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**Helge MAAKESTAD**

Oslo, Norway

**E-mail:** h [email protected]

**Received date:** November 09, 2009; **Revised date:** July 19, 2010

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this paper, we give general denitions of non-commutative jets in the local and global situation using square zero extensions and derivations. We study the functors Exank(A; I), where A is any k-algebra, and I is any left and right A-module and use this to construct ane non-commutative jets. We also study the Kodaira-Spencer class KS(L) and relate it to the Atiyah class.

In this paper, we give general denitions of non-commutative jets in the local and global
situation using square zero extensions and derivations. We study the functors , where *A* is any k-algebra, and *I* is any left and right A-module and use this to construct
affine non-commutative jets. In the nal section of the paper, we dene and prove basic
properties of the Kodaira-Spencer class
and relate it to the Atiyah class.

We give an elementary discussion of structural properties of square zero extensions of arbitrary
associative unital *k*-algebras. We introduce for any *k*-algebra *A* and any left and right
*A*-module *I* the set of isomorphism classes of square zero extensions of *A* by *I*
and show it is a left and right module over the center *C*(*A*) of *A*. This structure generalize
the structure as left *C*(*A*)-module introduced in [3]. We also give an explicit construction of in terms of cocycles. Finally, we give a direct construction of non-commutative
jets and generalized Atiyah sequences using derivations and square zero extensions.

Let in the following *k* be a fixed base field, and let

be an exact sequence of associative unital *k*-algebras with . Assume *s* is a map of
*k*-vector spaces with the following properties:

and

Such a section always exists since *B* and *A* are vector spaces over the field *k*. Note: *s* gives
the ideal *I* a left and right *A*-action.

**Lemma 2.1.** *There is an isomorphism:*

*of k-vector spaces.*

**Proof.** Define the following maps of vector spaces: and . It follows that and and the
claim of the proposition follows.

Dene the following element:

by

It follows that if and only if s is a ring homomorphism.

**Lemma 2.2.** *The map gives rise to an element *

**Proof. **We easily see that and for all . Moreover, for any , it follows that

Hence we get a well-defined element as claimed.

Dene the following product on (2.1)

We let denote the abelian group with product dened by (2.1).

**Proposition 2.3.** *The natural isomorphism:*

*of vector spaces is a unital ring isomorphism if and only if the following holds:*

for all .

**Proof.** We have dened two isomorphisms of vector spaces , :

and

We define a product on the direct sum using and :

Here, we define

and

One checks that

and

for all . It follows that the morphism is unital. Since and the following holds:

and

Hence, the multiplication is distributive over addition. Hence for an arbitrary section *s* of
*p* of vector spaces mapping the identity to the identity, it follows the multiplication dened
above always has a left and right unit and is distributive. We check when the multiplication
is associative:

Also,

It follows that the multiplication is associative if and only if the following equation holds for
the element *C*:

for all . The claim follows.

Let

(2.2)be the *cocycle condition*.

**Definition 2.4.** Let be the set of elements satisfying the
cocycle condition (2.2).

**Proposition 2.5.** *Equation (2.2) holds for all* :

**Proof.** We get,

and

We get

and the claim follows.

**Corollary 2.6.** *The morphism is an isomorphism of unital associative
k-algebras.*

**Proof.** This follows from Proposition 2.5 and Proposition 2.3.

Hence, there is always a commutative diagram of exact sequences:

where the middle vertical morphism is an isomorphism associative unital *k*-algebras.

Define the following left and right *A*-action on the ideal *I*:

where *s* is the section of *p* and . Recall .

**Proposition 2.7.** *The actions defined above give the ideal I a left and right A-module
structure. The structure is independent of choice of section s.*

**Proof.** One checks that for any and , the following holds:

Also,

since . It follows that , hence *I* is a left *A*-module. A similar argument
prove *I* is a right *A*-module. Assume *t* is another section of *p*. It follows that

since . It follows that . Similarly, hence *s* and *t* induce
the same structure of *A*-module on *I* and the proposition is proved.

We have proved the following theorem: let *A* be any associative unital *k*-algebra and let
*I* be a left and right *A*-module. Let be a morphism satisfying the cocycle
condition (2.2).

**Theorem 2.8.** *The exact sequence:*

*is a square zero extension of A with the module I. Moreover, any square zero extension of A
with I arise this way for some morphism satisfying equation (2.2).*

**Proof.** The proof follows from the discussion above.

Let

with and and

with and be square zero extensions of associative *k*-algebras *A, B*
with left and right modules *I, J*. This means the sequences are exact and the following holds . A triple (*w, u, v*) of maps of *k*-vector spaces giving rise to a commutative
diagram of exact sequences:

is a morphism of extensions if *u* and *v* are maps of *k*-algebras and *w* is a map of left and
right modules. This means

for all and .

We say two square zero extensions:

and

are *equivalent* if there is an isomorphism of *k*-algebras making all diagrams
commute.

**Definition 2.9.** Let denote the set of all isomorphism classes of square zero
extensions of *A* by *I*.

**Theorem 2.10.** *Let C(A) be the center of A. The set is a left and right module
over C(A). Moreover, there is a bijection:*

*of sets.*

**Proof.** We first prove that is a left and right *C(A)*-module. Let . This means are elements satisfying the cocycle condition
(2.2). let be elements. Define *aC, Ca* as follows:

and

We see

hence . Similarly, one proves hence we have defined a left
and right action of *C(A)* on the set *. Given * define

One checks that hence has an addition operation. One checks the following hold:

hence the set is a left and right *C(A)*-module. Dene the following map: let be an equivalence class of a square zero extension. Define

by

We prove this gives a well-defined map of sets. Assume are two elements in . Note: we use brackets to denote isomorphism classes of extensions. The two extensions are equivalent if and only if there is an isomorphism:

of *k*-algebras such that all diagrams are commutative. This means that

for all . We get

This gives the equality:

for all . Hence , and the map *φ* is well defined. It is clearly an injective map. It is surjective by Theorem 2.8 and the claim of the
theorem follows.

Theorem 2.10 shows that there is a structure of left and right *C(A)*-module on the set
of equivalence classes of extensions . The structure as left *C(A)*-module agrees
with the one defined in [3].

Let . Let be defined by

One checks that for all .

**Definition 2.11.** Let be the subset of of maps for .

**Lemma 2.12.** *The set is a left and right sub C(A)-module.*

**Proof.** The proof is left to the reader as an exercise.

**Definition 2.13.** Let be the image of under the
bijection .

It follows that is a left and right sub *C(A)*-module. Recall the definition of the *Hochschild complex* as follows.

**Definition 2.14.** Let *A* be an associative *k*-algebra, and let *I* be a left and right *A*-module. Let . Let be defined as follows:

We let denote the *i*'th cohomology of this complex. It is the *i'th Hochschild
cohomology* of *A* with values in *I*.

**Proposition 2.15.** *There is an exact sequence:*

*of left and right C(A)-modules.*

**Proof.** The proof is left to the reader as an exercise.

**Example 2.16.** Characteristic classes of *L*-connections.

Let *A* be a commutative *k*-algebra and let be a Lie-Rinehart algebra.
Let *W* be a left *A*-module with an *L*-connection . In [6], we define a
characteristic class when *W* is of finite presentation, is
the open set, where *W* is locally free, and is the Lie-Rinehart cohomology of with values in . If *L* is locally free, it follows that , where
*U(L)* is the generalized universal enveloping algebra of *L*. There is an obvious structure of
left and right *U*(*L*)-module on and an isomorphism:

of abelian groups. The exact sequence 2.15 gives a sequence:

with *A* = *U*(*L*) and *I* = End_{}* _{k}*(

of the class:

we get a generalization of the characteristic class from [6] to arbitrary Lie-Rinehart algebras
*L*. This problem will be studied in a future paper on the subject (see [7]).

**Example 2.17.** Non-commutative Kodaira-Spencer maps.

Let *A* be an associative *k*-algebra, and let *M* be a left *A*-module. Let be the *module of first-order differential operators *on *A*. It is defined as follows: an element is in *D ^{1}*(

by

Here, , and . Since , we get a well-defined map. Let for
any and . It follows is an endomorphism of *M*.
We get

Hence,

for all and . The Hochschild complex gives a map:

and

It follows that we get a map:

We get an induced map:

**Lemma 2.18.** *The following holds*

**Proof.** The proof is left to the reader as an exercise.

One checks that . It follows that we get an induced map:

the *non-commutative Kodaira-Spencer map*.

**Lemma 2.19.** *Assume A is commutative. The following hold:*

**Proof.** We first prove (2.3): assume . By definition, this is if and only if
there are maps such that the following hold:

One checks that conditions (2.7) and (2.8) hold if and only if the following hold:

and

We claim . We get

Hence, and is a *k*-Lie algebra. It is an *A*-module since *g* is
*A*-linear, hence it is a Lie-Rinehart algebra. Claim (2.3) is proved. Claim (2.4) and (2.5)
follows from the proof of (2.3). Claim (2.6) is obvious and the lemma is proved.

The Lie-Rinehart algebra is the *linear Lie-Rinehart algebra* of *M*.

Let in the following *E* be a left and right *A*-module.

**Definition 2.20.** Let

be the *first-order I-jet bundle of E*.

Pick a derivation of left and right modules. This means that

for all . Let and define the following left *B ^{C}*-action on:

for any elements and .

**Proposition 2.21.** *The abelian group is a left B ^{C}-module if and only if 0 for all and .*

**Proof.** One easily checks that for any and the following hold:

Moreover,

It remains to check that . Let and . Let also . We get

We also get

It follows that

if and only if

and the claim of the proposition follows.

Note the abelian group is always a left *A*-module and there is an exact sequence
of left *A*-modules:

defining a characteristic class:

The class *c _{I}* (

with

Let be the smallest two-sided ideal containing *Im*(*C*), where is the cocycle defining *B ^{C}*. Let and . We get a square zero extension:

of *A* by the square zero ideal *I ^{C}*. It follows that as abelian group. Since in

Also *D ^{C}* is the largest quotient of

**Definition 2.22.** Let

be the *first-order I ^{C}-jet bundle of E*.

**Example 2.23.** First-order commutative jets.

Let be a commutative k-algebra, and let be the ideal of the diagonal. Let and . We get an exact sequence of left A-modules:

(2.9)It follows that with the following product:

hence the sequence (2.9) splits. Let be the first-order -of *E*. We
get an exact sequence of left *A*-modules:

Since the sequence (2.9) splits, it follows that is a lifting of *E* to the first-order jet .

In this section, we dene and prove some properties of Atiyah classes and Kodaira-Spencer classes.

Let *X* be any scheme defined over an arbitrary basefield *F* and let Pic(*X*) be the *Picard
group* of *X*. Let be the following subsheaf of abelian groups: for any open set , the group is the multiplicative group of units in . Define for any open
set the following morphism:

defined by

where *d* is the universal derivation and .

**Lemma 3.1.** *The following holds*:

for

**Proof.** The proof is left to the reader as an exercise.

Hence, defines a map of sheaves of abelian groups. The map dlog induces a map on cohomology

and by definition

Let be any sub -module, and let be the quotient sheaf. We get a derivation:

by composing with the universal derivation. We get a canonical map:

and we let

be the image of under this map.

**Definition 3.2.** The class is the *first Chern class* of the line bundle . The class is the *generalized first Chern clas*s of

Let be any -module and consider the following sequence of sheaves of abelian groups:

where

as sheaf of abelian groups. Let *s* be a local section of , and let be a local section
of over some open set *U*. Make the following definition:

It follows that the sequence

is a short exact sequence of sheaves of abelian groups. It is called the *Atiyah-Karoubi sequence*.

**Definition 3.3.** An -connection is a map:

of sheaves of abelian groups with

**Proposition 3.4.** *The Atiyah-Karoubi sequence is an exact sequence of left -modules. It
is left split by an -connection.*

**Proof.** We first show that it is an exact sequence of left -modules. The -module structure is twisted by the derivation *d*, hence we must verify that this gives a well-defined left -structure on . Let be a local section of , and let *s*, *t* be
local sections of . We get the following calculation:

It follows that is a left -module and the sequence is left exact. Assume that

is a left splitting. It follows that for *e* a local section of . It follows that is a generalized connection and the theorem is proved.

Note: If , we get that is the first-order jet bundle of and the exact
sequence above specializes to the well-known *Atiyah sequence*:

The Atiyah sequence is left split by a connection:

The -module is *the generalized first-order jet bundle of* .

**Definition 3.5.** The characteristic class:.

is called the *Atiyah class* of .

The class is defined for an arbitrary -module and an arbitrary sub module .

Assume that is a line bundle on *X*. We get isomorphisms:

We get a morphism:

**Proposition 3.6. ***The following holds*:

*Hence, the Atiyah class calculates the generalized first Chern class of a line bundle.*

**Proof.** Let . It is well known that calculates the rst Chern class . From
this, the claim of the proposition follows.

Let *T _{X}* be the tangent sheaf of

which satisfies

It follows that is a subsheaf of Lie algebras - the *Kodaira-Spencer sheaf* of .

Define for any local sections *a*, *b* of , of and *e* of the following:

**Lemma 3.7.** *It follows that *.

and the lemma is proved.

**Lemma 3.8.** *The following formula holds:*

*for all local sections a, b, and* .

**Proof.** We get

and the lemma is proved.

Let be the *linear Lie-Rinehart algebra* of . Let have
the following left -module structure:

Here, *a*, , are local sections of , and . We twist the trivial structure on the element *L*. We get a sequence of sheaves of abelian
groups:

where *i* and *p* are the canonical maps. An -linear map:

satisfying

is a *-connection* on .

**Proposition 3.9.** *The sequence defined above is an exact sequence of left -modules. It
is left split by a -connection .*

**Proof.** We need to check that has a well-defined left -module structure. By
definition,

We get

and it follows that the sequence is a left exact sequence of -modules. If

is a section, it follows that . One checks that is a -connection, and the theorem is proved.

**Definition 3.10.** We get a characteristic class:

the Kodaira-Spencer class of .

Assume that is locally free and is a line bundle on *X*. Assume also
that for some submodule .We get the following calculation:

We get a map:

of sheaves.

**Proposition 3.11.** *The following holds: there is an equality:*

*in* . *Hence the Kodaira-Spencer class calculates the class *.

**Proof.** The proof is left to the reader as an exercise.

We get the following diagram expressing the relationship between the characteristic classes dened above:

The following equation holds in :

- Andre M (1974) Homologie des algebrescommutatives. Die Grundlehren der mathematischenWissenschaften206, Springer-Verlag, Berlin.
- Atiyah M (1957) Complex analytic connections in fibre bundles.Trans Amer Math Soc85: 181-207.
- Grothendieck A (1964) Elements de geometriealgebrique IV. Etude locale des schemas et des morphismes de schemas, InstHautes Etudes SciPubl Math 20: 101-355.
- Karoubi M (1987) Homologiecyclique et K theorie.Asterisqueno 147- 148.
- Maakestad H (2005) A note on principal parts on projective space and linear representations.ProcAmer Math Soc133: 349-355.
- Maakestad H(2007) Chern classes and Lie-Rinehart algebras Indag Math New Ser 18: 589-599.
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