1. Introduction

The notions of connection and covariant derivative come from Riemannian geometry, where there is no distinction between them (just different points of view). We begin by doing a brief historical review of these notions.

In 1869 the German mathematician and physicist Elwin Bruno Christoffel (1829- 1900) published Über die Transformation der homogenen Differentialausdrücke zweiten Grades in which he introduced the famous symbols
(Christoffel’s notation was/ ^{8}. What Christoffel did not realize was the fact that the symbols he had discovered themselves determine a connection on the
-module of sections of the tangent bundle,
denote the
-algebra of all infinitely differentiable real-valued functions on the differential manifold
).

It was noticed after by the Italian mathematicians Gregorio RicciCurbastro (1853-1925) and Tullio Levi-Civita (1873-1941) (in ^{26}) that the Christoffel symbols obtained from the Riemannian metric could be used to create a “coordinate free” differential calculus. This extension of the Differential Calculus allowed a modification of partial differentiation to spaces with Riemannian metric (i.e. non-Euclidean) by means of a covariant derivative.

In fact, they defined the covariant derivative of a vector field in local coordinates) to be: An -linear endomorphism of (the sections of the tangent bundle associated to the n-dimensional differential manifold ) defined locally by the endomorphisms where the k-th component of is given by In the Euclidean space with the usual Riemannian metric, the sum is zero (since all the Christoffel symbols are zero) and hence the covariant derivative is just an ordinary partial derivative.

In the 1920’s the French mathematician Élie Joseph Cartan (1869-1951) developed a new notion of connection, the projective and conformal connections (see ^{5} and ^{6} .

In 1950 the French mathematician Jean-Louis Koszul gave an algebraic framework for regarding a connection as a differential operator. The Koszul connection was both more general than that of Levi-Civita, and was easier to work with because it finally was able to eliminate (or at least to hide) the awkward Christoffel symbols from the connection formalism ^{24}.

It was Cartan’s student Charles Ehresmann (1905-1979) who finally untangled and successfully classified all of the generalized and specific connections which had emerged in the first half of the 20th century ^{15}. He began this endeavor in an effort to understand Cartan’s connections from a global point of view. Toward this end Ehresmann introduced the concept of fiber bundles (independently of Whitney and Steenrod). Ehresmann published his first notes on the subject during the period 1941-1944 in which he defines locally trivial principal bundles and their associated fiber bundles (also locally trivial). In his 1943 paper Sur les espaces fibrés associés une variété différentiable a manifold is defined by means of an atlas of local charts for the first time ^{14}.

For example, in ^{4} an (affine) connection on the differential manifold
is defined as a mapping
satisfy the following three properties for every
:

Since any vector field can be considered as a derivation in
(cf. ^{22}). So
above can be thought as an
-linear homomorphism from
to
) satisfying the third condition. Thus, it is very natural to study this notion in a purely algebraic context, we will do that in Subsection 2.1, where we call covariant derivative such kind of homomorphism (see also ^{20} and ^{23}). In particular, we can consider covariant derivatives for other vector bundles *E* on
. Related to this see example 2.6.

On the other hand, the notion of connection has appears in physics from time to time. For example, in 1954 the Chinese physicist Chen-Ning Franklin Yang (1922) and the American physicist Robert L. Mills (1927--1999) proposed a Gauge theory that allows us to understand the interaction proton-neutron. In their mathematical formulation, appear the now called Yang-Mills functional, which it is defined over the set of connection of a fix vector bundle. For example, if you consider the trivial complex vector bundle
with a given hermitian product on its sections, then we can deduce the Maxwell's equations of electrodynamics ^{28}.

However the notion of connection reappears again in the context of noncommutative geometry (^{9}, ^{10}). In fact, Connes introduced the notion of noncommutative differentials, which was used in place of Kähler differentials (in the definition of connection given in Subsection 2.1) by Krämer ^{25}, to conclude that the existence of such connections, it is verified exactly for projective modules over not (necessarily) commutative rings (cf. Theorem 5.2).

2. Connection and Covariant Derivative over modules^{3}

Let *k* be an algebraically closed field of characteristic 0, *A* a commutative *k*-algebra, and *M* an *A*-module (module means left module).

A derivation from *A* to *M* is a *k-*linear map
.

One can readily verify under this definition that

Set
be the *A*-module of derivation from *A* to *M* and
.

The Kähler differential module,
. The module of Kähler differential is an *A-*module
together with an universal derivation
which is universal in the following sense: for any *A-*module *M*, for any derivation
, there exists a unique *A-*module homomorphism
.

**2.1. Example**. Let
be the ring of polynomials in *n* variables.Then

Follows from the universal property that is the canonical basis of the free ).

**2.2. Remarks.** Let A be a commutative k- algebra.

1. The module of Kähler differential,
, exists (see ^{21}, ^{13}).

2. it is an isomorphism of A- modules. In particular,

If is a finitely generate A-module having as a set of generators, then generated as A-module.

**2.1. Connection & Covariant Derivative**

We define a connection on M to be a K- linear homomorphism

satisfying (𝑑is the universal derivation).

We denote by the vector space of k-linear endomorphisms of M. We will consider the notion of a covariant derivative on M, as an A- linear homomorphism

**2.3. Proposition**. Let A be a commutative k- algebra. If the A- module M has a connection, then M has a covariant derivative.

**Proof.** Let
be a connection on M. Now, we use the universal property of tensor products to associate to each
an A- linear homomorphism
comes from the universal property of the Kähler differential module.

Now define defines a covariant derivative on M.

Thus we concluded easily that.

**2.4. Corollary.** Let 𝐴be a commutative k- algebra. If the A- module M does not admit a covariant derivative, then neither admit connections.

So an important question here is to know what class of modules admit connections. For example, given by the usual derivation defines a connection on the non-projective -module (t). On the other hand, as we shall see in the next proposition, every projective module has a connection.

**2.5. Proposition.** Let A be a commutative k- algebra and M an A- module. Set
. Then

1. Every projective A- module admits a connection.

3. If defines a right action of , which is free and transitive. Thus is an affine space. In particular, can be identified with

**Proof.** We prove this theorem by considering two cases. We first established that free modules have a connection. If F is a free A- module with basis
, then define
only for a finite set of i). It is easy to conclude that
is a connection on F.

Now, let M be a projective A-module. Thus we can consider an A-module 𝑁such that M x N is a free A-module. Hereafter, let / be a connection on M x N and define

where p_{
1
} is the projection from
. It is a straightforward verification to show that
is a connection on M.

The verification on second statement we left to the reader.

To prove the third statement, first note that defines a connection on M for every and . Now, follows from the definitions that defines a free right action of on To verify that this action is transitive, let us consider , we get the k- linear homomorphism given by . So, now it is sufficient to prove that the k-linear homomorphism is also an A-linear homomorphism. In fact,

Finally, fix , it is a bijection.

**2.6. Example. The**
**- module**
**admits a connection**

Let be a compact differentiable manifold and be the commutative - algebra of all infinitely differentiable real-valued functions on . For a differentiable (or smooth) vector bundle E on , set be the space of differentiable sections.

Next, we remember the correspondence established by Swan in ^{30} (details may be found in ^{27}) between

In fact, it is proven that: is a finitely generated projective - module. Conversely, if be a finitely generated projective - module, then there exists a vector bundle E such that .

Since every projective module admits a connection (cf. Prop. 2.5) and consequently a covariant derivative (cf. Prop. 2.3). Then the -module admits a connection, as well as a covariant derivative.

3. Some Counterexamples

We consider the commutative given by

with the usual operations of addition and multiplication on the polynomial ring.

Since each can be written in a unique way as by

here f’_{
1
} and f’_{
2
} denote the usual derivative of the polynomials f_{
1
} and f_{
2
} , respectively.

**3.1. Lemma**. Let
as in (3.2). Then 1.
. In particular, it is verified that
. 2.
3.
is a minimal set of generators of the
. 4. If

**Proof.** The first statement follows from the definition of
together with the fact that
space with basis

It is a straightforward verification to show that belong to Now consider and set

Having in mind that xy=0, we concluded that . Thus substituting (3.3) in the last equation, we get that

Thus from (3.3) we obtain that . Therefore .

Now suppose that / is a generator of the . Thus we have that

We can write and substitute it in (3.5). Then keeping in mind that (cf. first statement), we conclude that

So evaluating the expression for in (3.6) at 𝑦, we get that which implies that . Now let us to evaluate the expression for in (3.6) at x and y, respectively. Then we obtain

Since , then from the last equality above we get that , thus the first equality in (3.7) becomes . The last equation implies that . Thus we have arrived at a contradiction. So the third statement is proved.

The last item follows from the other statements already proved in this lemma.

2. The pair verifies the universal property of . In particular, we conclude that as -modules.

3. The elements as an -module. Moreover

**Proof.** It is a straightforward verification to show that d_{
A
} is a derivation. On the other hand, if N is an A- module and
Note that
is an
-linear homomorphism such that
. So it induces an
-linear homomorphism
After an easy computation we conclude that
is the unique
-linear homomorphism such that
.

Now, observe that applying the universal property to the -module and 𝑑 (the universal derivation). We obtain an -linear isomorphism such that .From where we deduce easily the third statement.

Finally, let us assume that , so applying the inverse of the -linear isomorphism above, we obtain that which implies that for some in . Writing we achieve that:

Since it is a basis of the - vector space , we conclude using the equations above that . Thus we have arrived to a contradiction. In a similar way we can show that

**3.1. A module without covariant derivative.**

Let be the algebra defined in (3.1) and set .

Let us suppose that there exists a covariant derivative on M. For simplicity, we will use the notation instead of . Assume that:

Since (see Lemma 3.1), we obtain using (3.9) that

Then follows from (3.10) and (3.8) that for some .

Now setting , follows from (3.11) that

Now have in mind one more time that (cf. (3.2)) to conclude from (3.8) and (3.9) that:

Then follows from (3.8) that

Thus follows from the second equality in (3.12) that and consequently that (after substitution of in the first equality in (3.12)). Thus we have arrived at a contradiction.

**3.2. A module with covariant derivative and without connection**

Let be the commutative -algebra defined in (3.1).

. The -module has a covariant derivative.

Define for (with as in (3.2)). Note that each is an -linear homomorphism and that is contained in the kernel of both homomorphisms. Thus they induce -linear endomorphisms from , which (keeping in mind the isomorphism in lemma 3.1) determine the - linear endomorphisms such that

Now we consider defined by . Note that is an A- linear homomorphism such that (since ). Thus, using again the isomorphism in lemma 3.1, we conclude that induces an A- linear homomorphism .

Now, an easy verification shows that for any Having in mind that and doing some more computations, we concluded that is a covariant derivative on .

. The -module does not have a connection.

Suppose that M admits a connection Thus we can write

Now remember that (cf. lemma 3.1). Therefore

Furthermore, if we multiply both sides of Eq. (3.13) by 𝑦 (and have in mind that ), we get:

Since (cf. third statement in Lemma 3.2), after substitution Eq. (3.15) becomes

Substituting (3.16) in Eq. (3.14), we arrive to equation:

On the other hand, since is a set of generators of the - module , it is a set of generators of as an - module, which verified that

Next, we apply the universal property of tensor products to get the - linear homomorphism such that

Now applying to Eq. (3.17), we obtain:

So evaluating the homomorphism in (3.19) a t , we get (using (3.18) that . Now, the last statement in lemma 3.2 assures us that . Thus we have arrived at a contradiction.

4. Equivalent notions and obstruction^{1}
^{13}
^{21}

Eriksen has proven in Proposition 4.7 (at p. 51 in ^{18} that the notions of connections and covariant derivative are equivalent when we consider regular k- algebras of finite type. For the reader's convenience, we review Eriksen's proof giving a few more details in the next proposition.

**4.1. Lemma.** Let A be a commutative k -algebra of finite type then
are finitely presented A- modules.

**Proof.** We first prove that
is a finitely presented A- module. In fact, if
is a set of generators of the k- algebra A then
is a set of generators of the A -module
. Thus it is a finitely generated A- module, and therefore also finitely presented, since A is a Noetherian ring.

In order to prove the second statement, we recall that
, for some polynomial algebra
in n variables and some ideal I in S since A is k- algebra of finite type. So, follows from Lemma 2.1.2 in ^{3} that
, which is a submodule of
. Since A is Noetherian, it is a finitely generated A- module, and therefore also finitely presented.

**4.2. Proposition.** Let A be a regular, commutative k - algebra of finite type. Then, every covariant derivative induces a connection. In fact, this establishes a correspondence between the notions of connection and covariant derivative.

**Proof.** Having in mind the isomorphism in (2.1) which associated to each derivation
the A- linear homomorphism
. Next, we use the universal property of tensor products to conclude that there exists a unique A -linear homomorphism

We propose in what follows to show that is an isomorphism. In order to do that, we are going to verify that, the localization of at the prime ideal p of A is an isomorphism for any prime ideal in A

We begin by noting that:

(4.1)

To arrive to the isomorphism in the last line above, we use the hypothesis of regularity of A and denote by n_{
p
} the Krull dimension of A_{
p
}

Now, Prop. 16.9 (in ^{13}) assured us that
as A_{
p
} modules. So, keeping in mind that
is an A- module finitely presented, we can apply Prop. 2.10 (in ^{13}) to concluded that
Now, follows from the universal property of the module of Kähler differentials,
, that
as A_{
p-
} modules. Moreover, using again the regularity of 𝐴, we have:

Thus from (4.2) we have

Now, having in mind that
- modules, and that
is also an A- module finitely presented, we can apply one more time Prop. 2.10 (in ^{13}) and (4.3) to get:

Now let us consider
be a basis of the free A_{
p
} -module
be a basis of the free A_{
p
} - module
.

Right now, if
are the unique elements in A_{
p
} such that
(here 𝑑 is the universal derivation for
). Then follows from the isomorphisms in (4.1) that the element
is taken in
.

Hereafter follows from the isomorphism in (4.4) that the element
determine an A_{
p
} -linear homomorphism such that
(have in mind that
for every
).

Finally, note that the isomorphism in (4.2) takes
. So
is a basis of the free A_{
p
} - module
, so
is a change of basis matrix of the free A_{
p
} - module
, which allows to conclude that
is an isomorphism for every prime ideal p of A. Therefore
is an isomorphism.

Now let be a given covariant derivative on M. For each A straightforward verification shows that is an A - linear homomorphism.

Next define . We leave to the reader the task of verifying that is a connection on M.

**4.1 Hochschild cohomology and obstruction.**

Let A be a commutative k-algebra and let Q be an A- bimodule. We define Hochschild-cohomology on A with values in Q as the cohomology of the following complex:

Let and so on. Let us also consider the coboundary operators be given by

In fact, , so an easy verification shows that . Also, we have that Therefore, (the elements in are usually named it inner derivations).

Thus, we have the following exact sequence:

Let us now consider
. Note that and (the last isomorphism is proved at p. 170 in ^{7}), and the exact sequence (4.5) takes the form:

The next proposition was proved by Eriksen (at p. 5 in ^{17}). In fact, we bring out the part what we are interested in.

**4.3. Proposition**. There exists an obstruction
, which is canonical and has the property that:

There exists a connection on M if and only if .

**Proof.** Let us consider
is the universal derivation. It's verified that
. Now makes
. Note that, if there is a connection
on M then
, which allows to conclude that
(from the exactness in (4.6)). Reciprocally, if
, so again from the exactness in (4.6), we have that there exists
such that
, so
is a connection on M.

5. Connections defined via noncommutative differentials

The result listed here can be found in Krähmer's Lecture 1 ^{25} and ^{29}

Here we considerer a k- algebra A not necessarily commutative. Now, if is an A- bimodule, then we say that is a derivation, if , easily we conclude that is k-linear).

Next we consider as an A- bimodule (with the A- actions given by right multiplication in the second tensor component).

Let the multiplication map determined by . Note that is an homomorphism of A- bimodules, so is an A- bimodule. Note that

is a derivation.

Krähmer proved in ^{25} that:

**5.1. Proposition**. Notation as above. It is verified that:

1. generated as an A- bimodule.

2. satisfied the following universal property: If L is any A- bimodule and is a derivation with , then factors uniquely through d, that is, there is a unique A- bimodule homomorphism .

We define a connection on M to be an k -linear homomorphism

Krähmer also established the equivalence between

**5.2. Theorem.** The connections on an A- module M correspond bijectively to A -linear splittings of the action map
. In particular, M admits a connection if and only if it is projective.

**Proof.** See the Theorem at p. 13 in ^{25} or Corollary 3.4 in ^{29}.

**5.1. The motivation for considering noncommutative differentials**

One of the major advances of science in the 20th century was the discovery of a mathematical formulation of quantum mechanics by Heisenberg in 1925. From a mathematical point of view, transition from classical mechanics to quantum mechanics amounts to, among other things, passing from the commutative algebra of classical observables to the noncommutative algebra of quantum mechanical observables. Recall that in classical mechanics an observable (e.g. energy, position, momentum, etc.) is a function on a manifold called the phase space of the system. Immediately after Heisenbergs work, ensuing papers by Dirac ^{12} and Born-Heisenberg-Jordan ^{2}, made it clear that a quantum mechanical observable is a (selfadjoint) operator on a Hilbert space called the state space of the system. Thus the commutative algebra of functions on a space is replaced by the noncommutative algebra of operators on a Hilbert space.

In the early 80's, Alain Connes (^{9},^{10})realized that a similar procedure can in fact be applied to areas of mathematics where the classical notions of space (e.g. measure space, or a smooth space) can not be used and can be replaced by a new idea of space, represented by a noncommutative algebra. His motivation can from the correspondence:

Which was established in the late 40's by the two Russian mathematicians Gelfand and Naimark (in ^{19}).

So when Noncommutative Geometry meets General Relativity a natural question arises about a satisfactory generalization of the notion of a linear connection. In order to do that Connes ^{9} has defined connections on modules using noncommutative differential forms. After this notion was extend to bimodules by Cuntz--Quillen in ^{11}.

**5.3. Remark.** The seed of the notion of connection coming from 1869 paper by Christofell ^{8} has grown to be a great tree. One of his latest fruit appear in theoretical physics around 1964 and reappear in the real word at CERN as the Higgs boson of the Standard Model on July 2012 (see ^{16}).