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.
An important example is for R a field and S a unital algebra over R (such as the coordinate ring of an affine variety).
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.
Another construction proceeds by letting I be the ideal in the tensor product
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.
For example, for a single polynomial in a single variable, 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.
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).
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.
is a smooth variety (or scheme), then the relative cotangent sequence proves
Differential forms of higher degree are defined as the exterior powers (over
The de Rham complex enjoys an additional multiplicative structure, the wedge product This turns the de Rham complex into a commutative differential graded algebra.
Algebraic de Rham cohomology was introduced by Grothendieck (1966a).
can be computed as the cohomology of the complex of abelian groups which is, termwise, the global sections of the sheaves
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
By way of comparison, the algebraic de Rham cohomology groups of
For example, if then as shown above, the computation of algebraic de Rham cohomology gives explicit generators
[4] Other counterexamples can be found in algebraic plane curves with isolated singularities whose Milnor and Tjurina numbers are non-equal.
[5] A proof of Grothendieck's theorem using the concept of a mixed Weil cohomology theory was given by Cisinski & Déglise (2013).
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
The tangent bundle of a smooth variety X is, by definition, the dual of the cotangent sheaf
Periods are, broadly speaking, integrals of certain arithmetically defined differential forms.
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
Composing these isomorphisms yields two rational vector spaces which, after tensoring with
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.
If L / K is a finite extension with rings of integers R and S respectively then the different ideal δL / K, which encodes the ramification data, is the annihilator of the R-module ΩR/S:[9] Hochschild homology is a homology theory for associative rings that turns out to be closely related to Kähler differentials.
of an algebra of a smooth variety is isomorphic to the de-Rham complex