In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.
Let R and S be commutative rings and φ : R → S be 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). Kähler differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed in purely algebraic terms. This observation can be turned into a definition of the module
of differentials in different, but equivalent ways.
Definition using derivations
An R-linear derivation on S is an R-module homomorphism to an S-module M with the image of R in its kernel, satisfying the Leibniz rule . The module of Kähler differentials is defined as the S-module for which there is a universal derivation . As with other universal properties, this means that d is the best possible derivation in the sense that any other derivation may be obtained from it by composition with an S-module homomorphism. In other words, the composition with d provides, for every S-module M, an S-module isomorphism
One construction of ΩS/R and d proceeds by constructing a free S-module with one formal generator ds for each s in S, and imposing the relations
- dr = 0,
- d(s + t) = ds + dt,
- d(st) = s dt + t ds,
for all r in R and all s and t in S. The universal derivation sends s to ds. The relations imply that the universal derivation is a homomorphism of R-modules.
Definition using the augmentation ideal
and the universal derivation is the homomorphism d defined by
This construction is equivalent to the previous one because I is the kernel of the projection
Thus we have:
Then may be identified with I by the map induced by the complementary projection
This identifies I with the S-module generated by the formal generators ds for s in S, subject to d being a homomorphism of R-modules which sends each element of R to zero. Taking the quotient by I2 precisely imposes the Leibniz rule.
Examples and basic facts
For any commutative ring R, the Kähler differentials of the polynomial ring are a free S-module of rank n generated by the differentials of the variables:
Kähler differentials are compatible with extension of scalars, in the sense that for a second R-algebra R′ and for , there is an isomorphism
Given two ring homomorphisms , there is a short exact sequence of T-modules
If for some ideal I, the term vanishes and the sequence can be continued at the left as follows:
A generalization of these two short exact sequences is provided by the cotangent complex.
The latter sequence and the above computation for the polynomial ring allows the computation of the Kähler differentials of finitely generated R-algebras . Briefly, these are generated by the differentials of the variables and have relations coming from the differentials of the equations. For example, for a single polynomial in a single variable,
Kähler differentials for schemes
Because Kähler differentials are compatible with localization, they may be constructed on a general scheme by performing either of the two definitions above on affine open subschemes and gluing. However, the second definition has a geometric interpretation that globalizes immediately. In this interpretation, 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). Moreover, it extends to a general morphism of schemes by setting to be the ideal of the diagonal in the fiber product . The cotangent sheaf , together with the derivation defined analogously to before, is universal among -linear derivations of -modules. If U is an open affine subscheme of X whose image in Y is contained in an open affine subscheme V, then the cotangent sheaf restricts to a sheaf on U which is similarly universal. It is therefore the sheaf associated to the module of Kähler differentials for the rings underlying U and V.
Similar to the commutative algebra case, there exist exact sequences associated to morphisms of schemes. Given morphisms and of schemes there is an exact sequence of sheaves on
Also, if is a closed subscheme given by the ideal sheaf there is an exact sequence of sheaves on
Finite separable field extensions
If is a finite field extension, then if and only if is separable. Consequently, if is a finite separable field extension and is a smooth variety (or scheme), then the relative cotangent sequence
Cotangent modules of a projective variety
Given a projective scheme , its cotangent sheaf can be computed from the sheafification of the cotangent module on the underlying graded algebra. For example, consider the complex curve
then we can compute the cotangent module as
Morphisms of schemes
Consider the morphism
in . Then, using the first sequence we see that
Higher differential forms and algebraic de Rham cohomology
de Rham complex
As before, fix a map . Differential forms of higher degree are defined as the exterior powers (over ),
The derivation extends in a natural way to a sequence of maps
satisfying This is a cochain complex known as the de Rham complex.
The de Rham complex enjoys an additional multiplicative structure, the wedge product
de Rham cohomology
The hypercohomology of the de Rham complex of sheaves is called the algebraic de Rham cohomology of X over Y and is denoted by or just if Y is clear from the context. (In many situations, Y is the spectrum of a field of characteristic zero.) Algebraic de Rham cohomology was introduced by Grothendieck (1966). It is closely related to crystalline cohomology.
As is familiar from coherent cohomology of other quasi-coherent sheaves, the computation of de Rham cohomology is simplified when X = Spec S and Y = Spec R are affine schemes. In this case, because affine schemes have no higher cohomology, can be computed as the cohomology of the complex of abelian groups
which is, termwise, the global sections of the sheaves .
To take a very particular example, suppose that is the multiplicative group over Because this is an affine scheme, hypercohomology reduces to ordinary cohomology. The algebraic de Rham complex is
The differential d obeys the usual rules of calculus, meaning The kernel and cokernel compute algebraic de Rham cohomology, so
and all other algebraic de Rham cohomology groups are zero. By way of comparison, the algebraic de Rham cohomology groups of are much larger, namely,
Grothendieck's comparison theorem
If X is smooth over there is a natural comparison map
between the Kähler (i.e., algebraic) differential forms on X and the smooth (i.e., have derivatives of all orders) differential forms on , the complex manifold associated to X. This map need not be an isomorphism. However, the induced map
between algebraic and smooth de Rham cohomology is an isomorphism, as was first shown by Grothendieck (1966). A proof using the concept of a Weil cohomology was given by Cisinski & Déglise (2013). Counter examples in the singular case can be found with non-Du Bois singularities. For example, the graded ring with where and — The details are spelled out on mathoverflow. Other counterexamples can be found in algebraic plane curves with isolated singularities whose Milnor and Tjurina numbers are non-equal.
is a line bundle or, equivalently, a divisor. It is referred to as the canonical divisor. The canonical divisor is, as it turns out, a dualizing complex and therefore appears in various important theorems in algebraic geometry such as Serre duality or Verdier duality.
Classification of algebraic curves
For curves, this purely algebraic definition agrees with the topological definition (for ) as the "number of handles" of the Riemann surface associated to X. There is a rather sharp trichotomy of geometric and arithmetic properties depending on the genus of a curve, for g being 0 (rational curves), 1 (elliptic curves), and greater than 1 (hyperbolic Riemann surfaces, including hyperelliptic curves), respectively.
Tangent bundle and Riemann–Roch theorem
The tangent bundle of a smooth variety X is, by definition, the dual of the cotangent sheaf . The Riemann–Roch theorem and its far-reaching generalization, the Grothendieck–Riemann–Roch theorem, contain as a crucial ingredient the Todd class of the tangent bundle.
Unramified and smooth morphisms
The sheaf of differentials is related to various algebro-geometric notions. A morphism of schemes is unramified if and only if is zero. A special case of this assertion is that for a field k, is separable over k iff , which can also be read off the above computation.
Periods are, broadly speaking, integrals of certain, arithmetically defined differential forms. The simplest example of a period is , which arises as
On the other hand, the right hand cohomology group is isomorphic to de Rham cohomology of the complex manifold associated to X, denoted here Yet another classical result, de Rham's theorem, asserts an isomorphism of the latter cohomology group with singular cohomology (or sheaf cohomology) with complex coefficients, , which by the universal coefficient theorem is in its turn isomorphic to Composing these isomorphisms yields two rational vector spaces which, after tensoring with become isomorphic. Choosing bases of these rational subspaces (also called lattices), the determinant of the base-change matrix is a complex number, well defined up to multiplication by a rational number. Such numbers are periods.
Algebraic number theory
In algebraic number theory, Kähler differentials may be used to study the ramification in an extension of algebraic number fields. If L / K is a finite extension with rings of integers O and o respectively then the different ideal δL / K, which encodes the ramification data, is the annihilator of the O-module ΩO/o:
Hochschild homology is a homology theory for associative rings which turns out to be closely related to Kähler differentials. This is because of the Hoschild-Kostant-Rosenberg theorem which states that the Hochschild homology of an algebra of a smooth variety is isomorphic to the de-Rham complex for a field of characteristic . There is a derived enhancement of this theorem which states that the Hochschild homology of a dga is isomorphic to the derived de-Rham complex.
The de Rham–Witt complex is, in very rough terms, an enhancement of the de Rham complex for the ring of Witt vectors.
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