Homological algebra

Chain complexes, exact sequences and homology as the measure of obstruction; Tor and Ext as derived functors — conservation laws and what blocks them.

Chain two linear maps so that the second annihilates the output of the first — the image of stage one lands exactly in the “kernel” of stage two. When that containment is tight, nothing is hidden. When it almost holds but not quite, the leftover is real, structural content that no single stage explains: think of the independent current loops a circuit graph admits beyond the obvious branches. Homological algebra is the bookkeeping of that leftover: it turns “what got lost” and “what cannot be undone” into a computable group.

Everything below lives in an additive world — objects and maps add, and the governing equation is linear. So read every invariant here as a first-order accountant, not a constitutive law. It sees superposition and structural (yes/no) obstructions; it is blind to saturation, hysteresis, and bifurcations. That caveat is not a footnote — it is the precise scope of the whole chapter.

Complexes and the law \(d\circ d=0\)

The basic object is a sequence of modules strung together by maps that vanish in pairs. A chain complex \((M_\bullet,d)\) is a list of objects \(M_n\) with differentials \(d_n\colon M_n\to M_{n-1}\) satisfying

\[ d_{n-1}\circ d_n = 0. \]

The map \(d\) is one step of a staged process; a cochain complex is the same idea with the index going up. The law \(d\circ d=0\) looks austere but says something simple: applying two consecutive stages annihilates everything. Equivalently, \(\operatorname{im} d_n \subseteq \ker d_{n-1}\) — whatever stage \(n\) produces is invisible to stage \(n-1\).

Homology measures the gap

Inside each \(M_n\) live two subgroups. The cycles \(Z_n=\ker d_n\) are what is conserved at stage \(n\). The boundaries \(B_n=\operatorname{im} d_{n+1}\) are what arrived from the previous stage — already accounted for. The law \(d\circ d=0\) guarantees \(B_n\subseteq Z_n\), so the quotient makes sense:

\[ H_n \;=\; \frac{Z_n}{B_n} \;=\; \frac{\ker d_n}{\operatorname{im} d_{n+1}}. \]

This is homology: the conserved content that the previous stage does not explain. It is exactly the slack in the containment \(\operatorname{im}\subseteq\ker\). If \(H_n=0\) for all \(n\), image equals kernel everywhere and the complex is exact — perfectly transparent, no leftover. If \(H_n\neq 0\), you have found an obstruction.

Two facts make this powerful rather than ad hoc. Homology is a functor — it transports along maps of complexes, so it is the prototypical invariant (this is the functoriality that lets an invariant deploy across systems, P7/P13). And it is robust: it is unchanged by the “same up to correction” relations we meet below.

Exact sequences: conservation laws

The cleanest complex is one that is exact. A short exact sequence (SES) is the workhorse:

\[ \begin{CD} 0 @>>> L @>i>> M @>p>> N @>>> 0. \end{CD} \]

Exactness at each spot says precisely three things: \(i\) is injective (\(L\) embeds), \(p\) is surjective (\(N\) is hit), and \(\operatorname{im} i=\ker p\). Read together, \(L\cong\ker p\) and \(N\cong M/L\) — so \(M\) is “\(L\) glued to \(N\).” It is a conservation law: \(M\) is exactly its sub-object \(L\) plus its quotient \(N\), nothing more, nothing less. As the figure shows, homology measures the gap that appears the moment that gluing is not tight.

The decisive question is how \(L\) and \(N\) are glued. The SES splits if there is a section \(s\colon N\to M\) with \(p\circ s=\mathrm{id}_N\) (equivalently a retraction of \(i\)). Then

\[ M \;\cong\; L\oplus N, \]

a direct sum — two decoupled subsystems you can pull apart. If no such section exists the sequence is non-split: \(L\) and \(N\) are fused so that separating them destroys structure. The non-split possibilities are classified by a single group, \(\operatorname{Ext}^1(N,L)\), with the split case sitting at \(0\). (Over a field every SES splits — there is nothing to glue, which is exactly why fields are too simple to carry this phenomenon.)

The connecting map: obstructions propagate

Now stack two short exact sequences and join them by vertical maps \(f,g,h\), with exact rows:

\[\begin{CD} 0 @>>> L @>>> M @>>> N @>>> 0 \\ @. @VVfV @VVgV @VVhV @. \\ 0 @>>> L' @>>> M' @>>> N' @>>> 0 \end{CD}\]

The snake lemma asserts a single exact sequence threading the kernels and cokernels of the verticals:

\[ 0\to\ker f\to\ker g\to\ker h\xrightarrow{\;\delta\;}\operatorname{coker} f\to\operatorname{coker} g\to\operatorname{coker} h\to 0. \]

The interesting arrow is the connecting homomorphism \(\delta\), and it is built by a diagram chase that is worth narrating: take \(n\in\ker h\), lift it to the hidden middle \(M\), push it down to \(M'\), observe that it must come from a unique \(l'\in L'\), and set \(\delta(n)=[l']\). The output is forced, but it is only reachable by passing through the unobserved object \(M\).

Derived functors, Tor and Ext: principled loss-accounting

Many natural transformations break exactness, and that break is where information leaks. A functor is left-exact if it preserves \(0\to L\to M\to N\), right-exact if it preserves \(L\to M\to N\to 0\), and exact if both. Two examples carry most of the weight: \(\operatorname{Hom}(A,-)\) and \(\operatorname{Hom}(-,A)\) are left-exact; the tensor product \(-\otimes A\) is right-exact. The end that breaks is exactly where the SES stops being a conservation law — non-exactness is information loss under a transformation (P2, P7).

Derived functors repair the break by keeping a complete record of it. First model the object by simple pieces: a projective resolution \(\cdots\to P_1\to P_0\to A\to 0\) replaces \(A\) by free-like projective modules (a projective is a summand of a free module — maps onto it always lift). Injective resolutions are the dual, built from objects into which maps always extend. Any two resolutions are equivalent up to the correction relations below, so the construction is canonical. Then apply the functor \(F\) to the resolution (not to \(A\)) and take homology: \(L_iF(A)=H_i(F(P_\bullet))\) for right-exact \(F\), and \(R^iF\) dually. You recover \(F\) in degree \(0\) (\(L_0F\cong F\)), and \(F\) is exact precisely when all higher derived functors vanish.

The headline is \(\operatorname{Ext}^1\). It is in bijection with the extensions \(0\to L\to M\to N\to 0\) — the ways to glue \(N\) and \(L\) through a hidden middle — with the split (decoupled) extension being the zero class. So the snake-lemma story and the derived-functor story meet here: \(\operatorname{Ext}^1(N,L)\) is the group of ways things can fail to come apart.

Sameness up to correction — and a warning for QC

Two maps of complexes are chain homotopic, \(f\simeq g\), when \(f-g=dh+hd\) for some \(h\) — “equal up to a correction term that washes out in homology.” Homotopic maps induce equal maps on \(H_n\). A quasi-isomorphism is a map that induces an isomorphism on all homology: the two complexes carry the same essential information, even though the map itself is generally not invertible. That last clause is the sharpest formal model of P2: irreversibility with information preserved. A resolution \(P_\bullet\to A\) is exactly such a quasi-isomorphism.

The disciplined home for this is the derived category \(D(\mathsf A)\), built by formally inverting the quasi-isomorphisms (the homotopy category \(K(\mathsf A)\) takes the first step, identifying homotopic maps). All of this lives inside an abelian category — the axiomatization of a “well-behaved domain” (zero object, biproducts, every mono a kernel, every epi a cokernel) where kernels, cokernels, exact sequences and homology all make sense. By Freyd–Mitchell, any small abelian category embeds into a module category, so you may legitimately chase elements in a diagram.

Recap

• A complex obeys \(d\circ d=0\), i.e. \(\operatorname{im}\subseteq\ker\); homology \(H_n=\ker d_n/\operatorname{im} d_{n+1}\) is the slack in that containment — the obstruction, the latent state (P3, P2).
• A short exact sequence \(0\to L\to M\to N\to 0\) is a conservation law: \(M\) is \(L\) glued to \(N\). It splits (mixture, \(M\cong L\oplus N\)) or is non-split (compound) — and \(\operatorname{Ext}^1(N,L)\) classifies the gluing.
• The connecting map \(\delta\) (snake lemma → long exact sequence) computes an observable effect through an unobserved middle — obstructions propagate (P3).
• Derived functors (\(\operatorname{Tor}\), \(\operatorname{Ext}\)) are graded loss-accounting for functors that break exactness; \(\operatorname{Ext}^1\) counts irreducible couplings — the candidate emergence invariant (RD8).
• A quasi-isomorphism preserves all homology without being invertible — irreversibility with preserved information (P2); the derived category inverts these formally.
• Everything is additive: these invariants see linear/structural obstructions only. Over a field \(\operatorname{Ext}^1=0\) and every SES splits — the math itself tells you when a domain is too simple to carry emergence.