Kähler differential
Encyclopedia
In mathematics
, Kähler differentials provide an adaptation of differential form
s to arbitrary commutative ring
s or scheme
s.
in the 1930s. It was adopted as standard, in commutative algebra
and algebraic geometry
, somewhat later, following the need to adapt methods from geometry over the complex number
s, and the free use of calculus
methods, to contexts where such methods are not available.
Let R and S be commutative rings and φ:R → S a ring homomorphism
. An important example is for R a field
and S a unital algebra
over R (such as the coordinate ring of an affine variety).
An R-linear derivation on S is a morphism of R-modules
with R in its kernel, and satisfying Leibniz rule 
. The module of Kähler differentials is defined as the R-linear derivation 
that factors all others.
over R, where Ω1S/R is an S-module
, which is a purely algebraic analogue of the exterior derivative
. This means that d is a homomorphism of R-modules such that
for all s and t in S, and d is the best possible such derivation in the sense that any other derivation may be obtained from it by composition with an S-module homomorphism.
The actual construction of Ω1S/R and d can proceed by introducing formal generators ds for s in S, and imposing the relations
for all s and t in S.
Another construction proceeds by letting I be the ideal in the tensor product
, defined as the kernel
of the multiplication map:
, given by 
. Then the module of Kähler differentials of "S" can be equivalently defined by :Ω1S/R = I/I2, together with the morphism
Use in algebraic geometry
Geometrically, in terms of affine schemes, I represents the ideal defining the diagonal
in the fiber product of Spec(S) with itself over Spec(S) → Spec(R). This construction therefore has a more geometric flavor, in the sense that the notion of first infinitesimal neighbourhood of the diagonal is thereby captured, via functions vanishing modulo
functions vanishing at least to second order (see cotangent space for related notions).
For any S-module M, the universal property of Ω1S/R leads to a natural isomorphism
where the left hand side is the S-module of all R-linear derivations from S to M. As in the case of adjoint functors
(though this isn't an adjunction), this is more than just an isomorphism of modules; it commutes with S-module homomorphisms M → M and hence is an isomorphism of functors.
To get ΩpS/R, the Kähler p-forms for p > 1, one takes the R-module exterior power of degree p. The behaviour of the construction under localization of a ring
(applied to R and S) ensures that there is a geometric notion of sheaf
of (relative) Kähler p-forms available for use in algebraic geometry, over any field
R.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Kähler differentials provide an adaptation of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s to arbitrary commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s or scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s.
Presentation
The idea was introduced by Erich KählerErich Kähler
was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...
in the 1930s. It was adopted as standard, in commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, somewhat later, following the need to adapt methods from geometry over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, and the free use of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
methods, to contexts where such methods are not available.
Let R and S be commutative rings and φ:R → S a ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
. An important example is for R a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
and S a unital algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
over R (such as the coordinate ring of an affine variety).
An R-linear derivation on S is a morphism of R-modules
with R in its kernel, and satisfying Leibniz rule . The module of Kähler differentials is defined as the R-linear derivation that factors all others.Construction
The idea is now to give a universal construction of a derivationDerivation (abstract algebra)
In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...
- d:S → Ω1S/R
over R, where Ω1S/R is an S-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, which is a purely algebraic analogue of the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
. This means that d is a homomorphism of R-modules such that
- d(st) = s dt + t ds
for all s and t in S, and d is the best possible such derivation in the sense that any other derivation may be obtained from it by composition with an S-module homomorphism.
The actual construction of Ω1S/R and d can proceed by introducing formal generators ds for s in S, and imposing the relations
- dr = 0 for r in R,
- d(s + t) = ds + dt,
- d(st) = s dt + t ds
for all s and t in S.
Another construction proceeds by letting I be the ideal in the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
, defined as the kernelKernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of the multiplication map:
, given by . Then the module of Kähler differentials of "S" can be equivalently defined by :Ω1S/R = I/I2, together with the morphismUse in algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
Geometrically, in terms of affine schemes, I represents the ideal defining the diagonalDiagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
in the fiber product of Spec(S) with itself over Spec(S) → Spec(R). This construction therefore has a more geometric flavor, in the sense that the notion of first infinitesimal neighbourhood of the diagonal is thereby captured, via functions vanishing modulo
Modulo
In the mathematical community, the word modulo is often used informally. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
functions vanishing at least to second order (see cotangent space for related notions).
For any S-module M, the universal property of Ω1S/R leads to a natural isomorphism
where the left hand side is the S-module of all R-linear derivations from S to M. As in the case of adjoint functors
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
(though this isn't an adjunction), this is more than just an isomorphism of modules; it commutes with S-module homomorphisms M → M and hence is an isomorphism of functors.
To get ΩpS/R, the Kähler p-forms for p > 1, one takes the R-module exterior power of degree p. The behaviour of the construction under localization of a ring
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
(applied to R and S) ensures that there is a geometric notion of sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of (relative) Kähler p-forms available for use in algebraic geometry, over any field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
R.