Alphabeta MathTensor Coherence and Algebraic Descent
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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.
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✓ 9 results · all verified · 3 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Scalars, tensor powers, the empty tensor, opposite algebras and finite sums
Lemma
Associator naturality, pentagon, unit triangle and symmetry hexagons on elementary tensors
Lemma
Tensoring injections and the kernel of a tensor product of quotient maps over a field
Lemma
Coefficient separation for an independent family of vectors, with explicit Choice assumptions
Lemma
Finite tensor duality and basis-independent coevaluation
Lemma
The free associative R-algebra on a set and descent of relations
Lemma
Presentation base change and transport of explicit bases to commutative specializations
Lemma
Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields
Lemma
Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace
Lemma
The left regular module and its tensor powers detect linear and tensor identities
Sources
- Keith Conrad, Tensor products (University of Connecticut expository notes, 60 pp.)
- The CRing Project, open-source commutative algebra text (2016 PDF; Chapter 13)
- George M. Bergman, An Invitation to General Algebra and Universal Constructions (Springer Universitext; author's revised PDF v3.4, April 30, 2020)