A comparison isomorphism that relates Betti (singular/topological) cohomology of a suitable complex algebraic variety or manifold with its algebraic or analytic de Rham cohomology, often after tensoring with the complex numbers, identifying cycle integrals with periods of differential forms.

Principle

Integration of algebraic differential forms over topological cycles induces an isomorphism between the de Rham cohomology (computed from forms) and Betti cohomology with complex coefficients for smooth proper varieties (and more general settings with hypotheses), underlying Hodge theory and period relations.

Demonstration

For a smooth projective complex curve, the comparison identifies singular cohomology H^1(X(C), C) with the algebraic de Rham H^1_dR(X)⊗C; explicit bases of holomorphic differentials integrated over a homology basis produce the classical period matrix.

Misapplication

Assuming an integral (Z-)comparison without tensoring by C or applying the isomorphism to singular spaces lacking the required smoothness/properness; confusing Betti–de Rham with étale–de Rham comparisons valid in different arithmetic contexts.

Consequence

Connects topological invariants and algebraic/differential structures, enables computation of periods and Hodge structures, and provides the bridge used in transcendence questions and comparison of motivic realizations.

Reversal

The converse—recovering de Rham forms uniquely from Betti classes without analytic structure or coefficients—fails without the de Rham complex; one needs the differential form data to produce the isomorphism rather than purely topological input.

Boundary

Holds under hypotheses such as smoothness and properness (or with suitable growth/coefficients modifications); it is not generally true for arbitrary singular varieties or schemes over noncomplex fields without replacing Betti cohomology by an appropriate analogue.

Semantic Tension

Tension exists between Betti–de Rham comparison and other comparison theorems (e.g., étale–de Rham, p-adic comparison): each identifies different cohomological realizations under various base fields and topologies, so one must choose the correct comparison for the context.

Synthesis

The Betti–De Rham comparison equates topological cohomology classes with algebraic differential forms via integration, yielding an isomorphism (after suitable tensoring) that makes geometric topology and algebraic de Rham calculus two faces of the same cohomological invariants.