Engineering · Mathematics · Advanced Algebra Handbook
Homology, Chain Complexes and Derived Functors
Homological algebra measures failure of exactness. Complexes, homology, derived constructions and cohomology turn extension and lifting problems into computable invariants, but the direction and degree of every map must be tracked carefully. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Executive summary
This chapter develops homology, chain complexes and derived functors as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
Write the complex or short exact sequence with every object and arrow.
Verify consecutive differentials compose to zero.
Compute cycles, boundaries and the quotient that defines homology.
When deriving a functor, choose the permitted resolution and track degrees consistently.
Use long exact sequences to transport information between related objects.
For cohomological classifications, distinguish cocycles, coboundaries and equivalence classes.
Core definitions
Definition
A projective resolution of a module M is an exact sequence, · · · →Pn →Pn−1 →· · · →P1 →P0 →M →0, in which each module Pn is projective. A free resolution is a projective resolution in which each module Pn is free.
Definition
A complex9 (C•, d•) is a sequence of modules and maps, for every n ∈Z, C• = · · · →Cn+1 dn+1 −→Cn dn −→Cn−1 →· · · , in which dndn+1 = 0 for all n. The maps dn are called differentiations. Usually, we will shorten the notation (C•, d•) to C•. Note that the equation dndn+1 = 0 is equivalent to im dn+1 ⊆ker dn.
Definition
If R is a ring, then the category of all complexes of left R-modules is denoted by R-Comp; if the ring R is understood, then we will omit the prescript R. The category Comp is a preadditive category (that is, the Hom’s are abelian groups and the distributive laws hold whenever possible) if we define ( f + g)n = fn + gn for all n ∈Z. The following definitions imitate the construction of homology groups of triangulated spaces that we described in Section 10.1.
Definition
We call Hn( f ) the induced map, and we usually denote it by fn∗, or even by f∗.
Definition
If · · · →P2 →P1 d1 −→P0 →A →0 is a projective resolution of a module A, then its deleted projective resolution is the complex PA = · · · →P2 →P1 →P0 →0. Similarly, if 0 →A →E0 d0 −→E1 →E2 →· · · is an injective resolution of a module A, then a deleted injective resolution is the complex EA = 0 →E0 →E1 →E2 →· · · . In either case, deleting A loses no information: A ∼= coker d1 in the first case, and A ∼= ker d0 in the second case. Of course, a deleted resolution is no longer exact: H0(PA) = ker(P0 →{0})/ im d1 = P0/ im d1 ∼= A. We know that a module has many presentations, and so the next result is fundamental.
Definition
A functor T : RMod →RMod, of either variance, preserves multiplications if T (µr): T A →T A is multiplication by r for all r ∈Z(R). Tensor product and Hom preserve multiplications. We claim that if T preserves multiplications, then LnT also preserves multiplications; that is, LnT (µr) = multiplication by r. Since T preserves multiplications, the terms of the chain map T ˇµ are multiplication by r, and so the induced maps in homology are also multiplication by r: (T ˇµ)∗: zn + im T dn+1 ↦(T ˇµn)zn + im T dn+1 = rzn + im T dn+1, where zn ∈ker T dn. ◀
Definition
If A is a right R-module and T = A⊗R , define torR n (A, ) = LnT . Thus, if QB = · · · →Q2 d2 −→Q1 d1 −→Q0 →0 is the chosen deleted projective resolution of a module B, then torR n (A, B) = Hn(A ⊗R QB) = ker(1A ⊗dn) im(1A ⊗dn+1). The domain of torR n (A, ) is RMod, the category of left R-modules; its target is Ab, the category of abelian groups but, as before, its target may be smaller (if R = Q, for example) or larger [if R = ZG, for every Z-module can be viewed as a (trivial) R-module]. Derived Functors One of the nice theorems of homological algebra is, for all A and B (and for all R and n), that TorR n (A, B) ∼= torR n (A, B). There are now several points to discuss. The definition of LnT assumes that a choice of deleted projection resolution of each module has been made. Does LnT depend on this choice? And, once we dispose of this question (the answer is that LnT does not depend on the choice), how can we use these functors? Assume that new choices ΔPA of deleted projective resolutions have been made, and let us denote the left derived functors arising from these new choices by ΔLnT .
Definition
If T = HomR( , C), define extn R( , C) = RnT . Thus, if · · · →P2 d2 −→P1 d1 −→P0 →0 is the chosen deleted projective resolution of a module A, then extn R(A, C) = Hn(HomR(PA, C) = ker(dn+1)∗ im(dn)∗, where (dn)∗: HomR(P′ n, C) →HomR(Pn, C) is defined, as usual, by (dn)∗: f ↦f dn. The same phenomenon that holds for Tor holds for Ext: for all A and C (and for all R and n), Extn R(A, C) ∼= extn R(A, C). The same proof that shows that Tor is independent of the variable resolved also works theorem, we will dispense with the two notations for Ext. Assume that new choices ΔPA of deleted projective resolutions have been made, and let us denote the right derived functors arising from these new choices by ΔRnT .
Principal results and structural facts
Key result
displayed an exact sequence of free left ZQ-modules F3 d3 −→F2 d2 −→F1 d1 −→F0, where Fn is the free Q-module with basis Qn. The module F0 = ZQ is free on the generator 1, and the map d1 : F1 →ZQ is given by d1 : [x] ↦x −1.
Key result
Let R and A be rings, and let T : RMod →AMod be an exact additive functor. Then T commutes with homology; that is, for every complex (C•, d•) ∈ R-Comp and for every n ∈Z, there is an isomorphism Hn(T C•, T d•) ∼= T Hn(C•, d•).
Key result
If 0• →C′ • i −→C• p −→C′′ • →0• is an exact sequence of complexes, then there is an exact sequence of modules · · · →Hn+1(C′′ •) ∂n+1 −→Hn(C′ •) i∗ −→Hn(C•) p∗ −→Hn(C′′ •) ∂n −→Hn−1(C′ •) →· · · . Homology Functors
Key result
Given a commutative diagram of complexes with exact rows, 0• C′ • i f C• p g C′′ • h 0• 0• A′ • j A• q A′′ • 0• there is a commutative diagram of modules with exact rows, · · · Hn(C′ •) i∗ f∗ Hn(C•) p∗ g∗ Hn(C′′ •) ∂ h∗ Hn−1(C′ •) f∗ · · · · · · Hn(A′ •) j∗ Hn(A•) q∗ Hn(A′′ •) ∂′ Hn−1(A′ •) · · ·
Key result
Given a pair of rings R and S and an additive covariant functor T : RMod →SMod, then LnT : RMod →SMod is an additive covariant functor for every n.
Key result
Let T : RMod →SMod be an additive covariant functor. If P is a projective module, then LnT (P) = {0} for all n ≥1. In particular, if A and P are right R-modules, with P projective, and if B and Q are left R-modules, with Q projective, then TorR n (P, B) = {0} and TorR n (A, Q) = {0} for all n ≥1. Derived Functors
Key result
If T : RMod →SMod is a covariant additive functor, then the functor L0T is right exact.
Key result
Given a commutative diagram of modules having exact rows, A′ f i A p g A′′ h 0 C′ j C q C′′ 0 Derived Functors there is, for all n, a commutative diagram with exact rows TorR n (A′, B) f∗ i∗ TorR n (A, B) p∗ g∗ TorR n (A′′, B) h∗ ∂n TorR n−1(A′, B) f∗ TorR n (C′, B) j∗ TorR n (C, B) q∗ TorR n (C′′, B) ∂′ n TorR n−1(C′, B) There is a similar diagram if the first variable is fixed.
Key result
Given a pair of rings R and S, and an additive covariant functor T : RMod →SMod, then, for each n, the functors RnT and ΔRnT are naturally equivalent. In particular, for all A, (RnT )A ∼= (ΔRnT )A, and so these modules are independent of the choice of (deleted) injective resolution of A.
Key result
If T : RMod →SMod is a covariant additive functor, then the functor R0T is left exact.
Key result
Given a commutative diagram of modules having exact rows, A′ f i A p g A′′ h 0 C′ j C q C′′ 0 there is, for all n, a commutative diagram with exact rows Extn R(B, A′) f∗ i∗ Extn R(B, A) p∗ g∗ Extn R(B, A′′) h∗ ∂n Extn+1 R (B, A′) f∗ Extn R(B, C′) j∗ Extn R(B, C) q∗ Extn R(B, C′′) ∂′n Extn+1 R (B, C′) Finally, we discuss derived functors of contravariant functors T . If we define right derived functors RnT , in terms of deleted resolutions C• for which T C• is on the right, then we start with a deleted projective resolution PA, for then the contravariance of T puts T PA on the right.13 Given an additive contravariant functor T : RMod →SMod, where R and S are rings, we are now going to construct, for all n ∈Z, its right derived functors RnT : RMod → SMod. Choose, once for all, a deleted projective resolution PA of every module A, form the complex T PA, and take homology: RnT (A) = Hn(T PA) = ker T dn+1 im T dn . If f : A →A′, define RnT ( f ): RnT (A′) →RnT (A) as we did for left derived functors. By the comparison theorem, there is a chain map ˇf : PA →P′ A′ over f , unique to homotopy, which induces a map RnT ( f ): Hn(T P′ A′) →Hn(T PA), namely, (T ˇfn)∗, in homology. 13If we were interested in left derived functors of a contravariant T , but we are not, then we would use injective resolutions. Derived Functors
Key result
Given a pair of rings R and S, and an additive contravariant functor T : RMod →SMod, then, for each n, the functors RnT and ΔRnT are naturally equivalent. In particular, for all A, (RnT )A ∼= (ΔRnT )A, and so these modules are independent of the choice of (deleted) projective resolution of A.
Key result
If T : RMod →SMod is a contravariant additive functor, then the functor R0T is left exact.
Key result
Given a commutative diagram of modules having exact rows, A′ f i A p g A′′ h 0 C′ j C q C′′ 0 there is, for all n, a commutative diagram with exact rows Extn R(A′′, B) p∗ Extn R(A, B) i∗ Extn R(A′, B) ∂n Extn+1 R (A′′, B) Extn R(C′′, B) h∗ q∗ Extn R(C, B) g∗ j∗ Extn R(C′, B) f ∗ ∂n′ Extn+1 R (C′′, B) h∗ Remark. When T is a covariant functor, then we call the ingredients of LnT chains, cycles, boundaries, and homology. When T is contravariant, we often add the prefix “co,” and the ingredients of RnT are usually called cochains, cocycles, coboundaries, and cohomology. Unfortunately, this clear distinction is blurred because the Hom functor is contravariant in one variable but covariant in the other. In spite of this, we usually use the “co” prefix for the derived functors Extn of Hom. ◀ Derived functors are one way to construct functors like Ext and Tor. Indeed, derived functors will rarely be mentioned in the sequel.
Source-grounded examples
Worked source example
A complex is an exact sequence if and only if all its homology groups are {0}: that is, Hn(C•) = {0} for all n. Thus, the homology groups measure the deviation of a complex from being an exact sequence. An exact sequence is often called an acyclic complex; acyclic means “no cycles”; that is, no cycles that are not boundaries. ◀
Worked source example
If r ∈Z(R) is a central element in a ring R, and if A is a left R-module, then µr : A →A, defined by µr : A ↦r A, is an R-map. We call µr multiplication by r.
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
| Failure mode | Control |
|---|---|
| Forgetting to verify that a differential squares to zero. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Reversing homological and cohomological grading. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Confusing cycles with homology classes. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Using a resolution that lacks the required projective or injective property. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Dropping connecting morphisms from a long exact sequence. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
Verification checklist
- The ambient set, ring, field, group, module or category has been stated.
- Every operation and map used is well-defined in that setting.
- The hypotheses of each structural result have been checked before use.
- Representatives, coordinates or generators have not been confused with the underlying object.
- Existence and uniqueness have been separated where both matter.
- The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about homology, chain complexes and derived functors?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.