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On jets, extensions and characteristic classes I

In this paper we give general definitions 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 relate affine non-commutative jets to liftings of modules. We also study the Kodaira-Spencer class KS(L) and relate it to the Atiyah class.


Introduction
In this paper we give general definitions of non-commutative jets in the local and global situation using square zero extensions and derivations. We study the functors Exan k (A, I) where A is any k-algebra and I is any left and right A-module and use this to relate affine non-commutative jets to liftings of modules. In the final section of the paper we define and prove basic properties of the Kodaira-Spencer class KS(L) and relate it to the Atiyah class.

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 Exan k (A, I) 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 Exan k (A, I) 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 i(I) 2 = 0. Let i : I → B and p : B → A denote the morphisms. Assume s is a map of k-vector spaces with the following properties: s(1) = 1 and p • s = id . Such a section always exist since B and A are vector spaces over the field k. Note: s gives the ideal I a left and right A-action. It follows thatC = 0 if and only if s is a ring homomorphism.
Lemma 2.2. The mapC gives rise to an element C ∈ Hom k (A ⊗ k A, I).
Hence we get a well defined element C ∈ Hom k (A ⊗ k A, I) as claimed.
Define the following product on I ⊕ A: (u, x) × (v, y) =(uy + xv + C(x, y), xy). (2.2.1) We let I ⊕ C A denote the abelian group I ⊕ A with product defined by 2.2.1.
of vector spaces is a unital ring isomorphism if and only if the following holds: for all x, y, z ∈ A.
Here we define uy = us(y) and xv = s(x)v. One checks that φ(1) = (1 − sp(1), 1) = (0, 1) = 1 and 1(u, x) = (u, x)1 = (u, x) for all (u, x) ∈ I ⊕ A. It follows the morphism φ is unital. Since C(x + y, z) = C(x, z) + C(y, z) and C(x, y + z) = C(x, y) + C(x, z) the following holds: . 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 defined above always has a left and right unit and is distributive. We check when the multiplication is associative.
It follows the multiplication is associative if and only if the following equation holds for the element C: xC(y, z) − C(xy, z) + C(x, yz) − C(x, y)z = 0 for all x, y, z ∈ A. The claim follows.
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 x ∈ A, u ∈ I. Recall I 2 = 0.
Proposition 2.7. The actions defined above give the ideal I a left and right Amodule structure. The structure is independent of choice of section s.
Proof. One checks that for any x, y ∈ A and u, v ∈ I the following holds: Also (xy)u − x(yu) = s(xy)u − s(x)s(y)u = (s(xy) − s(x)s(y))u = 0 since I 2 = 0. It follows (xy)u = x(yu) 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 Similarly us(x) = ut(x) 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 C : A⊗ k A → I be a morphism satisfying the cocycle condition 2.3.1.
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 C ∈ Hom k (A ⊗ k A, I) satisfying Equation 2.3.1.
Proof. The proof follows from the discussion above.
with j : J → F and q : F → B 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: i(I) 2 = j(J) 2 = 0. 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 x, y ∈ I and a ∈ A. We say two square zero extensions Proof. We first prove exan k (A, I) is a left and right C(A)-module. Let C, D ∈ exan k (A, I). This means C, D ∈ Hom k (A ⊗ k A, I) are elements satisfying the cocycle condition 2.3.1. let a, b ∈ C(A) ⊆ A be elements. Define aC, Ca as follows: Similarly one proves Ca ∈ exan k (A, I) hence we have defined a left and right action of C(A) on the set exan k (A, I). Given C, D ∈ exan k (A, I) define (C + D)(x, y) = C(x, y) + D(x, y). One checks that C + D ∈ exan k (A, I) hence exan k (A, I) has an addition operation. One checks the following hold: This gives the equality 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 there is a structure of left and right C(A)-module on the set of equivalence classes of extensions Exan k (A, I). The structure as left C(A)-module agrees with the one defined in [3].
Let φ ∈ Hom k (A, I). Let C φ ∈ Hom k (A ⊗ k A, I) be defined by One checks that C φ ∈ exan k (A, I) for all φ ∈ Hom k (A, I).
Definition 2.11. Let exan inn k (A, I) be the subset of exan k (A, I) of maps C φ for φ ∈ Hom k (A, I).
We let HH i (A, I) denote the i'th cohomology of this complex. It is the i'th Hochschild cohomology of A with values in I. , End k (A)) 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]).  (A) and a, b ∈ A. The Hochschild complex gives a map Proof. We first prove 2.19.1: Assume g(δ) = g(η) = 0. By definition this is if and only if there are maps φ, ψ ∈ End k (M ) such that the following holds: Proof. One easily checks that for any a, b ∈ B C and l, j ∈ J 1 I (E) the following hold: (a + b)i = ai + bi a(i + j) = ai + aj. Moreover 1i = i. It remains to check that a(bi) = (ab)i. Let a = (v, y) ∈ B C and b = (u, x) ∈ B C . Let also i = (w ⊗ e, f ) ∈ J 1 I (E). We get a(bi) = (v, y)((u, x)(w ⊗ e, f )) = (vx ⊗ f + yu ⊗ f + yxw ⊗ e + d(yx) ⊗ f, yxf ).
We also get It follows that (ab)i − a(bi) = 0 if and only if C(y, x) ⊗ f = 0, and the claim of the Proposition follows.

Note the abelian group J 1 I (E) is always a left A-module and there is an exact sequence of left A-modules
The class c I (E) has the property that c I (E) = 0 if and only if E has an I-connection: Let J ⊆ I ⊆ B C be the smallest two sided ideal containing Im(C) where C : A ⊗ k A → I is the cocycle defining B C . Let D C = B C /J and I C = I/J. We get a square zero extension of A by the square zero ideal I C . It follows D C = I C ⊕ A as abelian group. Since C(x, y) = 0 in I C it follows D C has a well defined associative multiplication defined by (u, x)(v, y) = (uy + xv, xy).
Also D C is the largest quotient of B C such that the ring homomorphism B C → D C fits into a commutative diagram of square zero extensions Example 2.23. First order commutative jets.
Let k → A be a commutative k-algebra and let I ⊆ A ⊗ k A be the ideal of the diagonal. Let J 1 A = A ⊗ A/I 2 and Ω 1 A = I/I 2 . We get an exact sequence of left A-modules A ⊕ A with the following product: (ω, a)(η, b) = (ωa + bη, ab) hence the sequence 2.23.1 splits. Let J 1 A (E) = Ω 1 A ⊗ E ⊕ E be the first order Ω 1 A -jet of E. We get an exact sequence of left A-modules Since the sequence 2.23.1 splits it follows J 1 A (E) is a lifting of E to the first order jet J 1 A .

Atiyah classes and Kodaira-Spencer classes
In this section we define 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 E be any O X -module and consider the following sequence of sheaves of abelian groups:  Proof. We first show it is an exact sequence of left O X -modules. The O X -module structure is twisted by the derivation d, hence we must verify that this gives a well defined left O X -structure on J 1 F (E). Let ω = (x ⊗ e, f ) be a local section of J 1 F (E) and let s, t be local sections of O X . We get the following calculation: is a left splitting. It follows s(e) = (∇(e), e) for e a local section of E. It follows ∇ is a generalized connection and the Theorem is proved.
is the first order jet bundle of E and the exact sequence above specialize to the well known Atiyah sequence: is the generalized first order jet bundle of E. The class AT(E) is defined for an arbitrary O X -module E and an arbitrary sub module I ⊆ Ω 1 X . Assume E = L ∈ Pic(X) is a line bundle on X. We get isomorphisms We get a morphism φ : Ext 1 OX (L, L ⊗ F ) → H 1 (X, F ). Proposition 3.6. The following hold: Hence the Atiyah class calculates the generalized first Chern class of a line bundle.
Proof. Let I = 0. It is well known that AT(L) calculates the first Chern class c 1 (L). From this the claim of the Proposition follows.
Let T X be the tangent sheaf of X. It has the property that for any open affine set U = Spec(A) ⊆ X the local sections T X (U ) equals the module Der F (A) of derivations of A. Let V E ⊆ T X be the subsheaf of local sections ∂ of T X with the following property: The section ∂ ∈ T X (U ) lifts to a local section ∇(∂) of End F (E| U ) with the following property: It follows V E ⊆ T X is a sheaf of Lie-Rinehart algebras -the Kodaira-Spencer sheaf of E. Define for any local sections a, b of O X ,∂ of V E and e of E the following: L(a, ∂)(e) = a∇(∂)(e) − ∇(a∂)(e). Here a, φ and ∂ are local sections of O X , End OX (E) and V E . We twist the trivial O X structure on End OX (E) ⊕ V E with the element L. We get a sequence of sheaves of abelian groups Proposition 3.9. The sequence defined above is an exact sequence of left O Xmodules. It is left split by a V E -connection ∇.
Proof. We need to check that LR(V E ) has a well defined left O X -module structure. By definition a(φ, ∂) = (aφ + L(a, ∂), a∂).
Definition 3.10. We get a characteristic class KS(E) ∈ Ext 1 OX (V E , End OX (E)) the Kodaira-Spencer class of E.
Assume V E is locally free and E = L ∈ Pic(X) is a line bundle on X. Assume also V * E = F = Ω 1 X /I for some submodule I. We get the following calculation: OX (O X , End OX (L) ⊗ F ) → H 1 (X, F ). We get a map ψ : Ext 1 OX (V E , End OX (L)) → H 1 (X, F ) of sheaves. Proof. The proof is left to the reader as an exercise.
We get the following diagram expressing the relationship between the characteristic classes defined above: