The Tor Functor

Q: What does Tor mean over a non-commutative ring?

Torn(M, N) for a right module M and left module N is an abelian group, symmetric in the sense that resolving either variable gives the same answer. It is not a module unless extra structure is available.

Executive summary

Symmetric, and it measures torsion

Tor_n(M, N) is computed by resolving either variable projectively and tensoring; the two answers agree, and Tor is symmetric — a stronger statement than the balance of Ext, since both variables play the same role. Over ℤ, Tor₁ is exactly a torsion product, which is where the name comes from, and it is the correction term in both the Künneth formula and the universal coefficient theorem for homology.

Learning objectives

Compute Tor from a projective resolution.
State the symmetry and balance properties.
Apply the vanishing criteria involving flatness.
Reproduce the standard Tor computations over ℤ.
Explain the torsion interpretation.

Section 01Definition and symmetry

AlgorithmTor from a projective resolutionin: modules M, N → out: Tor_n(M, N)

Choose a projective resolution P_• ↠ M.
Delete M and apply − ⊗ N, giving a chain complex … → P₁ ⊗ N → P₀ ⊗ N → 0. Covariant, so the arrows keep their direction.
Tor_n(M, N) is the n-th homology of that complex.
Tor₀(M, N) = M ⊗ N, recovering the original functor.
Resolving N instead gives a canonically isomorphic answer.

Why Tor is symmetric but Ext is not

Both variables of tensor are covariant and both are resolved by projectives, so the double complex used in the proof is symmetric under exchange. Ext has one contravariant and one covariant variable, resolved by projectives and injectives respectively, so no such symmetry exists.

Section 02Vanishing and flatness

Vanishing criteria for Tor
Condition	Consequence
M flat	Tor_n(M, −) = 0 for all n ≥ 1
M projective	Same, since projective implies flat
Λ a PID	Tor_n = 0 for n ≥ 2
M or N torsion-free over a PID	Tor₁ = 0
Λ semisimple	Tor_n = 0 for n ≥ 1

Tor detects flatness exactly

M is flat if and only if Tor₁(M, N) = 0 for every N, and it suffices to test N = Λ/J for finitely generated ideals J. This gives a workable criterion where the definition by exactness does not.

Section 03Computations over ℤ

Using the length-one resolution of ℤ/mℤ:

Tor₁(ℤ/mℤ, A) = A[m], Tor_n = 0 for n ≥ 2

Standard Tor groups over ℤ
Tor₁(M, N)	Value
Tor₁(ℤ/m, ℤ/n)	ℤ/gcd(m, n)
Tor₁(ℤ/m, ℤ)	0 — ℤ is torsion-free
Tor₁(ℚ, A)	0 — ℚ is flat
Tor₁(A, B) for f.g. A, B	Depends only on the torsion of both

The name is literal

Over ℤ, Tor₁(M, N) depends only on the torsion subgroups of M and N, and vanishes if either is torsion-free. In the Künneth formula it is exactly the term recording the interaction of torsion in the two factors.

Section 04Long exact sequences

A short exact sequence in either variable gives a long exact sequence, running in the homological direction with decreasing index:

… → Tor₁(M, N″) → M ⊗ N′ → M ⊗ N → M ⊗ N″ → 0

Because tensor is right exact, the sequence terminates in 0 on the right rather than beginning with 0 on the left — the mirror of the Ext sequences, and a useful check that a sequence has been written the right way round.

Direction discipline

Ext sequences increase the index and start with 0; Tor sequences decrease the index and end with 0. Mixing them produces sequences that appear well formed and are wrong. Writing the degree-0 term first and extending from there is the reliable habit.

ReferenceFrequently asked questions

Why is Tor written with a subscript and Ext with a superscript?

Because Tor is a left derived functor, so its index decreases along the complex — homology convention. Ext is a right derived functor with increasing index — cohomology convention. The notation encodes which kind of derivation produced it.

Does Tor commute with direct limits?

Yes, in both variables, because tensor does and homology commutes with filtered colimits. This is used constantly to reduce statements about arbitrary modules to finitely generated ones.

What does Tor mean over a non-commutative ring?

Tor_n(M, N) for a right module M and left module N is an abelian group, symmetric in the sense that resolving either variable gives the same answer. It is not a module unless extra structure is available.

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Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review The Tor Functor. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat The Tor Functor as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—section, functor, symmetry, vanishing, computations—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.