1. Introduction and Preliminaries
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Elie Cartan in 1899; it allows for a natural, metric-independent generalization of Stokes’ theorem, Gauss’s theorem, and Green’s theorem from vector calculus. If a k-form is thought of as measuring the flux through an infinitesimal k-parallelotope, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)-parallelotope.
In order to prove the results regarding algebraic clusters and their coordinate rings, one of the methods is to study the Kahler module of differential operators. This notion of studying the Kahler module may reduce questions about algebras to questions regarding module theory.
Differential forms are ubiquitous in modern mathematical physics, and their relevance for computations has increasingly been realized. Differential forms in mathematical physics have been studied by C. von Westenholz. An example or two will give the flavor of the subject. First, let M be an n-dimensional smooth differentiable manifold that is thought of as the configuration space of a mechanical system with n degrees of freedom. Each point of M has a neighborhood with a local coordinate system
. When the system is in motion, we need not only the coordinates
of a point of M, but also the momentum vector
at q. Thus, we are led to the phase space, or cotangent bundle of M, which is denoted
. This space already has an interesting structure: the differential form of degree one with local expression:
which is really a global quantity on
. Its exterior derivative:
is automatically a global quantity on
, and an exterior differential form of degree two (skew-symmetric covariant two-tensor) [
1].
In this study, we searched the homological features of differential operators and the Kahler modules of second-degree exterior derivations. Let X be a commutative algebra to an algebraically closed field k with characteristic 0. Let
and
symbolize the Kahler module of nth-degree derivations and standard nth degree k-derivation of X, respectively. The double
has the universal functioning feature that for any X-module N and any high derivation
of degree
, there is only an X-homomorphism
; so
.
is produced by the set
. Therefore, if X is parochially produced k-algebra, in turn,
will be parochially produced X-module. Let X and Y be commutative algebras onto an algebraically closed field k with the characteristic 0. Then,
is a commutative ring with unit by describing:
where
and
. Let U and V be ideals of X and Y, respectively. If
and
are standard homomorphisms of k-algebras, there exists an k-algebra isomorphism:
Northcott [
2].
2. Background Material
All of the rings considered in this paper are commutative with identity, and all of the fields are of characteristic zero, unless otherwise stated. Let’s say that X is a commutative algebra onto an algebraically closed field k of characteristic 0, and M is an X-module. For any non-negative integer n, by the universal nth degree differential operator on M, we denote a pair
that is composed of an R-module
and a differential operator
such that for any nth degree differential operator D from M to an randomly X-module N, there is only an X-module homomorphism α from
to N, which satisfied
. The module
is named the Kahler module of the nth degree differential operators on M [
3]. For the case M = X,
is produced by the cluster
. Therefore, if X is parochially produced k-algebra, then
will be a parochially produced X-module.
Let M and N be X-modules. A bilinear function
is named and alternating if
for any
. Let
be the tensor multiplication of M with itself, and let G be the submodule of
that is produced by the member of the form
where
. Bear in mind the next factor module:
The module
is called to be the second exterior power of M [
4].
Lemma 1. Let T be an X-module andbe a linear alternating map. Then, there exists an R-module homomorphism; so, the subsequent diagram:commutes. Erdogan [4]. Proposition 1. Let M be an X-module, T be a submodule of M, andbe a submodule ofproduced by the cluster.
Then, there is an X-module isomorphism:Erdogan [4]. Proposition 2. Assume thatis the Kahler module of differential operators of degree a onwith the universal differential.
Then, there exists only the X-module homomorphism: So, the next diagram:commutes. Erdogan [4]. Remark 1. Let X be a k-algebra andbe an X-module homomorphism, as given in Proposition 2. In the circumstances, we have an exact sequence of X-modules as follows:where i is the inclusion map, and p is the natural surjection:Erdogan [4]. Lemma 2. Let X be a commutative k-algebra. We presume thatis the Kahler module of derivations of X with the universal derivation.
In the present case, the function:is a differential operator of degree 1 overwhere[5]. Proposition 3. There is a split exact sequence of X-modules:Hart [6]. Under the favor of this test, the conditions and are found in the following result.
Theorem 1. Take into consideration the affine k-algebras X and Y. Let U be an ideal of X, andbe the standard nth degree k-derivation of X. Imagine that P is a submodule ofthat is produced by all of the members of the style.
In that case, the sequence:is an exact sequence of-modules [
7].
Proposition 4. Let U and V be ideals of X and Y, in return. At that rate, there is an exact sequence:of-modules [8]. Theorem 2. Take into consideration the affine k-algebras X and Y. Let U and V be ideals of X and Y, respectively, and say thatand P is a submodule ofthat is produced by all of the members of the style, whereis the standard nth degree k-derivation of.
At that, the sequence:is an exact sequence of-modules [
8].
Proposition 5. Assume thatandare polynomial algebras, and let U and V be the ideals produced by membersand membersof X and Y, respectively, and.
Hence, K is produced by cluster:Olgun and Erdogan [8]. Corollary 1. Letandbe the nth degree Kahler derivation operators. Therefore,is produced by:Olgun and Erdogan [8]. Theorem 3. Conceive affine k-algebras X and Y. Let U and V be ideals of X and Y, respectively., and say that. Given that P is a submodule ofproduced by all of the members of style, whereis the standard nth degree k-derivation of. Hence:
(ii)Olgun and Erdogan [8]. 3. Main Results
Throughout this section, X and Y affine k-algebras. Let U and V be the ideals of X and Y, respectively, and let’s say that
. In this section, we will be studying the second-degree exterior derivation of Kahler modules
on
. Let us firstly mention that there exists an exact sequence:
of
-modules where Q is a free
-module, and T is the submodule of Q [
8].
Definition 1. Letbe a commutative k-algebra. Let’s suppose thatis the Kahler module of derivations ofwith the universal derivation.
So, the function:is a differential operators of degree 1 over.
Let
be a commutative k-algebra and
be the Kahler module of second-degree derivations of
. Also, let
be the Kahler module of differential operators of a degree less than or equal to 1 on
with a universal differential operator
. There exists
; so, the next diagram:
commutes, and
.
Let
. By the universal feature of
, there is a
-module homomorphism:
So, the next diagram:
commutes, and
.
Remark 2. Letbe an affine k-algebra. The function:is an isomorphism of-modules whereand. Here,is a free-module with bases, andis a submodule of.
Theorem 4. Letbe an affine k-algebra. At that rate:is an exact sequence of modules.
Proof of Theorem 4. It is enough to indicate that the sequence is exact at
. Consider
is produced by
for
,
and that
□
This is to say that
is included in the
. Hence, we have a reduced function:
Assume that
and
in Remark 2. The cluster:
Since
is a free
-module with bases:
we can define a function:
Therefore, if
is a producing set for K, we have:
Hence,
. So, q reduces an
-module homomorphism:
It is nearly explicit that and are the identities, and so, Therefore, the sequence is exact.
Proposition 6. Letbe a local k-algebra of dimension 1. Then,is a regular ring if.
Proof of Proposition 6. Assume that
be a local k-algebra of dimension 1. So,
is a free
-module of rank 1, and
On the other hand, suppose that
. Let
be the maximal ideal of
, where
is the maximal ideal of X and
is the maximal ideal of Y. Then, we have:
□
Since:
is a vectorspace onto the
, it instantly follows that either:
or:
Then, ; so, from Nakayama’s lemma, , which is a discrepancy. Therefore, we get , and is a free -module of rank 1 just as we want. Here, , which is the number of members of a minimal producing set of .
Theorem 5. Letbe an affine k-algebra of dimension 1. Letbe as over. Then,is a regular k-algebra ifis a surjective-module homomorphism.
Proof of Theorem 5. We have seen that if
is regular of dimension 1, then
is an isomorphism. On the other hand, assume that
is surjective. Then, by the exact sequence in Proposition 6, we see that
Let
be the maximal ideal of
. Then:
So, we attain that is regular just as we want. □
Theorem 6. Letbe an affine domain of dimensionSuppose thatis a projective-module. Then,is regular.
Proof of Theorem 6. We need to see that is projective. Due to , it follows that just as we want. □
Now, we will give an example with regard to our above results.
Example 1. LetandLetandbe ideals of X and Y, respectively, and let’s say thatLet Q be the-module produced by:and let T be a submodule of Q produced by: Now, the rank of
with bases:
The rank of
with bases:
And finally, the rank of
with bases:
So, we have the exact sequence:
of
-modules. Therefore, we have also seen that the projective dimension of
; that is,
4. Discussion
Many studies related to the exterior derivation of Kahler modules have been done by researchers. Especially, a large work area related to first-degree exterior derivations of Kahler modules was created. In this paper, we investigated some homological properties of second-exterior derivations of Kahler modules. We believe that the results we found are useful, particularly in future works. Therefore, now whether or not to calculate higher degrees of exterior derivation of Kahler modules, that is the third degree, the fourth degree or the higher degree of exterior derivation, of Kahler modules comes to mind. Furthermore, we think that future research ought to relate to the following questions:
- (1)
Can we calculate a higher degree of exterior derivation of Kahler modules?
- (2)
Under which conditions can we calculate higher-degree orders of exterior derivation of Kahler modules?
- (3)
Which Kahler modules have finite projective dimensions?
5. Conclusions
We know that exterior derivation has an important place not only in mathematical physics, but also in commutative algebra. Study on the structure of the exterior derivation of Kahler modules over presents interesting aspects for both mathematical physics and commutative algebra. The second exterior derivation of Kahler modules over has never been studied before in detail. So, in this paper, by using the concept of the first degree of exterior derivation of Kahler modules, we searched the new approach for ways to discover the homological features of the second degree of exterior derivation of Kahler modules. Finally, we investigated some interesting properties of the algebras of Kahler modules.