Classification of Lie algebras of differential operators

TL;DR

This paper classifies D-Lie algebras with projective canonical quotients, introduces a functor for their classification, and explores connections and Chow-operators relating to algebraic cycles.

Contribution

It introduces a classification framework for D-Lie algebras with projective quotients and relates connections on these algebras to algebraic cycles and correspondences.

Findings

Classified D-Lie algebras with projective canonical quotients.

Established a functor $F_f(-)$ for classification based on 2-cocycles.

Connected $ ilde{L}$-connections to algebraic cycles via Chow-operators.

Abstract

In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of $\tilde{L}$-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient $L$ of a D-Lie algebra $\tilde{L}$ and use this to classify D-Lie algebras where $L$ is projective as $A$-module. We define for any 2-cocycle $f\in \operatorname{Z}^2(\operatorname{Der}_k(A),A)$ a functor $F_{f}(-)$ from the category of $A/k$-Lie-Rinehart algebras to the category of D-Lie algebras and classify D-Lie algebras with projective canoncial quotient using the functor $F_{f}(-)$. We prove a similar classification for non-abelian extensions of D-Lie algebras. We classify $\tilde{L}$-connections in the case when the canonical quotient $L$ of $\tilde{L}$ is projective as $A$-module. Any $\tilde{L}$-connection is determined by a 2-cocycle $f\in \operatorname{Z}^2(\operatorname{Der}_k(A),A)$ and an $L$-connection $(E,\nabla)$. We introduce the correspondence and Chow-operator of an $\tilde{L}$-connection. The aim of this construction is to relate connections on D-Lie algebras to algebraic cycles an the category of correspondences. The Chow-operator cannot be defined for an ordinary connection on an $A/k$-Lie-Rinehart algebra. It depends in a non-trivial way on the right $A$-module structure on $\tilde{L}$ and the canonical quotient $A/k$-Lie-Rinehart algebra $L$ has no such structure.

Equations

$$[u,cv]=c[u,v]+\tilde{\pi }(u)(c)v$$

$$uc=cu+\tilde{\pi }(u)(c)D$$

$$U^{\otimes }:\underline{D-Lie}\to \underline{Rings}$$

$$U^{\rho }:\underline{D-Lie}\to \underline{Rings}.$$

$$F_1:\underline{\mathrm{Mod}}(\tilde{L},Id)\cong \underline{\mathrm{Mod}}(U^{\otimes }(\tilde{L}))$$

$$F_2:\underline{\mathrm{Conn}}(\tilde{L},Id)\cong \underline{\mathrm{Mod}}(U^{\rho }(\tilde{L}))$$

$${\psi }_f:\underline{\mathrm{Conn}}(L)\cong \underline{\mathrm{Mod}}(U^{\otimes }(L(f^{\alpha }))$$

$${\mathrm{Ext}}_{U^{\otimes }(L(f^{\alpha }))}^i({\psi }_f(V),{\psi }_f(W))\cong {\mathrm{Ext}}_{\underline{\mathrm{Conn}}(L)}^i(V,W).$$

$$U_J^{\otimes }(\tilde{L})\cong U$$

$$\underline{\mathrm{Mod}}(U)\cong \underline{\mathrm{Mod}}(\tilde{L},Id,J)$$

$$\rho :{\mathrm{Sym}}_A^{\ast }({\tilde{L}}^{\ast })\to Gr(U^{\otimes }(\tilde{L})).$$

$$0\to {\mathrm{End}}_A(E)\to \mathrm{End}(\tilde{L},E)\to \tilde{L}\to 0$$

$$I(\rho ,{\sigma }_1,{\sigma }_2):{\mathrm{CH}}^{\ast }(X)\to {\mathrm{CH}}^{\ast }(X)$$

$$C_f:\underline{\mathrm{Conn}}(L,\mathrm{End})\to \underline{\mathrm{Conn}}(L(f^{\alpha }))$$

$$R_{{\rho }_{\nabla }}(u,v)(e)=R_{\nabla \circ i}(x,y)(e)-f^{\alpha }(u,v)e.$$

$$c(a,x):=(ca,cx)$$

$$(a,x)c:=(ac+x(c),cx)$$

$$\varphi :{\mathrm{D}}^1(A,g)\to {\mathrm{D}}^1(A,f)$$

$${\pi }_f\circ \varphi ={\pi }_g\text{ and }\varphi (z_g)=z_f.$$

$$\varphi (a,x)=\varphi (a,0)+\varphi (0,x)=a{z}_f+({\varphi }_1(x),x)=(a+{\varphi }_1(x),x)$$

$$[(a,x),(b,y)]:=(\alpha (x)(b)-\alpha (y)(a)+g^{\alpha }(x,y),[x,y])$$

$${\pi }_g:L(g^{\alpha })\to {\mathrm{Der}}_k(A)$$

$${\pi }_g(a,x):=\alpha (x).$$

$${\alpha }^{\ast }:H^2({\mathrm{Der}}_k(A),A)\to H^2(L,A).$$

$$\overline{g^{\alpha }}=\overline{f^{\alpha }}\in {\alpha }^{\ast }{H}^2({\mathrm{Der}}_k(A),A)\subseteq H^2(L,A).$$

$${\alpha }_g(a,x)=(a+{\alpha }_1(x),\alpha (x))$$

$${\alpha }_g:L(g^{\alpha })\to {\mathrm{D}}^1(A,f)$$

$${\alpha }_g(a,x)=(a+{\alpha }_1(x),\alpha (x)).$$

$$\varphi (a,x)=(a+{\varphi }_1(x),{\varphi }_2(x))$$

$$[u,cv]=c[u,v]+\pi (u)(c)v$$

$$uc=cu+\pi (u)(c)D.$$

Full text

1. 1 Introduction
2. 2 Classification of morphisms of $\operatorname{D}$-Lie algebras
3. 3 Classification of $\operatorname{D}$-Lie algebras with projective canonical quotient
4. 4 Classification of connections on $\operatorname{D}$-Lie algebras with projective canonical quotient

1. Introduction

A $\operatorname{D}$-Lie algebra $\tilde{L}$ is a refinement of the notion of an $A/k$-Lie-Rinehart algebra: It is a $k$-Lie algebra and left $\operatorname{P}^{1}_{A/k}$-module, where $\operatorname{P}^{1}_{A/k}$ is the first order module of principal parts of $A/k$. The Lie-product on $\tilde{L}$ satisfies the anchor condition

[TABLE]

for all $u,v\in\tilde{L}$ and $c\in A$. Here $\tilde{\pi}:\tilde{L}\rightarrow\operatorname{Der}_{k}(A)$ is a map of $k$-Lie algebras and left $\operatorname{P}$-modules. Hence if we view $\tilde{L}$ as a left $A$-module and $k$-Lie algebra, it follows the pair $(\tilde{L},\tilde{\pi})$ is an $A/k$-Lie-Rinehart algebra. There is a canonical central element $D\in\tilde{L}$ with the property that

[TABLE]

holds for $u\in\tilde{L}$ and $c\in A$. We may define the notion of a connection $\rho:\tilde{L}\rightarrow\operatorname{Diff}^{1}(E)$ where $E$ is a left $A$-module. In the papers [8] and [9] general properties of the category of $\operatorname{D}$-Lie algebras are introduced and studied: Universal enveloping algebras of $\operatorname{D}$-Lie algebras, cohomology and homology of connections on $\operatorname{D}$-Lie algebras. Let $\underline{\operatorname{D-Lie}}$ denote the category of $\operatorname{D}$-Lie algebras and morphisms. In [8] Theorem 3.5 and 3.18 we prove the following result:

Theorem 1.1**.**

There are covariant functors

[TABLE]

with the following property: For any $\operatorname{D}$-Lie algebra $\tilde{L}$ there are exact equivalences of categories

[TABLE]

with the property that $F_{1}$ and $F_{2}$ preserves injective and projective objects.

We used the associative unital rings $\operatorname{U}^{\otimes}(\tilde{L})$ and $\operatorname{U}^{\rho}(\tilde{L})$ to define the cohomology and homology of an arbitrary $\tilde{L}$-connection $(E,\rho)$. By Lemma 3.19 in [8] there is for any 2-cocycle $f\in\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$ an exact equivalence of categories

[TABLE]

preserving injective and projective objetcs. Here $L$ is an $A/k$-Lie-Rinehart algebra and $\underline{\operatorname{Conn}}(L)$ is the category of $L$-connections and morphisms of connections. Hence we may use the associative unital ring $\operatorname{U}^{\otimes}(L(f^{\alpha}))$ to define the cohomology and homology of any $L$-connection, flat or non-flat. Using $\psi_{f}$ we get for any pair of $L$-connections $V,W$ and any integer $i\geq 0$ isomorphisms

[TABLE]

Hence the associative ring $\operatorname{U}^{\otimes}(L(f^{\alpha}))$ may be used to calculate the ”true” $\operatorname{Ext}$-groups of $V$ and $W$. Theorem 1.1 was one of the main reasons for the introduction of the notion $\operatorname{D}$-Lie algebra.

Given an almost commutative ring $U$ with filtration $U_{i}$ where $U$ is generated by $U_{1}$, there is a pre-$\operatorname{D}$-Lie algebra $\tilde{L}$ and a 2-sided ideal $J\subseteq\operatorname{U}^{\otimes}(\tilde{L})$ and an isomorphism

[TABLE]

of filtered associative rings. Hence there is an exact equivalence of categories

[TABLE]

preserving injective and projective objects, where $\underline{\operatorname{Mod}}(U)$ is the category of left $U$-modules and $\underline{\operatorname{Mod}}(\tilde{L},Id,J)$is the category of $\tilde{L}$-connections with $J$-curvature equal to zero. Most rings of differential operators appearing in the litterature are almost commutative and generated by $U_{1}$. As special cases it follows the ring $\operatorname{Diff}(\mathcal{L})$ of differential operators on a linebundle $\mathcal{L}$ on $A$ and the generalized universal enveloping algebra $\operatorname{U}(A,L,f)$ where $L$ is an $A/k$-Lie-Rinehart algebra and $f$ is a 2-cocycle on $L$ with values in $A$, may be realized as $\operatorname{U}^{\otimes}_{J}(\tilde{L})$ (see [8], Example 3.35). The rings $\operatorname{U}^{\otimes}(\tilde{L})$ and $\operatorname{U}^{\otimes}_{J}(\tilde{L})$ have canonical filtrations $\operatorname{U}^{\otimes}(\tilde{L})^{i},\operatorname{U}^{\otimes}_{J}(\tilde{L})^{i}$ and the associated graded rings are commutative. Let $\tilde{L}^{*}:=\operatorname{U}^{\otimes}(\tilde{L})^{1}/\operatorname{U}^{\otimes}(\tilde{L})^{0}$. There is a canonical surjective map of graded $A$-algebras

[TABLE]

The map $\rho$ is not always an isomorphism in the case when $\tilde{L}$ is a projective $A$-module. Hence the PBW-theorem does not always hold for $\operatorname{U}^{\otimes}(\tilde{L})$. If $(U,U_{i})$ is an almost commutative ring with $L:=U_{1}/U_{0}$ a projective $A$-module and the canonical map $\eta:\operatorname{Sym}_{A}^{*}(L)\rightarrow Gr(U)$ an isomorphism, it follows there is an isomorphism of associative rings $U\cong\operatorname{U}(A,L,f)$ where $L:=U_{1}/U_{0}$ and $f$ a 2-cocycle for $L$ with values in $A$. Hence a left $U$-module $A$ corresponds to a connection $\nabla:L\rightarrow\operatorname{End}_{k}(E)$ of curvature type $f$. It follows the k’th Chern class of $E$ is determined by the first Chern class. A similar PBW-theorem for the universal ring $\operatorname{U}^{\otimes}(\tilde{L})$ would put similar conditions on all Chern classes of all finite rank projective $A$-modules. Hence the study of the associative ring $\operatorname{U}^{\otimes}(\tilde{L})$ and the PBW-theorem for $\operatorname{U}^{\otimes}(\tilde{L})$ will have applications in the study of the Chern classes and Chern character of $A$.

Note: The category $\underline{\operatorname{Conn}}(L)$ of connections on an $A/k$-Lie-Rinehart algebra $L$ is a small abelian category, hence the Freyd-Mitchell full embedding theorem gives an equivalence $\phi$ of $\underline{\operatorname{Conn}}(L)$ with a sub category of $\underline{\operatorname{Mod}}(R)$ where $R$ is an associative ring. The equivalence $\phi$ does not preserve injective and projective objects, hence we cannot use $\phi$ to define the cohomology and homology of a connection. Theorem 1.1 gives a geometric construction of cohomology and homology groups of any $\tilde{L}$- and $L$-connection where $\tilde{L}$ is any $\operatorname{D}$-Lie algebra and $L$ is any $A/k$-Lie-Rinehart algebra. The construction is functorial and can be done for any scheme and any sheaf of Lie-Rinehart algebras.

The aim of this paper is to classify maps of $\operatorname{D}$-Lie algebras. We also classify $\operatorname{D}$-Lie algebras and non-abelian extensions of $\operatorname{D}$-Lie algebras with projective canonical quotient in terms of a functor $F_{f^{\alpha}}$ introduced in [8].

Let $(L,\alpha)$ and $(L^{\prime},\alpha^{\prime})$ be $A/k$-Lie-Rinehart algebras and let $f,g\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be two 2-cocycles. Let $F_{g}(L):=L(g^{\alpha})$ and $F_{f}(L^{\prime}):=L^{\prime}(f^{\alpha^{\prime}})$ be the $\operatorname{D}$-Lie algebras introduced in [8], Theorem 2.7. In Theorem 2.3 we prove the following:

Theorem 1.2**.**

There is a map of $\operatorname{D}$-Lie algebras $\phi:L(g^{\alpha})\rightarrow L^{\prime}(f^{\alpha^{\prime}})$ if and only if $\overline{g^{\alpha}}=\overline{f^{\alpha}}$ in $\alpha^{*}\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$. In this case there is an equality between the set of maps of $\operatorname{D}$-Lie algebras $\phi:L(g^{\alpha})\rightarrow L^{\prime}(f^{\alpha^{\prime}})$ and the set of maps of $A/k$-Lie-Rinehart algebras $\psi_{2}:L\rightarrow L^{\prime}$.

In Theorem 3.9 we prove the following:

Theorem 1.3**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(L,\alpha)$ be the canonical quotient $A/k$-Lie-Rinehart algebra of $\tilde{L}$. Assume $L$ is projective as left $A$-module. There is an isomorphism $\tilde{L}\cong L(f^{\alpha})$ as $\operatorname{D}$-Lie algebras where $L(f^{\alpha}):=F_{f^{\alpha}}(L)$ and $F_{f^{\alpha}}$ is the functor from Example 3.3. Hence $\tilde{L}$ is uniquely determined up to isomorphism by the canonical quotient $(L,\alpha)$ and the 2-cocycle $f\in\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$.

In Corollary 3.10 we classify maps of $\operatorname{D}$-Lie algebras between two $\operatorname{D}$-Lie algebras $\tilde{L}_{1}$ and $\tilde{L}_{2}$.

In Corollary 4.20 we get a classification of non-abelian extensions of $\operatorname{D}$-Lie algebras in terms of the functor $F_{f^{\alpha}}$.

In the paper we introduce the category of $\operatorname{D}$-Lie algebras, connections on $\operatorname{D}$-Lie algebras using the module of principal parts. This gives an equivalent definition of a $\operatorname{D}$-Lie algebra to the one introduced in [8]. We prove in Theorem 4.19 that for any $\tilde{L}$-connection $(E,\rho)$ there is an extension of $\operatorname{D}$-Lie algebras

[TABLE]

with the property that 1.3.1 splits in the category of $\operatorname{D}$-Lie algebras if and only if $E$ has a flat $\tilde{L}$-connection $\rho^{\prime}:\tilde{L}\rightarrow\operatorname{Diff}^{1}(E)$. Theorem 4.19 generalize a similar result proved in [9] for non-abelian extensions of $A/k$-Lie-Rinehart algebras.

Using the $\operatorname{P}$-module structure on the $\operatorname{D}$-Lie algebra $\tilde{L}$ we define in Definition 4.6 the correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ associated to a connection and degeneracy locies $\sigma_{1}$ and $\sigma_{2}$. The correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ induce an endomorphism

[TABLE]

of the Chow-group $\operatorname{CH}^{*}(X)$. The correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ and Chow-operator $I(\rho,\sigma_{1},\sigma_{2})$ depend in a non-trivial way on the $\operatorname{P}$-module structure of $\tilde{L}$ and $\operatorname{P}$-linearity of the $\tilde{L}$-connection $\rho$ and cannot be defined for an ordinary $L$-connection where $(L,\alpha)$ is an $A/k$-Lie-Rinehart algebra (see Example 4.3). There is no non-trivial right $A$-module structure on $L$.

We also prove in Theorem 4.15 the following result relating the category of connections on an $A/k$-Lie-Rinehart algebra $L$ to the category of connections on the $\operatorname{D}$-Lie algebra $L(f^{\alpha})$:

Theorem 1.4**.**

Let $(L,\alpha)$ be an $A/k$-Lie-Rinehart algebra and let $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be a 2-cocycle. There is an equivalence of categories

[TABLE]

from the category $\underline{\operatorname{Conn}}(L,\operatorname{End})$ of $(L,\psi)$-connections $\nabla$, to the category $\underline{\operatorname{Conn}}(L(f^{\alpha}))$ of $L(f^{\alpha})$-connections $\rho$. Let $u:=az+x,v:=bz+y\in L(f^{\alpha})$ and let $e\in E$. Let $(E,\nabla)$ be an $L$-connection and let $\rho_{\nabla}:=C_{f}(\nabla)$. The following holds:

[TABLE]

For any $L(f^{\alpha})$-connection $(E,\rho)$ there is an $(L,\psi)$-connection $(E,\nabla)$ with $C_{f}(E,\nabla)=(E,\rho)$.

Hence there is for any 2-cocycle $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ a functorial way to define an $L(f^{\alpha})$-connection $C_{f}(E,\nabla)$ from an $L$-connection $(L,\alpha)$. As a Corollary we are able to classify $\tilde{L}$-connections on $\operatorname{D}$-Lie algebras with projective canonical quotient. In Corollary 4.16 we prove the following:

Corollary 1.5**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(E,\rho)$ be an $\tilde{L}$-connection. Assume the canonical quotient $L$ of $\tilde{L}$ is a projective $A$-module. It follows any $\tilde{L}$-connection $(E,\rho)$ is on the form $C_{f}(E,\nabla)$ where $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ and $(E,\nabla)$ is an $(L,\psi)$-connection for $\psi\in\operatorname{End}_{A}(E)$. If $\rho(D)=I$ it follows we may choose $(E,\nabla)$ to be an $L$-connection.

2. Classification of morphisms of $\operatorname{D}$-Lie algebras

Let in the following $A$ be a fixed commutative $k$-algebra where $k$ is a fixed commutative unital ring. Let $f,g\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ bw two 2-cocycles and consider the two $k$-Lie algebras $\operatorname{D^{1}(A,f)}$ and $\operatorname{D^{1}(A,g)}$. Let $\operatorname{P}:=A\otimes_{k}A/I^{2}$ be the first order module of principal parts of $A/k$. The $k$-Lie algebra $\operatorname{D^{1}(A,f)}$ has the following left and right $A$-module structure:

[TABLE]

and

[TABLE]

for $(a,x)\in\operatorname{D^{1}(A,f)}$ and $c\in A$. Let $z_{f}:=(1,0)\in\operatorname{D}^{1}(A,f)$. It follows $u:=(a,x)$ that $uc=cu+x(c)z_{f}=cu+\pi_{f}(u)(c)z_{f}$. Similarly for $\operatorname{D^{1}(A,g)}$. Let $d:A\rightarrow A\otimes_{k}A$ be defined by $dc:=1\otimes c-c\otimes 1$. It follows $dc(u)=uc-cu=\pi_{f}(u)(c)z$. One checks that $zc=cz$ hence $dbdc(u)=db(\pi_{f}(u)(c)z=0$ hence $\operatorname{D^{1}(A,f)}$ is annihilated by $I^{2}\subseteq A\otimes_{k}A$ and it follows $\operatorname{D^{1}(A,f)}$ is a left $\operatorname{P}$-module. The two projection maps $\pi_{f}:\operatorname{D^{1}(A,f)}\rightarrow\operatorname{Der}_{k}(A)$ and $\pi_{g}:\operatorname{D^{1}(A,g)}\rightarrow\operatorname{Der}_{k}(A)$ and $\pi_{f},\pi_{g}$ are maps of $\operatorname{P}$-modules and $k$-Lie algeras. We want to classify maps

[TABLE]

of $\operatorname{P}$-modules and $k$-Lie algebras such that

[TABLE]

Lemma 2.1**.**

Let $\phi:\operatorname{D^{1}(A,g)}\rightarrow\operatorname{D^{1}(A,f)}$ be a map of $\operatorname{P}$-modules and $k$-Lie algebras satisfying condition 2.0.1. It follows $\phi(a,x)=(a+\phi_{1}(x),x)$ where $\phi_{1}\in\operatorname{C}^{1}(\operatorname{Der}_{k}(A),A)$, $g=f+d^{1}(\phi_{1})$ and $d^{1}:\operatorname{C}^{1}(\operatorname{Der}_{k}(A),A)\rightarrow\operatorname{C}^{2}(\operatorname{Der}_{k}(A),A)$ is the first differential in the Lie-Rinehart complex of $\operatorname{Der}_{k}(A)$. Hence the map $\phi$ exists if and only if $\overline{g}=\overline{f}\in\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$. If the map $\phi$ exists it is an isomorphism of $\operatorname{P}$-modules and $k$-Lie algebras with inverse $\psi:\operatorname{D^{1}(A,f)}\rightarrow\operatorname{D^{1}(A,g)}$ defined by $\psi(a,x):=(a-\phi_{1}(x),x)$.

Proof.

Let $\phi:\operatorname{D^{1}(A,g)}\rightarrow\operatorname{D^{1}(A,f)}$ be a map of $\operatorname{P}$-modules and $k$-Lie algebras satisfying 2.0.1 It follows for any $(a,x)\in\operatorname{D^{1}(A,g)}$ we get

[TABLE]

with $\phi_{1}\in\operatorname{C}^{1}(\operatorname{Der}_{k}(A),A)$. One checks the map $\phi$ is a map of $k$-Lie algebras if and only if $g=f+d^{1}(\phi_{1})$. The map $\phi$ in 2.1.1 is always a map of $\operatorname{P}$-modules. Hence the map $\phi$ exists if and only if $\overline{g}=\overline{f}\in\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$. If $\phi$ exists, an inverse $\psi$ is given by $\psi(a,x):=(a-\phi_{1}(x),x)$. The Lemma follows. ∎

Assume $(L,\alpha)$ is an $A/k$-Lie-Rinehart algebra and let $f,g\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be two 2-cocycles and let $f^{\alpha},g^{\alpha}\in\operatorname{Z}^{2}(L,A)$ be the pull-back cocycles. Let $F_{g}(L):=L(g^{\alpha}):=Az_{g}\oplus L$ be the $A/k$-Lie-Rinehart algebra defined in [8]. It has the following $k$-Lie product:

[TABLE]

for $(a,x),(b,y)\in L(g^{\alpha})$. There is a map

[TABLE]

defined by

[TABLE]

The pair $(L(g^{\alpha}),\pi_{g})$ is an $A/k$-Lie-Rinehart algebra.

Note: The map $\alpha$ induce a map of cohomology groups

[TABLE]

Let $Im(\alpha^{*}):=\alpha^{*}\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$ denote the image of the mape $\alpha^{*}$ in $\operatorname{H}^{2}(L,A)$.

Lemma 2.2**.**

There is a map $\alpha_{g}:L(g^{\alpha})\rightarrow\operatorname{D^{1}(A,f)}$ of $\operatorname{P}$-modules and $k$-Lie algebras with $\pi\circ\alpha_{g}=\pi_{g}$ if and only if there is an element $\alpha_{1}\in\operatorname{C}^{1}(L,A)$ with $g^{\alpha}=f^{\alpha}+d^{1}(\alpha_{1})$. The map $\alpha_{g}$ is defined as follows: $\alpha_{g}(a,x):=(a+\alpha_{1}(x),\alpha(x))\in\operatorname{D^{1}(A,f)}$. Hence there exists a 5-tuple $(L(g^{\alpha}),\alpha_{g},\pi_{g},[,],z_{g})$ which is a $\operatorname{D}$-Lie algebra if and only if there is an equality of cohomology classes

[TABLE]

Proof.

By definition the map $\alpha_{g}$ must look as follows:

[TABLE]

where $(a,x)\in L(g^{\alpha})$ and $\alpha_{1}\in\operatorname{C}^{1}(L,A)$. One checks the map $\alpha_{g}$ is a map of left $\operatorname{P}$-modules. The map $\alpha_{g}$ is a map of $k$-Lie algebras if and only if $g^{\alpha}=f^{\alpha}+d^{1}(\alpha_{1})$. Hence the 5-tuple $(L(g^{\alpha}),\alpha_{g},\pi_{g},[,],z_{g})$ is a $\operatorname{D}$-Lie algebra over $\operatorname{D^{1}(A,f)}$ if and only if $\overline{g^{\alpha}}=\overline{f^{\alpha}}$ in $\alpha^{*}\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$. The Lemma follows. ∎

We may now classify maps between arbitrary $\operatorname{D}$-Lie algebras. Let $(L,\alpha)$ and $(L^{\prime},\alpha^{\prime})$ be $A/k$-Lie-Rinehart algebras and let $f,g\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ with $g^{\alpha}=f^{\alpha}+d^{1}(\alpha_{1})$ for $\alpha_{1}\in\operatorname{C}^{1}(L,A)$. Hence there is by Lemma 2.2 a structure as $\operatorname{D}$-Lie algebra $\alpha_{g}:L(g^{\alpha})\rightarrow\operatorname{D^{1}(A,f)}$ given by the map $\alpha_{g}(a,x):=(a+\alpha_{1}(x),\alpha(x))$. The 5-tuple $(L(g^{\alpha}),\alpha_{g},\pi_{g},[,],z_{g})$ is a $\operatorname{D}$-Lie algebra over $\operatorname{D^{1}(A,f)}$ with $\pi_{g}(a,x):=\alpha(x)\in\operatorname{Der}_{k}(A)$.

Let $(L,\alpha)$ and $(L^{\prime},\alpha^{\prime})$ be $A/k$-Lie-Rinehart algebras and let $f,g\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be two 2-cocycles. Let $F_{g}(L):=L(g^{\alpha})$ and $F_{f}(L^{\prime}):=L^{\prime}(f^{\alpha^{\prime}})$ be the $\operatorname{D}$-Lie algebras introduced in [8], Theorem 2.7.

Theorem 2.3**.**

There is a map of $\operatorname{D}$-Lie algebras $\phi:L(g^{\alpha})\rightarrow L^{\prime}(f^{\alpha^{\prime}})$ if and only if $\overline{g^{\alpha}}=\overline{f^{\alpha}}$ in $\alpha^{*}\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$. In this case there is an equality between the set of maps of $\operatorname{D}$-Lie algebras $\phi:L(g^{\alpha})\rightarrow L^{\prime}(f^{\alpha^{\prime}})$ and the set of maps of $A/k$-Lie-Rinehart algebras $\psi_{2}:L\rightarrow L^{\prime}$.

Proof.

If there is an equality $\overline{g^{\alpha}}=\overline{f^{\alpha}}$ it follows there is a map

[TABLE]

defined by

[TABLE]

The map $\phi:L(g^{\alpha})\rightarrow L^{\prime}(f^{\alpha^{\prime}})$ must look as follows:

[TABLE]

where $\phi_{1}:L\rightarrow A$ and $\phi_{2}:L\rightarrow L^{\prime}$. Since $\alpha^{\prime}_{f}\circ\phi=\alpha_{g}$ it follows $\phi_{1}=\alpha_{1}$ and $\alpha^{\prime}\circ\phi_{2}=\alpha$. Hence $\phi_{2}$ is a map of $A/k$-Lie-Rinehart algebras. One checks the map $\phi$ is $\operatorname{P}$-linear and a map of $k$-Lie algebras. Conversely, for an arbitrary map $\phi_{2}:L\rightarrow L^{\prime}$ of $A/k$-Lie-Rinehart algebras it follows the map $\phi(a,x):=(a+\alpha_{1}(x),\phi_{2}(x))$ is a map of $\operatorname{D}$-Lie algebras. The Theorem follows. ∎

3. Classification of $\operatorname{D}$-Lie algebras with projective canonical quotient

In this section we define the notion of a $\operatorname{D}$-Lie algebra, the category of $\operatorname{D}$-Lie algebras and connections on $\operatorname{D}$-Lie algebras.

The module $\operatorname{Diff}^{1}(A)$ of first order differential operators on a commutative $k$-algebra $A$ is in a canonical way a $k$-Lie algebra and a left $\operatorname{P}$-module where $\operatorname{P}:=A\otimes_{k}A/I^{2}$ is the module of first order principal parts on $A/k$. By definition $\operatorname{Diff}^{1}(A):=\operatorname{Hom}_{A}(\operatorname{P},A)$ and it follows $\operatorname{Diff}^{1}(A)$ is in a canonical way a left and right $A$-module. There is a canonical projection map $\pi:\operatorname{Diff}^{1}(A)\rightarrow\operatorname{Der}_{k}(A)$ which is a map of $k$-Lie algebras and left $\operatorname{P}$-modules. There is a canonical inclusion of left and right $A$-modules and $k$-Lie algebras $\operatorname{Diff}^{1}(A)\subseteq\operatorname{End}_{k}(A)$ and the Lie product $[,]$ on $\operatorname{Diff}^{1}(A)$ satisfies the following formula:

[TABLE]

for all $u,v\in\operatorname{Diff}^{1}(A)$ and $c\in A$. There is a canonical element $D\in\operatorname{Diff}^{1}(A)$ with the property that

[TABLE]

The element $D$ is central in $\operatorname{Diff}^{1}(A)$ and $\pi(D)=0$. A $\operatorname{D}$-Lie algebra $\tilde{L}$ is a generalization of $\operatorname{Diff}^{1}(A)$ and we may speak the category of $\operatorname{D}$-Lie algebras, extensions and non-abelian extensions of $\operatorname{D}$-Lie algebras, cohomology of $\operatorname{D}$-Lie algebras, connections etc. Equation 3.0.1 is the equation defining an $A/k$-Lie-Rinehart algebra and any $\operatorname{D}$-Lie algebra $\tilde{L}$ has an underlying $A/k$-Lie-Rinehart algebra. Hence we may view a $\operatorname{D}$-Lie algebra as a refinement of the notion of a $A/k$-Lie-Rinehart algebra. A $\operatorname{D}$-Lie algebra is an $A/k$-Lie-Rinehart algebra with extra structure defined by the $\operatorname{P}$-module structure, the canonical element $D$ and Equations 3.0.1 and 3.0.2.

Note: We may consider the $A/k$-Lie-Rinehart algebra $\operatorname{Der}_{k}(A)\subseteq\operatorname{Diff}^{1}(A)$ and $\operatorname{Der}_{k}(A)$ has a canonical structure as $k$-Lie algebra and left $A$-module. It has no-non-trivial right $A$-module structure. To get a non-trivial right $A$-module structure, we must consider the abelian extension $\operatorname{Diff}^{1}(A)$ and this is one of the motivations for the introduction of the notion of a $\operatorname{D}$-Lie algebra. The $\operatorname{P}$-module structure on $\operatorname{Diff}^{1}(A)$ is canonical and is related to the notion of curvature of a connection.

Using the $\operatorname{P}$-module structure on $\tilde{L}$ we define the correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ associated to a connection and degeneracy locies $\sigma_{1}$ and $\sigma_{2}$. The correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ induce an endomorphism

[TABLE]

of the Chow-group $\operatorname{CH}^{*}(X)$.

The main Theorem in this section is Theorem 4.19 where we construct the non-abelian extension $\operatorname{End}(\tilde{L},E)$ of any $\tilde{L}$-connection $(E,\rho)$ in $\underline{\operatorname{Conn}}(\tilde{L},Id)$, and prove the canonical projection map of $\operatorname{D}$-Lie algebras

[TABLE]

splits in the category of $\operatorname{D}$-Lie algebras if and only if $E$ has a flat $\tilde{L}$-connection in $\underline{\operatorname{Conn}}(\tilde{L},Id)$. In [9] a similar result was proved for $A/k$-Lie-Rinehart algebras.

We also define the canonical quotient $A/k$-Lie-Rinehart algebra $(L,\pi_{L})$ of a $\operatorname{D}$-Lie algebra $\tilde{L}$ and classify the set of $\operatorname{D}$-Lie algebras with canonical quotient $L$. It follows from Theorem 3.9 the $\operatorname{D}$-Lie algebra $\tilde{L}$ is uniquely determined by $(L,\pi_{L})$ up to isomorphism when $L$ is projective as left $A$-module. Hence for any cohomology class $c=\overline{f}\in\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$ and any $A/k$-Lie-Rinehart algebra $(L,\alpha)$ with $L$ a projective $A$-module, there is a unique $\operatorname{D}$-Lie algebra $\tilde{\alpha}:\tilde{L}\rightarrow\operatorname{D^{1}(A,f)}$ with canonical quotient $(L,\pi_{L})$, defined by $\tilde{L}:=F_{f^{\alpha}}(L)$, where $F_{f^{\alpha}}$ is the functor defined and studied in [8].

Let in the following $A$ be a fixed commutative unital $k$-algebra where $k$ is a commutative unital ring. Let $\operatorname{Der}_{k}(A)$ be the $k$-Lie algebra of $k$-linear derivations of $A$. Let $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be a 2-cocycle and let $\operatorname{D^{1}(A,f)}:=A\oplus\operatorname{Der}_{k}(A)$ with the following $k$-Lie algebra structure and $A\otimes_{k}A$-module structure: Let $u:=(a,x),v:=(b,y)\in\operatorname{D^{1}(A,f)}$ and define

[TABLE]

Define for any element $c\in A$

[TABLE]

and

[TABLE]

It follows $\operatorname{D^{1}(A,f)}$ is a left $A\otimes_{k}A$-module. Define the map $\pi:\operatorname{D^{1}(A,f)}\rightarrow\operatorname{Der}_{k}(A)$ by $\pi(u):=\pi(a,x)=x$. Define for $x\in\operatorname{Der}_{k}(A)$ $xc:=cx$. It follows $\operatorname{Der}_{k}(A)$ is a left $A\otimes_{k}A$-module. Let $\operatorname{P}:=A\otimes_{k}A/I^{2}$ where $I$ is the kernel of the multiplication map. It follows $\operatorname{P}$ is the first order module of principal parts of $A/k$. Let $d:A\rightarrow\operatorname{P}$ be the universal derivation. Let $p(a):=1\otimes a$ and $q(a):=a\otimes 1$. It follows $d=p-q$. Let $D:=(1,0)\in\operatorname{D^{1}(A,f)}$.

Lemma 3.1**.**

The $k$-Lie algebra $\operatorname{D^{1}(A,f)}$ is in a canonical way a left $\operatorname{P}$-module. The map $\pi:\operatorname{D^{1}(A,f)}\rightarrow\operatorname{Der}_{k}(A)$ is a map of $k$-Lie algebras and left $\operatorname{P}$-modules. The following holds for all $u,v\in\operatorname{D^{1}(A,f)}$ and $c\in A$:

[TABLE]

The element $D$ is a central element with $\pi(D)=0$.

Proof.

Since $\operatorname{D^{1}(A,f)}$ is a left and right $A$-module with $(au)b=a(ub)$ for all $u\in\operatorname{D^{1}(A,f)}$ and $a,b\in A$ it follows $\operatorname{D^{1}(A,f)}$ is a left $A\otimes_{k}A$-module. One checks that for any element $w\in I^{2}$ it follows $wu=0$ hence $\operatorname{D^{1}(A,f)}$ is a left $\operatorname{P}$-module. The map $\pi$ is left and right $A$-linear. It follows $\pi$ is a map of $\operatorname{P}$-modules and $k$-Lie algebras. One checks Equation 3.1.1 and 3.1.2 holds and the Lemma is proved. ∎

The notion of a $\operatorname{D}$-Lie algebra is a generalization of the $k$-Lie algebra and left $\operatorname{P}$-module $\operatorname{D^{1}(A,f)}$ from Lemma 3.1.

Definition 3.2**.**

Let $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be a 2-cocycle and let $\operatorname{D^{1}(A,f)}$ be the $k$-Lie algebra and $\operatorname{P}$-module defined in Lemma 3.1. A 5-tuple $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ is a $\operatorname{D}$-Lie algebra if the following holds. $\tilde{L}$ is a $k$-Lie algebra and left $\operatorname{P}$-module. The element $\tilde{\alpha}$ is a map

[TABLE]

of $\operatorname{P}$-modules and $k$-Lie algebras. The element $\tilde{\pi}$ is a map

[TABLE]

of left $\operatorname{P}$-modules and $k$-Lie algebras with $\tilde{\pi}=\pi\circ\tilde{\alpha}$. The $k$-Lie product satisfies

[TABLE]

for all $u,v\in\tilde{L}$ and $a\in A$. The following holds for all $a\in A$:

[TABLE]

The element $D$ is a central element in $\tilde{L}$ with $\tilde{\pi}(D)=0$. Given two $\operatorname{D}$-Lie algebras $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ and $(\tilde{L}^{\prime},\tilde{\alpha}^{\prime},\tilde{\pi}^{\prime},[,],D^{\prime})$. A map of $\operatorname{D}$-Lie algebras is a map

[TABLE]

of left $\operatorname{P}$-modules and $k$-Lie algebras with $\phi(D)=D^{\prime}$ and $\tilde{\pi}^{\prime}=\tilde{\pi}\circ\phi$. Let $\operatorname{D}$-Lie denote the category of $\operatorname{D}$-Lie algebras and maps. An ideal $\tilde{I}$ in a $\operatorname{D}$-Lie algebra $\tilde{L}$ is a sub-$k$-Lie algebra and sub-$\operatorname{P}$-module $\tilde{I}\subseteq\tilde{L}$ such that $[\tilde{L},\tilde{I}]\subseteq\tilde{I}$. A $\operatorname{D}$-ideal $\tilde{I}$ is a sub-$k$-Lie algebra and sub-$\operatorname{P}$-module $\tilde{I}\subseteq\tilde{L}$ with $[\tilde{L},\tilde{I}]\subseteq\tilde{I}$ and with $D\notin\tilde{I}$.

Note: One checks Definition 3.2 gives the same notion as Definition 2.3 in [8].

Example 3.3**.**

Non-trivial examples: $\operatorname{D}$-Lie algebras and $A/k$-Lie-Rinehart algebras.

It follows from Lemma 3.1 that the 5-tuple $(\operatorname{D^{1}(A,f)},id,\pi,[,],D)$ is a $\operatorname{D}$-Lie algebra for any 2-cocycle $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$.

Given a $\operatorname{D}$-Lie algebra $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ it follows the left $A$-module $\tilde{L}$ and the map $\tilde{\pi}:\tilde{L}\rightarrow\operatorname{Der}_{k}(A)$ is an $A/k$-Lie-Rinehart algebra.

Given any 2-cocycle $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$, let $(L,\alpha)$ be an $A/k$-Lie-Rinehart algebra and let $f^{\alpha}\in\operatorname{Z}^{2}(L,A)$ be the pull back 2-cocycle. There is by Theorem 2.7 in [8] a functor

[TABLE]

from the category LR(A/k) of $A/k$-Lie-Rinehart algebras to the category $\operatorname{D}$-Lie of $\operatorname{D}$-Lie algebras, hence it is easy to give non-trivial examples of $\operatorname{D}$-Lie algebras. When $\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)\neq 0$ we get for any $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ a non-trivial functor $F_{f^{\alpha}}$. Since $F_{f^{\alpha}}(L)$ is independent of choice of representative for the class $c:=\overline{f^{\alpha}}\in\operatorname{H}^{2}(L,A)$ the following holds: There is for any two representatives $f,f^{\prime}$ for the class $c$ a canonical isomorphism $F_{f}(L)\cong F_{f^{\prime}}(L)$ of extensions of $A/k$-Lie-Rinehart algebras. Hence the two functors $F_{f}$ and $F_{f^{\prime}}$ are equal up to isomorphism. Hence we get from Theorem 2.7 in [8] a well defined functor $F_{c}$ for any cohomology class $c:=\overline{f^{\alpha}}\in\operatorname{H}^{2}(L,A)$:

[TABLE]

When the class $c$ is the zero class it follows the underlying $A/k$-Lie-Rinehart algebra of $F_{c}(L):=Az\oplus L$ is the trivial abelian extension of $L$ by $A$. Hence the cohomology group $\operatorname{H}^{2}(L,A)$ parametrize a large class of non-trivial functors $F_{c}$.

Lemma 3.4**.**

Let $\phi:\tilde{L}_{1}\rightarrow\tilde{L}_{2}$ be a map of $\operatorname{D}$-Lie algebras. It follows the kernel $ker(\phi)$ is a $\operatorname{D}$-ideal in $\tilde{L}_{1}$. Let $\tilde{I}:=I/I^{2}\subseteq\operatorname{P}$ be the module of differentials and let $\tilde{L}$ be any $\operatorname{D}$-Lie algebra. It follows $\tilde{I}\tilde{L}\subseteq\tilde{L}$ is an ideal. Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let J:=\{aD:\text{ with a\in A}\}\subseteq\tilde{L}. It follows $J$ is an ideal. Let $L:=\tilde{L}/J$. There is an exact sequence of $\operatorname{P}$-modules and $k$-Lie algebras

[TABLE]

There is a canonical map of left $A$-modules and $k$-Lie algebras $\alpha_{L}:L\rightarrow\operatorname{Der}_{k}(A)$ making $(L,\alpha_{L})$ into an $A/k$-Lie-Rinehart algebra. The sequence 3.4.1 is an exact sequence of $A/k$-Lie-Rinehart algebras. The ideal $J$ is a free left $A$-module on the element $D$.

Proof.

One checks that $ker(\phi)$ is a $\operatorname{P}$-submodule of $\tilde{L}_{1}$ and that $[\tilde{L}_{1},ker(\phi)]\subseteq ker(\phi)$. Since $D\notin ker(\phi)$ it follows $ker(\phi)$ is a $\operatorname{D}$-ideal. The same holds for $\tilde{I}$: $\tilde{I}$ is a $\operatorname{P}$-module and $[\tilde{L},\tilde{I}]\subseteq\tilde{I}$. For any element $u\in\tilde{L}$ and $c\in A$ it follows $uc=cu+\tilde{\pi}(u)(c)D$. It follows

[TABLE]

since $D$ is central. It follows $[\tilde{L},J]\subseteq J$. Hence $[J,J]\subseteq J$ and $J$ is a $k$-Lie algebra. We get

[TABLE]

hence

[TABLE]

since $Dc=cD$. Hence $J$ is a left $\operatorname{P}$-module. It follows the Sequence 3.4.1 is an exact sequence of $\operatorname{P}$-modules and $k$-Lie algebras. If $\overline{u}\in\tilde{L}/J$ is an element with $u\in\tilde{L}$ it follows $\overline{u}c-c\overline{u}=\tilde{\pi}(u)(c)D:=0$. Hence $\overline{u}c=c\overline{u}$ in $\tilde{L}/J$. Hence $\tilde{L}/J$ is trivially a left $\operatorname{P}$-module. Since $\tilde{\pi}(aD)=a\tilde{\pi}(D)=0$ it follows we get a canonical map

[TABLE]

and one checks $(\tilde{L}/J,\pi_{L})$ is an $A/k$-Lie-Rinehart algebra. The rest is clear and the Lemma follows. ∎

Definition 3.5**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra. The quotient $A/k$-Lie-Rinehart algebra $(L,\pi_{L})$ with $L:=\tilde{L}/J$ from Lemma 3.4 is the canonical quotient of $\tilde{L}$.

Example 3.6**.**

When the canonical quotient is projective.

Assume $L:=\tilde{L}/J$ where $J$ is the ideal defined in Lemma 3.4 and let $\pi_{L}:L\rightarrow\operatorname{Der}_{k}(A)$ be the anchor map. Assume $L$ is projective as left $A$-module and let $s$ be a left $A$-linear section of the canonical projection map $p:\tilde{L}\rightarrow L$. Define for $u,v\in L$ the following map

[TABLE]

by

[TABLE]

Define the map $\nabla_{s}:L\rightarrow\operatorname{End}_{k}(J)$ by

[TABLE]

Lemma 3.7**.**

It follows $(J,\nabla_{s})$ is a flat $L$-connection and $\psi_{s}\in\operatorname{Z}^{2}(L,(J,\nabla_{s}))$ where $\operatorname{C}^{p}(L,(J,\nabla_{s}))$ is the Lie-Rinehart complex of the flat connection $(J,\nabla_{s})$. If $s^{\prime}$ is another left $A$-linear splitting of $p$ it follows there is an equality of connections $\nabla_{s}=\nabla_{s^{\prime}}$. If $\psi_{s^{\prime}}$ is the 2-cocycle associated to $s^{\prime}$ it follows there is an element $\rho\in\operatorname{C}^{1}(L,(J,\nabla_{s}))$ with $\psi_{s^{\prime}}=\psi_{s}+d^{1}_{\nabla_{s}}(\rho)$. Hence there is an equality of cohomology classes

[TABLE]

Proof.

One checks that $\nabla_{s}$ is a flat $L$-connection on $J$ and that $\psi\in\operatorname{Z}^{2}(L,(J,\nabla_{s}))$. Assume $s^{\prime}=s+\rho$ where $\rho\in\operatorname{Hom}_{A}(L,J)$. We get for any element $u\in L$ and $x\in J$ the following:

[TABLE]

since $J$ is an abelian $k$-Lie algebra. It moreover follows

[TABLE]

[TABLE]

[TABLE]

since $J$ is an abelian Lie algebra and hence $[\rho(u),\rho(v)]=0$ for all $u,v\in L$. The Lemma follows. ∎

Define the map

[TABLE]

by

[TABLE]

Make the following definition: $J\oplus^{(\phi,\psi)}L$ is the left $A$-module $J\oplus L$ with the following right $A$-module structure: Given $z:=(u,x)\in J\oplus L$ and $c\in A$ define

[TABLE]

We get since $\overline{u}c=c\overline{u}$ the following:

[TABLE]

[TABLE]

Hence

[TABLE]

Since $\tilde{\pi}=\pi_{L}\circ p$ and $p\circ s=Id$ it follows $\tilde{\pi}(s(\overline{u}))=\pi_{L}(p(s(\overline{u}))=\pi_{L}(\overline{u})$. It follows $zc=cz+\pi_{L}(\overline{u})(c)(D,0)=cz+\pi_{L}(\overline{u})(c)\tilde{D}$ where $\tilde{D}:=(D,0)$.

Let $z:=(x,\overline{u}),w:=(y,\overline{v})\in J\oplus L$ and define the following product:

[TABLE]

It follows the product $[,]$ is a $k$-Lie product on $J\oplus L$. Define the map

[TABLE]

by

[TABLE]

It follows $\rho$ is an isomorphism of $\operatorname{P}$-modules and $k$-Lie algebras. Define the map

[TABLE]

by

[TABLE]

Define $\pi_{J}:J\oplus L\rightarrow\operatorname{Der}_{k}(A)$ by $\pi_{J}(u,x):=\pi_{L}(x)$. Associated to a left $A$-linear splitting $s$ of $p:\tilde{L}\rightarrow L$ we get by Lemma 3.7 a unique flat connection $\nabla_{s}:L\rightarrow\operatorname{End}_{k}(J)$, a unique cohomology class $\overline{\psi_{s}}\in\operatorname{H}^{2}(L,(J,\nabla_{s}))$ and a 5-tuple $(J\oplus L,\alpha_{J},\pi_{J},[,],\tilde{D})$.

Proposition 3.8**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(L,\alpha)$ be the canonical quotient $A/k$-Lie-Rinehart algebra of $\tilde{L}$. Assume $L$ is projective as left $A$-module. The 5-tupe $(J\oplus L,\alpha_{L},\pi_{L},[,],\tilde{D})$ constructed above is a $\operatorname{D}$-Lie algebra and there is an isomorphism $J\oplus L\cong\tilde{L}$ of $\operatorname{D}$-Lie algebras.

Proof.

The proof follows from the construction and calculations above. ∎

Theorem 3.9**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(L,\alpha)$ be the canonical quotient $A/k$-Lie-Rinehart algebra of $\tilde{L}$. Assume $L$ is projective as left $A$-module. There is an isomorphism $\tilde{L}\cong L(f^{\alpha})$ as $\operatorname{D}$-Lie algebras where $L(f^{\alpha}):=F_{f^{\alpha}}(L)$ and $F_{f^{\alpha}}$ is the functor from Example 3.3. Hence $\tilde{L}$ is uniquely determined by the canonical quotient $(L,\alpha)$ and the 2-cocycle $f\in\operatorname{H}^{2}(\operatorname{Der}_{k}(A),A)$.

Proof.

Let $s$ be a left $A$-linear splitting of the canonical projection map $p:\tilde{L}\rightarrow L$. From Proposition 3.8 it follows the 5-tuple $(J\oplus L,\alpha_{L},\pi_{L},[,],\tilde{D})$ is a $\operatorname{D}$-Lie algebra and there is by 3.7.3 construction an isomorphism of $\operatorname{D}$-Lie algebras

[TABLE]

defined by

[TABLE]

Consider the map $\tilde{\alpha}:\tilde{L}\rightarrow\operatorname{D^{1}(A,f)}$. It looks as follows: $\tilde{\alpha}(u)=\alpha_{1}(u)I+\tilde{\pi}(u)\in A\oplus\operatorname{Der}_{k}(A)$ with $\alpha_{1}\in\operatorname{Hom}_{A}(\tilde{L},A)$

Let $z:=(aD,\overline{u})$ and $z^{\prime}:=(bD,\overline{v})$. It follows

[TABLE]

[TABLE]

The Lie product on $J\oplus L$ is defined as follows:

[TABLE]

with

[TABLE]

from equation 3.6.1.

Here $g(\overline{u},\overline{v})\in\operatorname{Z}^{2}(L,A)$ is a 2-cocycle. We get

[TABLE]

[TABLE]

It follows

[TABLE]

if and only if

[TABLE]

hence $\rho$ is a map of $k$-Lie algebras if and only if

[TABLE]

and $\alpha_{1}\circ s\in\operatorname{Hom}_{A}(L,A)$. Hence $\rho$ is a map of $k$-Lie algebras if and only if there is an isomorphism of $\operatorname{D}$-Lie algebras

[TABLE]

It follows there is an isomorphism $\tilde{L}\cong L(f^{\alpha})$ of $\operatorname{D}$-Lie algebras and the Theorem follows. ∎

Corollary 3.10**.**

Let $(\tilde{L}_{i},\tilde{\alpha}_{i},\tilde{\pi}_{i},[,],D_{i})$ be $\operatorname{D}$-Lie algebras for $i=1,2$ with projective canonical quotients $(L_{i},\alpha_{i})$ for $i=1,2$. There is an equality between the set of maps of $\operatorname{D}$-Lie algebras $\phi:\tilde{L}_{1}\rightarrow\tilde{L}_{2}$ and the set of maps of $A/k$-Lie-Rinehart algebras $\phi^{*}:L_{1}\rightarrow L_{2}$.

Proof.

By Theorem 3.9 there are isomorphisms $\tilde{L}_{i}\cong L(f^{\alpha_{i}})$ for $i=1,2$. It follows again from Theorem 3.9 that the set of maps of $\operatorname{D}$-Lie algebras between $\tilde{L}_{1}$ and $\tilde{L}_{2}$ equal the set of maps of $A/k$-Lie-Rinehart algebras between $L_{1}$ and $L_{2}$ and the Corollary is proved. ∎

Example 3.11**.**

The $\operatorname{D}$-Lie algebra associated to an $A/k$-Lie-Rinehart algebra.

Let $(L,\alpha)$ be an $A/k$-Lie-Rinehart algebra and let $f^{\alpha}\in\operatorname{Z}^{2}(L,A)$ be the 2-cocycle associated to a 2-cocycle $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$. In [8], Theorem 2.8 we constructed a functor

[TABLE]

by

[TABLE]

where $L(f^{\alpha}):=Az\oplus L$ is the abelian extension of $L$ by the free rank one $A$-module on the symbol $z$. Define for any $\operatorname{D}$-Lie algebra $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ the following: $G(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D):=(L,\pi_{L})$ where $(L,\pi_{L})$ is the canonical quotient of $\tilde{L}$. Since any map of $\operatorname{D}$-Lie algebras $\phi:\tilde{L}\rightarrow\tilde{L}^{\prime}$ satisfies $\phi(D)=D^{\prime}$ where $D^{\prime}\in\tilde{L}^{\prime}$ is the canonical central element, we get a canonical map of $A/k$-Lie-Rinehart algebras

[TABLE]

where $(L^{\prime},\pi_{L^{\prime}})$ is the canonical quotient of $\tilde{L}^{\prime}$. Hence we get a functor

[TABLE]

Theorem 3.9 says that in the case when the canonical quotient $L$ of $\tilde{L}$ is a projective $A$-module, it follows $G(F(L,\alpha))\cong(L,\alpha)$ and $F(G(\tilde{L}))\cong\tilde{L}$ are isomorphisms.

4. Classification of connections on $\operatorname{D}$-Lie algebras with projective canonical quotient

In this section we classify connections on a $\operatorname{D}$-Lie algebra $\tilde{L}$ with projective canonical quotient $(L,\alpha)$. We prove in Theorem 4.15 there is a 2-cocycle $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ and an equivalence of categories

[TABLE]

We use the equivalence $C_{f}$ in 4.0.1 to classify arbitrary $\tilde{L}$-connections in Corollary 4.16.

We also introduce the correspondence and Chow-operator of an $\tilde{L}$-connection $(E,\rho)$.

Lemma 4.1**.**

Let $E$ be a left $A$-module and let $\operatorname{Diff}^{1}(E)$ be the module of first order differential operators on $E$. It follows $\operatorname{Diff}^{1}(E)$ is a left $\operatorname{P}$-module and $k$-Lie algebra. There is a map

[TABLE]

defined by

[TABLE]

where $I$ is the identity operator. It follows $\psi(\partial,ab)=a\psi(\partial,b)+\psi(\partial,a)b$ for all $a,b\in A$ and $\partial\in\operatorname{Diff}^{1}(E)$.

Proof.

A differential operator $\partial\in\operatorname{Diff}^{1}(E)$ is by definition an operator $\partial\in\operatorname{End}_{k}(E)$ with $[[\partial,aI]bI]=0$ for all elements $a,b\in A$ where $I$ is the identity operator. The module of differential operators $\operatorname{Diff}^{1}(E)$ has a left $A\otimes_{k}A$-module structure defined by $(a\otimes b.\partial)(e):=a\partial(be)$. It follows

[TABLE]

It follows $da.db.\partial:=[[\partial,bI],aI]=0$ hence for any element $w\in I^{2}$it follows $w\partial=0$ and it follows $\operatorname{Diff}^{1}(E)$ is a left $\operatorname{P}$-module. One checks the product

[TABLE]

defines a $k$-Lie algebra structure on $\operatorname{Diff}^{1}(E)$. The rest is trivial and the Lemma follows. ∎

Definition 4.2**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $E$ be a left $A$-module. An $\tilde{L}$-connection $\rho$ is a map

[TABLE]

of left $\operatorname{P}$-modules. The curvature of a connection $(\rho,E)$ is the map

[TABLE]

defined by

[TABLE]

Given two $\tilde{L}$-connections $(E,\rho_{E})$ and $(F,\rho_{F})$, a map of $\tilde{L}$-connections is a map of left $A$-modules

[TABLE]

with $\rho_{F}(u)\circ\phi=\phi\circ\rho_{E}(u)$ for all $u\in\tilde{L}$. Let $\underline{\operatorname{Conn}}(\tilde{L})$ denote the category of $\tilde{L}$-connections and maps of connections. Let $\underline{\operatorname{Conn}}(\tilde{L},Id)$ denote the category of $\tilde{L}$ connections $(\rho,E)$ with $\rho(D)=Id_{E}\in\operatorname{End}_{A}(E)$.

Note: It follows $\underline{\operatorname{Conn}}(\tilde{L},Id)$ is a full sub category of $\underline{\operatorname{Conn}}(\tilde{L})$.

Note: A connection $\rho:\tilde{L}\rightarrow\operatorname{Diff}^{1}(E)$ in $\underline{\operatorname{Conn}}(\tilde{L},Id)$ is in particular an $A\otimes_{k}A$-linear map with $\rho(D)=Id_{E}$.

Example 4.3**.**

The degeneracy loci and correspondence associated to a connection.

A connection in the sense of Definition 4.2 is a map of left and right $A$-modules

[TABLE]

and we may associate to $\rho$ several types of correspondences.

Definition 4.4**.**

Given a connection $\rho\in\operatorname{Hom}_{\operatorname{P}}(\tilde{L},\operatorname{Diff}^{1}(E))$. Let $I(\rho)\subseteq\operatorname{P}:=A\otimes_{k}A/I^{2}$ be the annihilator ideal of the element $\rho$. Let $Z_{\operatorname{P}}(\rho):=V(I(\rho))\subseteq\operatorname{Spec}(\operatorname{P})$ be the correspondence of $\rho$

By definition $I(\rho)$ is the set of elements in $x\in\operatorname{P}$ with $x\rho=0$. The ideal $I(\rho)$ gives rise to an ideal $J(\rho)\subseteq A\otimes_{k}A$ containing the square of the diagonal $I^{2}$. We get in a canonical way a correspondence $Z(\rho):=V(J(\rho))\subseteq\operatorname{Spec}(A\otimes_{k}A):=X\times X$ where $X:=\operatorname{Spec}(A)$. Hence the connection $\rho$ gives in a canonical way rise to a correspondence $Z(\rho)$ on $X$.

The left $\operatorname{P}$-module $\operatorname{Diff}^{1}(E)$ is projective as left and right $A$-module when $A$ is a regular ring of finite type over a field and $E$ a finite rank projective $A$-module. Hence when $\tilde{L}$ is projective as left and right $A$-module it follows a connection

[TABLE]

is a map of left and right $A$-modules that are projective as left and right $A$-modules. Hence a connection is a geometric object and we may use $\rho$ to define a correspondence on $X\times X$. For a classical connection

[TABLE]

it is not immediate how to do this, since $\nabla$ is a map of $k$-vector spaces and not $A$-modules. We may view an ordinary connection as an $A$-linear map

[TABLE]

where $L$ is an $A/k$-Lie-Rinehart algebra satisfying $\nabla(x)(ae)=a\nabla(x)(e)+x(a)e$ and it is not immediate how to define a correspondence from $\nabla$.

We may associate to $\rho$ the degeneracy locies $D_{\sigma}^{l}(\rho)$ and $D_{\sigma}^{r}(\rho)$ where $l$ and $r$ refer to the degeneracy loci of $\rho$ as a left and right $A$-linear map. Here $D_{\sigma}^{l}(\rho)$ and $D_{\sigma}^{r}(\rho)$ are defined using local trivializations of the map $\rho$. Locally the map $\rho$ is a matrix $M$ with coefficients in a commutative ring $B$ and we may use minors of $M$ of a given size to define an ideal in $B$ associated to $\sigma$ as done in [2]. It is possible to do this in a way that is intrinsic and does not depend of the choice of local trivialization of the map $\rho$. We get for any two $\sigma_{1},\sigma_{2}$ a correspondence $Z(\rho,\sigma_{1},\sigma_{2}):=D_{\sigma}^{l}(\rho)\times D_{\sigma}^{r}(\rho)\subseteq X\times X$. Hence we may associate different types of correspondences to the connection $\rho$. If $X$ is smooth over $k$ we get for each connection $\rho$ and each correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ an endomorphism

[TABLE]

defined by

[TABLE]

where $p,q:X\times X\rightarrow X$ are the projection maps, $\operatorname{CH}^{*}(X)$ is the Chow group of $X$ and $Z(\rho,\sigma_{1},\sigma_{2})\cap\alpha)$ is the intersection product. We get a similar construction for any reasonable cohomology theory $\operatorname{H}(-)$ equipped with a cycle map. One would like to relate the correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ and operation $I(\rho,\sigma_{1},\sigma_{2})$ to the Chern classes of $(E,\rho)$. Assume $\gamma:\operatorname{CH}^{*}(X\times_{k}X)\rightarrow\operatorname{H}^{*}(X\times_{k}X)$ is a cycle map and let $\alpha\in\operatorname{H}^{*}(X)$ be a cohomology class. we get an operator

[TABLE]

defined by

[TABLE]

Example 4.5**.**

Algebraic cycles and the Gauss-Manin connection.

If $\operatorname{H}^{*}(-)$ is a Weil cohomology theory, there are Lefschetz operators

[TABLE]

defined for any $i=0,\ldots,dim(X)$ and the operator $\Lambda$ is conjectured to be induced by an algebraic cycle $Z\subseteq X\times_{k}X$. The cycle $[Z]$ of the sub-scheme $Z$ induce an operator

[TABLE]

defined by

[TABLE]

It has been conjectured that when $X\subseteq\mathbb{P}^{n}_{k}$ is a smooth projective variety over an algebraically closed field, the Lefschetz operator $\Lambda$ is induced by an operator on the form $I_{\operatorname{H}}(Z)$ for some closed sub-scheme $Z\subseteq X\times_{k}X$. One wants to construct non-trivial cycle classes $\beta\in\operatorname{CH}^{*}(X\times_{k}X)$ and calculate the operator $I_{\operatorname{H}}(\beta)$. The Chow group $\operatorname{CH}^{*}(X\times_{k}X)$ is hard to calculate and there are no general formulas for it. One also wants a construction of all Weil-cohomology theories $\operatorname{H}(-)$. If one could realize a Weil cohomology theory $\operatorname{H}^{*}(-)$ as the cohomology $\operatorname{H}^{*}(\tilde{L},-)$ of a $\operatorname{D}$-Lie algebra $\tilde{L}$ as defined in [8], Definition 3.23, one could approach conjectures on algebraic cycles for smooth projective families of varieties. One has to develop the formalism of the Gauss-Manin connection for the cohomology theory $\operatorname{H}^{*}(\tilde{L},-)$ in the language of $\operatorname{D}$-Lie algebras. In previous papers (see [7]) I have delevoped a formalism aimed at making explicit calculations of such connections. If one could realize the action of $\Lambda$ on a Weil cohomology $\operatorname{H}^{*}(X)$ of the total space $X$ of a smooth projective family $\pi:X\rightarrow S$, as the action of $\nabla(x)$ where $x$ is a vector field on $S$ and $\nabla$ the Gauss-Manin connection, this could be a first step in the direction of determining if $\Lambda$ is induced by an algebraic cycle. The vector field $x$ and the Gauss-Manin connection $\nabla$ are algebraic objects, hence we would get an algebraic construction of $\Lambda$.

Definition 4.6**.**

Let $Z(\rho,\sigma_{1},\sigma_{2})$ be the correspondence of $\rho$ of type $(\sigma_{1},\sigma_{2})$. Let $I(\rho,\sigma_{1},\sigma_{2})$ be the Chow-operator of $\rho$ of type $(\sigma_{1},\sigma_{2})$.

When using the notion of a $\operatorname{D}$-Lie algebra $\tilde{L}$, the derivation property of the connection $\rho:\tilde{L}\rightarrow\operatorname{Diff}^{1}(E)$ is encoded in the right $A$-linearity of the map $\rho$. Hence the correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ encodes properties of the left $A$-linearity of the map $\rho$ and the derivation property (the right $A$-linearity) of the map $\rho$. Hence the correspondence $Z(\rho,\sigma_{1},\sigma_{2})$ and the endomorphism $I(\rho,\sigma_{1},\sigma_{2})$ dependes on the connection $\rho$ is a non-trivial way. It is not clear how to make a similar definition with connections on the form 4.4.1 and 4.4.2 depending in a non-trivial way on the derivation property of the connection. Hence in the case of an ordinary connection or a connection $(E,\nabla)$ on an $A/k$-Lie-Rinehart algebra $L$ it is essential we work with the associated $\operatorname{D}$-Lie algebra $F_{f}(L):=L(f^{\alpha})$ and $L(f^{\alpha})$-connection $C_{f}(E,\nabla)$ for some 2-cocycle $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ if we want to define the correspondence and Chow-operator of $(E,\nabla)$.

Example 4.7**.**

Restricted Lie-Rinehart algebras, logarithmic derivaties and Picard groups.

Let $B$ be a commutative ring over a field $k$ characteristic $p>0$ and let $(L,\alpha)$ be a restricted $B/k$-Lie-Rinehart algebra. Let $A:=ker(L)$ be the kernel of $L$ in $B$. The ring $B$ is a purely inseparable Galois extension of $A$ if $B$ is a finitely generated and projective $A$-module and $B[L]=\operatorname{Hom}_{A}(B,B)$ in the sense of [17]. Yuan proves in [17] the existence of an exact sequence

[TABLE]

where $\operatorname{Log}(B/A)$ is the logarithmic derivative group of $B/A$, $\operatorname{H}^{2}_{res}(L,B)$ is the restricted Lie-Rinehart cohomology group of $L$ with values in $B$, $\operatorname{Br}(B/A)$ the Brauer group of $B/A$ and $\operatorname{Pic}(B),\operatorname{Pic}(A)$ the Picard groups of $B$ and $A$. Hence in characteristic $p>0$, the restricted version of classical $B/k$-Lie-Rinehart cohomology calculates Picard groups and Brauer groups. One wants to generalize the exact sequence 4.7.1 to the case of restricted $\operatorname{D}$-Lie algebras.

Example 4.8**.**

$\operatorname{D}$-Lie algebras, groupoid schemes and algebraic stacks.

Recall the following from [1], page 140: Let $T:=(X,Y,s_{1},s_{2},t,p,i)$ be a groupoid scheme with $X$ the scheme of arrows and $Y$ the scheme of objects. We say $T$ is a schematic equivalence relation if the map $(s_{1},s_{2}):X\times X\rightarrow Y\times Y$ is a monomorphism. There is the following result:

Proposition 4.9**.**

Assume $Y$ is smooth over a field $k$ of characteristic zero. There is a one-to-one correspondence between locally trivial sheaves of $\mathcal{O}_{Y}/k$-Lie-Rinehart algebras $(\mathcal{L},\alpha)$ and formal infinitesimal groupoid schemes with $Y$ as a scheme of objects. Ynder this correspondence schematic equivalence relations corresponds to sub-sheaves of $\mathcal{O}_{Y}/k$-Lie-Rinehart algebras of the tangent sheaf $T_{Y/k}$.

Proof.

See [5], VIII.1.1.5. ∎

Hence there is a close connection between $A/k$-Lie-Rinehart algebras, moduli spaces and algebraic stacks. If $\mathcal{F}\subseteq T_{Y/k}$ is a locally trivial finite rank sub $\mathcal{O}_{Y}$-module and sheaf of $k$-Lie algebras, and the corresponding equivalence relation $R\subseteq Y\times_{k}Y$ is an etale equivalence relation, we may form the stack quotient $[Y/\mathcal{F}]$. The quotient $[Y/\mathcal{F}]$ is a Deligne-Mumford stack. The sheaf of universal enveloping algebras $\mathcal{U}(\mathcal{O}_{Y},\mathcal{F})$ is in a natural way a sheaf of rings on $[Y/\mathcal{F}]$ - the structure sheaf $\mathcal{O}_{[Y/\mathcal{F}]}:=\mathcal{U}(\mathcal{O}_{Y},\mathcal{F})$ of the stack $[Y/\mathcal{F}]$. A flat $\mathcal{F}$-connection $(\mathcal{E},\nabla)$ which is a quasi coherent sheaf of $\mathcal{O}_{Y}$-modules, is in a natural way a quasi coherent sheaf of $\mathcal{O}_{[Y/\mathcal{F}]}$-modules. Any Deligne-Mumford stack $[Y/R]$ arise by Proposition 4.9 from an involutive sub-bundle $\mathcal{F}\subseteq T_{Y/k}$. Hence if we are given the scheme of objects $Y$ of a DM-stack we may construct the equivalence relation $R$ on $Y$ using a bundle on the form of $\mathcal{F}$. The structure sheaf $\mathcal{O}_{[Y/R]}$ is a sheaf of filtered almost commutative rings, hence the ”ringed space” $([Y/R],\mathcal{O}_{[Y/R]})$ may be viewed as a non-commutative ringed space. Hence connections arise naturally when studying algebraic stacks, quasi coherent sheaves on algebraic stacks and non-commutative ringed spaces.

Note: Associated to any $\operatorname{D}$-Lie algebra $\tilde{L}$ which is of finite rank and projective as left $A$-module, we get a formal groupoid scheme over $Y:=\operatorname{Spec}(A)$ using the underlying $A/k$-Lie-Rinehart algebra of $\tilde{L}$ and Proposition 4.9.

Example 4.10**.**

$\operatorname{L}$-functions and cristalline cohomology.

Let $S:=\operatorname{Spec}(\mathcal{O}_{K})$ where $K$ is an algebraic number field and let $f:X\rightarrow S$ be a regular scheme of finite type over $S$of dimension $d$. There is an equality of $\operatorname{L}$-functions

[TABLE]

where $\operatorname{L}(X,s)$ is the global $\operatorname{L}$-function of the scheme $X$ and $\operatorname{L}(X(p),s)$ is the $\operatorname{L}$-function of the fiber $X(p):=f^{-1}(p)$ for a closed point $p\in S$. The residue field $\kappa(p)$ is a finite field of characteristic $q>0$ and the fiber $X(p)$ is a scheme of finite type over $\kappa(p)$. Hence there is an equality

[TABLE]

where $Z(X(p),t)$ is the Weil zeta function of $X(p)$. By the work of Kedlaya (see [6]) it follows the Weil conjectures can be proved using a p-adic cohomology theory $\operatorname{H}^{*}(-)$. In fact Kedlaya has proved the Weil conjectures for an arbitrary scheme $X$ of finite type over a finite field with no condition on smoothness or projectivity on $X$. If the Ext-group $\operatorname{Ext}^{*}(V,W)$ of two connections $V,W$ calculate cristalline cohomology (see [5]), it may be we can use $\operatorname{Ext}^{*}(V,W)$ to prove the Weil conjectures for any scheme $X$ of finite type over a finite field. If the Weil zeta function $Z(X(p),t)$ can be calculated using the Ext group $\operatorname{Ext}^{*}(V,W)$, it may be we can use such a description in the study of the global $\operatorname{L}$-function $\operatorname{L}(X,s)$ via the product formula in 4.10.1. The Ext-group is defined in complete generality and may be defined for connections on the family $X$. One has to calculate explicit examples to check if this idea leads to interesting constructions and results. If $X$ is a regular scheme the following is conjectured in [12]:

[TABLE]

Here

[TABLE]

is the $\operatorname{K}$-theoretic Euler characteristic of $X$ as defined in [12]. The conjecture in 4.10.3 is referred to as a conjecture due to Lichtenbaum, Deligne, Bloch, Beilinson and others in Wiles official problem description [16] for the BSD-conjecture. If it can be proved the Ext-group $\operatorname{Ext}^{*}(V,W)$ calculate cristalline cohomology of the fibers $X(p)$ for all primes $p\neq(0)$, it might be the group $\operatorname{Ext}^{*}(V,W)$ can be used in the study of Conjecture 4.10.3.

Lemma 4.11**.**

The following holds for an $\tilde{L}$-connection $\rho:\tilde{L}\rightarrow\operatorname{Diff}^{1}(E)$:

[TABLE]

for all elements $a\in A,u\in\tilde{L}$ and $e\in E$. It follows $\rho(D)\in\operatorname{End}_{A}(E)$. The curvature $R_{\rho}$ defines a map

[TABLE]

Assume $P:\tilde{L}\rightarrow\operatorname{End}_{A}(E)$ is an $\operatorname{P}$-linear map. It follows $\rho^{\prime}:=\rho+P$ is an $\tilde{L}$-connection. If $P(D)=0$ it follows $\rho^{\prime}(D)=Id_{E}$.

Proof.

Since $\rho$ is $\operatorname{P}$-linear it follows $\rho$ is $A\otimes_{k}A$-linear. It follows $\rho(au)=a\rho(u)$ and $\rho(ua)=\rho(u)a$. We get

[TABLE]

Since $Da=aD$ it follows $\rho(D)\in\operatorname{End}_{A}(E)$. The second statement holds since the element $[\rho(u),\rho(v)]:=\rho(u)\rho(v)-\rho(v)\rho(u)\in\operatorname{Diff}^{1}(E)$. Since $\rho$ is $\operatorname{P}$-linear it follows $\rho^{\prime}:=\rho+P$ is $\operatorname{P}$-linear. If $P(D)=0$ it follows $\rho^{\prime}(D)=\rho(D)+P(D)=Id_{E}$. The Lemma follows. ∎

Hence the notion of an $\tilde{L}$-connection introduced in Definition 4.2 agrees with the notion introduced in the paper [8].

Example 4.12**.**

The $L(f^{\alpha})$-connection associated to an $(L,\psi)$-connection.

Definition 4.13**.**

Let $(L,\alpha)$ an $A/k$-Lie-Rinehart algebra and let $\nabla:L\rightarrow\operatorname{End}_{k}(E)$ be an $(L,\psi)$-connection where $\psi\in\operatorname{End}_{A}(E)$. This means

[TABLE]

for all $a\in A,e\in E$ and $x\in L$. Let $\underline{\operatorname{Conn}}(L,\operatorname{End})$ denote the category of $(L,\psi)$-connections and morphisms. The endomorphism $\psi$ may vary. Let $\underline{\operatorname{Conn}}(L)$ denote the category of ordinary $L$-connections and morphisms of $L$-connections.

Note: If $\nabla:L\rightarrow\operatorname{End}_{k}(E)$ is an ordinary connection and $\psi\in\operatorname{End}_{A}(E)$ it follows $\nabla\circ\psi$ is an $(L,\psi)$-connection.

Recall the following construction: Let $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be a 2-cocycle and let $F_{f}(L):=(L(f^{\alpha}),\alpha_{f},\pi_{f},[,],z)$ be the $\operatorname{D}$-Lie algebra associated to $L$ and $f$. Define the following map:

[TABLE]

by

[TABLE]

Let $C_{f}(E,\nabla):=(E,\rho_{\nabla})$.

Let $(F,\nabla^{\prime})$ be an $(L,\psi^{\prime})$-connection with $\psi^{\prime}\in\operatorname{End}_{A}(E)$ and let $\phi:(E,\nabla)\rightarrow(F,\nabla^{\prime})$ be a map of connections. Define the following map

[TABLE]

It follows the map $C_{f}(\phi):(E,\rho_{\nabla})\rightarrow(F,\rho_{\nabla^{\prime}})$ is a map of $L(f^{\alpha})$-connections. Let $i:L\rightarrow L(f^{\alpha})$ be the canonical left $A$-linear map and let $(E,\rho)$ be an $L(f^{\alpha})$-connection:

[TABLE]

Let $\nabla_{\rho}:=\rho\circ i$. It follows

[TABLE]

is an $(L,\psi)$-connection. Let $R_{f}(E,\rho):=(E,\nabla_{\rho})$.

Lemma 4.14**.**

Let $(L,\alpha)$ be an $A/k$-Lie-Rinehart algebra and let $(E,\nabla)$ be an $L$-connection. Let $(F,\rho)$ be an $L(f^{\alpha})$-connection The construction above define functors

[TABLE]

by

[TABLE]

and

[TABLE]

by

[TABLE]

It follows $C_{f}\circ R_{L}=Id$ and $R_{f}\circ C_{f}=Id$ hence $C_{f}$ and $R_{f}$ are equivalences of categories for any 2-cocycle $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$

Proof.

The proof follows immediately from the constructions above. ∎

Theorem 4.15**.**

Let $(L,\alpha)$ be an $A/k$-Lie-Rinehart algebra and let $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ be a 2-cocycle. Lemma 4.14 gives an equivalence of categories

[TABLE]

from the category $\underline{\operatorname{Conn}}(L,\operatorname{End})$ of $(L,\psi)$-connections, to the category $\underline{\operatorname{Conn}}(L(f^{\alpha}))$ of $L(f^{\alpha})$-connections $\rho$ for any 2-cocycle $f$. Let $u:=az+x,v:=bz+y\in L(f^{\alpha})$ and let $e\in E$. Let $(E,\nabla)$ be an $L$-connection and let $\rho_{\nabla}:=C_{f}(\nabla)$. The following holds:

[TABLE]

For any $L(f^{\alpha})$-connection $(E,\rho)$ there is an $(L,\psi)$-connection $(E,\nabla)$ with $C_{f}(E,\nabla)=(E,\rho)$. If $\rho(z)=Id$ we may chose $(E,\nabla)\in\underline{\operatorname{Conn}}(L,Id)$.

Proof.

One checks for any $(L,\psi)$-connection $(E,\nabla)$ the corresponding map $\rho_{\nabla}$ is a map

[TABLE]

of left $\operatorname{P}$-modules. Moreover for any map of $(L,\psi)$-connections $\phi:(E,\nabla)\rightarrow(F,\nabla^{\prime})$ it follows the map

[TABLE]

is a map of $L(f^{\alpha})$-connections with $C_{f}(\phi\circ\psi)=C_{f}(\phi)\circ C_{f}(\psi)$. By definition $C_{f}\circ R_{f}=Id$ and $R_{f}\circ C_{f}=Id$ and the first claim follows. The statement on the curvature follows from Lemma 2.20 in [8]. In particular given an $L(f^{\alpha})$-connection $(E,\rho)$ it follows $(E,\rho)\cong C_{f}(R_{f}(E,\rho))$ and $R_{f}(E,\rho)\in\underline{\operatorname{Conn}}(L,\operatorname{End})$. The Theorem follows. ∎

We may classify $\tilde{L}$-connections in terms of $(L,\psi)$-connections in the case when the canonical quotient $L$ of $\tilde{L}$ is a projective $A$-module.

Corollary 4.16**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(E,\rho)$ be an $\tilde{L}$-connection. Assume the canonical quotient $L$ of $\tilde{L}$ is a projective $A$-module. It follows any $\tilde{L}$-connection $(E,\rho)$ is on the form $C_{f}(E,\nabla)$ where $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$ and $(E,\nabla)$ is an $(L,\psi)$-connection for some $\psi\in\operatorname{End}_{A}(E)$. If $\rho(D)=I$ it follows there is an $L$-connection $(E,\nabla)$ with $C_{f}(E,\nabla)=(E,\rho)$.

Proof.

By Theorem 3.9 there is an isomorphism $\tilde{L}\cong L(f^{\alpha})$ for $f\in\operatorname{Z}^{2}(\operatorname{Der}_{k}(A),A)$. The Corollary now follows from Theorem 4.15. ∎

Example 4.17**.**

Non-abelian extensions of $\operatorname{D}$-Lie algebras.

Definition 4.18**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(\rho,E)$ be an $\tilde{L}$-connection with $\rho(D)=Id_{E}$. Let $(\operatorname{End}(\tilde{L},E),\alpha_{E},\pi_{E},[,],\tilde{D})$ be the $\operatorname{D}$-Lie algebra constructed in Section 4 in [8].

The following Theorem generalize Theorem 2.14 from [9] from a connection on an $A/k$-Lie-Rinehart algebra to the case of a connection on a $\operatorname{D}$-Lie algebra:

Theorem 4.19**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(\rho,E)$ be an $\tilde{L}$-connection with $\rho(D)=Id_{E}$. Let $\operatorname{End}(\tilde{L},E)$ be the non-abelian extension of $\tilde{L}$ with $\rho$ from Definition 4.18. There is a canonical flat connection

[TABLE]

defined by $\rho^{1}(\phi,u):=\phi+\rho(u)$. The exact sequence

[TABLE]

is split in the category of $\operatorname{D}$-Lie algebras if and only if $E$ has a flat $\tilde{L}$-connection.

Proof.

By Proposition 4.9 in [8] we get an extensins of $\operatorname{D}$-Lie algebras

[TABLE]

where $\operatorname{End}(\tilde{L},E):=\operatorname{End}_{A}(E)\oplus\tilde{L}$ with $\tilde{D}:=(0,D)$ and $p_{E}:\operatorname{End}(\tilde{L},E)\rightarrow\tilde{L}$ defined by $p_{E}(\phi,u):=u\in\tilde{L}$. Define $c(\phi,u):=(c\phi,cu)$ and $(\phi,u)c:=(\phi c,uc)$ and define

[TABLE]

by

[TABLE]

and

[TABLE]

by

[TABLE]

Define moreover for any $z:=(\phi,u),w:=(\psi,v)\in\operatorname{End}(\tilde{L},E)$

[TABLE]

It follows from Proposition 4.9 in [8] that $\operatorname{End}(\tilde{L},E)$ is an extension of $\tilde{L}$ by the $A$-Lie algebra $\operatorname{End}_{A}(E)$. Asssume $s:\tilde{L}\rightarrow\operatorname{End}(\tilde{L},E)$ is a section of the map $p_{E}$. Hence $p_{E}\circ s=Id_{\tilde{L}}$ and $s$ is a map of $\operatorname{D}$-Lie algebras. It follows $s$ is $\operatorname{P}$-linear, a map of $k$-Lie algebras and $s(D)=(0,D)$. It follows $s(u)=(P(u),u)$ where $P:\tilde{L}\rightarrow\operatorname{End}_{A}(E)$ is a $\operatorname{P}$-linear map with $P(D)=0$. It follows from Lemma 4.11 the map $\rho^{\prime}:=\rho+P$ is an $\tilde{L}$-connection $\rho^{\prime}:\tilde{L}\rightarrow\operatorname{Diff}^{1}(E)$. One checks that the map $s$ is a map of $k$-Lie algebras if and only if $R_{\rho^{\prime}}=0$, hence the sequence 4.19.1 is split in the category of $\operatorname{D}$-Lie algebras if and only if $E$ has a flat $\tilde{L}$-connection. The Theorem is proved. ∎

Note: In [9], Theorem 2.4 a result similar to Theorem 4.19 is proved for $A/k$-Lie-Rinehart algebras. Note moreover that by Theorem 4.19 we may view any $\tilde{L}$-connection $(\rho,E)$ as a representation of the $k$-Lie algebra $\operatorname{End}(\tilde{L},E)$. Since the induced connection $\rho^{!}$ is flat, it follows $\rho^{!}$ is a map of $k$-Lie algebras.

Corollary 4.20**.**

Let $(\tilde{L},\tilde{\alpha},\tilde{\pi},[,],D)$ be a $\operatorname{D}$-Lie algebra and let $(\rho,E)$ be an $\tilde{L}$-connection with $\rho(D)=Id_{E}$. It follows the canonical quotient of $\operatorname{End}(\tilde{L},E)$ is isomorphic to the $A/k$-Lie-Rinehart algebra $\operatorname{End}(L,E)$ where $(L,\pi_{L})$ is the canonical quotient of $\tilde{L}$. If $L$ is projective there is an isomorphism $\operatorname{End}(\tilde{L},E)\cong\operatorname{End}(L(f^{\alpha}),E)$, where $L(f^{\alpha})=F_{f^{\alpha}}(L)$ and $F_{f^{\alpha}}$ is the functor from Example 3.3.

Proof.

The proof follows from Theorem 4.19 and Theorem 3.9 since $\operatorname{End}(\tilde{L},E):=\operatorname{End}_{A}(E)\oplus\tilde{L}$ and $\tilde{D}:=(0,D)$. ∎