Exact Functors and Projective Modules
Handbook guide to exact functors and projective modules with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Exact 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.
Projective Modules
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.
Definitions and Comments We are going to investigate the behavior of the hom
We are going to investigate the behavior of the hom and tensor functors when presented with an exact sequence.
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.
Left Exactness of HomR(M,
Left Exactness of HomR(M, ) Suppose that one has a short exact sequence f g 0 → A → B → C → 0 (1) We apply the covariant hom functor F=HomR(M, ) to the sequence, dropping the last term on the right. We will show that the sequence Ff Fg 0 → FA → FB → FC (2) is exact. A functor that behaves in this manner is called left exact. We must show that the transformed sequence is exact at FA and FB.
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.
Left Exactness of HomR(
Left Exactness of HomR( , N) The contravariant hom functor G = HomR( , N) is a functor on the opposite category, so before applying it to the sequence (1), we must reverse all the arrows. Thus left-exactness of G means that the sequence Gg Gf 0 → GC → GB → GA (3) is exact. Again one has three steps. (a) Gg is monic.
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 Exactness of the Functors M⊗R
Right Exactness of the Functors M⊗R and ⊗RN If we apply the functor H = M ⊗R to the exact sequence f g 0 → A → B → C → 0 , we will show that the sequence Hf Hg HA → HB → HC → 0 (4) is exact. A similar result holds for ⊗R N. A functor that behaves in this way is called right exact. Once again, there are three items to prove.
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.
Definition
A functor that is both left and right exact is called exact. Thus an exact functor is one that maps exact sequences to exact sequences. One has already seen one example, the localization functor (Section 8.5, Problems 4 and 5). If we ask under what conditions the hom and tensor functors become exact, we are led to the study of projective, injective and flat modules, to be considered later in the chapter.
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.
Definition
The R-module P is projective if given f : P →N, and g : M → N surjective, there exists h : P →M (not necessarily unique) such that the diagram is commutative, that is, f = gh. We sometimes say that one has “lifted” f to h. The definition may look obscure, but the condition described is a familiar property of free modules.
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.
Theorem
The following conditions on the R-module P are equivalent. (1) P is projective. (2) The functor HomR(P, ) is exact. (3) Every short exact sequence 0 →M →N →P →0 splits.
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.
Corollary
The direct sum P = ⊕Pj is projective if and only if each Pj is projective.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Theorem
A module M over a principal ideal domain R is projective if and only if it is free.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Examples
1. A vector space over a field k is a free k-module, hence is projective. 2. A finite abelian group G is not a projective Z-module, because it is not free. [If g ∈G and n = |G|, then ng = 0, so g can never be part of a basis.] 3. If p and q are distinct primes, then R = Zpq = Zp ⊕Zq.
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.
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 supplied worked solutions for this section repeatedly test kernel, module, homomorphism, exact, injective, functor, Tor. These checks are used here as verification themes rather than copied as answer text.
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 |
|---|---|---|
| 10.4 | Exact Functors | 206–208 |
| 10.5 | Projective Modules | 209–210 |
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.