Homological algebra · The Lumenary

Lens

MotivationH

Measuring what exactness loses

Homological algebra began in topology. To detect the "holes" of a space, Poincaré counted closed loops and surfaces (cycles) that do not bound anything, treating those that bound (boundaries) as trivial. Algebraically this is a sequence of groups and maps $\cdots \to C_2 \xrightarrow{d_2} C_1 \xrightarrow{d_1} C_0 \to 0$ with $d \circ d = 0$ — "the boundary of a boundary is empty" — and the invariant is the quotient $\ker d / \operatorname{im} d$ , the homology.

The same structure appears throughout algebra, for a different reason. The functors $\Hom_R(M, -)$ , $\Hom_R(-, N)$ and $- \otimes_R N$ are not exact: they turn some short exact sequences into sequences that are exact only at one end (see modules). Homological algebra measures the failure precisely. Each such functor $F$ comes with a sequence of derived functors — $\mathrm{Ext}^n$ for $\Hom$ , $\mathrm{Tor}_n$ for $\otimes$ — that repair the lost exactness in a long exact sequence. These groups answer concrete questions:

$\mathrm{Ext}^1(A, B)$ classifies the ways of building a module $E$ with submodule $B$ and quotient $A$ ;
$\mathrm{Tor}_1^\Z(A, \Z/n)$ is the $n$ -torsion of $A$ — hence the name;
the homology of $\R P^2$ with coefficients in $\mathbb{F}_2$ differs from that with coefficients in $\Q$ , by an amount computed by $\mathrm{Tor}$ .

This lesson assumes modules, exact sequences and functors. The snake lemma is proved in abelian categories; here we use it. Everything works in any abelian category with enough projectives (or injectives); for concreteness we work with modules over a ring $R$ .

IntuitionH

Cycles, boundaries and resolutions

Take the boundary of a triangle: vertices $u, v, w$ and oriented edges $[uv], [vw], [uw]$ . Let $C_1 = \Z^3$ (edges) and $C_0 = \Z^3$ (vertices), with $d([uv]) = v - u$ . The chain $z = [uv] + [vw] - [uw]$ goes once around the triangle and has $d(z) = 0$ : it is a cycle. With no 2-cell to fill the triangle, it is not a boundary, and $H_1 = \ker d_1 = \Z z$ detects the hole. Fill the triangle with a 2-cell $F$ , $d_2(F) = z$ , and $H_1$ becomes $0$ .

The same quotient measures failure of exactness. A sequence $A \to B \to C$ with composite $0$ is exact exactly when its homology $\ker / \operatorname{im}$ vanishes. To measure how far a functor $F$ is from exact, homological algebra replaces a module $M$ by a projective resolution $\cdots \to P_1 \to P_0 \to M \to 0$ — an exact sequence of projective (for instance free) modules: generators, relations among the generators, relations among the relations, and so on. The resolution is a complex with homology only $M$ in degree $0$ . Apply $F$ to the $P_i$ and take homology: if $F$ were exact, the result would again be just $F(M)$ in degree $0$ ; the higher homology groups $L_nF(M)$ measure exactly what $F$ fails to preserve. The comparison theorem below says the answer does not depend on which resolution we chose.

DefinitionsH

Definitions

Definition— chain complex, homology

A chain complex $C_\bullet$ is a sequence of modules and homomorphisms $\cdots \to C_{n+1} \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1} \to \cdots$ with $d_n \circ d_{n+1} = 0$ for all $n$ . Its cycles are $Z_n = \ker d_n$ , its boundaries $B_n = \operatorname{im} d_{n+1} \subseteq Z_n$ , and its homology is $H_n(C) = Z_n / B_n$ . A cochain complex has differentials raising the degree, $d^n : C^n \to C^{n+1}$ , and cohomology $H^n = \ker d^n / \operatorname{im} d^{n-1}$ . A complex is exact (acyclic) if all its homology vanishes.

Definition— chain map, quasi-isomorphism, chain homotopy

A chain map $f : C_\bullet \to D_\bullet$ is a family $f_n : C_n \to D_n$ with $d f_n = f_{n-1} d$ ; it sends cycles to cycles and boundaries to boundaries, so it induces $H_n(f) : H_n(C) \to H_n(D)$ . It is a quasi-isomorphism if every $H_n(f)$ is an isomorphism. Two chain maps $f, g$ are chain homotopic, $f \simeq g$ , if there are maps $s_n : C_n \to D_{n+1}$ with

f_n - g_n = d_{n+1} s_n + s_{n-1} d_n \quad \text{for all } n.

A homotopy equivalence is a chain map $f$ with a chain map $g$ such that $gf \simeq \mathrm{id}$ and $fg \simeq \mathrm{id}$ .

Definition— short exact sequence of complexes

A sequence of chain maps $0 \to A_\bullet \xrightarrow{f} B_\bullet \xrightarrow{g} C_\bullet \to 0$ is short exact if $0 \to A_n \to B_n \to C_n \to 0$ is exact for every $n$ .

Definition— projective and injective resolutions

A projective resolution of $M$ is a complex $P_\bullet$ of projective modules ( $P_n = 0$ for $n < 0$ ) with a map $\varepsilon : P_0 \to M$ such that $\cdots \to P_1 \to P_0 \xrightarrow{\varepsilon} M \to 0$ is exact. Every module has one: take $P_0$ free on generators of $M$ , then $P_1$ free on generators of $\ker \varepsilon$ , and so on. An injective resolution $0 \to N \to I^0 \to I^1 \to \cdots$ is the dual notion, with the $I^n$ injective; every module has one. The projective dimension of $M$ is the shortest length of a projective resolution (possibly $\infty$ ); the global dimension of $R$ is the supremum of the projective dimensions of all modules.

Definition— derived functors; Tor and Ext

Let $F$ be an additive functor from $R$ -modules to abelian groups. If $F$ is covariant and right exact, its left derived functors are $L_nF(M) = H_n(F(P_\bullet))$ for a projective resolution $P_\bullet \to M$ . If $F$ is contravariant and left exact, its right derived functors are $R^nF(M) = H^n(F(P_\bullet))$ . In particular

\mathrm{Tor}_n^R(M, N) = H_n(P_\bullet \otimes_R N), \qquad \mathrm{Ext}^n_R(M, N) = H^n\bigl(\Hom_R(P_\bullet, N)\bigr).

Equivalently, $\mathrm{Ext}^n_R(M, N) = H^n(\Hom_R(M, I^\bullet))$ for an injective resolution $N \to I^\bullet$ , and $\mathrm{Tor}$ may be computed by resolving either variable ("balance"; Weibel, §2.7). For non-commutative $R$ , $M$ is a right and $N$ a left module in $\mathrm{Tor}$ .

Definition— extension; equivalence

An extension of $A$ by $B$ is a short exact sequence $0 \to B \to E \to A \to 0$ . Two extensions $E, E'$ are equivalent if there is a homomorphism $E \to E'$ making both squares commute (it is then an isomorphism by the five lemma). The split extension is $0 \to B \to B \oplus A \to A \to 0$ .

Worked examplesH

Examples

Example— A complex with torsion homology

The real projective plane is a square with boundary word $abab$ : one 2-cell $F$ , two edges $a, b$ , and two vertices $v, w$ ( $a$ runs from $v$ to $w$ , $b$ from $w$ to $v$ ). Its cellular chain complex is

0 \to \Z \xrightarrow{\ d_2\ } \Z^2 \xrightarrow{\ d_1\ } \Z^2 \to 0, \qquad d_2(F) = 2a + 2b, \quad d_1(a) = w - v, \quad d_1(b) = v - w.

Then $\ker d_1 = \Z(a + b)$ and $\operatorname{im} d_2 = \Z \cdot 2(a + b)$ , so $H_1 \cong \Z/2$ ; $H_2 = \ker d_2 = 0$ ; and $H_0 = \Z^2 / \Z(w - v) \cong \Z$ . With $\mathbb{F}_2$ coefficients, $d_2 = 0$ and $d_1$ has rank $1$ , so $H_n(\R P^2; \mathbb{F}_2) = \mathbb{F}_2$ for $n = 0, 1, 2$ . (Formalised in part below; computed for four surfaces in the second experiment.)

Example— Resolutions

Over $\Z$ : $0 \to \Z \xrightarrow{n} \Z \to \Z/n \to 0$ . Every abelian group has a free resolution of length $\leq 1$ (subgroups of free abelian groups are free), so $\Z$ has global dimension $1$ .
Over $k[x]$ : $0 \to k[x] \xrightarrow{x - a} k[x] \to k[x]/(x - a) \to 0$ .
Over $k[x, y]$ , the Koszul resolution of $k = k[x, y]/(x, y)$ :

0 \to k[x, y] \xrightarrow{\begin{pmatrix} -y \\ x \end{pmatrix}} k[x, y]^2 \xrightarrow{(x \ \ y)} k[x, y] \to k \to 0.

Over $\Z/4$ , the module $\Z/2$ has the infinite periodic resolution $\cdots \xrightarrow{2} \Z/4 \xrightarrow{2} \Z/4 \xrightarrow{2} \Z/4 \to \Z/2 \to 0$ , and no finite one: $\Z/4$ has infinite global dimension.

Example— Tor and Ext over ℤ

Resolve $\Z/m$ by $0 \to \Z \xrightarrow{m} \Z \to 0$ . Tensoring with $N$ gives $N \xrightarrow{m} N$ , and $\Hom(-, N)$ gives $N \xrightarrow{m} N$ as well, so

\mathrm{Tor}_0(\Z/m, N) = N/mN, \quad \mathrm{Tor}_1(\Z/m, N) = N[m] = \{x : mx = 0\}, \quad \mathrm{Hom}(\Z/m, N) = N[m], \quad \mathrm{Ext}^1(\Z/m, N) = N/mN,

and all higher groups vanish. For $N = \Z/n$ each of these is cyclic of order $\gcd(m, n)$ ; for $N = \Z$ : $\mathrm{Tor}_1(\Z/m, \Z) = 0$ and $\mathrm{Ext}^1(\Z/m, \Z) \cong \Z/m$ . In general $\mathrm{Tor}_1^\Z(\Z/n, A)$ is the $n$ -torsion subgroup of $A$ — the origin of the name "Tor".

Example— Tor and Ext over k[x] and ℤ/4

Over $R = k[x]$ , with $S_a = k[x]/(x - a)$ , the resolution above gives $S_b \xrightarrow{x - a} S_b$ , which is multiplication by $b - a$ on $k$ . So $\mathrm{Tor}_1^{k[x]}(S_a, S_b) \cong \mathrm{Ext}^1_{k[x]}(S_a, S_b) \cong k$ if $a = b$ , and $0$ otherwise: the two points must coincide to interact. Over $\Z/4$ , the periodic resolution gives $\mathrm{Tor}_n^{\Z/4}(\Z/2, \Z/2) \cong \Z/2$ for every $n \geq 0$ : derived functors over a ring of infinite global dimension need not stop.

ProofH

The main theorems

Theorem— Theorem 1 (long exact homology sequence)

A short exact sequence of chain complexes $0 \to A_\bullet \xrightarrow{f} B_\bullet \xrightarrow{g} C_\bullet \to 0$ induces an exact sequence

\cdots \to H_n(A) \xrightarrow{f_*} H_n(B) \xrightarrow{g_*} H_n(C) \xrightarrow{\ \partial\ } H_{n-1}(A) \xrightarrow{f_*} H_{n-1}(B) \xrightarrow{g_*} H_{n-1}(C) \to \cdots

in which the connecting homomorphism is $\partial[c] = [a]$ where $c = g(b)$ and $d b = f(a)$ .

Proof

Apply the snake lemma to the map of short exact sequences $d_n : (0 \to A_n \to B_n \to C_n \to 0) \to (0 \to A_{n-1} \to B_{n-1} \to C_{n-1} \to 0)$ . Its kernel part says that $0 \to Z_n(A) \to Z_n(B) \to Z_n(C)$ is exact; applied to $d_{n+1}$ , its cokernel part says that $A_n/B_n(A) \to B_n/B_n(B) \to C_n/B_n(C) \to 0$ is exact. Since $d \circ d = 0$ , the differential induces maps $\bar d : C_n / B_n(C) \to Z_{n-1}(C)$ , and similarly for $A$ and $B$ , giving a commutative diagram with exact rows

\begin{array}{ccccccc} A_n/B_n(A) & \to & B_n/B_n(B) & \to & C_n/B_n(C) & \to & 0 \\ \downarrow \bar d & & \downarrow \bar d & & \downarrow \bar d & & \\ 0 \to Z_{n-1}(A) & \to & Z_{n-1}(B) & \to & Z_{n-1}(C). & & \end{array}

The kernel of $\bar d : C_n/B_n \to Z_{n-1}$ is $Z_n / B_n = H_n(C)$ and its cokernel is $Z_{n-1}/B_{n-1} = H_{n-1}(C)$ , and likewise for $A$ and $B$ . The snake lemma now gives the exact sequence $H_n(A) \to H_n(B) \to H_n(C) \xrightarrow{\partial} H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C)$ , and these six-term pieces overlap to form the long exact sequence. Unwinding the snake's construction: lift a cycle $c$ to $b \in B_n$ ; then $g(db) = dc = 0$ , so $db = f(a)$ for a unique $a$ , which is a cycle because $f(da) = ddb = 0$ and $f$ is injective; and $\partial[c] = [a]$ .

∎

Theorem— Theorem 2 (homotopy invariance)

If $f \simeq g : C_\bullet \to D_\bullet$ , then $H_n(f) = H_n(g)$ for all $n$ . Hence a homotopy equivalence is a quasi-isomorphism, and additive functors, which preserve chain homotopies, send homotopy equivalences to homotopy equivalences.

Proof

For a cycle $z \in Z_n(C)$ , $f(z) - g(z) = d s(z) + s(dz) = d s(z)$ is a boundary, so $[f(z)] = [g(z)]$ . If $F$ is additive, applying it to $f - g = ds + sd$ gives $F(f) - F(g) = F(d)F(s) + F(s)F(d)$ .

∎

Theorem— Theorem 3 (comparison theorem)

Let $P_\bullet \to M$ be a complex of projective modules over $M$ (that is, $P_n = 0$ for $n < 0$ and $\cdots \to P_1 \to P_0 \to M \to 0$ is a complex), and $Q_\bullet \to N$ a resolution (exact, not necessarily projective). Then every homomorphism $\varphi : M \to N$ lifts to a chain map $f : P_\bullet \to Q_\bullet$ compatible with the augmentations, and any two such lifts are chain homotopic.

Proof

Existence. $\varphi \varepsilon_P : P_0 \to N$ lifts through the surjection $\varepsilon_Q : Q_0 \to N$ because $P_0$ is projective; call the lift $f_0$ . Suppose $f_0, \dots, f_n$ have been built with $d f_k = f_{k-1} d$ (and $\varepsilon_Q f_0 = \varphi \varepsilon_P$ ). The map $f_n d : P_{n+1} \to Q_n$ lands in $\ker(Q_n \to Q_{n-1})$ (with $Q_{-1} = N$ ): indeed $d f_n d = f_{n-1} d d = 0$ , or $\varepsilon_Q f_0 d = \varphi \varepsilon_P d = 0$ when $n = 0$ . By exactness of $Q_\bullet \to N$ , that kernel is the image of $d : Q_{n+1} \to Q_n$ , so projectivity of $P_{n+1}$ gives $f_{n+1} : P_{n+1} \to Q_{n+1}$ with $d f_{n+1} = f_n d$ .

Uniqueness up to homotopy. Let $f, g$ be two lifts and $h = f - g$ , a lift of $0$ . We build $s_n : P_n \to Q_{n+1}$ with $h_n = d s_n + s_{n-1} d$ , starting from $s_{-1} = 0$ . Since $\varepsilon_Q h_0 = 0$ , $h_0$ lands in $\ker \varepsilon_Q = \operatorname{im}(Q_1 \to Q_0)$ ; lift to $s_0$ with $d s_0 = h_0$ . Inductively, $h_n - s_{n-1} d$ lands in $\ker(d : Q_n \to Q_{n-1})$ , because

d(h_n - s_{n-1} d) = h_{n-1} d - d s_{n-1} d = (h_{n-1} - d s_{n-1}) d = s_{n-2} d d = 0,

and so, by exactness and projectivity of $P_n$ , it equals $d s_n$ for some $s_n : P_n \to Q_{n+1}$ .

∎

Theorem— Corollary 4 (derived functors are well defined)

Any two projective resolutions of $M$ are homotopy equivalent, by a homotopy equivalence lifting $\mathrm{id}_M$ that is unique up to homotopy. Consequently $L_nF(M)$ , $\mathrm{Tor}_n$ and $\mathrm{Ext}^n$ do not depend on the resolution, up to canonical isomorphism, and are functorial in $M$ . Moreover:

$L_0F = F$ for right exact $F$ , so $\mathrm{Tor}_0(M, N) = M \otimes N$ ; and $\mathrm{Ext}^0(M, N) = \Hom(M, N)$ ;
$L_nF(P) = 0$ for $n \geq 1$ if $P$ is projective; so $\mathrm{Tor}_n(P, N) = \mathrm{Ext}^n(P, N) = 0$ ;
a short exact sequence $0 \to M' \to M \to M'' \to 0$ gives long exact sequences

\cdots \to \mathrm{Tor}_1(M'', N) \to M' \otimes N \to M \otimes N \to M'' \otimes N \to 0,

0 \to \Hom(M'', N) \to \Hom(M, N) \to \Hom(M', N) \to \mathrm{Ext}^1(M'', N) \to \mathrm{Ext}^1(M, N) \to \cdots.

Proof

If $P_\bullet, Q_\bullet$ resolve $M$ , Theorem 3 gives lifts $f : P \to Q$ and $g : Q \to P$ of $\mathrm{id}_M$ ; $gf$ and $\mathrm{id}_P$ both lift $\mathrm{id}_M$ , so $gf \simeq \mathrm{id}_P$ by uniqueness, and likewise $fg \simeq \mathrm{id}_Q$ . By Theorem 2, $F(P)$ and $F(Q)$ have canonically isomorphic homology. (1): $P_1 \to P_0 \to M \to 0$ is exact and $F$ is right exact, so $H_0(F(P)) = \operatorname{coker} F(P_1 \to P_0) \cong F(M)$ ; dually for $\Hom$ . (2): $P$ resolves itself ( $P_0 = P$ , $P_n = 0$ otherwise). (3): The horseshoe lemma builds projective resolutions $P'$ , $P$ , $P''$ forming a short exact sequence of complexes $0 \to P' \to P \to P'' \to 0$ that is split in each degree (since $P''_n$ is projective). Additive functors preserve split exact sequences, so $0 \to F(P') \to F(P) \to F(P'') \to 0$ is short exact, and Theorem 1 gives the long exact sequence (Weibel, Lemma 2.2.8 and Theorem 2.4.6).

∎

Theorem— Theorem 5 (Ext¹ classifies extensions)

There is a bijection between equivalence classes of extensions $0 \to B \to E \to A \to 0$ and $\mathrm{Ext}^1_R(A, B)$ , under which the split extension corresponds to $0$ . In particular every extension of $A$ by $B$ splits if and only if $\mathrm{Ext}^1(A, B) = 0$ .

Proof

Choose a short exact sequence $0 \to K \xrightarrow{\iota} P \xrightarrow{\pi} A \to 0$ with $P$ projective. Its long exact $\mathrm{Ext}(-, B)$ sequence, with $\mathrm{Ext}^1(P, B) = 0$ , gives

\mathrm{Ext}^1(A, B) \;\cong\; \operatorname{coker}\bigl(\iota^* : \Hom(P, B) \to \Hom(K, B)\bigr).

From extensions to classes. Given $0 \to B \xrightarrow{j} E \xrightarrow{q} A \to 0$ , lift $\pi$ through $q$ to $\beta : P \to E$ (projectivity). Then $q \beta \iota = \pi\iota = 0$ , so $\beta \iota = j \varphi$ for a unique $\varphi : K \to B$ . Another lift is $\beta + j\psi$ with $\psi : P \to B$ , changing $\varphi$ to $\varphi + \psi\iota$ ; so the class $\Theta(E) = [\varphi]$ is well defined, and equivalent extensions give the same class.

From classes to extensions. Given $\varphi : K \to B$ , let $E_\varphi$ be the pushout $(B \oplus P)/\{(\varphi(k), -\iota(k)) : k \in K\}$ , with $B \to E_\varphi$ , $b \mapsto [b, 0]$ , and $E_\varphi \to A$ , $[b, p] \mapsto \pi(p)$ . A diagram chase shows $0 \to B \to E_\varphi \to A \to 0$ is exact, and replacing $\varphi$ by $\varphi + \psi\iota$ gives an equivalent extension, via the map $E_{\varphi + \psi\iota} \to E_\varphi$ , $[b, p] \mapsto [b + \psi(p), p]$ .

They are inverse. For $E_\varphi$ , the lift $\beta(p) = [0, p]$ satisfies $\beta\iota(k) = [0, \iota(k)] = [\varphi(k), 0]$ , so $\Theta(E_\varphi) = [\varphi]$ . Conversely, for an extension $E$ with lift $\beta$ and class $[\varphi]$ , the map $E_\varphi \to E$ , $[b, p] \mapsto j(b) + \beta(p)$ , is well defined and compatible with both ends, hence an equivalence. Finally, $\Theta(E) = 0$ iff $\varphi = \psi\iota$ for some $\psi$ , iff the lift $\beta - j\psi$ vanishes on $K$ and so factors through $A$ as a section of $q$ , iff $E$ splits.

∎

Under this bijection, $\mathrm{Ext}^1(\Z/2, \Z) \cong \Z/2$ has two elements: the split extension $\Z \oplus \Z/2$ and $0 \to \Z \xrightarrow{2} \Z \to \Z/2 \to 0$ . The group structure on $\mathrm{Ext}^1$ corresponds to the Baer sum of extensions (Baer, 1934). Higher $\mathrm{Ext}^n$ classify $n$ -fold exact sequences $0 \to B \to E_n \to \cdots \to E_1 \to A \to 0$ up to a suitable equivalence (Yoneda).

Theorem— Theorem 6 (three landmarks; stated)

Universal coefficient theorem. For a chain complex $C_\bullet$ of free abelian groups and an abelian group $G$ there are natural short exact sequences, split but not naturally,

0 \to H_n(C) \otimes G \to H_n(C \otimes G) \to \mathrm{Tor}_1^\Z(H_{n-1}(C), G) \to 0.

Hilbert's syzygy theorem (1890). Every finitely generated module over $k[x_1, \dots, x_n]$ has a free resolution of length at most $n$ ; the global dimension of $k[x_1, \dots, x_n]$ is exactly $n$ .
Auslander–Buchsbaum–Serre (1956). A Noetherian local ring is regular if and only if it has finite global dimension.

Proofs: Hatcher, Algebraic Topology, Theorem 3A.3, or Weibel, Theorem 3.6.2 (universal coefficients); Weibel, §4.3–4.4, and Eisenbud, Chapter 19 (syzygies and regular rings). The third theorem is a striking use of homological algebra in commutative algebra: it shows at once that localisations of regular local rings are regular, which had resisted direct attack.

Explore

Explore: chain complexes of glued squares

Each surface below is a square with its sides glued in pairs. Its cellular chain complex has one 2-cell $F$ , one 1-cell for each letter, and one 0-cell for each class of glued corners, and the boundary of $F$ is read off the word: $d_2(F)$ is the sum of the letters, with sign $-1$ for an inverse. So:

torus $aba^{-1}b^{-1}$ : $d_2(F) = a + b - a - b = 0$ , so $H_2 = \Z$ and $H_1 = \Z^2$ ;
Klein bottle $aba^{-1}b$ : $d_2(F) = 2b$ , so $H_2 = 0$ and $H_1 = \Z \oplus \Z/2$ ;
projective plane $abab$ : $d_2(F) = 2a + 2b$ , so $H_2 = 0$ and $H_1 = \Z/2$ ;
sphere $aa^{-1}bb^{-1}$ : $d_2(F) = 0$ and three vertices, so $H_2 = \Z$ , $H_1 = 0$ .

Walk the flatlander across the seams. Exactly on the non-orientable surfaces it comes back mirror-reversed — and exactly there $d_2(F) \neq 0$ , so the 2-cell is not a cycle over $\Z$ and $H_1$ acquires the torsion $\Z/2$ . Over $\mathbb{F}_2$ the coefficient $2$ vanishes, and every one of these closed surfaces has $H_2(X; \mathbb{F}_2) = \mathbb{F}_2$ , as the universal coefficient theorem predicts: for the Klein bottle and $\R P^2$ this class comes from $\mathrm{Tor}_1(\Z/2, \mathbb{F}_2) = \mathbb{F}_2$ .

Gluing words and boundary maps

Where the flag flips, the boundary of the 2-cell is not zero.

Keys: ↑ forward, ↓ back, ← and → turn, Space walks or stops. You can also drag it.

Projective planea b a b

Orientable: No
Euler characteristic: χ = 1

Gluing turns the 4 corners into V = 2, the 4 sides into E = 2 and the square into F = 1: V − E + F = 2 − 2 + 1 = 1

Same way round as at the start.

Sides crossed so far: 0

Both pairs of opposite sides are glued with a twist. This is the real projective plane: non-orientable, with χ = 1. Like the Klein bottle, it cannot sit in 3D space without passing through itself.

The second experiment builds these chain complexes from the words and computes all their homology groups.

Formal mathematicsF

Homological algebra in Lean and Mathlib

We begin with the chain complex of $\R P^2$ as integer matrices, verify $d_1 d_2 = 0$ , and prove that $H_1$ contains an element of order exactly $2$ : the cycle $a + b$ is not a boundary, but twice it is.

rp2_d2Lean 4Formally verified

/-- The cellular boundary `d₂ : ℤ → ℤ²` of the real projective plane, glued from a square with
boundary word `a b a b`: the 2-cell runs along `a` twice and `b` twice, so `d₂(F) = 2a + 2b`. -/
def rp2_d2 : Matrix (Fin 2) (Fin 1) ℤ := !![2; 2]

Axioms used: propext, Quot.soundformal/Lumenary/HomologicalAlgebra.leanTry it in the Lean web editor

rp2_d1Lean 4Formally verified

/-- The cellular boundary `d₁ : ℤ² → ℤ²` (edges `a, b` to vertices `v, w`): `a` runs from `v` to
`w` and `b` from `w` back to `v`, so `d₁(a) = w - v` and `d₁(b) = v - w`. Columns are edges. -/
def rp2_d1 : Matrix (Fin 2) (Fin 2) ℤ := !![-1, 1; 1, -1]

Axioms used: propext, Quot.soundformal/Lumenary/HomologicalAlgebra.leanTry it in the Lean web editor

rp2_ddLean 4Proved here from basic factsFormally verified

/-- **`d ∘ d = 0`**: the boundary of the boundary vanishes, `d₁(2a + 2b) = 2(w - v) + 2(v - w)`.
A direct matrix computation. -/
theorem rp2_dd : rp2_d1 * rp2_d2 = 0 := by
  ext i j
  fin_cases i <;> fin_cases j <;> simp [rp2_d1, rp2_d2, Matrix.mul_apply, Fin.sum_univ_two]