Module homomorphisms induce tensor-product homomorphisms functorially

Depends on

• Universal property of the tensor product for balanced maps into abelian groups
• Module homomorphism and isomorphism, kernel, image and cokernel

Used by

• Graded Eilenberg-Watts respects bicategorical coherence Corollary
• The left-to-right exact equivalence need not preserve the identity Counterexample
• Coalgebra and coalgebra map over a field Definition
• Picard group of a scheme Definition
• Subcoalgebras, coideals, and quotient coalgebras Definition
• The induced map ΛᵏT on exterior powers Definition
• The Morita bicategory of rings and bimodules Definition
• A counital coideal quotient of the two-by-two comatrix coalgebra Example
• Natural transformations between tensor composites are governed by bimodule maps Example
• The kernel end and coend distinguish the regular and co-regular bimodules Example
• Associator naturality, pentagon, unit triangle and symmetry hexagons on elementary tensors Lemma
• Bilinear right exact functors are determined by their value on the regular modules Lemma
• Canonical free presentations force the comparison to be an isomorphism Lemma
• Coefficient separation for an independent family of vectors, with explicit Choice assumptions Lemma
• Convolution of linear maps is associative and unital Lemma
• Counit uniqueness and finite Sweedler calculus Lemma
• Every comodule element lies in a finite-dimensional subcomodule Lemma
• Finite comatrix coalgebras exist Lemma
• Finite Eilenberg–Watts kernels: explicit end and coend universal maps Lemma
• Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent Lemma
• Group algebras, finite-group function algebras, and polynomial Hopf algebras Lemma
• Homogeneous free presentations prove the graded comparison is an isomorphism Lemma
• Kernels, images, quotients, and matrix coefficients of comodules Lemma
• Nakayama kernels give well-defined adjoint functors Lemma
• Tensor-Hom adjunction for bimodules over arbitrary unital rings Lemma
• Tensoring defines a schematic pseudofunctor with interchange Lemma
• Tensoring injections and the kernel of a tensor product of quotient maps over a field Lemma
• The canonical comparison to the tensor functor of F(A) is balanced and natural Lemma
• The dual-basis isomorphism for a finitely generated projective bimodule Lemma
• The functor M⊗_A- is additive, right exact, and preserves direct sums over an arbitrary unital ring Lemma
• The Morita data satisfy the bicategory coherence axioms Lemma
• The opposite Deligne product is the category of finite bimodules Lemma
• The tensor product of two coalgebras is a coalgebra Lemma
• The two standard bialgebra compatibility formulations are equivalent Lemma
• Universal finite projective cohomology complex over any base Lemma
• Complexification is a functor on real vector spaces and real-linear maps Proposition
• The projective Nakayama pairing and the symmetric-algebra specialization Proposition
• Categorical Eilenberg–Watts equivalences for finite linear categories Theorem
• Coideal quotients, coalgebra-map kernels, and sums of subcoalgebras Theorem
• Every projective module over a commutative ring is flat Theorem

…and 12 more results.

Dependency tree · two levels

10 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

• H. Miller, Lectures on Algebraic Topology I, Sections 20-21 (standard reference, not scraped)