Derived Functors — Ext, Tor — Mathematics

⚡ Key Takeaways

Derived functors are the canonical machinery for measuring the failure of a functor to be exact. Ext and Tor are the derived functors of $\mathrm{Hom}$ and $\otimes$ respectively: $\mathrm{Ext}^n_R(A,B)$ measures obstructions to lifting/extending homomorphisms and classifies extensions, while $\mathrm{Tor}_n^R(A,B)$ measures torsion-like obstructions to exactness of tensoring. They apply throughout homological algebra, algebraic topology, algebraic geometry, and number theory. Introduced in the 1940s–1950s (Eilenberg, MacLane, Cartan, Eilenberg, Tor).

💡 The Core Insight

An exact functor preserves all the information in a short exact sequence. A non-exact functor loses information — but the loss is measurable and natural. Derived functors are precisely the sequence of "correction terms" $L_0F, L_1F, L_2F, \dots$ (or $R^0F, R^1F, \dots$ ) that splice the broken sequence back into an infinite long exact sequence. Ext and Tor are the two most important instances, governing extensions, deformations, torsion, and cohomology everywhere.

DefinitionResolutions

Let $R$ be a ring (associative, unital; modules are left modules unless stated). For an $R$ -module $A$ :

A projective resolution is an exact sequence

1

\quad \cdots \to P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \xrightarrow{\varepsilon} A \to 0

with each $P_i$ projective.

An injective resolution is an exact sequence

2

\quad 0 \to A \xrightarrow{\eta} I^0 \xrightarrow{d^0} I^1 \xrightarrow{d^1} I^2 \to \cdots

with each $I^j$ injective. Derivation hint: Existence of projective resolutions follows from the fact that every module is a quotient of a free (hence projective) module; iterate on the kernel of $\varepsilon$ . Injective resolutions exist because $R\text{-Mod}$ has enough injectives (Baer's criterion + injective envelopes).

DefinitionLeft and Right Derived Functors

Let $F: \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories with enough projectives/injectives. For $F$ right exact (e.g. $A \otimes_R -$ ), the left derived functors are

3

\quad L_nF(A) = H_n\big(F(P_\bullet)\big), \qquad P_\bullet \to A \text{ a projective resolution}

For $F$ left exact (e.g. $\mathrm{Hom}_R(A,-)$ ), the right derived functors are

4

\quad R^nF(A) = H^n\big(F(I^\bullet)\big), \qquad A \to I^\bullet \text{ an injective resolution}

Derivation hint: Independence of the chosen resolution is the Comparison Theorem: any two projective resolutions are chain homotopy equivalent, and additive functors preserve chain homotopies, so homology is well-defined up to canonical isomorphism.

DefinitionExt and Tor

For $R$ -modules $A,B$ :

5

\quad \mathrm{Ext}^n_R(A,B) = R^n\big(\mathrm{Hom}_R(A,-)\big)(B) = H^n\big(\mathrm{Hom}_R(P_\bullet, B)\big)

6

\quad \mathrm{Tor}_n^R(A,B) = L_n\big(A \otimes_R -\big)(B) = H_n\big(P_\bullet \otimes_R B\big)

where $P_\bullet \to A$ is a projective resolution (for Ext using the contravariant $\mathrm{Hom}_R(-,B)$ applied to a projective resolution of $A$ ; for Tor applied to a flat/projective resolution of $A$ ). Derivation hint: $\mathrm{Hom}_R(P_\bullet,B)$ is a cochain complex because $\mathrm{Hom}_R(-,B)$ is contravariant, reversing the chain differentials into coboundaries; $P_\bullet \otimes_R B$ stays a chain complex because $-\otimes B$ is covariant.

TheoremLong Exact Sequences

Let $0 \to A' \to A \to A'' \to 0$ be a short exact sequence of $R$ -modules. Then there are natural long exact sequences:

7

\quad \cdots \to \mathrm{Tor}_n^R(A',B) \to \mathrm{Tor}_n^R(A,B) \to \mathrm{Tor}_n^R(A'',B) \xrightarrow{\partial} \mathrm{Tor}_{n-1}^R(A',B) \to \cdots

8

\quad \cdots \to \mathrm{Ext}^n_R(A'',B) \to \mathrm{Ext}^n_R(A,B) \to \mathrm{Ext}^n_R(A',B) \xrightarrow{\partial} \mathrm{Ext}^{n+1}_R(A'',B) \to \cdots

ending with $\mathrm{Tor}_0^R(-,B) = -\otimes_R B$ and $\mathrm{Ext}^0_R(-,B) = \mathrm{Hom}_R(-,B)$ . Derivation hint: Apply the Horseshoe Lemma to build a short exact sequence of projective resolutions $0 \to P'_\bullet \to P_\bullet \to P''_\bullet \to 0$ (degreewise split), apply $F$ , and use the Snake Lemma / long exact homology sequence of the resulting short exact sequence of complexes.

TheoremBalancing

There are natural isomorphisms, for all $n$ :

9

\quad \mathrm{Tor}_n^R(A,B) \cong L_n(A\otimes_R -)(B) \cong L_n(-\otimes_R B)(A)

10

\quad \mathrm{Ext}^n_R(A,B) \cong R^n\mathrm{Hom}_R(A,-)(B) \cong R^n\mathrm{Hom}_R(-,B)(A)

i.e. one may resolve either variable. Derivation hint: Form the double complex $P_\bullet \otimes_R Q_\bullet$ (resp. $\mathrm{Hom}_R(P_\bullet, Q^\bullet)$ ) from resolutions of both variables. Both spectral sequences of the double complex degenerate (rows/columns are exact except at the augmentation), forcing both filtrations to compute the total homology — hence the two answers agree.

ProofConstruction of Tor via projective resolution — well-definedness

Step 1. Choose a projective resolution $P_\bullet \xrightarrow{\varepsilon} A$ . Apply $- \otimes_R B$ to the deleted resolution $P_\bullet$ (omit $A$ ), obtaining the chain complex

11

\quad \cdots \to P_2 \otimes_R B \xrightarrow{d_2\otimes 1} P_1 \otimes_R B \xrightarrow{d_1 \otimes 1} P_0 \otimes_R B \to 0.

Define $\mathrm{Tor}_n^R(A,B) = \ker(d_n\otimes 1)/\mathrm{im}(d_{n+1}\otimes 1)$ . Derivation hint: We delete $A$ before applying $F$ because $F$ is not exact; keeping $A$ would force the homology at degree $0$ to vanish artificially.

Step 2. $\mathrm{Tor}_0^R(A,B) = \mathrm{coker}(d_1\otimes 1) = P_0\otimes B / \mathrm{im}(P_1\otimes B)$ . Right-exactness of $-\otimes B$ applied to $P_1 \to P_0 \to A \to 0$ gives $P_0 \otimes B / \mathrm{im} \cong A\otimes_R B$ . Derivation hint: Right-exact functors preserve cokernels; the cokernel of $P_1\to P_0$ is $A$ .

Step 3. Independence of resolution. If $Q_\bullet \to A$ is another projective resolution, the Comparison Theorem yields chain maps $f: P_\bullet \to Q_\bullet$ and $g: Q_\bullet \to P_\bullet$ lifting $\mathrm{id}_A$ , unique up to chain homotopy. Then $gf \simeq \mathrm{id}$ , $fg \simeq \mathrm{id}$ . Applying $-\otimes B$ preserves chain homotopies (additivity), so $f\otimes 1, g\otimes 1$ are mutually inverse on homology. Derivation hint: A chain homotopy $s$ with $df+fd' = \mathrm{id}-gf$ tensors to $(d\otimes 1)(s\otimes 1)+(s\otimes 1)(d'\otimes 1)=\mathrm{id}-(g\otimes1)(f\otimes1)$ .

Step 4. Naturality. A module map $B\to B'$ induces a chain map of complexes $P_\bullet \otimes B \to P_\bullet \otimes B'$ , hence functoriality; similarly in $A$ by lifting maps to resolutions. $\square$

∎

mathematician · Samuel Eilenberg & Saunders Mac Lane | 1940s–1950s

Eilenberg (1913–1998) and Mac Lane (1909–2005) created homological algebra and category theory in concert. The very symbol "Tor" comes from torsion: $\mathrm{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/n, A)$ recovers the $n$ -torsion of $A$ . "Ext" abbreviates extension, reflecting the classification of module extensions by $\mathrm{Ext}^1$ . Their work, culminating in the 1956 Cartan–Eilenberg Homological Algebra (with Henri Cartan), axiomatized derived functors and unified previously ad hoc constructions across topology, group cohomology, and Lie algebra cohomology into a single framework.

🧠 Intuition

Think of an exact functor as a perfectly transparent lens: a short exact sequence in goes to a short exact sequence out, no distortion. A non-exact functor is a cracked lens — it preserves most structure but introduces blur at the boundary (it preserves left- or right-exactness only). The derived functors are the catalog of every crack, organized by "depth" $n$ . For Tor: tensoring with $B$ can collapse injections. If $A' \hookrightarrow A$ is injective but $A'\otimes B \to A\otimes B$ is not, the kernel of the latter is detected by $\mathrm{Tor}_1^R(A'',B)$ . The classic example: $\mathbb{Z}\xrightarrow{\times n}\mathbb{Z}$ is injective, but tensoring with $\mathbb{Z}/n$ gives the zero map — the lost injectivity surfaces as $\mathrm{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/n,\mathbb{Z}/n)=\mathbb{Z}/n$ . For Ext: $\mathrm{Hom}_R(-,B)$ fails to be surjective on lifts. Given $A\twoheadleftarrow$ surjection, not every map into $B$ extends; $\mathrm{Ext}^1$ measures the obstruction. Geometrically, $\mathrm{Ext}^1(A,B)$ is the set of "ways to glue $B$ and $A$ into a single module" (extensions) modulo equivalence — nonzero Ext means non-split gluings exist.

Figure

Building Tor/Ext: delete the object, resolve, apply functor, take homology

ExampleTor over

\mathbb{Z}

:

\mathrm{Tor}_n^{\mathbb{Z}}(\mathbb{Z}/m, \mathbb{Z}/k)

Given: compute all $\mathrm{Tor}_n^{\mathbb{Z}}(\mathbb{Z}/m, \mathbb{Z}/k)$ , $m,k \ge 2$ .

Step 1 — Resolve $\mathbb{Z}/m$ . Since $\mathbb{Z}$ is a PID, a free resolution is short:

12

\quad 0 \to \mathbb{Z} \xrightarrow{\times m} \mathbb{Z} \to \mathbb{Z}/m \to 0.

Derivation hint: the kernel of $\mathbb{Z}\twoheadrightarrow \mathbb{Z}/m$ is $m\mathbb{Z}\cong\mathbb{Z}$ , free, so the resolution stops at length 1.

Step 2 — Delete and tensor with $\mathbb{Z}/k$ :

13

\quad 0 \to \mathbb{Z}/k \xrightarrow{\times m} \mathbb{Z}/k \to 0.

Derivation hint: $\mathbb{Z}\otimes\mathbb{Z}/k=\mathbb{Z}/k$ ; the map $\times m$ tensors to multiplication by $m$ on $\mathbb{Z}/k$ .

Step 3 — Homology.

$\mathrm{Tor}_0 = \mathrm{coker}(\times m) = \mathbb{Z}/k \big/ m\mathbb{Z}/k = \mathbb{Z}/\gcd(m,k)$ .
$\mathrm{Tor}_1 = \ker(\times m \text{ on } \mathbb{Z}/k) = \{x : mx \equiv 0 \bmod k\} = \mathbb{Z}/\gcd(m,k)$ .
$\mathrm{Tor}_{n} = 0$ for $n\ge 2$ (resolution has length 1).

Boxed answer:

\boxed{\;\mathrm{Tor}_0^{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z}/k)=\mathrm{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z}/k)=\mathbb{Z}/\gcd(m,k),\quad \mathrm{Tor}_{\ge 2}=0\;}

ExampleExt and extensions:

\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z})

Given: compute $\mathrm{Ext}^n_{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z})$ and interpret.

Step 1 — Same resolution $0\to\mathbb{Z}\xrightarrow{\times m}\mathbb{Z}\to\mathbb{Z}/m\to 0$ .

Step 2 — Apply $\mathrm{Hom}_{\mathbb{Z}}(-,\mathbb{Z})$ (contravariant, deleted resolution):

14

\quad 0 \to \mathrm{Hom}(\mathbb{Z},\mathbb{Z}) \xrightarrow{(\times m)^*} \mathrm{Hom}(\mathbb{Z},\mathbb{Z}) \to 0,\quad\text{i.e.}\quad 0\to\mathbb{Z}\xrightarrow{\times m}\mathbb{Z}\to 0.

Derivation hint: $\mathrm{Hom}(\mathbb{Z},\mathbb{Z})\cong\mathbb{Z}$ , and precomposition with $\times m$ becomes multiplication by $m$ (placed in cohomological degrees 0 and 1).

Step 3 — Cohomology.

$\mathrm{Ext}^0=\ker(\times m)=0$ .
$\mathrm{Ext}^1=\mathrm{coker}(\times m)=\mathbb{Z}/m$ .
$\mathrm{Ext}^{\ge 2}=0$ .

Step 4 — Interpretation. $\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z})\cong\mathbb{Z}/m$ classifies extensions $0\to\mathbb{Z}\to E\to\mathbb{Z}/m\to 0$ . The class $r\in\mathbb{Z}/m$ corresponds to $E$ with the relation pulling $\mathbb{Z}/m$ "tighter": $r=0$ gives the split $\mathbb{Z}\oplus\mathbb{Z}/m$ , while $r=1$ gives $E=\mathbb{Z}$ (sitting as $\mathbb{Z}\xrightarrow{\times m}\mathbb{Z}\to\mathbb{Z}/m$ ).

Boxed answer:

\boxed{\;\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z})\cong\mathbb{Z}/m,\quad \mathrm{Ext}^0=\mathrm{Ext}^{\ge2}=0\;}

⚠️ Frequent errors

Forgetting to delete the object. You apply $F$ to the deleted resolution $P_\bullet$ (without $A$ ), not to $P_\bullet \to A$ . Including $A$ makes $H_0$ trivially zero and destroys the computation.
Ext is not symmetric. $\mathrm{Ext}^n_R(A,B)\ne\mathrm{Ext}^n_R(B,A)$ in general; the two variables play asymmetric roles (one contravariant, one covariant), even though balancing lets you resolve either.
Tor depends on the ring. $\mathrm{Tor}_n^R$ is a $\mathbb{Z}$ -module (or $R$ -module if $R$ commutative) but the answer changes drastically with $R$ . The superscript/subscript ring is not decoration.
Vanishing $\mathrm{Ext}^1$ ≠ splitting of everything. $\mathrm{Ext}^1(A,B)=0$ means every extension of $A$ by that specific $B$ splits; projectivity of $A$ is the stronger statement $\mathrm{Ext}^1(A,-)=0$ for all $B$ .
Long exact sequence direction. For Ext, the contravariant variable reverses arrows: $0\to A'\to A\to A''\to 0$ yields $\cdots\mathrm{Ext}^n(A'')\to\mathrm{Ext}^n(A)\to\mathrm{Ext}^n(A')\to\mathrm{Ext}^{n+1}(A'')\cdots$ — note $A''$ leads.

CounterexampleFlat ≠ projective; Tor-acyclic ≠ flat in all senses

$\mathbb{Q}$ is a flat $\mathbb{Z}$ -module (so $\mathrm{Tor}_n^{\mathbb{Z}}(\mathbb{Q},-)=0$ for $n\ge1$ ) but $\mathbb{Q}$ is not projective over $\mathbb{Z}$ (it is not a direct summand of any free abelian group; it is divisible and torsion-free but not free). This shows: 1. Tor-vanishing detects flatness, not projectivity. $\mathrm{Tor}_1^R(M,-)=0 \iff M$ flat. 2. $\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$ is in fact uncountable ( $\cong \mathbb{A}_f/\mathbb{Q}$ -ish, an enormous group) — $\mathbb{Q}$ being flat does not make its Ext vanish. Flatness and Ext-acyclicity are independent conditions.

connection · Where Ext and Tor sit in the homological web

Group Cohomology: $H^n(G,M)=\mathrm{Ext}^n_{\mathbb{Z}G}(\mathbb{Z},M)$ and $H_n(G,M)=\mathrm{Tor}_n^{\mathbb{Z}G}(\mathbb{Z},M)$ — group (co)homology is literally Ext/Tor over the group ring.
Sheaf Cohomology: $H^n(X,\mathcal{F})=R^n\Gamma(\mathcal{F})$ , the right derived functors of global sections; local Ext sheaves $\mathcal{E}xt^n$ globalize $\mathrm{Ext}$ .
<a class="wiki-link" href="/wiki/universal-coefficient-theorem">Universal Coefficient Theorem</a>: $H_n(X;A)$ relates to $H_n(X;\mathbb{Z})$ via $\mathrm{Tor}_1^{\mathbb{Z}}$ ; cohomology via $\mathrm{Ext}^1_{\mathbb{Z}}$ .
Künneth Theorem: the homology of a tensor product of complexes carries a $\mathrm{Tor}_1$ correction term.
Spectral Sequences: balancing and composite functors ( $R^n(G\circ F)$ , the Grothendieck spectral sequence) are governed by spectral sequences of double complexes.
Derived Category: Ext $^n(A,B)=\mathrm{Hom}_{D(R)}(A,B[n])$ — Ext groups are morphisms (with shifts) in the derived category, the modern home of derived functors.

🔬 Concrete uses across mathematics and beyond

⏳ History

- 1934 — Reidemeister, Schreier: torsion in homology of complexes hints at the role of "Tor"-like corrections in topology.

- 1942 — Eilenberg & Mac Lane: introduce the group

\mathrm{Ext}(A,B)

to compute homology of spaces with twisted coefficients; the Universal Coefficient Theorem named.

- 1943–1945 — Eilenberg & Mac Lane: systematic study of

\mathrm{Ext}

and the bifunctor "Tor" (torsion product); cohomology of groups via

\mathrm{Ext}_{\mathbb{Z}G}

.

- 1947 — Hopf, Eckmann: independent group (co)homology constructions clarified as derived functors.

- 1948 — D.K. Harrison / Mac Lane: extension-theoretic interpretation of

\mathrm{Ext}^1

solidified (Yoneda's later

n

-extension viewpoint).

- 1956 — Cartan & Eilenberg, Homological Algebra: the definitive axiomatization of derived functors, projective/injective resolutions, balancing, the long exact sequences — birth of the modern subject.

- 1957 — Grothendieck, Tôhoku paper: derived functors in arbitrary abelian categories with enough injectives;

\delta

-functors; the Grothendieck spectral sequence.

- 1960s — Yoneda:

\mathrm{Ext}^n

realized as equivalence classes of

n

-fold extensions, with composition product (Yoneda product), making

\mathrm{Ext}^\bullet

a graded ring.

- 1967 — Verdier (under Grothendieck): the derived category, recasting Ext as Hom in

D(R)

and unifying all derived functors as total derived functors.

open-problem · Boundaries, generalizations, and hard cases

Infinite global dimension: for general (non-regular, non-hereditary) rings the global dimension is infinite, so $\mathrm{Ext}^n$ and $\mathrm{Tor}_n$ persist for all $n$ ; controlling these (e.g. the finitistic dimension conjecture for finite-dimensional algebras, still open) is genuinely hard.
Non-balanced settings: in categories without enough projectives or injectives (e.g. some categories of sheaves, or non-abelian settings), classical resolutions fail; one passes to derived categories and uses K-injective/K-projective resolutions, or model-categorical fibrant/cofibrant replacements.
Bar resolutions and computability: for group rings $\mathbb{Z}G$ with $G$ infinite, explicit resolutions exist (bar resolution) but computation of $\mathrm{Ext}^*_{\mathbb{Z}G}(\mathbb{Z},\mathbb{Z})$ (= group cohomology) can be wildly intractable; cohomology of arithmetic and mapping class groups is largely unknown in high degrees.
Tate (co)homology: for finite groups / Gorenstein rings one can splice Ext and Tor across the dimension 0 boundary into doubly infinite Tate cohomology $\hat{H}^n$ , $n\in\mathbb{Z}$ — a refinement requiring complete resolutions.
Derived $\otimes$ and Tor amplitude: the derived tensor product $\otimes^{\mathbb{L}}$ packages all Tor; understanding Tor-amplitude and perfect complexes underlies modern (derived) algebraic geometry and is an active research interface.

Derived Functors — Ext, Tor

Formal Statement

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also