Kähler differential

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, following the need to adapt methods from geometry over the complex numbers, and the free use of calculus methods, to contexts where such methods are not available.

Definition

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). 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

\Omega _{S/R}.

of differentials in different, but equivalent ways.

Definition using derivations

An R-linear derivation on S is a map ${\mathrm d}\colon S\to M$ to an S-module M with R in its kernel, satisfying Leibniz rule $\mathrm {d} (fg)=f\,\mathrm {d} g+g\,\mathrm {d} f$ . The module of Kähler differentials is defined as the R-linear derivation ${\mathrm d}\colon S\to \Omega _{{S/R}}$ that factors all others. 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. Symbolically, this can be written as an isomorphism

\operatorname {Der}_{R}(S,M)\cong \operatorname {Hom}_{S}(\Omega _{{S/R}},M),\,

which, as in the case of adjoint functors (though this isn't an adjunction), is natural (or functorial) in M.

The actual construction of Ω_S/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.

Definition using the augmentation ideal

Another construction proceeds by letting I be the ideal in the tensor product $S\otimes _{R}S$ , defined as the kernel of the multiplication map: $S\otimes _{R}S\to S$ , given by $\Sigma s_{i}\otimes t_{i}\mapsto \Sigma s_{i}\cdot t_{i}$ . Then the module of Kähler differentials of "S" can be equivalently defined by^[1] $Ω S / R = I / I 2$ , together with the morphism

{\mathrm d}s=1\otimes s-s\otimes 1.\,

To see that this construction is equivalent to the previous one, note that I is the kernel of the projection $S\otimes _{R}S\to S\otimes _{R}R$ , given by $\Sigma s_{i}\otimes t_{i}\mapsto \Sigma s_{i}\cdot t_{i}\otimes 1$ . Thus we have:

S\otimes _{R}S\equiv I\oplus S\otimes _{R}R.\,

Then $S\otimes _{R}S/S\otimes _{R}R$ may be identified with I, by the map induced by the complementary projection which is given by $\Sigma s_{i}\otimes t_{i}\mapsto \Sigma s_{i}\otimes t_{i}-\Sigma s_{i}\cdot t_{i}\otimes 1$ .

Thus this map identifies I with the S module generated by the formal generators ds for s in S, subject to the first two relations given above (with the second relation strengthened to demanding that d is R-linear). The elements set to zero by the final relation map to precisely I² in I.

Examples and basic facts

For any commutative ring R, Kähler differentials of the polynomial ring $S=R[t_{1},\dots ,t_{n}]$ is given by a free S-module of rank n:

\Omega _{R[t_{1},\dots ,t_{n}]/R}^{1}=\bigoplus _{i=1}^{n}R[t_{1},\dots t_{n}]\,dt_{i}.

Kähler differentials are compatible with base change and localization in the sense that for a second R-algebra R' and $S':=R'\otimes _{R}S$ , there is an isomorphism

\Omega _{S/R}\otimes _{S}S'=\Omega _{S'/R'}.

If W is a multiplicative system in S, then there is an isomorphism

\Omega _{S/R}[W^{-1}]=\Omega _{S[W^{-1}]/R}.

Given two ring homomorphisms $R\to S\to T$ , there is a short exact sequence

T\otimes _{S}\Omega _{S/R}\to \Omega _{T/R}\to \Omega _{T/S}\to 0.

If $T=S/I$ for some ideal I, the term $\Omega _{T/S}$ vanishes and the sequence can be continued at the left as follows:

I/I^{2}\,{\stackrel {[f]\mapsto df\otimes 1}{\,}}\longrightarrow T\otimes _{S}\Omega _{S/R}\to \Omega _{T/R}\to 0.

A generalization of these two short exact sequences is provided by the cotangent complex introduced by Illusie. The latter sequence and the above computation for the polynomial ring allows to compute Kähler differentials of finitely generated R-algebras $T=R[t_{1},\dots ,t_{n}]/(f_{1},\dots ,f_{m})$ . For example, for a single polynomial in a single variable,

\Omega _{R[t]/(f)/R}=R[t]\,dt/(f,df).

Kähler differentials for schemes

The second definition above allows the following geometric reinterpretation 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). 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. For a map of schemes $f:X\to Y$ , there is thus a sheaf of $\mathcal O_X$ -modules, also referred to as the cotangent sheaf,

\Omega _{X/Y},

which is the sheaf associated to the S-module $\Omega _{S/R}$ if f is the map corresponding to the ring homomorphism $R\to S$ above.

Higher differential forms and algebraic de Rham cohomology

de Rham complex

As before, fix a map $X\to Y$ . Differential forms of higher degree are defined as the exterior powers (over $\mathcal O_X$ ),

\Omega _{X/Y}^{n}:=\bigwedge ^{n}\Omega _{X/Y}.

The derivation ${\mathcal {O}}_{X}\to \Omega _{X/Y}$ extends in a natural way to a sequence of maps

0\to {\mathcal {O}}_{X}\,{\stackrel {d}{\to }}\,\Omega _{X/Y}^{1}\,{\stackrel {d}{\to }}\,\Omega _{X/Y}^{2}\,{\stackrel {d}{\to }}\,\cdots

satisfying $d\circ d=0$ , i.e., a cochain complex known as de Rham complex.

The de Rham complex enjoys additional multiplicative structure: the wedge product

\Omega _{X/Y}^{n}\otimes \Omega _{X/Y}^{m}\to \Omega _{X/Y}^{n+m}

turns the de Rham complex into a commutative graded differential algebra. It also has a coalgebra structure inherited from the one on the exterior algebra.^[2]

de Rham cohomology

The hypercohomology of the de Rham complex of sheaves is called the algebraic de Rham cohomology of X over Y and denoted by

H_{dR}^{n}(X/Y)

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, $H_{dR}^{n}(X/Y)$ can be computed as the cohomology of the complex of abelian groups

0\to S\,{\stackrel {d}{\to }}\,\Omega _{S/R}^{1}\,{\stackrel {d}{\to }}\,\Omega _{S/R}^{2}\,{\stackrel {d}{\to }}\,\cdots

which are, termwise, the global sections of the sheaves $\Omega _{X/Y}^{r}$ .

Grothendieck's comparison theorem

If X is smooth over $Y=\operatorname {Spec} \mathbf {C}$ , there is a natural comparison map

\Omega _{X/\mathbf {C} }^{r}(X)\to \Omega _{X^{an}}^{r}(X^{an})

between the Kähler (i.e., algebraic) differential forms on X and the smooth (i.e., $C^{\infty }$ ) differential forms on $X^{an}$ , the complex manifold associated to X. This map need not be an isomorphism. However, the induced map

H_{dR}^{n}(X/\mathbf {C} )\to H_{dR}^{n}(X^{an})

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).

Applications

Canonical divisor

If X is a smooth variety over a field k, then $\Omega _{X/k}$ is a vector bundle (i.e., locally free $\mathcal O_X$ -module) of rank equal to the dimension of X. This implies, in particular, that

\omega _{X/k}:=\bigwedge ^{\dim X}\Omega _{X/k}

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 over a field k of dimension d is defined as the dimension

g:=\dim H^{0}(X,\Omega _{X/k}^{d}).

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 (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 $\Omega _{X/k}$ . 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 $f:X\to Y$ of schemes is unramified if and only if $\Omega _{X/Y}$ is zero.^[3] A special case of this assertion is that for a field k, $K:=k[t]/f$ is separable over k iff $\Omega _{K/k}=0$ , which can also be read off the above computation.

A map f (of finite type) is a smooth morphism if it is flat and if $\Omega _{X/Y}$ is a locally free $\mathcal O_X$ -module. The computation of $\Omega _{R[t_{i}]/R}$ above corresponds to the fact that the projection from affine space, $\mathbb {A} _{R}^{n}\to Spec(R)$ , is smooth.

Periods

Periods are, broadly speaking, integrals of certain, arithmetically defined differential forms.^[4] The simplest example of a period is $2\pi i$ , which arises as $\int _{S^{1}}{\frac {dz}{z}}=2\pi i.$

Kähler differential forms or 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

H_{dR}^{n}(X/\mathbf {Q} )\otimes _{\mathbf {Q} }\mathbf {C} =H_{dR}^{n}(X\otimes _{\mathbf {Q} }\mathbf {C} /\mathbf {C} ).

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 theorem, a consequence of the Poincaré lemma, asserts an isomorphism of the latter cohomology group with singular cohomology (or sheaf cohomology) with complex coefficients, $H^{n}(X^{an},\mathbf {C} )$ , which by the universal coefficient theorem is in its turn isomorphic to $H^{n}(X^{an},\mathbf {Q} )\otimes _{\mathbf {Q} }\mathbf {C}$ . 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 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:^[5]

\delta _{L/K}=\{x\in O:x\,\mathrm {d} y=0{\text{ for all }}y\in 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.

References

↑ Hartshorne (1977, p. 172)
↑ Laurent-Gengoux, C.; Pichereau, A.; Vanhaecke, P. (2013). Poisson structures. §3.2.3: Springer. ISBN 978-3-642-31090-4.
↑ Milne, James, Etale cohomology, Proposition I.3.5 ; the map f is supposed to be locally of finite type for this statement.
↑ André, Yves (2004). Une introduction aux motifs. Partie III: Société Mathématique de France.
↑ Neukirch (1999, p. 201)

Cisinski, D.-C.; Déglise, F. (2013). "Mixed Weil cohomologies". Adv. Math. arXiv:0712.3291. doi:10.1016/j.aim.2011.10.021.
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Grothendieck, A. (1966), Letter to J. Tate (PDF) .
Grothendieck, Alexander (1968), "Crystals and the de Rham cohomology of schemes", in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L.; et al., Dix Exposés sur la Cohomologie des Schémas (PDF), Advanced studies in pure mathematics, 3, Amsterdam: North-Holland, pp. 306–358, MR 0269663
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Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge University Press.
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Rosenlicht, M. (1976). "On Liouville's theory of elementary functions". Pacific J. Math. 65: 485–492. doi:10.2140/pjm.1976.65.485. Zbl 0318.12107.
Fu, G.; et al. (2011). "Some remarks on Kähler differentials and ordinary differentials in nonlinear control systems". Systems and Control Letters. 60: 699–703. doi:10.1016/j.sysconle.2011.05.006.

External links

A thread devoted to the relation on algebraic and analytic differential forms
Differentials (Stacks project)

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