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Canonical static concept record

Tor functor

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Summary

The left-derived functors $\operatorname{Tor}^R_n(M,N)$ of the tensor product over a ring.

Record metadata

Canonical name: Tor functor
Concept ID: tor_functor
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Homological algebra
Domains: Homological algebra, Algebra, Category theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Derived functor

Carrier(s)

right $R$ -module $M$ and left $R$ -module $N$

Data

graded groups $\operatorname{Tor}^R_n(M,N)$

Axioms / constraints

computation from a projective or flat resolution in one argument

Canonically induces

long exact Tor sequences
obstructions to flatness

Notes

For a commutative ring both inputs may be written as $R$ -modules; over a noncommutative ring their handedness matters.

Concept sources

Wikipedia — Tor functor — Tor functor · encyclopedic reference · source ID wp-tor-functor
nLab — Tor — Tor · mathematical reference · source ID nlab-tor-functor
Stacks Project — Tor groups and flatness — Tor groups and flatness · mathematical reference · source ID stacks-tor-groups

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Derived category Tor functor

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: derived_category_to_tor
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: homology of the derived tensor product

Authored explanation

In derived categories of modules, the homology of $M\otimes_R^{\mathbf L}N$ computes $\operatorname{Tor}^R_n(M,N)$ , with grading conventions stated explicitly.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Tor functor — Tor functor · encyclopedic reference · source ID wp-tor-functor
nLab — Tor — Tor · mathematical reference · source ID nlab-tor-functor
Stacks Project — Tor groups and flatness — Tor groups and flatness · mathematical reference · source ID stacks-tor-groups

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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