$(S,R)$-bimodules and commuting left and right scalar actions

Depends on

• Unital left and right modules over a ring; unqualified module means left module

Used by

• Additive cocontinuous module functors admit right adjoints Corollary
• Composition of Deligne kernels is balanced tensor product Corollary
• Eilenberg-Watts is a schematic equivalence of Hom categories Corollary
• Exact finite tensor functors have projective right-module kernels Corollary
• Exact module tensor functors correspond to right-flat bimodules Corollary
• Finite Eilenberg–Watts is a biequivalence Corollary
• Finite one-sided exactness is equivalent to the existence of the corresponding adjoint Corollary
• Graded bimodule maps classify shift-compatible transformations Corollary
• The center is Morita invariant, via natural endomorphisms of the identity Corollary
• A Deligne kernel need not be one external tensor factor Counterexample
• The left-to-right exact equivalence need not preserve the identity Counterexample
• Associative graded algebras, bimodules, and internal shifts Definition
• Cohomological left and right differential graded modules Definition
• Enveloping algebra and the bimodule–module dictionary Definition
• Restriction of scalars and extension of scalars S⊗_RM along a ring homomorphism R→ S Definition
• The Morita bicategory of rings and bimodules Definition
• Natural transformations between tensor composites are governed by bimodule maps Example
• The kernel end and coend distinguish the regular and co-regular bimodules Example
• The matrix-ring Morita pair with explicit tensor inverses Example
• FALSE: the left and right internal homs agree in every monoidal category False statement
• F(A) is a (B,A)-bimodule for every additive functor F Lemma
• Finite Eilenberg–Watts kernels: explicit end and coend universal maps Lemma
• Finite module duality is exact with commuting bimodule actions Lemma
• Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent Lemma
• Homogeneous right multiplication reconstructs the graded kernel action Lemma
• Induction commutes with an external tensor factor Lemma
• Internal shifts are autoequivalences and commute with the graded tensor product Lemma
• Nakayama kernels give well-defined adjoint functors Lemma
• Tensor-Hom adjunction for bimodules over arbitrary unital rings 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 projective Nakayama pairing and the symmetric-algebra specialization Proposition
• The function model of induction agrees with the tensor-product model k[G]⊗_k[H]W Remark
• A commuting outer scalar action descends to a tensor product Theorem
• Categorical Eilenberg–Watts equivalences for finite linear categories Theorem
• Eilenberg-Watts theorem for arbitrary unital rings Theorem
• Finite Eilenberg–Watts for right exact linear functors Theorem
• Finite left exact functors are Hom functors with dual bimodule kernels Theorem

…and 3 more results.

Dependency tree · two levels

3 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

• Stacks Project, Section 10.12: Tensor products (standard reference, not scraped)