Engineering · Mathematics · Abstract Algebra

Computing Ext, Tor and Base Change

Handbook guide to computing ext, tor and base change with core definitions, structural results, reasoning methods and verification checks.

Approx. 10 min read

Handbook scope. This handbook article develops computing ext, tor and base change as a connected part of abstract algebra. The supplied source treats the topic through the sequence Some Properties of Ext and Tor; Base Change in the Tensor Product. The presentation below preserves that dependency: definitions come first, then structural results, constructions and calculation methods, followed by checks that expose the hypotheses most likely to be missed. Proofs from the source are condensed to proof strategies rather than reproduced line-for-line, while theorem statements, algebraic relationships and decision conditions are retained in technical form.

Section S6: pp. 235–236Section S7: pp. 237–238

2source sections integrated

9formal results and definitions distilled

4source pages in the primary theory range

How the topic fits together

Some Properties of Ext and Tor

This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.

Base Change in the Tensor Product

This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.

Core definitions and structural results

The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.

Corollary · S5.8

Corollary

If N is an R-module, the following conditions are equivalent. (a) N is injective; (b) Extn(M, N) = 0 for all n ≥1 and all modules M; (c) Ext1(M, N) = 0 for all modules M.

Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.

Example · S6.1

Example We will calculate Extn

We will calculate Extn Z(Zm, B) for an arbitrary abelian group B. To ease the notational burden slightly, we will omit the subscript Z in Ext and Hom, and use = (most of the time) when we really mean ∼=. One has the following projective resolution of Zm: m 0 → Z → Z → Zm → 0 where the m over the arrow indicates multiplication by m. Switching to a deleted resolution and applying the contravariant hom functor, we get m 0 → Hom(Z, B) → Hom(Z, B) → 0 ↑ Hom(Zm, B) But by (9.4.1), one has HomR(R, B) ∼= B (1) and the above diagram becomes m 0 → B → B → 0 (2) By (S5.3), Ext0(Zm, B) = Hom(Zm, B).

Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.

Proposition · S6.2

Proposition Extn

Extn Z(A, B) = 0 for all n ≥2 and all abelian groups A and B.

Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.

Lemma · S6.3

Lemma Ext0

Ext0 Z(Z, B) = HomZ(Z, B) = B and Ext1 Z(Z, B) = 0.

Example · S6.4

Example We will compute TorZ

We will compute TorZ n(Zm, B) for an arbitrary abelian group B. As before, we drop the superscript Z and write = for ∼=. We use the same projective resolution of Zm as in (S6.1), and apply the functor ⊗B. Since R ⊗R B ∼= B by (8.7.6), we reach diagram (2) as before.

Proposition · S6.5

Proposition TorZ

TorZ n(A, B) = 0 for all n ≥2 and all abelian groups A and B.

Lemma · S6.6

Lemma Tor1(Z, B) = Tor1(A, Z) = 0; Tor0(Z, B) = Z ⊗B = B.

Tor1(Z, B) = Tor1(A, Z) = 0; Tor0(Z, B) = Z ⊗B = B.

Result · S6.7

Finitely generated abelian groups We will show how to compute Extn(A, B) and

Finitely generated abelian groups We will show how to compute Extn(A, B) and Torn(A, B) for arbitrary finitely generated abelian groups A and B. By (4.6.3), A and B can be expressed as a finite direct sum of cyclic groups. Now Tor commutes with direct sums: Torn(A, ⊕r j=1Bj) = ⊕r j=1Torn(A, Bj). [The point is that if Pj∗is a projective resolution of Bj, then the direct sum of the Pj∗is a projective resolution of ⊕jBj, by (10.5.4). Since the tensor functor is additive on direct sums, by (8.8.6(b)), the Tor functor will be additive as well.

Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.

Proposition · S7.3

Proposition A[X] ⊗A M ∼= M[X]

A[X] ⊗A M ∼= M[X] where the elements of M[X] are of the form a0m0+a1Xm1+a2X2m2+· · ·+anXnmn, ai ∈ A, mi ∈M, n = 0, 1, . . ..

Quick-reference relationships

(b) Extn(M, N) = 0 for all n ≥1 and all modules M;

We will calculate Extn Z(Zm, B) for an arbitrary abelian group B.

To ease the notational burden slightly, we will omit the subscript Z in Ext and Hom, and use = (most of the time) when we really mean ∼=.

One has the following projective resolution of Zm: m 0 → Z → Z → Zm → 0 where the m over the arrow indicates multiplication by m.

Extn Z(A, B) = 0 for all n ≥2 and all abelian groups A and B.

Problem-solving workflow

Fix the coefficient ring and variance

State whether modules are left/right modules and whether a functor is covariant or contravariant.

Write the maps, not just the objects

Kernels, images, exactness and universal properties depend on the actual homomorphisms.

Use the appropriate universal property

Direct sums, products, tensor products, projectives, injectives and limits are best handled by their mapping property.

Check exactness at each position

Verify image equals kernel rather than relying on the appearance of a diagram.

Choose a resolution only when needed

Derived constructions should be tied to projective or injective resolutions and independence from the chosen resolution.

Test naturality and compatibility

For induced maps, ensure compositions and commutative squares behave as required.

Worked-solution emphasis from the supplied source

The source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.

Common mistakes and boundary conditions

Forgetting the coefficient ring when comparing modules.
Assuming tensor product preserves every exact sequence.
Confusing direct sum with direct product for infinite families.
Reading exactness from a diagram without checking image equals kernel.

Verification checklist

State the ambient algebraic structure and operation before applying a theorem.
Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
Distinguish a definition from a theorem that follows from it.
Check whether a map is well-defined before using its kernel, image, inverse or induced map.
Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.

Source coverage map

Source section	Subject	PDF pages analysed
S6	Some Properties of Ext and Tor	235–236
S7	Base Change in the Tensor Product	237–238

Related Mathematics pages

Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.