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 and be commutative rings and be a ring homomorphism. An important example is for a field and a unital algebra over (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
\OmegaS/R
of differentials in different, but equivalent ways.
d:S\toM
d(fg)=fdg+gdf
\OmegaS/R
d:S\to\OmegaS/R
\operatorname{Hom}S(\OmegaS/R,M)\xrightarrow{\cong}\operatorname{Der}R(S,M).
One construction of and proceeds by constructing a free -module with one formal generator for each in, and imposing the relations
for all in and all and in . The universal derivation sends to . The relations imply that the universal derivation is a homomorphism of -modules.
S ⊗ RS
\begin{cases}S ⊗ RS\toS\ \sumsi ⊗ ti\mapsto\sumsi ⋅ ti\end{cases}
Then the module of Kähler differentials of can be equivalently defined by
\OmegaS/R=I/I2,
and the universal derivation is the homomorphism defined by
ds=1 ⊗ s-s ⊗ 1.
This construction is equivalent to the previous one because is the kernel of the projection
\begin{cases}S ⊗ RS\toS ⊗ RR\ \sumsi ⊗ ti\mapsto\sumsi ⋅ ti ⊗ 1\end{cases}
Thus we have:
S ⊗ RS\equivI ⊕ S ⊗ RR.
Then
S ⊗ RS/S ⊗ RR
\sumsi ⊗ ti\mapsto\sumsi ⊗ ti-\sumsi ⋅ ti ⊗ 1.
This identifies with the -module generated by the formal generators for in, subject to being a homomorphism of -modules which sends each element of to zero. Taking the quotient by precisely imposes the Leibniz rule.
S=R[t1,...,tn]
1 | |
\Omega | |
R[t1,...,tn]/R |
=
n | |
oplus | |
i=1 |
R[t1,...tn]dti.
Kähler differentials are compatible with extension of scalars, in the sense that for a second -algebra and for
S'=R' ⊗ RS
\OmegaS/R ⊗ SS'\cong\OmegaS'/R'.
As a particular case of this, Kähler differentials are compatible with localizations, meaning that if is a multiplicative set in, then there is an isomorphism
W-1\OmegaS/R\cong
\Omega | |
W-1S/R |
.
Given two ring homomorphisms
R\toS\toT
\OmegaS/R ⊗ ST\to\OmegaT/R\to\OmegaT/S\to0.
If
T=S/I
\OmegaT/S
I/I2\xrightarrow{[f]\mapstodf ⊗ 1}\OmegaS/R ⊗ ST\to\OmegaT/R\to0.
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 -algebras
T=R[t1,\ldots,tn]/(f1,\ldots,fm)
\Omega(R[t]/(f))\cong(R[t]dt ⊗ R[t]/(f))/(df)\congR[t]/(f,df/dt)dt.
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, represents the ideal defining the diagonal in the fiber product of with itself over . 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\toY
l{I}
X x YX
\OmegaX/Y=l{I}/l{I}2
d:l{O}X\to\OmegaX/Y
f-1l{O}Y
l{O}X
Similar to the commutative algebra case, there exist exact sequences associated to morphisms of schemes. Given morphisms
f:X\toY
g:Y\toZ
X
*\Omega | |
f | |
Y/Z |
\to\OmegaX/Z\to\OmegaX/Y\to0
Also, if
X\subsetY
l{I}
\OmegaX/Y=0
X
l{I}/l{I}2\to\OmegaY/Z|X\to\OmegaX/Z\to0
If
K/k
1 | |
\Omega | |
K/k |
=0
K/k
K/k
\pi:Y\to\operatorname{Spec}(K)
\pi*\Omega
1 | |
K/k |
\to
1 | |
\Omega | |
Y/k |
\to
1 | |
\Omega | |
Y/K |
\to0
proves
1 | |
\Omega | |
Y/k |
\cong
1 | |
\Omega | |
Y/K |
Given a projective scheme
X\in\operatorname{Sch}/k
\operatorname{Proj}\left( | \Complex[x,y,z] |
(xn+yn-zn) |
\right)=\operatorname{Proj}(R)
then we can compute the cotangent module as
\OmegaR/\Complex=
R ⋅ dx ⊕ R ⋅ dy ⊕ R ⋅ dz | |
nxn-1dx+nyn-1dy-nzn-1dz |
Then,
\OmegaX/\Complex=\widetilde{\OmegaR/\Complex
Consider the morphism
X=\operatorname{Spec}\left(
\Complex[t,x,y] | |
(xy-t) |
\right)=\operatorname{Spec}(R)\to\operatorname{Spec}(\Complex[t])=Y
in
\operatorname{Sch}/\Complex
\widetilde{R ⋅ dt}\to\widetilde{
R ⋅ dt ⊕ R ⋅ dx ⊕ R ⋅ dy | |
ydx+xdy-dt |
hence
\OmegaX/Y=\widetilde{
R ⋅ dx ⊕ R ⋅ dy | |
ydx+xdy |
As before, fix a map
X\toY
lOX
n | |
\Omega | |
X/Y |
:=wedgen\OmegaX/Y.
The derivation
lOX\to\OmegaX/Y
0\tol{O}X\xrightarrow{d}
1 | |
\Omega | |
X/Y |
\xrightarrow{d}
2 | |
\Omega | |
X/Y |
\xrightarrow{d} …
satisfying
d\circd=0.
The de Rham complex enjoys an additional multiplicative structure, the wedge product
n | |
\Omega | |
X/Y |
⊗
m | |
\Omega | |
X/Y |
\to
n+m | |
\Omega | |
X/Y |
.
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.
The hypercohomology of the de Rham complex of sheaves is called the algebraic de Rham cohomology of over and is denoted by
n | |
H | |
dR(X |
/Y)
n | |
H | |
dR(X) |
As is familiar from coherent cohomology of other quasi-coherent sheaves, the computation of de Rham cohomology is simplified when and are affine schemes. In this case, because affine schemes have no higher cohomology,
n | |
H | |
dR(X |
/Y)
0\toS\xrightarrow{d}
1 | |
\Omega | |
S/R |
\xrightarrow{d}
2 | |
\Omega | |
S/R |
\xrightarrow{d} …
which is, termwise, the global sections of the sheaves
r | |
\Omega | |
X/Y |
To take a very particular example, suppose that
X=\operatorname{Spec}\Q\left[x,x-1\right]
\Q.
\Q[x,x-1]\xrightarrow{d}\Q[x,x-1]dx.
The differential obeys the usual rules of calculus, meaning
d(xn)=nxn-1dx.
0(X) | |
\begin{align} H | |
dR |
&=\Q
1(X) | |
\\ H | |
dR |
&=\Q ⋅ x-1dx \end{align}
and all other algebraic de Rham cohomology groups are zero. By way of comparison, the algebraic de Rham cohomology groups of
Y=\operatorname{Spec}Fp\left[x,x-1\right]
0(Y) | |
\begin{align} H | |
dR |
&=opluskFp ⋅ xkp
1(Y) | |
\\ H | |
dR |
&=opluskFp ⋅ xkp-1dx \end{align}
Since the Betti numbers of these cohomology groups are not what is expected, crystalline cohomology was developed to remedy this issue; it defines a Weil cohomology theory over finite fields.
If is a smooth complex algebraic variety, there is a natural comparison map of complexes of sheaves
\bullet | |
\Omega | |
X/\Complex |
(-)\to
\bullet | |
\Omega | |
Xan |
((-)an)
between the algebraic de Rham complex and the smooth de Rham complex defined in terms of (complex-valued) differential forms on
Xan
\ast | |
H | |
dR(X/\Complex) |
\cong
an) | |
H | |
dR(X |
from algebraic to smooth de Rham cohomology (and thus to singular cohomology by de Rham's theorem). In particular, if X is a smooth affine algebraic variety embedded in , then the inclusion of the subcomplex of algebraic differential forms into that of all smooth forms on X is a quasi-isomorphism. For example, if
X=\{(w,z)\in\C2:wz=1\}
then as shown above, the computation of algebraic de Rham cohomology gives explicit generators for
0 | |
H | |
dR |
(X/\C)
1 | |
H | |
dR |
(X/\C)
Counter-examples in the singular case can be found with non-Du Bois singularities such as the graded ring
k[x,y]/(y2-x3)
y
\deg(y)=3
\deg(x)=2
A proof of Grothendieck's theorem using the concept of a mixed Weil cohomology theory was given by .
If is a smooth variety over a field, then
\OmegaX/k
lOX
\omegaX/k:=wedge\dim\OmegaX/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.
The geometric genus of a smooth algebraic variety of dimension over a field is defined as the dimension
g:=\dimH0(X,
d | |
\Omega | |
X/k |
).
For curves, this purely algebraic definition agrees with the topological definition (for
k=\Complex
The tangent bundle of a smooth variety is, by definition, the dual of the cotangent sheaf
\OmegaX/k
The sheaf of differentials is related to various algebro-geometric notions. A morphism
f:X\toY
\OmegaX/Y
K:=k[t]/f
\OmegaK/k=0
A morphism of finite type is a smooth morphism if it is flat and if
\OmegaX/Y
lOX
\Omega | |
R[t1,\ldots,tn]/R |
n | |
A | |
R |
\to\operatorname{Spec}(R)
Periods are, broadly speaking, integrals of certain arithmetically defined differential forms. The simplest example of a period is
2\pii
\int | |
S1 |
dz | |
z |
=2\pii.
Algebraic de Rham cohomology is used to construct periods as follows: For an algebraic variety defined over
\Q,
n | |
H | |
dR(X |
/\Q) ⊗ \Q\Complex=
n | |
H | |
dR(X |
⊗ \Q\Complex/\Complex).
Xan
an). | |
H | |
dR(X |
Hn(Xan,\Complex)
Hn(Xan,\Q) ⊗ \Q\Complex.
\Complex
In algebraic number theory, Kähler differentials may be used to study the ramification in an extension of algebraic number fields. If is a finite extension with rings of integers and respectively then the different ideal, which encodes the ramification data, is the annihilator of the -module :
\deltaL/K=\{x\inR:xdy=0forally\inR\}.
Hochschild homology is a homology theory for associative rings that turns out to be closely related to Kähler differentials. This is because of the Hochschild-Kostant-Rosenberg theorem which states that the Hochschild homology
HH\bullet(R)
\bullet | |
\Omega | |
R/k |
k
0
The de Rham–Witt complex is, in very rough terms, an enhancement of the de Rham complex for the ring of Witt vectors.