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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On jets, extensions and characteristic classes I

Helge MAAKESTAD

Oslo, Norway
E-mail: h [email protected]

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

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Abstract

In this paper, we give general de nitions 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.

1 Introduction

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 image, 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 image and relate it to the Atiyah class.

2 Jets, liftings, and small extensions

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 image 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 image 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

image

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

image

and

image

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:

image

of k-vector spaces.

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

Dene the following element:

image

by

image

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

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

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

image

Hence we get a well-defined element image as claimed.

Dene the following product on image image (2.1)

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

Proposition 2.3. The natural isomorphism:

image

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

image

for all image.

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

image

and

image

We define a product on the direct sum image using image and image:

image image

Here, we define

image

and

image

One checks that

image

and

image

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

image

and

image

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:

image

Also,

image

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

image

for all image. The claim follows.

Let

image (2.2)

be the cocycle condition.

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

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

Proof. We get,

image

and

image

We get

image

and the claim follows.

Corollary 2.6. The morphism image 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:

image

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

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

image

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

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 image and image, the following holds:

image

Also,

image

since image. It follows that image, 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

image

since image. It follows that image. Similarly, image 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 image be a morphism satisfying the cocycle condition (2.2).

Theorem 2.8. The exact sequence:

image

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 image satisfying equation (2.2).

Proof. The proof follows from the discussion above.

Let

image

with image and image and

image

with image and image 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 image. A triple (w, u, v) of maps of k-vector spaces giving rise to a commutative diagram of exact sequences:

image

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

image

for all image and image.

We say two square zero extensions:

image

and

image

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

Definition 2.9. Let image 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 image is a left and right module over C(A). Moreover, there is a bijection:

image

of sets.

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

image

and

image

We see

image

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

image

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

image

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

image

by

image

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

image

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

image

for all image. We get

image

This gives the equality:

image

for all image. Hence image, 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 image. The structure as left C(A)-module agrees with the one defined in [3].

Let image. Let image be defined by

image

One checks that image for all image.

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

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

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

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

It follows that image 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 image. Let image be defined as follows:

image

We let image 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:

image

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

image

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

image

with A = U(L) and I = Endk(A). If we can construct a lifting:

image

of the class:

image

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 image be the module of first-order differential operators on A. It is defined as follows: an element image is in D1(A) if and only if image for all image. Define the following map:

image

by

image

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

image

Hence,

image

for all image and image. The Hochschild complex gives a map:

image

and

image

It follows that we get a map:

image

We get an induced map:

image

Lemma 2.18. The following holds image

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

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

image

the non-commutative Kodaira-Spencer map.

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

image (2.3) image (2.4) image (2.5) image is the maximal Lie-Rinehart algebra satisfying (2.5). (2.6)

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

image (2.7) image (2.8)

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

image

and

image

We claim image. We get

image

Hence, image and image 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 image is the linear Lie-Rinehart algebra of M.

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

Definition 2.20. Let

image

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

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

image

for all image. Let image and define the following left BC-action onimage:

image

for any elements image and image.

Proposition 2.21. The abelian group image is a left BC-module if and only if image0 for all image and image.

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

image

Moreover,

image

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

image

We also get

image

It follows that

image

if and only if

image

and the claim of the proposition follows.

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

image

defining a characteristic class:

image

The class cI (E) has the property that cI (E) = 0 if and only if E has an I-connection:

image

with

image

Let image be the smallest two-sided ideal containing Im(C), where image is the cocycle defining BC. Let image and image. We get a square zero extension:

image

of A by the square zero ideal IC. It follows that image as abelian group. Since image in IC, it follows that DC has a well-defined associative multiplication defined by

image

Also DC is the largest quotient of BC such that the ring homomorphism image fits into a commutative diagram of square zero extensions:

image

Definition 2.22. Let

image

be the first-order IC-jet bundle of E.

Example 2.23. First-order commutative jets.

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

image (2.9)

It follows that image with the following product:

image

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

image

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

3 Atiyah classes and Kodaira-Spencer classes

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 image be the following subsheaf of abelian groups: for any open set image, the group image is the multiplicative group of units in image. Define for any open set image the following morphism:

image

defined by

image

where d is the universal derivation and image.

Lemma 3.1. The following holds:

image

for image

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

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

image

and by definition

image

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

image

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

image

and we let

image

be the image of image under this map.

Definition 3.2. The class image is the first Chern class of the line bundle image. The class image is the generalized first Chern class of image

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

image

where

image

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

image

It follows that the sequence

image

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

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

image

of sheaves of abelian groups with

image

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

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

image

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

image

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

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

image

The Atiyah sequence is left split by a connection:

image

The image-module imageis the generalized first-order jet bundle of image.

Definition 3.5. The characteristic class:.

image

is called the Atiyah class of image.

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

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

image

We get a morphism:

image

Proposition 3.6. The following holds:

image

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

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

Let TX be the tangent sheaf of X. It has the property that for any open affine set image the local sections TX(U) equal the module image of derivations of A. Let image be the subsheaf of local sections image of TX with the following property: the section image lifts to a local section image of image with the following property:

image

which satisfies

image

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

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

image

Lemma 3.7. It follows that image.

image

and the lemma is proved.

Lemma 3.8. The following formula holds:

image

for all local sections a, b, and image.

Proof. We get

image

and the lemma is proved.

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

image

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

image

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

image

satisfying

image

is a -connection on image.

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

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

image

We get

image

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

image

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

Definition 3.10. We get a characteristic class:

image

the Kodaira-Spencer class of image.

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

image

We get a map:

image

of sheaves.

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

image

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

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:

image

The following equation holds in image:

image

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