nα,D,αTq,nβ,E,β, Tp+q,n(α,β),E D,(α,β)=Tp+q,n(α,β),G ◦F,(α,β)=Tq,nβ,E,βTp,nα,D,α. Let us write H=[F, G] =F◦G◦F∗◦G∗. From Lemma 2.1.7 we know that (H) ≥1+2. Let us prove that: (5) Tp+q,n(α,β),H,(μ,λ)= 0 whenever (μ, λ) =(0, 0) and |μ| < 1or |λ| < 2. By using (1), (2) and the generators of type (iii) of Iagain, we obtain: Tp+q,n(α,β),H,(μ,λ)=···= Tp+q,n(α,β),F,(μ,0) Tp+q,n(α,β),G,(0,λ)Tp+q,n(α,β),F ∗,(μ,0) Tp+q,n(α,β),G∗,(0,λ)= Tp,nα,D,μTq,nβ,E,λTp,nα,D∗,μ Tq,nβ,E∗,λ ,(35) where both sums are indexed by the (μ, μ, λ, λ)such that μ+μ =μand λ+λ =λ. If μ =0and 0 <|λ|then Tp+q,n(α,β),H,(0,λ)=···= λ+λ=λ Tq,nβ,E,λTq,nβ,E∗,λ =Tq,nβ,E◦E∗,λ =Tq,nβ,I,λ =0, by using generators of type (iii), (ii) of I. In a similar way, we have that Tp+q,n(α,β),H,(μ,0) = 0 whenever 0 <|μ|. Assume now that μ =0and λ =0. If |μ| < 1or |λ| < 2, then all the summands in (35)vanish by (c) (remember that (D∗) =(D)and (E∗) =(E)) and so Tp+q,n(α,β),H,(μ,λ)=0. (6) By using F◦G =H◦(G◦F)and the generators of type (iii) of Iwe have: Tp+q,n(α,β),F ◦G,(α,β)= α+α=α β+β=β Tp+q,n(α,β),H,(α,β)Tp+q,n(α,β),G ◦F,(α,β). Hence: Tp+q,n(α,β),F ◦G,(α,β)−Tp+q,n(α,β),G ◦F,(α,β)= |α|+|β|>0 Tp+q,n(α,β),H,(α,β)Tp+q,n(α,β),G ◦F,(α,β) (c) = |α|+|β|≥(H) Tp+q,n(α,β),H,(α,β)Tp+q,n(α,β),G ◦F,(α,β) (4),(5) = |α|≥1,|β|≥2 Tp+q,n(α,β),H,(α,β)Tq,nβ,E,β Tp,nα,D,α , where all the indexes (α, α, β, β)in the above sums satisfy α+α =αand β+β =β, and so, by (4): deg "Tp,nα,D,α Tq,nβ,E,β −Tq,nβ,E,β Tp,nα,D,α#= deg Tp+q,n(α,β),F ◦G,(α,β)−Tp+q,n(α,β),G ◦F,(α,β)≤ max ,deg Tp+q,n(α,β),H,(α,β)+deg"Tq,nβ,E,β #+ deg (Tp,nα,D,α )-=
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 357 max *|α|+|β| (α,β)(H)+|β| β (E)+|α| α (D)+≤ max *|α|+|β| (H)+|β| (E)+|α| (D)+≤ max *|α|+|β| 1+2 +|β| 2 +|α| 1 +≤max *|α|+|β| 1+2 +|β| 2 +|α| 1 +< |α+α| 1 +|β+β| 2 =|α| 1 +|β| 2 =deg(Tp,nα,D,α)+deg"Tq,nβ,E,β#, where the max’s are taken over the α, α ∈Npand β, β ∈Nqsuch that α+α =α, β+β =β, |α| ≥1and |β| ≥2, and the last (strict) inequality comes from Lemma 3.3.9.2 Lemma 3.3.9. Let 1, 2≥1be integers. For any integers a, b, a, b ≥0with a≥1, b≥2we have: a+b 1+2+a 1+b 2<a+a 1+b+b 2. Proof. We have a+b 1+2 +a 1 +b 2 ≤max *a 1 ,b 2 ++a 1 +b 2 < a 1 +b 2 +a 1 +b 2 ≤ a+a 1 +b+b 2 .2 3.4. The case of HS-smooth algebras Our first goal is to define a canonical map of graded A-algebras from the divided power algebra of the module of f-integrable k-derivations (see Definitions 1.4.3 and 2.3.1) of Ato the graded ring of UA/k. We will closely follow the procedure in [11, §2.2] (see also section 2.3). Proposition 3.4.1. For each integer m ≥1the group homomorphism σ◦Υ1 m:HS k(A;m)−→ Ugr(gr UA/k;m) vanishes on ker τm,1and its image is contained in Em(gr UA/k). Proof. Let us consider the combinatorial substitution maps ι1, ι2:A[[s]]m→A[[s1, s2]](m,m)given by ιi(s) =si, i =1, 2, and the substitution map ϕ :A[[s]]m→A[[s1, s2]]mgiven by ϕ(s) =s1+s2. Notice that in ιi=ιiand in ϕ =ϕ(see Proposition 1.3.5). An element r∈U(gr UA/k; m) belongs to Em(gr UA/k)if and only if (ι1•r)(ι2•r) =ϕ•r(see 1.4.1). Let D∈HSk(A; m)be a HS-derivation, and let us denote r=(σ◦Υ1 m)(D), E=ϕ•D, F= (ι1•D)◦(ι2•D)and H=E◦F∗. It is clear that H(1,0) =H(0,1) =0and so (H) >1. Then, deg "T1,tm,H,(i,j)#≤deg "T1,tm,H,(i,j)#=i+j (i,j)(H)≤i+j (H)<i+j for all (i, j)with 0 <i +j≤m, and so (σ◦Υ1 m)(H)=σ i+j≤m T1,tm,H,(i,j)si 1sj 2= i+j≤m σi+j"T1,tm,H,(i,j)#si 1sj 2=1.(36) We deduce that:
358 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 ϕ•r=(inϕ)•"σ"Υ1 m(D)##() =σ"ϕ•Υ1 m(D)#=σ"Υ2 m(E)#=σ"Υ2 m(H◦F)#= σ"Υ2 m(H)Υ2 m(F)#(36) =σ"Υ2 m(F)#=σ"Υ2 m(ι1•D)Υ2 m(ι2•D)#= σ"(ι1•Υ1 m(D)) (ι2•Υ1 m(D))#=σ"(ι1•Υ1 m(D))#σ"(ι2•Υ1 m(D))#() = ((in ι1)•r) ((in ι2)•r)=(ι1•r)(ι2•r), where equalities ()come from Proposition 1.3.10, and so r=(σ◦Υ1 m)(D) ∈Em(gr UA/k). On the other hand, if D∈ker τm,1, then (D) >1and we can proceed as before with Hand deduce that (σ◦Υ1 m)(D) =1. 2 Corollary 3.4.2. There is a natural system of A-linear maps χm:IDer k(A;m)−→ Em(gr UA/k),m≥1, such that for m≥mthe following diagram is commutative: IDerk(A;m)Em(gr UA/k) IDerk(A;m)Em(gr UA/k). χm incl. trunc. χm (37) Moreover, the system above induces a natural A-linear map χ:IDer f k(A) −→ E(gr UA/k). Proof. Since IDerk(A; m)is by definition the image of the group homomorphism τm,1:HS k(A;m)→HSk(A;1)≡Derk(A), we deduce from Proposition 3.4.1 that the group homomorphism σ◦Υ1 minduces a natural group homomorphism χm:IDer k(A; m) −→ Em(gr UA/k). If δ∈IDerk(A; m), then χm(δ) =m i=0 σi(T1,m,D,i)siwhere D∈HSk(A; m)is any m-integral of δ, i.e. D1=δ. Then, for each a ∈A, a•Dis an m-integral of aδ and χm(aδ)= m i=0 σi(T1,m,a•D,i)si() = m i=0 σi⎛ ⎝ i j=0 ajT1,m,D,j⎞ ⎠si= = m i=0 σi"aiT1,m,D,i#si= m i=0 σi(T1,m,D,i)(as)i=aχm(δ), where equality ()comes from generators of type (v) of I, and so χmis A-linear (remember that the A-action on exponential type series is given by substitutions s → as, a ∈A, see (23)). The commutativity of (37) comes from the commutativity of the following diagram (σand the Υp Δare compatible with truncations): HSk(A;m)Em(gr UA/k) HSk(A;m)Em(gr UA/k). σ◦Υ1 m trunc. trunc. σ◦Υ1 m The map χis simply the inverse limit of the χm.2
L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 359 Corollary 3.4.3. There is a natural map ϑ:Γ AIDerf k(A) −→ gr UA/k of graded A-algebras such that the following diagram is commutative: ΓAIDerf k(A)grUA/k gr DA/k, ϑ ϑf A/k gr υ(38) where ϑf A/k is the map defined in (32)and υis defined in Proposition 3.3.3. Proof. Let us denote γ:δ∈IDerf k(A)−→ ∞ n=0 γn(δ)sn∈E(ΓAIDerf k(A)) the canonical map (see 1.4.3). The existence of ϑcomes from the universal property of γ. Namely, there is a unique map of A-algebras ϑ:Γ AIDerf k(A) −→ gr UA/k such that χ=E(ϑ)◦γ. More explicitly, for each δ∈IDerf k(A)and for each D∈HSk(A; m)such that D1=δ, we have ϑ(γm(δ)) =σm(T1,m,D,m). In particular, ϑis graded. The commutativity of the diagram (38)is a consequence of the commutativity of the diagram IDerf k(A)E(gr UA/k) E(gr DA/k), χ χE(gr υ) where χis the inverse limit of the maps χm:IDer k(A; m) →Em(gr DA/k), m ≥1, defined in [11, Corollary (2.7)]. 2 Proposition 3.4.4. Assume that IDerf k(A) =Der k(A). Then, the map ϑ:Γ AIDerf k(A)−→ gr UA/k is surjective. Proof. The A-algebra gr UA/k is generated by the σd(Tq,∇,E,β) for q≥1, ∇ ∈CI (Nq), β∈∇, E∈ HSq k(A; ∇), E=I, d =|β| β(E). After 3.3.1, we may assume that ∇ =nβand so β(E) =(E). Let us call m =ht(∇). Let {δs, s ∈s}be a system of generators of the A-module Derk(A). Since IDerk(A; m) =Der k(A), for each s ∈sthere exists Ds∈HSk(A; m)which is an m-integral of δs. By considering some total ordering <on s, we can define D∈HSs k(A; m)as the external product (see Definition 1.2.5) of the ordered family {Ds, s ∈s}, i.e. D0=Idand for each α∈N(s), α=0, Dα=Ds1 αs1◦··· ◦Dse αsewith supp α={s1<···<s e}. After [13, Theorem 1], there exists a substitution map ϕ0:A[[s]]m→A[[t1, ..., tq]]∇such that E=ϕ0•D. Moreover, it is clear that we can take ord(ϕ0) =(E). Since ∇is finite, condition (17) in [13, Proposition 2] implies that the set {s ∈s | ϕ0(s) =0}is finite. Let us call {s1<···<s p}this set. We have a factorization of substitution maps:
360 L. Narváez Macarro / Journal of Pure and Applied Algebra 224 (2020) 320–361 A[[s]]mA[[t1,...,t q]]∇ A[[s1,...,s p]]m ϕ0 ϕ1 ϕ where ϕ1(s) =0if s =si, ϕ1(si) =siand ϕ(si) =ϕ0(si). Then we have E=ϕ0•D=ϕ•Fwith F=ϕ1•D=Ds1···Dsp∈HSp k(A; (m, ..., m)). We obviously have ord(ϕ) =ord(ϕ0) =(E)and so Cβ(ϕ, α) = 0 whenever |α|(E) >|β|. So, Tq,∇,E,β = |α|≤m |α|≤|β| Cβ(ϕ, α)Tp,m,F,α = |α|≤m |α|(E)≤|β| Cβ(ϕ, α)T1,m,Ds1,α1T1,m,Ds2,α2···T1,m,Dsp,αp, σd(Tq,∇,E,β)= |α|=d Cβ(ϕ, α) p . j=1 σαj"T1,m,Dsj,αj#=ϑ⎛ ⎝ |α|=d Cβ(ϕ, α) p . j=1 γαj(δj)⎞ ⎠ and we deduce that ϑis surjective. 2 Remark 3.4.5. In the proof of the above proposition we have used the Axiom of Choice in order to consider a total ordering on s. This could be avoided when Derk(A)is a finitely generated A-module. In general, we could also avoid the Axiom of Choice by proving directly a convenient variant of Theorem 1 of [13]. Theorem 3.4.6. If Ais a HS-smooth k-algebra, then the natural map υ:UA/k −→ DA/k is an isomorphism of filtered k-algebras. Proof. It is enough to prove that gr υ:grUA/k −→ gr DA/k is an isomorphism of graded A-algebras. Since Ais a HS-smooth k-algebra, we have ϑf A/k :Γ AIDerf k(A) ∼ −→ gr DA/k and from Corollary 3.4.3 we deduce that ϑis injective. The surjectivity of ϑcomes from Proposition 3.4.4.2 Corollary 3.4.7. If Ais a HS-smooth k-algebra, then the category of left (resp. right) HS-modules over A/k is isomorphic to the category of left (resp. right) DA/k-modules. 3.5. Further developments and questions Question 3.5.1. With the hypotheses of the preceding section, it is easy to see that the map Υ1 1:HS k(A;1)≡Derk(A)−→ U(UA/k;1)≡UA/k is k-linear, compatible with Lie brackets and satisfies Leibniz rule. So, it induces a k-algebra map from the enveloping algebra of the Lie-Rinehart algebra Derk(A)([15]) to UA/k. The paper [14]is devoted to prove that this map is an isomorphism whenever Q ⊂k, and so HS-modules and classical integrable connections coincide in characteristic 0. Question 3.5.2. Assume that Ais a HS-smooth k-algebra and ΩA/k is a projective A-module of rank d. In an article in preparation we study how the operations in Proposition 3.2.6, the pre-HS-module structure on ΩA/k (see Proposition 3.1.7) and Proposition 3.1.2 give rise to a right HS-module structure on the dualizing module ωA/k =Ω d A/k.