Symmetric, and it measures torsion
Torn(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
- 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.
- Torn(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.
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
| Condition | Consequence |
|---|---|
| M flat | Torn(M, −) = 0 for all n ≥ 1 |
| M projective | Same, since projective implies flat |
| Λ a PID | Torn = 0 for n ≥ 2 |
| M or N torsion-free over a PID | Tor1 = 0 |
| Λ semisimple | Torn = 0 for n ≥ 1 |
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(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 |
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:
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.
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?
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.
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