Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Rationalization is exact and commutes with singular homology
Statement
Assume AC. For every abelian group A, A⊗_Z Q is its positive-integer localization: every element is a/s and a/s=0 iff a is killed by some positive integer. Thus A⊗Q=0 iff A is torsion. Rationalization is exact and commutes with direct sums. Naturally for every space Y, H_j(Y;Z)⊗Q≅H_j(Y;Q); for every rational vector space V, H_j(Y;V)≅H_j(Y;Q)⊗_Q V.
Facts & Assumptions
Given: AC; an abelian group , a space , and a rational vector space ; the positive integers as the multiplicative set with localization .
For a commutative ring , a multiplicative set and an -module , the localization is the tensor product ; every element of is a fraction , and if and only if some positive integer kills (Localisation of a module at a multiplicative subset, Localisation of modules is extension of scalars).
Localization is exact and commutes with direct sums and quotients for the multiplicative set of positive integers (Localisation of modules is exact, Localisation commutes with quotient modules and arbitrary direct sums).
Singular chains and homology with coefficients are the free construction on simplices, natural in the space (The singular chain complex and singular homology).
Under AC every vector space has a basis and every independent set extends to one (The Axiom of Choice, Zorn's lemma gives a basis between any linearly independent set and any spanning set containing it: if with independent and , there is a basis of with ).
Proof
Apply the published localization/tensor theorem with Z and the nonzero positive integers, whose ring localization is Q. In the fraction definition equality of a/s with 0/1 means u a=0 for some positive u. A finite sum of tensors a_t⊗(r_t/s_t) has a common denominator, and hence is a single fraction, so this criterion applies to every element. Exactness and direct sums are precisely the published localization statements.
For a chain complex C, apply exact rationalization to 0→Z_j(C)→C_j→B_{j−1}(C)→0 and 0→B_j(C)→Z_j(C)→H_j(C)→0. It identifies the cycles and boundaries in C⊗Q with the rationalizations of the original cycles and boundaries, hence identifies homology. The basis of singular simplices gives the literal chain isomorphism C_(Y;Z)⊗Q=C_(Y;Q), compatible with every continuous map. Finally choose a basis of V under AC. Tensoring a rational complex with V is a direct sum of copies of that complex. Kernels and images of its coordinate differential are direct sums of kernels and images, since elements have finite support. This proves the V assertion; the natural tensor map, not the auxiliary basis, supplies the isomorphism.
Depends on
- Localisation of a module at a multiplicative subset
- Localisation of modules is exact
- Localisation of modules is extension of scalars
- Localisation commutes with quotient modules and arbitrary direct sums
- The singular chain complex and singular homology
- Zorn's lemma gives a basis between any linearly independent set and any spanning set containing it: if $L \subseteq S \subseteq V$ with $L$ independent and $\operatorname{span}(S) = V$, there is a basis $B$ of $V$ with $L \subseteq B \subseteq S$
- The Axiom of Choice
Used by
- Finite-range rational homology vanishing implies rational homotopy vanishing Corollary
- First rational Hurewicz after killing lower torsion homotopy Lemma
- Rational cohomology of K(Z,n) through weak CW fiber comparison Lemma
- Rational homotopy comparison with one endpoint surjection implies homology comparison Lemma
- Rational Hurewicz for arbitrary wedges of high-dimensional spheres Lemma
- Rational sphere homotopy below the first unstable degree Lemma
- Torsion Eilenberg–Mac Lane spaces are rationally acyclic Lemma
- Products of complex projective spaces span rational oriented bordism Proposition
- Rational Hurewicz for highly connected CW complexes Theorem
- Rational oriented bordism is detected by Pontryagin numbers Theorem
Dependency tree · two levels
38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Altman and Kleiman, A Term of Commutative Algebra, 13th edition (standard reference, not scraped)