Kähler differential

Definition

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 d : S → M to an S-module M with R in its kernel, satisfying the Leibniz rule d ( f g ) = f d g + g d f . The module of Kähler differentials is defined as the S-module Ω S / R for which there is a universal derivation d : S → Ω S / R . 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 Ω_S/R and d proceeds by constructing a free R-module with one formal generator ds for each s in S, and imposing the relations

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

Another construction proceeds by letting I be the ideal in the tensor product S ⊗ R S defined as the kernel of the multiplication map S ⊗ R S → S given by Σ s i ⊗ t i ↦ Σ s i ⋅ t i . Then the module of Kähler differentials of S can be equivalently defined by Ω_S/R = I / I², and the universal derivation is the homomorphism d defined by

To see that this construction is equivalent to the previous one, note that I is the kernel of the projection S ⊗ R S → S ⊗ R R given by Σ s i ⊗ t i ↦ Σ s i ⋅ t i ⊗ 1 . Thus we have:

Then S ⊗ R S / S ⊗ R R may be identified with I by the map induced by the complementary projection Σ s i ⊗ t i ↦ Σ s i ⊗ t i − Σ s i ⋅ t i ⊗ 1 . 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 S = R [ t 1 , … , t n ] 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 S ′ = R ′ ⊗ R S , there is an isomorphism

As a particular case of this, Kähler differentials are compatible with localizations, meaning that if W is a multiplicative set in S, then there is an isomorphism

Given two ring homomorphisms R → S → T , there is a short exact sequence of T-modules

If T = S / I for some ideal I, the term Ω T / S 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 T = R [ t 1 , … , t n ] / ( f 1 , … , f m ) . 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 f : X → Y by setting I to be the ideal of the diagonal in the fiber product X × Y X . The cotangent sheaf Ω X / Y = I / I 2 , together with the derivation d : O X → Ω X / Y defined analogously to before, is universal among f − 1 O Y -linear derivations of O X -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.

de Rham complex

As before, fix a map X → Y . Differential forms of higher degree are defined as the exterior powers (over O X ),

The derivation O X → Ω X / Y extends in a natural way to a sequence of maps

satisfying d ∘ d = 0 . This is a cochain complex known as the de Rham complex.

The de Rham complex enjoys an additional multiplicative structure, the wedge product

This turns the de Rham complex into a commutative differential graded algebra. It also has a coalgebra structure inherited from the one on the exterior algebra.

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 H dR n ( X ) 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, H dR n ( X / Y ) can be computed as the cohomology of the complex of abelian groups

which is, termwise, the global sections of the sheaves Ω X / Y r .

To take a very particular example, suppose that X = Spec Q[x, x⁻¹] is the multiplicative group over Q. 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 d ( x n ) = n x n − 1 d x . The kernel and cokernel compute algebraic de Rham cohomology, so H dR 0 ( X ) = Q , H dR 1 ( X ) = Q ⋅ ( d x / x ) , and all other algebraic de Rham cohomology groups are zero. By way of comparison, the algebraic de Rham cohomology groups of Y = Spec F_p[x, x⁻¹], are much larger, namely, H dR 0 ( Y ) = ⨁ k ∈ Z F p ⋅ x k p and H dR 1 ( Y ) = ⨁ k ∈ Z F p ⋅ x k p − 1 d x .

Grothendieck's comparison theorem

If X is smooth over Y = Spec ⁡ C , there is a natural comparison map

between the Kähler (i.e., algebraic) differential forms on X and the smooth (i.e., C ∞ ) differential forms on X an , 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).

Canonical divisor

If X is a smooth variety over a field k, then Ω X / k is a vector bundle (i.e., a locally free O X -module) of rank equal to the dimension of X. This implies, in particular, that

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

The geometric genus of a smooth algebraic variety X of dimension d over a field k is defined as the dimension

For curves, this purely algebraic definition agrees with the topological definition (for k = C) 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 Ω X / k . 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 f : X → Y of schemes is unramified if and only if Ω X / Y is zero. A special case of this assertion is that for a field k, K := k [ t ] / f is separable over k iff Ω K / k = 0 , which can also be read off the above computation.

A morphism f of finite type is a smooth morphism if it is flat and if Ω X / Y is a locally free O X -module of appropriate rank. The computation of Ω R [ t 1 , … , t n ] / R above shows that the projection from affine space A R n → S p e c ( R ) is smooth.

Periods

Periods are, broadly speaking, integrals of certain, arithmetically defined differential forms. The simplest example of a period is 2 π i , which arises as ∫ S 1 d z z = 2 π i .

Algebraic de Rham cohomology is used to construct periods as follows: For an algebraic variety X defined over Q, the above-mentioned compatibility with base-change yields a natural isomorphism

On the other hand, the right hand cohomology group is isomorphic to de Rham cohomology of the complex manifold X an associated to X, denoted here H dR n ( X an ) . 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, H n ( X an , C ) , which by the universal coefficient theorem is in its turn isomorphic to H n ( X an , Q ) ⊗ Q C . Composing these isomorphisms yields two rational vector spaces which, after tensoring with C, 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 δ_{L / K}, which encodes the ramification data, is the annihilator of the O-module Ω_O/o:

Related notions

Hochschild homology is a homology theory for associative rings which turns out to be closely related to Kähler differentials. The de Rham–Witt complex is, in very rough terms, an enhancement of the de Rham complex for the ring of Witt vectors.