Higher derivations of modules and the Hasse-Schmidt module
aa r X i v : . [ m a t h . A C ] J u l HIGHER DERIVATIONS OF MODULES AND THEHASSE-SCHMIDT MODULE
CHRISTOPHER CHIU AND LUIS NARV ´AEZ MACARRO
Abstract.
In this paper we revisit Ribenboim’s notion of higher derivationsof modules and relate it to the recent work of De Fernex and Docampo on thesheaf of differentials of the arc space. In particular, we derive their formulafor the K¨ahler differentials of the Hasse-Schmidt algebra as a consequence ofthe fact that the Hasse-Schmidt algebra functors commute.
Higher derivations of modules were introduced in [8] in analogy to higher deriva-tions of rings to provide a similar notion of a map carrying “infinitesimal informa-tion”. In particular, there too exists a universal object parametrizing such higherderivations of modules, which we call the
Hasse-Schmidt module . This construc-tion was implicitly considered in [2] when establishing a formula for the sheaf ofdifferentials of jet and arc spaces. More precisely, the main statement behind theaforementioned formula in the affine case is the following:
Theorem ([2], Theorem 5.3) . Let A be a k -algebra. For n ∈ N ∪ {∞} denote by HS nk ( A ) the n -th Hasse-Schmidt algebra of A . Then there exists an A -module Q n such that Ω HS nk ( A ) /k = Ω A/k ⊗ A Q n . As the n -th Hasse-Schmidt algebra HS nk ( A ) parametrizes infinitesimal data on A up to order n (i.e. n -jets on A ), this formula suggests that tangents (i.e. infini-tesimal data up to order 1) of n -jets on A can be recovered from some higher orderoperation on tangents on A . We aim to make this idea precise by observing thatΩ A/k ⊗ A Q n is just the n -th Hasse-Schmidt module of Ω A/k . This was essentiallyproven in [2] but missing in [8]; we will derive it here by considering Q n as a dualbimodule, respectively, as a colimit of duals in the case n = ∞ . The formula itselfcan thus be seen as a consequence of the (elementary) fact that the Hasse-Schmidtalgebra functors commute, see Lemma 1.9. The final ingredient is the followingtheorem, which connects the Hasse-Schmidt algebra to the Hasse-Schmidt moduleby means of the symmetric algebra. Theorem.
Let M be an A -module and denote by HS nA/k ( M ) the n -th Hasse-Schmidt module of M . Then Sym A n ( HS nA/k ( M )) ≃ HS nk (Sym A ( M )) . Mathematics Subject Classification.
Primary 13N05, 13N15; Secondary 14F10.
Key words and phrases.
Higher derivations of modules, Hasse-Schmidt derivations, Jet andArc spaces.The research of the first author was supported by project P-31338 of the Austrian ScienceFund (FWF). The research of the second author was partially supported by MTM2016-75027-Pand FEDER..
Similar to how Hasse-Schmidt algebras of finite order can be glued to give the jetspace X n of an algebraic variety X , gluing Hasse-Schmidt modules gives a sheaf ofmodules over X n . In particular, this construction yields a whole class of “universal”vector bundles on X n .The basic outline of this paper is as follows: in Section 1 we will collect well-known facts about higher derivations and extend them to graded higher derivations.In Section 2 we will briefly touch on the notion of dualizable bimodules. In Section 3we discuss higher derivations of modules and the Hasse-Schmidt module as well asits relationship to the Hasse-Schmidt algebra, including proving the aforementionedresult as Theorem 3.8. Finally, in Section 4 we will briefly talk about globalizingthe construction in Section 3 to obtain sheaves on jet and arc spaces.Throughout this paper, k will denote an arbitrary commutative ring. If R is anyother ring, we will write R [[ t ]] n := R [[ t ]] / ( t n +1 ) for n ∈ N and R [[ t ]] ∞ = R [[ t ]]. Thecategory of k -algebras will be denoted by Alg k ; furthermore, we write Alg k, gr forthe category of N -graded k -algebras (where we regard k with the trivial grading).A morphism in Alg k, gr will always be a graded map of degree 0.1. Higher derivations and the Hasse-Schmidt algebra
In this section we will briefly recall some well-established facts about higherderivations (also called Hasse-Schmidt derivations) and the universal object asso-ciated to them, which we call the Hasse-Schmidt algebra and denote by HS nk ( A ).Our main references here are [10] and [6], [7]. Furthermore, we will extend thesedefinitions to graded rings and show that the corresponding universal object canbe obtained by endowing HS nk ( A ) with an induced (natural) grading.1.1. Higher derivations. If A , C are k -algebras and n ∈ N ∪ {∞} , then a higherderivation D = ( D i ) ni =0 : A → C of order n is a collection of k -linear maps D i : A → C such that D is a map of k -algebras and the higher Leibniz rules are satisfied: D i ( ab ) = X k + l = i D k ( a ) D l ( b ) . We write D ∈ HS nk ( A, C ). There exist bijections(1) HS nk ( A, C ) ≃ Hom
Alg k ( A, C [[ t ]] n ) , which are natural in A and C . The image of D under this map will be denoted by γ D .If C has in addition an A -algebra structure, then we write HS nA/k ( A, C ) for thesubset of all higher derivations D such that D : A → C is the structure map a a · C . The natural bijections Eq. (1) restrict to(2) HS nA/k ( A, C ) ≃ Hom ◦ Alg k ( A, C [[ t ]] n ) , where the right-hand side denotes the subset of k -algebra maps γ : A → C [[ t ]] n such that γ modulo ( t ) equals the structure map A → C .For any k -algebra A the functor HS nk ( A, − ) is representable by a k -algebra HS nk ( A ), the Hasse-Schmidt algebra, which comes equipped with a universal higherderivation d A = ( d A,i ) : A → HS nk ( A ). The map A → HS nk ( A )[[ t ]] n correspondingto d A under Eq. (1) will be denoted by γ A . By definition for every k -algebra C there exists natural bijections(3) Hom Alg k ( HS nk ( A ) , C ) ≃ HS nk ( A, C ) ≃ Hom
Alg k ( A, C [[ t ]] n ) , IGHER DERIVATIONS OF MODULES AND THE HASSE-SCHMIDT MODULE 3 given by ϕ ∈ Hom
Alg k ( HS nk ( A ) , C ) e ϕ ◦ γ A ∈ Hom
Alg k ( A, C [[ t ]] n ) , where e ϕ : HS nk ( A )[[ t ]] n → C [[ t ]] n is the t -linear extension of ϕ . We write ϕ D for the map HS nk ( A ) → C corresponding to a higher derivation D ∈ HS nk ( A, C ): D i = ϕ D ◦ d A,i for all i . For convenience’s sake we will often write A n := HS nk ( A ).Similarly, for every A -algebra C we obtain bijections(4) Hom Alg A ( HS nk ( A ) , C ) ≃ HS nA/k ( A, C ) ≃ Hom ◦ Alg k ( A, C [[ t ]] n ) , which are natural in C . Remark . The assignment A HS nk ( A ) yields a functor Alg k → Alg k ; inparticular, any k -algebra map f : A → A ′ gives rise to a natural map f n : HS nk ( A ) → HS nk ( A ′ ) such that f n ◦ d A,i = d A ′ ,i for all i . Remark . For m > n there exist natural maps π ∗ m,n : HS nk ( A ) → HS mk ( A )obtained from the truncations C [[ t ]] m → C [[ t ]] n . We will refer to the maps π ∗ m,n as co-truncation maps . Remark . The co-truncation maps π ∗ m,n give rise to a directed system and wehave HS ∞ k ( A ) ≃ lim −→ n HS nk ( A ). Indeed, this follows directly from the universalproperty of the Hasse-Schmidt algebra, as we have natural bijectionsHom Alg k ( A, C [[ t ]]) ≃ Hom
Alg k ( A, lim ←− n C [[ t ]] n ) ≃ lim ←− n Hom
Alg k ( A, C [[ t ]] n ) . The Hasse-Schmidt algebra HS nk ( A ) can be constructed as a quotient of thepolynomial algebra A [ x ( i ) | x ∈ A, i = 1 , . . . , n ]by the ideal generated by( x + y ) ( i ) − x ( i ) − y ( i ) , x, y ∈ A (5) ( xy ) ( i ) − X k + l = i x ( k ) y ( l ) , x, y ∈ Aa ( i ) , a ∈ k, for i = 1 , . . . , m ; note that we identify x (0) with x ∈ A . See [10] for more details.In this presentation the universal higher derivation d nA is given by d A ( x ) = n X i =0 x ( i ) t i . Remark . There exists a natural N -grading of HS nk ( A ) given by deg( x ( i ) ) = i for x ∈ A , which we will refer to as the structural grading of HS nk ( A ). Indeed, noticethat the system of equations (5) is homogeneous with respect to deg( x ( i ) ) = i .Moreover, for any k -algebra map f : A → A ′ the natural map f n : HS nk ( A ) → HS nk ( A ′ ) from Remark 1.1 is graded. CHRISTOPHER CHIU AND LUIS NARV´AEZ MACARRO
Graded higher derivations.
Let us first consider the case n ∈ N , i.e.of higher derivations of finite order. Let k be a ring and A = L i ∈ N A i and C = L i ∈ N C i graded k -algebras (where we regarded k with the trivial grading;in particular the sets of homogeneous elements A i and C i are k -modules). We calla higher derivation D = ( D i ) i : A → C of order n graded if every component D i is graded (of degree 0), that is, we have D i ( A j ) ⊂ C j for all j ∈ N . The set of allsuch D will be denoted by HS nk, gr ( A, C ). Note that C [[ t ]] n is a free C -module ofrank n + 1 and thus carries a natural grading induced by C . It is then immediatethat the natural isomorphism Eq. (1) restricts toHS nk, gr ( A, C ) ≃ Hom
Alg k, gr ( A, C [[ t ]] n ) . We claim that there exists an N -grading on HS nk ( A ), different from the structuralgrading introduced in Remark 1.4, such that, for every N -graded k -algebra C , thenatural bijections Eq. (3) restrict to natural bijectionsHom Alg k, gr ( HS nk ( A ) , C ) ≃ HS nk, gr ( A, C ) ≃ Hom
Alg k, gr ( A, C [[ t ]] n ) . In particular γ A : A → HS nk ( A )[[ t ]] n will be map of graded k -algebras. We call thisgrading the induced grading of HS nk ( A ). To construct the induced grading, notethat, if A = L i ∈ N A i , then the presentation given by Eq. (5) can be refined suchthat A is given as the quotient of A [ x ( i ) | x ∈ A j , j ∈ N , i = 1 , . . . , n ]by the ideal generated by( x + y ) ( i ) − x ( i ) − y ( i ) , x, y ∈ A j (6) ( xy ) ( i ) − X k + l = i x ( k ) y ( l ) , x, y ∈ A j a ( i ) , a ∈ k, for i = 1 , . . . , m and j ∈ N . We define the induced grading by setting deg( x ( i ) ) := j for x ∈ A j . Note that the system Eq. (6) is homogeneous with respect to thisgrading, so it is well-defined. The k -module ( HS nk ( A )) i of elements of degree i isgenerated by the set of products { x ( j )1 · · · x ( j r ) r | x l ∈ A i l , i + . . . + i r = i } . Note that ( HS nk ( A )) = HS nk ( A ) and ( HS nk ( A )) i is a module over HS nk ( A ).Now to see the claim above, if ϕ D : HS nk ( A ) → C is graded, then ϕ ( x ( j ) ) ∈ C i for x ∈ A i . The map γ D corresponding to ϕ D : A → C [[ t ]] n under Eq. (3) is givenby x n X j =0 x ( j ) t j , and thus is graded. The other direction follows in analogy. Remark . Under the above hypotheses, for any D ∈ HS nk, gr ( A, C ), we consider D = ( D i ) i , with D i : A → C the degree 0 part of D i : A → C . It is clear that D ∈ HS nk ( A , C ).In fact, taking the Hasse-Schmidt algebra gives rise to a functor Alg k, gr → Alg k, gr : IGHER DERIVATIONS OF MODULES AND THE HASSE-SCHMIDT MODULE 5
Theorem 1.6.
Let n ∈ N and A = L i A i be a graded k -algebra (where we consider k with the trivial grading). Then the assignment A HS nk ( A ) yields a functor Alg k, gr → Alg k, gr satisfying the following: for every N -graded k -algebra C thereexist bijections (7) Hom Alg k, gr ( HS nk ( A ) , C ) ≃ Hom
Alg k, gr ( A, C [[ t ]] n ) , natural in A and C .Remark . Similarly, if C is a graded A -algebra and one defines HS nA/k, gr ( A, C )to be the subset of graded higher derivations D : A → C such that D : A → C isthe structure map, then there exist natural bijections(8) Hom Alg A, gr ( HS nk ( A ) , C ) ≃ HS nA/k, gr ( A, C ) ≃ Hom ◦ Alg k, gr ( A, C [[ t ]] n ) . Note that HS nk ( A ) is a graded A -algebra by construction.Let us now consider the case n = ∞ . Clearly the co-truncation maps π ∗ m,n : HS nk ( A ) → HS mk ( A ), m > n preserve the gradings on both sides. By Remark 1.3we obtain a grading on HS ∞ k ( A ) coming from its direct limit structure. How-ever, note that there exists no result which is directly analogous to Theorem 1.6,since, for C a graded k -algebra, the power series ring C [[ t ]] does no longer carrya grading compatible with that of C itself. In fact, one would need to interpretHom Alg k, gr ( A, C [[ t ]]) as the set of all k -algebra maps ϕ : A → C [[ t ]] such that ϕ ( A i ) ⊂ C i [[ t ]] for all i ≥ Remark . The structural grading of HS nk ( A ) (see Remark 1.4) together withits induced grading yields an N -bigrading of HS nk ( A ). Indeed, observe that in thepresentation given by Eq. (6) this bigrading is defined via deg( x ( j ) ) = ( i, j ) for x ∈ A i . Moreover, for f : A → A ′ a map between graded k -algebras, the inducedmap f n : HS nk ( A ) → HS nk ( A ′ ) clearly respects the bigrading. Hence we can view A HS nk ( A ) as a functor from N -graded k -algebras to N -bigraded k -algebras.The following lemma states that the functors HS nk ( − ) commute with each other.Moreover, the composition of two such functors can also be seen as a universalobject for 2-variate higher derivations; note that for p -variate higher derivationsthe corresponding Hasse-Schmidt algebra is N p -graded. For more details on the p -variate case and the proof of the lemma see [5, Corollary 2.3.12]. Lemma 1.9.
For all n, m ∈ N ∪ {∞} there are natural isomorphisms of bigraded HS nk ( A ) -algebras (9) HS nk ( HS mk ( A )) ≃ HS ( n,m ) k ( A ) ≃ HS mk ( HS nk ( A )) ∗ , where, for any bigraded algebra B , we denote by B ∗ the bigraded algebra obtainedfrom interchanging the order of gradings on B . Moreover, if we call: • d mA,i : A → HS mk ( A ) , i ≤ m , the components of the universal higher deriva-tion d mA ∈ HS mk ( A, HS mk ( A )) , and • d ( n,m ) A, ( i,j ) : A → HS ( n,m ) k ( A ) , ( i, j ) ≤ ( n, m ) , the components of the universalhigher derivation d ( n,m ) A ∈ HS ( n,m ) k ( A, HS ( n,m ) k ( A )) ,then the isomorphisms in (9) are determined by d n HS mk ( A ) ,i ◦ d mA,j ≡ d ( n,m ) A, ( i,j ) ≡ d m HS nk ( A ) ,j ◦ d nA,i . CHRISTOPHER CHIU AND LUIS NARV´AEZ MACARRO Dualizable bimodules and limits
Let A be a ring and M a (left) A -module, then the (right) A -module M ∨ :=Hom A ( M, A ) is commonly called the dual module of M . In general, the dual ofa module is not well-behaved, as for example M might not be reflexive , i.e. thenatural map M → ( M ∨ ) ∨ might not be an isomorphism. For a module M to be dualizable we will thus require it to have a dual object in the (stronger) categoricalsense of [4]. In this section we will recall some well-known and elementary factsabout dualizable (bi)modules, which will be used to establish the existence of theHasse-Schmidt module in Section 3.Let us start by fixing some notation. If M and N are both right A -modules, wedenote the set of right A -homomorphisms M → N by Hom ( − ,A ) ( M, N ). Similarly,if M and N are left B -modules, we write Hom ( B, − ) ( M, N ) for the set of left B -homomorphisms M → N . This choice of notation will allow us to be precise aboutthe type of structure considered when dealing with bimodules.Let A , B be (not necessarily) commutative rings and P a ( B, A )-bimodule. Wesay that P is left dualizable if there exists an ( A, B )-bimodule Q and bimodulehomomorphisms η : A → Q ⊗ B P, θ : P ⊗ A Q → B such that the diagrams A ⊗ A Q η ⊗ id Q / / ≃ (cid:15) (cid:15) ( Q ⊗ B P ) ⊗ A Q ≃ (cid:15) (cid:15) Q ⊗ B B Q ⊗ B ( P ⊗ A Q ) id Q ⊗ θ o o and P ⊗ A A id P ⊗ η / / ≃ (cid:15) (cid:15) P ⊗ A ( Q ⊗ B P ) ≃ (cid:15) (cid:15) B ⊗ B P ( P ⊗ A Q ) ⊗ B P, θ ⊗ id P o o commute, with the vertical arrows being the associators and unitors. The map θ is called the evaluation and the map η the coevaluation . Furthermore, the ( A, B )-bimodule Q is unique up to isomorphism and will be referred to as the left dual of P . If there is no ambiguity we will write P ∨ = Q and Q ∨ = P . Note that P being left dualizable is equivalent to saying that the functor − ⊗ B P from right B -modules to right A -modules is right adjoint to − ⊗ A Q ; in particular, by tensor-homadjunction we have a natural isomorphism P ≃ Hom ( − ,B ) ( Q, B ) . We will now recall this classical result, which characterizes dualizable bimodulesas those who are finite projective.
Theorem 2.1.
Let P be a ( B, A ) -bimodule. Then P is left dualizable if and only ifit is finitely generated projective as a left B -module. In particular, we have bijections Hom ( − ,A ) ( M, N ⊗ B P ) ≃ Hom ( − ,B ) ( M ⊗ A P ∨ , N ) , which are natural in M and N . IGHER DERIVATIONS OF MODULES AND THE HASSE-SCHMIDT MODULE 7
Remark . If a functor G is left adjoint to − ⊗ B P , then G is automatically ofthe form − ⊗ A Q for some ( A, B )-bimodule Q (see for example [9, Theorem 3.60]).Thus P being left dualizable is equivalent to − ⊗ B P being right adjoint. Remark . If C is a monoidal category (for example, the category of modulesover a commutative ring R ), then a dualizable object X of C is an adjoint tothe morphism corresponding to X in the delooping 2-category B C . Similarly, abimodule P is left dualizable if the corresponding 1-morphism in the bicategory ofalgebras, bimodules and intertwiners (as introduced in [1]) is a right adjoint.For the purposes of the next section we need to consider the case where P is acofiltered limit of ( B, A )-bimodules P i , which are finitely generated projective asleft B -modules. In general, P will not be finite projective over B itself, so theredoes not exist a (left) dual of P in the sense discussed above. However, the functorlim ←− i ( − ⊗ B P i ) does have a left adjoint, which just follows from the fact that rightadjoints commute with limits. We summarize this fact in the following corollary: Corollary 2.4.
Let P = lim ←− i P i be a cofiltered limit of ( B, A ) -bimodules P i whichare finitely generated projective over B . Then the left duals P ∨ i form a filteredsystem. Furthermore, we have bijections Hom ( − ,A ) ( M, lim ←− i N ⊗ B P i ) ≃ Hom ( − ,B ) ( M ⊗ A lim −→ i P ∨ i , N ) , which are natural in M and N . Higher derivations of modules and the Hasse-Schmidt module
Higher derivations of modules were first introduced in [8], where they were de-fined with respect to a given higher derivation of rings. The existence of a universalobject, which we call Hasse-Schmidt module, parametrizing such higher derivationswas already established there; however, it also appeared implicitly in more detailin [2]. In this section our aim is to provide a top-down view of the constructionof the Hasse-Schmidt module and how it relates to the Hasse-Schmidt algebra bymeans of Theorem 3.8, which is the main result of this section.Throughout this section we will write A n := HS nk ( A ) for the Hasse-Schmidtalgebra of a k -algebra A . All rings considered here are assumed to be commutative;in particular, we do not have to distinguish between left and right actions.Let now A and C be k -algebras and D = ( D i ) ni =0 : A → C be a higher derivationof length n over k . A higher derivation over D , ∆ = (∆ i ) ni =0 : M → N , where M ∈ Mod A , N ∈ Mod C , is a collection of k -linear maps ∆ i : M → N satisfying∆ i ( a · m ) = X k + l = i D k ( a ) · ∆ l ( m ) , a ∈ A, m ∈ M. The set of all such maps is denoted by HS nD ( M, N ). We have isomorphismsHS nD ( M, N ) ≃ Hom A ( M, N [[ t ]] n ) , which are natural in M and N . Note that the A -module structure on N [[ t ]] n abovecomes from scalar restriction through γ D : A → C [[ t ]] n , with D i = ϕ D ◦ d nA,i . If weconsider N as an A n -module via restriction by ϕ D : A n → C , then we see thatHS nD ( M, N ) ≃ HS nd nA ( M, N ) . CHRISTOPHER CHIU AND LUIS NARV´AEZ MACARRO
Let us first treat the case n ∈ N . We consider A n [[ t ]] n as a ( A n , A )-bimodule, wherethe left action is just the usual multiplication and the right action is induced by γ A . Then we have an isomorphism of right A -modules N [[ t ]] n ≃ N ⊗ A n A n [[ t ]] n . Note that A n [[ t ]] n is free of rank n + 1 over A n . By Theorem 2.1 we getHS nd nA ( M, N ) ≃ Hom A ( M, N ⊗ A n A n [[ t ]] n ) ≃ Hom A n ( M ⊗ A ( A n [[ t ]] n ) ∨ , N ) , where ( A n [[ t ]] n ) ∨ ≃ Hom ( − ,A n ) ( A n [[ t ]] n , A n ) is the left dual of A n [[ t ]] n . Using thefact that extension of scalars is left adjoint to restriction, we obtainHS nD ( M, N ) ≃ Hom C ( M ⊗ A ( A n [[ t ]] n ) ∨ ⊗ A n C, N )Now, in the case n = ∞ , we have that A ∞ [[ t ]] = lim ←− n A ∞ [[ t ]] n is the limit of aprojective system of ( A ∞ , A )-bimodules which are finite free over A ∞ . Arguing asbefore and applying Corollary 2.4 yieldsHS ∞ D ( M, N ) ≃ Hom C ( M ⊗ A lim −→ n ( A ∞ [[ t ]] n ) ∨ ⊗ A ∞ C, N ) . Thus we have proven the following result:
Theorem 3.1 ([8, § . Let A and C be k -algebras and D : A → C be a higherderivation of order n ∈ N ∪ {∞} . Then, for any A -module M , the functor HS nD ( M, − ) : Mod C → Set is representable by a module HS nA/k ( M ) ⊗ A n C , where we call the A n -module HS nA/k ( M ) the n -th Hasse-Schmidt module of M . Moreover, we have HS nA/k ( M ) ≃ M ⊗ A ( A n [[ t ]] n ) ∨ for n ∈ N and HS ∞ A/k ( M ) ≃ M ⊗ A lim −→ n ( A ∞ [[ t ]] n ) ∨ , where ( A ∞ [[ t ]] n ) ∨ is the (left) dual of the finite free A ∞ -module A ∞ [[ t ]] n .Remark . The module HS nA/k ( M ) comes attached with a universal higher deriva-tion ∆ M : M → HS nA/k ( M ) over d nA : A → A n . We want to give an explicit descrip-tion of ∆ M in the case where M = A , i.e. HS nA/k ( M ) = ( A n [[ t ]] n ) ∨ . To that avail,if n ∈ N , observe that the A -module map α D : A → HS nA/k ( M )[[ t ]] n correspondingto ∆ M is just the coevaluation. Thus, if we take the standard basis t i , i = 0 , . . . , n for A n [[ t ]] n over A n and let t [ i ] , i = 0 , . . . , n , denote the dual basis, the map α D isgiven by(10) 1 A n X i =0 t [ i ] t i . Note that if n = ∞ then the images of t [ i ] , i ∈ N , in lim −→ n ( A ∞ [[ t ]] n ) ∨ form a basisover A ∞ . Hence the universal higher derivation ∆ A : A → HS nA/k ( M ) is given by(11) (∆ M ) i (1 A ) = t [ i ] . IGHER DERIVATIONS OF MODULES AND THE HASSE-SCHMIDT MODULE 9
Remark . For n ∈ N , since A n [[ t ]] n is finite free over A n , we have that HS nA/k ( M ) ≃ M ⊗ A A n [[ t ]] n as A n -modules. Indeed, choosing the same basis as in Remark 3.2, this isomorphismis given identifying A n [[ t ]] n with its dual via t i t [ i ] . Remark . As A n [[ t ]] n is free for n ∈ N ∪ {∞} , if M is projective respectively freethen so is HS nA/k ( M ). Remark . Alternatively one could consider triples (
C, D, N ), with C a k -algebra, D ∈ HS nk ( A, C ) and N a C -modules, a suitable notion of morphism between suchtriples and the functor which associates to each such triple the set HS nD ( M, N ).Then the argument given before shows that this functor is represented by the triple( A n , d nA , HS nA/k ( M )). Remark . We have natural isomorphisms HS ∞ A/k ( M ) ≃ lim −→ n HS nA/k ( M ). Indeed,this follows just as in Remark 1.3 fromHom A ( M, N [[ t ]]) ≃ Hom A ( M, lim ←− n N [[ t ]] n ) ≃ lim ←− n Hom A ( M, N [[ t ]] n ) , where N is an A n -module.We will refer to [8] for a basic introduction to the theory of higher derivationsof modules. As one fact not included there, let us mention that the Hasse-Schmidtmodule behaves well under base change. Lemma 3.7.
Let A → A ′ be a k -algebra map and M an A -module. Then we have HS nA ′ /k ( M ⊗ A A ′ ) ≃ HS nA/k ( M ) ⊗ A n A ′ n . Proof.
Follows immediately from the description in Theorem 3.1. (cid:3)
The following theorem is the main result of this section and relates the Hasse-Schmidt module to the Hasse-Schmidt algebra by means of the symmetric algebra.
Theorem 3.8.
For each n ∈ N ∪ {∞} there are isomorphisms of N -graded A n -algebras Sym A n ( HS nA/k ( M )) ≃ HS nk (Sym A ( M )) , which are natural in M . Note that on the left side we are considering the naturalgrading on the symmetric algebra, while on the right hand side we are taking theinduced grading of HS nk (Sym A ( M )) .Remark . By considering HS nA/k ( M ) as a graded A n -module similar to Re-mark 1.4 the isomorphism in Theorem 3.8 extends to one of N -bigraded algebras. Proof.
Consider first the case n ∈ N and let B = L i ≥ B i be an N -graded A n -algebra, where we consider A n with the trivial grading. Then each B i is in particularan A n -module. Recall that B [[ t ]] n is a free B -module of rank n + 1 and thus has anatural N -grading given by ( B [[ t ]] n ) i := B i [[ t ]] n . We obtain natural bijectionsHom Alg gr An (Sym A n ( HS nA/k ( M )) , B ) ≃ Hom
Mod An ( HS nA/k ( M ) , B ) ≃≃ Hom
Mod A ( M, B [[ t ]] n ) ≃ Hom
Alg gr A (Sym A ( M ) , B [[ t ]] n ) . Here again we consider A with the trivial grading. If ρ : A n → B is the map definingthe A n -algebra structure on B , then the A -algebra (resp. A -module) structure on B [[ t ]] n (resp. B [[ t ]] n ) is given by ˜ ρ ◦ d nA , where ˜ ρ : A n [[ t ]] → B [[ t ]] is obtained from ρ by t -linear extension. We claim thatHom Alg gr A (Sym A ( M ) , B [[ t ]] n ) ≃ Hom
Alg gr An ( HS nk (Sym A ( M )) , B ) . Indeed, an element α of the left-hand side is given by a triangleSym A ( M ) α / / B [[ t ]] n A O O ˜ ρ ◦ d nA rrrrrrrrrrr where α = ( α i ) i ∈ N is graded of degree 0. By Theorem 1.6 we obtain a triangle of k -algebra maps HS nk (Sym A ( M )) α ∗ / / B HS nk ( A ) O O ρ qqqqqqqqqqqq , where α ∗ and ρ is graded. Conversely, every such triangle is obtained by one of theform above. This proves the claim for n ∈ N .We are thus left with the case n = ∞ . Taking colimits of the isomorphism forfinite n we obtainlim −→ n HS nk (Sym A ( M )) ≃ lim −→ n Sym A n ( HS nA/k ( M )) ≃ Sym A ∞ (lim −→ n HS nA/k ( M )) . As we have both HS ∞ k ( B ) = lim −→ n HS nk ( B ) and HS ∞ A/k ( M ) = lim −→ n HS nA/k ( M ) theresult follows. (cid:3) Remark . Applying Theorem 3.8 in the case M = A and n ∈ N yields thefollowing:(12) HS nA/k ( A ) ≃ ( A n [[ t ]] n ) ∨ ≃ [ HS nk (Sym A ( A ))] . To make this isomorphism explicit, let e := 1 A . Then HS nk (Sym A ( A )) ≃ A n [ e ( i ) | i = 0 , . . . , n ] . Thus the A n -submodule of elements of degree 1 is generated by e ( i ) , i = 0 , . . . , n .Using the same basis as in Remark 3.2 we see that the isomorphism in Eq. (12) isgiven by t [ i ] e ( i ) . Now, since the A -action on HS nA/k ( A ) ≃ ( A n [[ t ]] n ) ∨ is given by precomposition, weobtain an induced A -action on [ HS nk (Sym A ( A ))] given by a · e ( i ) = i X j =0 a ( i − j ) e ( j ) . Compare this with the construction of P n given in [2, Section 4]. It is not clearhow to obtain this A -action naturally by just considering HS nk (Sym A ( A )).Using Theorem 3.8 we will recover the formula in [2, Theorem 5.3] as a directconsequence of the fact that the Hasse-Schmidt algebra functors commute (seeLemma 1.9). IGHER DERIVATIONS OF MODULES AND THE HASSE-SCHMIDT MODULE 11
Theorem 3.11.
For all n ∈ N ∪ {∞} there is a natural isomorphism Ω A n /k ≃ HS nA/k (Ω A/k ) ≃ Ω A/k ⊗ A HS nA/k ( A ) . of A n -modules. Additionally, if n ∈ N , we have Ω A n /k ≃ Ω A/k ⊗ A A n [[ t ]] n . Proof.
Note that Sym B (Ω B/k ) ≃ HS k ( B ) for any k -algebra B . Thus Lemma 1.9implies that we have isomorphisms of graded algebras HS nk (Sym A (Ω A/k )) ≃ HS nk ( HS k ( A )) ≃ HS k ( HS nk ( A )) ≃ Sym A n (Ω A n /k ) . Thus the statement follows from Theorem 3.8 for M = Ω A/k and taking the ho-mogeneous part of degree 1 on each side. The second assertion just follows fromRemark 3.3. (cid:3)
Remark . A similar argument allows us to consider (usual) k -derivations d : A → M with M an A -module as higher derivations in the above sense. To thatend, note thatDer k ( A, M ) ≃ Hom A (Ω A/k , M ) ≃ Hom
Alg A , gr (Sym A (Ω A/k ) , Sym A ( M )) . Since Sym A (Ω A/k ) ≃ HS k ( A ) as graded k -algebras, by Remark 1.7 we obtainDer k ( A, M ) ≃ Hom ◦ Alg k, gr ( A, Sym A ( M )[[ t ]] ) . Since every such map factors as A → HS k ( A )[[ t ]] → Sym A ( M )[[ t ]] we haveHom ◦ Alg k, gr ( A, Sym A ( M )[[ t ]] ) ֒ → Hom
Mod A ( A, Sym A ( M )[[ t ]] ) , where the A -action on the Sym A ( M )[[ t ]] is induced by d A . Thus we obtain ourclaim that Der k ( A, M ) ֒ → HS d A ( A, Sym A ( M )) . Remark . In a similar vein to Theorem 3.8, higher derivations of modules canbe seen as graded higher derivations of symmetric algebras. Namely, if A and C are k -algebras, M ∈ Mod A , N ∈ Mod C , D ∈ HS nk ( A, C ) and n ∈ N , we have:HS nD ( M, N ) ≃ Hom A ( M, N [[ t ]] n ) ≃ Hom
Alg A , gr (Sym A ( M ) , Sym C [[ t ]] n ( N [[ t ]] n )) ≃ Hom
Alg A , gr (Sym A ( M ) , Sym C ( N )[[ t ]] n ) ⊂ Hom
Alg k , gr (Sym A ( M ) , Sym C ( N )[[ t ]] n ) ≃ HS nk, gr (Sym A ( M ) , Sym C ( N )) , and so any ∆ ∈ HS nD ( M, N ) gives rise to a well-defined graded higher derivation ∆ ∈ HS nk, gr (Sym A ( M ) , Sym C ( N )). Notice that for the degree 0 part we have ∆ = D . Actually, the above procedure gives natural bijections { ( D, ∆) | D ∈ HS nk ( A, C ) , ∆ ∈ HS nD ( M, N ) } ≃ HS nk, gr (Sym A ( M ) , Sym C ( N )) . Since these bijections are compatible with truncations, they also hold for n = ∞ . Universal vector bundles on jet spaces
Let X be a scheme of finite type over a field k . For n ∈ N the n -th jet space X n of X is the k -scheme representing the functor Y Hom k ( Y × k Spec( k [[ t ]] n ) , X ) . The natural maps k [[ t ]] n +1 → k [[ t ]] n give rise to an inverse system X n +1 → X n .The arc space X ∞ of X is defined to be the limit X ∞ = lim ←− n X n , which is againa scheme. If X = Spec( A ) is affine, then so is X n for all n ∈ N ∪ {∞} . In fact,we have X n = Spec( A n ), with A n = HS nk ( A ) the n -th Hasse-Schmidt algebra. Forgeneral X the scheme X n is constructed by considering a covering of X by affineopens Spec( A ) and glueing the corresponding Hasse-Schmidt algebras A n to obtaina sheaf of O X -algebras, whose relative spectrum gives X n .Let n ∈ N . Then the n -th jet space X n comes equipped with a universal family γ : U n := X n × k Spec( k [[ t ]] n ) → X . Write ρ for the canonical projection U n → X n .If X = Spec( A ) is affine, then γ is the map induced by the homomorphism γ A : A → HS nk ( A )[[ t ]] n and ρ is induced by the inclusion HS nk ( A ) → HS nk ( A )[[ t ]] n .Let F be a coherent sheaf of O X -modules. For n ∈ N ∪ {∞} we will construct acoherent sheaf of O X n -modules F n by glueing the Hasse-Schmidt modules of F ( U )for U ⊂ X affine. As in the construction of X n , to be able to glue we use that S − HS nA/k ( M ) ≃ HS nS − A/k ( S − M ) for any multiplicative subset S ⊂ A , whichin turn follows from Lemma 3.7. Since for any W ⊂ X open affine the scheme W n ⊂ X n is open affine we obtain a coherent sheaf F on X n .Alternatively, as done in [2], for n ∈ N we may construct from Q n = HS nA/k ( A ) =( A n [[ t ]] n ) ∨ a sheaf Q n on U n = X n × k Spec( k [[ t ]] n ). It is then easy to check that F n verifies the following universal property: Theorem 4.1.
Let n ∈ N , X a scheme of finite type over k and X n the n -th jetspace of X . For any F ∈
Mod O X the functor G ∈
Mod O Xn Hom O X ( F , γ ∗ ρ ∗ G ) is represented by F n = ρ ∗ ( γ ∗ F ⊗ Q n ) . If F ∈
Mod O X is locally free, then so is F n ∈ Mod O Xn by Remark 3.4. In thiscase, the global version of Theorem 3.8 says the following: Lemma 4.2.
Let X be a scheme of finite type over a field k and n ∈ N . Thenthe assignment F 7→ F n gives a functor Coh( X ) → Coh( X n ) which preserves theproperty of being locally free. Moreover, if E → X is the vector bundle correspondingto a locally free F ∈
Coh( X ) , then its jet space E n → X n is the vector bundlecorresponding to F n ∈ Coh( X n ) .Remark . A similar result holds true for the arc space X ∞ of X , only in thiscase the universal family attached to X ∞ is of the form d U ∞ → X , where d U ∞ isthe formal completion of the scheme X ∞ × k Spec( k [[ t ]]) with respect to the closedsubscheme X ∞ × { } .The construction of the sheaf F n was used in [2] to provide a global variant ofTheorem 3.11, as well as in [3] to show how the jet space behaves under Nash blow-up. Our hope is that other, similarly useful, formulas can be obtained by studyinguniversal vector bundles on jet spaces, in particular in constructing compactifica-tions. IGHER DERIVATIONS OF MODULES AND THE HASSE-SCHMIDT MODULE 13
We finish with computing explicitly F n for an invertible sheaf F on P , showingthat even in this very easy case the resulting universal vector bundle has potentiallyinteresting structure. Example . Let X = P and and L an invertible sheaf on P , isomorphic to O ( d )for some d ∈ Z . Let U and U be the standard affine opens of P . We write U i = Spec( k [ t i ]), then L | U i is generated by e i and the transition map L | U i ∩ U j →L | U j ∩ U i is given by e t d e . For n ∈ N we want to describe explicitly thesheaf L n on ( P ) n . To that avail, note that ( P ) n has an affine covering given by( U i ) n = Spec( k [ t (0) i , . . . , t ( n ) i ]). The restriction L n | ( U i ) n is freely generated by thesections e ( j ) i for j = 0 , . . . , n . By functoriality of the Hasse-Schmidt module thetransition map is given by e ( j )0 ∆ j (cid:16) ( t (0)1 ) d e ( j )1 (cid:17) , where ∆ is the universal higher derivation associated to L n | ( U ) n . In particular,for d = 1, it is easy to see that the space of global sections of O (1) n is generatedby e ( j ) i , where i = 0 , j = 0 , . . . , n . This gives a notion of “coordinates” for( P ) n . The same obviously holds for any projective space P m , m ≥ References [1] J. B´enabou. Introduction to bicategories.
Reports of the Midwest Category Seminar, Lect.Notes in Math. 47 , Springer (1967), 1–77.[2] T. de Fernex, R. Docampo. Differentials on the arc space.
Duke Math. J. arXiv:1703.07505 ).[3] T. de Fernex, R. Docampo. Nash blow-ups of jet schemes.
Ann. Inst. Fourier
69, no. 6 (2019),2577–2588. ( arXiv:1712.00911 ).[4] A. Dold, D. Puppe. Duality, trace and transfer.
Proceedings of the International Conferenceon Geometric Topology (Warsaw, 1978) (1980), 81–102.[5] L. Narv´aez Macarro. A mini-course on Hasse-Schmidt derivations. Lecture notes, 2018.[6] P. Ribenboim. Higher derivations of rings, I.
Revue Roumaine de Math. Pures et Appl.
Revue Roumaine de Math. Pures et Appl.
Portugal. Math.
39, no. 1-4 (1980), 381–397.[9] C. Schommer-Pries. The Classification of Two-Dimensional Extended Topological Field The-ories.
PhD Dissertation 2009 . ( arXiv:1112.1000 ).[10] P. Vojta. Jets via Hasse-Schmidt Derivations.
Diophantine Geometry, Proceedings, Edizionidella Normale, Pisa (2007), 335–361. ( arXiv:0407113 ).(C. Chiu)
Fakult¨at f¨ur Mathematik, Universit¨at Wien, Oskar-Morgenstern-Platz 1,A-1090 Wien ( ¨Osterreich)
E-mail address : [email protected] (L. Narv´aez Macarro) Departamento de ´Algebra & Instituto de Matem´aticas (IMUS),Facultad de Matem´aticas, Universidad de Sevilla, Calle Tarfia s/n, E-41012 Sevilla(Espa˜na)
E-mail address ::