Derived Functors, Tor and Ext
Handbook guide to derived functors, tor and ext with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Derived Functors
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.
Proposition
Every module M has an injective resolution.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Left Derived Functors Suppose that F is a right exact functor from modules to
Left Derived Functors Suppose that F is a right exact functor from modules to modules. (In general, the domain and codomain of F can be abelian categories, but the example one has in mind is M ⊗R .) Given a short exact sequence 0 →A →B →C →0, we form deleted projective resolutions PA∗→A, PB∗→B, PC∗→C. It is shown in texts on homological algebra that it is possible to define chain maps to produce a short exact sequence of complexes as shown below. 0 → A → B → C → 0 ↑ ↑ ↑ 0 → PA∗ → PB∗ → PC∗ → 0 The functor F will preserve exactness in the diagram, except at the top row, where we only have FA →FB →FC →0 exact. But remember that we are using deleted resolutions, so that the first row is suppressed.
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Right Derived Functors Suppose now that F is a left exact functor from modules to
Right Derived Functors Suppose now that F is a left exact functor from modules to modules, e.g., HomR(M, ). One can dualize the discussion in (S5.1) by reversing the vertical arrows in the commutative diagram of complexes, and replacing projective resolutions such as PA∗by injective resolutions EA∗. The right derived functors of F are defined by taking the homology of F(E). Equivalently, (RnF)(A) = Hn[F(EA∗)] where the superscript n indicates that we are using right resolutions and the indices are increasing as we move away from the starting point.
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Lemma
If A is projective, then (LnF)(A) = 0 for every n > 0; if A is injective, then (RnF)(A) = 0 for every n > 0.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Definition
If F is the right exact functor M ⊗R , the left derived functor LnF is called TorR n (M, ). If F is the left exact functor HomR(M, ), the right derived functor RnF is called Extn R(M, ). It can be shown that the Ext functors can also be computed using projective resolutions and the contravariant hom functor. Specifically, Extn R(M, N) = [RnHomR( , N)](M).
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
If M is an R-module, the following conditions are equivalent. (i) M is flat; (ii) Torn(M, N) = 0 for all n ≥1 and all modules N; (iii) Tor1(M, N) = 0 for all modules N.
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.
Proposition
If M is an R-module, the following conditions are equivalent. (i) M is projective; (ii) Extn(M, N) = 0 for all n ≥1 and all modules N; (iii) Ext1(M, N) = 0 for all modules N.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Quick-reference relationships
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 |
|---|---|---|
| S5 | Derived Functors | 232–234 |
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.