Alphabeta MathTensor Products of Modules
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✓ 30 results · all verified · 7 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Free Modules, Exact Sequences, Projective and Injective Modules
- 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
- Rings, Subrings, Integral Domains and Fields
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Balanced maps from a right module and a left module, and bilinear maps over a commutative ring
Definition
The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
Theorem
Universal property of the tensor product for balanced maps into abelian groups
Corollary
Tensor products are unique up to a unique isomorphism carrying elementary tensors to elementary tensors
Proposition
Module homomorphisms induce tensor-product homomorphisms functorially
Proposition
A formula on elementary tensors defines a homomorphism exactly when its underlying pairing is balanced
Definition
(S,R)-bimodules and commuting left and right scalar actions
Theorem
A commuting outer scalar action descends to a tensor product
Theorem
Associativity of tensor products for compatible bimodules
Theorem
Over a commutative ring, M⊗_RN is an R-module with r(m⊗ n)=(rm)⊗ n=m⊗(rn)
Theorem
Symmetry and associativity isomorphisms for tensor products over a commutative ring
Theorem
The regular module is a tensor unit: R⊗_RN≅ N and M⊗_RR≅ M
Corollary
Finite iterated tensor products represent multilinear maps independently of parenthesization
Theorem
Tensor products commute with arbitrary direct sums
Theorem
The elementary tensors of two bases form the product basis of the tensor product
Corollary
Rᵐ⊗_RRⁿ≅ Rᵐⁿ with the product basis, and dim_F(V⊗_FW)=dim_FV dim_FW
Definition
The R-module Hom_R(M,N) over a commutative ring
Theorem
For finite-dimensional V, the canonical map V^*⊗_FW toHom_F(V,W) is an isomorphism
Corollary
Under End_F(V)≅ V^*⊗_FV, tensor contraction is the trace
Definition
Algebras over a commutative ring, central structure maps, and algebra homomorphisms
Theorem
The tensor product of R-algebras has multiplication (a⊗ b)(a'⊗ b')=aa'⊗ bb'
Theorem
Universal mapping property of the tensor product of commutative algebras
Theorem
Hom-tensor adjunction: Hom_R(M⊗_RN,P)congHom_R(M,Hom_R(N,P))
Theorem
Tensoring is right exact
Definition
The submodule IM generated by products of elements of an ideal I with elements of a module M
Corollary
M⊗_RR/I≅ M/IM naturally
Definition
Flat and faithfully flat modules and ring homomorphisms
Theorem
Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests
Theorem
Every projective module over a commutative ring is flat
Corollary
Under the stated choice boundary, free modules are projective and hence flat
Corollary
For flat M, one has IM∩ JM=(I∩ J)M
Theorem
A short exact sequence with flat quotient remains short exact after tensoring
Definition
Restriction of scalars and extension of scalars S⊗_RM along a ring homomorphism R→ S
Theorem
Extension of scalars is left adjoint to restriction of scalars
Corollary
Change of rings: N⊗_RM≅ N⊗_S(S⊗_RM)
Proposition
Extension of scalars carries flat modules to flat modules
Proposition
Flatness is transitive under a flat change of rings
Theorem
The character dual of a flat module is injective
Definition
Tensor algebra of a vector space
Theorem
Universal property of the tensor algebra
Definition
Symmetric algebra of a vector space
Theorem
Universal property of the symmetric algebra
Sources
- Stacks Project, Section 10.12: Tensor products
- C. Dennis, Week 1 recap on tensor products
- H. Miller, Lectures on Algebraic Topology I, Sections 20-21
- C. Dennis, Week 4 on tensor products and flatness
- W. Li, Commutative Algebra, Lectures 9-10
- M. Barr, Acyclic Models, Chapter 2
- Stacks Project, Section 10.39: Flat modules and flat ring maps
- Stacks Project, Lemma 10.39.5
- Stacks Project, Lemma 10.39.2
- Stacks Project, Lemma 10.39.12
- MIT 18.721 Algebraic Geometry notes, Section 2.1
- MIT 18.721 Algebraic Geometry notes, Lemma 2.1.35
- Stacks Project, Lemma 10.39.7
- Stacks Project, Lemma 10.39.4
- M. Barr, Acyclic Models, Proposition 5.17
- Etingof, MIT 18.745 notes, §12.1, printed pp. 69–70
- Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.1, printed pp. 71–72
- Etingof, MIT 18.745 notes, §13.1, printed pp. 74–75