Derived Functors — Ext, Tor

Derived functors are the canonical machinery for measuring the failure of a functor to be exact. Ext and Tor are the derived functors of

Hom

and

\otimes

respectively:

Ext_{R}^{n} (A, B)

measures obstructions to lifting/extending homomorphisms and classifies extensions, while

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).

Given an additive functor $F$ between abelian categories that is only half-exact, one repairs its failure to commute with short exact sequences by replacing objects with resolutions (acyclic, often projective or injective) and taking homology of $F$ applied to the resolution. The resulting derived functors $L_{n} F$ (left) and $R^{n} F$ (right) form a long exact sequence that bookkeeps exactly the homological obstructions $F$ destroys. Specializing to $F = Hom_{R} (A, -)$ or $Hom_{R} (-, B)$ gives $Ext_{R}^{n}$ ; specializing to $F = A \otimes_{R} -$ gives $Tor_{n}^{R}$ . A surprising and foundational fact is balancing: $Ext$ and $Tor$ can be computed by resolving either variable, and the answers agree canonically — a consequence of double complexes and spectral sequences.

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" $L_{0} F, L_{1} F, L_{2} F, \dots$ (or $R^{0} F, R^{1} F, \dots$ ) 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.

Formal Statement

Resolutions

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

\dots \to P_{2} d_{2} P_{1} d_{1} P_{0} ε A \to 0

with each

P_{i}

projective.

An injective resolution is an exact sequence

0 \to A η I^{0} d^{0} I^{1} d^{1} I^{2} \to \dots

with each

I^{j}

injective.

⟳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

ε

. Injective resolutions exist because

R -Mod

has enough injectives (Baer's criterion + injective envelopes).

Left and Right Derived Functors

Let $F : A \to B$ 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

L_{n} F (A) = H_{n} (F (P_{∙})), P_{∙} \to A a projective resolution

For $F$ left exact (e.g. $Hom_{R} (A, -)$ ), the right derived functors are

R^{n} F (A) = H^{n} (F (I^{∙})), A \to I^{∙} an injective resolution

⟳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.

Ext and Tor

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

Ext_{R}^{n} (A, B) = R^{n} (Hom_{R} (A, -)) (B) = H^{n} (Hom_{R} (P_{∙}, B))

Tor_{n}^{R} (A, B) = L_{n} (A \otimes_{R} -) (B) = H_{n} (P_{∙} \otimes_{R} B)

where $P_{∙} \to A$ is a projective resolution (for Ext using the contravariant $Hom_{R} (-, B)$ applied to a projective resolution of $A$ ; for Tor applied to a flat/projective resolution of $A$ ).

⟳

Hom_{R} (P_{∙}, B)

is a cochain complex because

Hom_{R} (-, B)

is contravariant, reversing the chain differentials into coboundaries;

P_{∙} \otimes_{R} B

stays a chain complex because

- \otimes B

is covariant.

Long 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:

\dots \to Tor_{n}^{R} (A^{'}, B) \to Tor_{n}^{R} (A, B) \to Tor_{n}^{R} (A^{''}, B) \partial Tor_{n - 1}^{R} (A^{'}, B) \to \dots

\dots \to Ext_{R}^{n} (A^{''}, B) \to Ext_{R}^{n} (A, B) \to Ext_{R}^{n} (A^{'}, B) \partial Ext_{R}^{n + 1} (A^{''}, B) \to \dots

ending with $Tor_{0}^{R} (-, B) = - \otimes_{R} B$ and $Ext_{R}^{0} (-, B) = Hom_{R} (-, B)$ .

⟳Apply the Horseshoe Lemma to build a short exact sequence of projective resolutions

0 \to P_{∙}^{'} \to P_{∙} \to P_{∙}^{''} \to 0

(degreewise split), apply

F

, and use the Snake Lemma / long exact homology sequence of the resulting short exact sequence of complexes.

Balancing

There are natural isomorphisms, for all $n$ :

Tor_{n}^{R} (A, B) ≅ L_{n} (A \otimes_{R} -) (B) ≅ L_{n} (- \otimes_{R} B) (A)

Ext_{R}^{n} (A, B) ≅ R^{n} Hom_{R} (A, -) (B) ≅ R^{n} Hom_{R} (-, B) (A)

i.e. one may resolve either variable.

⟳Form the double complex

P_{∙} \otimes_{R} Q_{∙}

(resp.

Hom_{R} (P_{∙}, Q^{∙})

) 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.

Construction of Tor via projective resolution — well-definedness

Step 1. Choose a projective resolution

P_{∙} ε A

. Apply

- \otimes_{R} B

to the deleted resolution

P_{∙}

(omit

A

), obtaining the chain complex

\dots \to P_{2} \otimes_{R} B d_{2} \otimes 1 P_{1} \otimes_{R} B d_{1} \otimes 1 P_{0} \otimes_{R} B \to 0.

Define

Tor_{n}^{R} (A, B) = ker (d_{n} \otimes 1) / im (d_{n + 1} \otimes 1)

⟳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.

Tor_{0}^{R} (A, B) = coker (d_{1} \otimes 1) = P_{0} \otimes B / 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 / im ≅ A \otimes_{R} B

⟳Right-exact functors preserve cokernels; the cokernel of

P_{1} \to P_{0}

A

Step 3. Independence of resolution. If

Q_{∙} \to A

is another projective resolution, the Comparison Theorem yields chain maps

f : P_{∙} \to Q_{∙}

and

g : Q_{∙} \to P_{∙}

lifting

id_{A}

, unique up to chain homotopy. Then

g f ≃ id

f g ≃ id

. Applying

- \otimes B

preserves chain homotopies (additivity), so

f \otimes 1, g \otimes 1

are mutually inverse on homology.

⟳A chain homotopy

s

with

df + f d^{'} = id - g f

tensors to

(d \otimes 1) (s \otimes 1) + (s \otimes 1) (d^{'} \otimes 1) = id - (g \otimes 1) (f \otimes 1)

Step 4. Naturality. A module map

B \to B^{'}

induces a chain map of complexes

P_{∙} \otimes B \to P_{∙} \otimes B^{'}

, hence functoriality; similarly in

A

by lifting maps to resolutions.

□

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: $Tor_{1}^{Z} (Z / n, A)$ recovers the $n$ -torsion of $A$ . "Ext" abbreviates extension, reflecting the classification of module extensions by $Ext^{1}$ . 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.

Building 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 $A^{'} ↪ A$ is injective but $A^{'} \otimes B \to A \otimes B$ is not, the kernel of the latter is detected by $Tor_{1}^{R} (A^{''}, B)$ . The classic example: $Z \times n Z$ is injective, but tensoring with $Z / n$ gives the zero map — the lost injectivity surfaces as $Tor_{1}^{Z} (Z / n, Z / n) = Z / n$ .

For Ext: $Hom_{R} (-, B)$ fails to be surjective on lifts. Given $A ↞$ surjection, not every map into $B$ extends; $Ext^{1}$ measures the obstruction. Geometrically, $Ext^{1} (A, B)$ is the set of "ways to glue $B$ and $A$ into a single module" (extensions) modulo equivalence — nonzero Ext means non-split gluings exist.

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

Worked Examples

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

Given: compute all

Tor_{n}^{Z} (Z / m, Z / k)

m, k \geq 2

. Step 1 — Resolve $Z / m$ . Since

Z

is a PID, a free resolution is short:

0 \to Z \times m Z \to Z / m \to 0.

⟳the kernel of

Z ↠ Z / m

m Z ≅ Z

, free, so the resolution stops at length 1.

Step 2 — Delete and tensor with $Z / k$ :

0 \to Z / k \times m Z / k \to 0.

⟳

Z \otimes Z / k = Z / k

; the map

\times m

tensors to multiplication by

m

Z / k

Step 3 — Homology.

$Tor_{0} = coker (\times m) = Z / k / m Z / k = Z / g cd (m, k)$ .
$Tor_{1} = ker (\times m on Z / k) = {x : m x \equiv 0 mod k} = Z / g cd (m, k)$ .
$Tor_{n} = 0$ for $n \geq 2$ (resolution has length 1).

Boxed answer:

Tor_{0}^{Z} (Z / m, Z / k) = Tor_{1}^{Z} (Z / m, Z / k) = Z / g cd (m, k), Tor_{\geq 2} = 0

Ext and extensions: $\mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z})$

Given: compute

Ext_{Z}^{n} (Z / m, Z)

and interpret. Step 1 — Same resolution

0 \to Z \times m Z \to Z / m \to 0

. Step 2 — Apply $Hom_{Z} (-, Z)$ (contravariant, deleted resolution):

0 \to Hom (Z, Z) (\times m)^{*} Hom (Z, Z) \to 0, i.e. 0 \to Z \times m Z \to 0.

⟳

Hom (Z, Z) ≅ Z

, and precomposition with

\times m

becomes multiplication by

m

(placed in cohomological degrees 0 and 1).

Step 3 — Cohomology.

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

Step 4 — Interpretation.

Ext_{Z}^{1} (Z / m, Z) ≅ Z / m

classifies extensions

0 \to Z \to E \to Z / m \to 0

. The class

r \in Z / m

corresponds to

E

with the relation pulling

Z / m

"tighter":

r = 0

gives the split

Z \oplus Z / m

, while

r = 1

gives

E = Z

(sitting as

Z \times m Z \to Z / m

). Boxed answer:

Ext_{Z}^{1} (Z / m, Z) ≅ Z / m, Ext^{0} = Ext^{\geq 2} = 0

Common Misconceptions

Frequent errors

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

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

Q

is a flat

Z

-module (so

Tor_{n}^{Z} (Q, -) = 0

for

n \geq 1

) but

Q

is not projective over

Z

(it is not a direct summand of any free abelian group; it is divisible and torsion-free but not free). This shows:

Tor-vanishing detects flatness, not projectivity. $Tor_{1}^{R} (M, -) = 0 ⟺ M$ flat.
$Ext_{Z}^{1} (Q, Z)$ is in fact uncountable ( $≅ A_{f} / Q$ -ish, an enormous group) — $Q$ being flat does not make its Ext vanish. Flatness and Ext-acyclicity are independent conditions.

Connections

Where Ext and Tor sit in the homological web

Group Cohomology: $H^{n} (G, M) = Ext_{Z G}^{n} (Z, M)$ and $H_{n} (G, M) = Tor_{n}^{Z G} (Z, M)$ — group (co)homology is literally Ext/Tor over the group ring.
Sheaf Cohomology: $H^{n} (X, F) = R^{n} Γ (F)$ , the right derived functors of global sections; local Ext sheaves $E x t^{n}$ globalize $Ext$ .
Universal Coefficient Theorem: $H_{n} (X; A)$ relates to $H_{n} (X; Z)$ via $Tor_{1}^{Z}$ ; cohomology via $Ext_{Z}^{1}$ .
Künneth Theorem: the homology of a tensor product of complexes carries a $Tor_{1}$ 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 $^{n} (A, B) = Hom_{D (R)} (A, B [n])$ — Ext groups are morphisms (with shifts) in the derived category, the modern home of derived functors.

Real-World Applications

Concrete uses across mathematics and beyond

Deformation theory (algebraic geometry): $Ext^{1}$ of the tangent sheaf classifies first-order deformations; $Ext^{2}$ houses obstructions. Used in moduli space construction (e.g. moduli of vector bundles, of curves).
Coherent sheaf computations / Serre duality: $Ext^{n} (F, ω_{X}) ≅ H^{d - n} (X, F)^{\lor}$ on smooth projective $X$ of dimension $d$ — Ext is the engine of duality theorems in computational algebraic geometry.
Number theory — Galois cohomology: $Ext / Tor$ over group rings of Galois groups (e.g. Selmer groups, Tate–Shafarevich groups, Iwasawa theory) measure arithmetic obstructions; central to BSD-type conjectures.
Topological data analysis (persistent homology): the structure theorem and the Universal Coefficient relations (Ext/Tor over $Z$ or fields) control how persistence modules behave under coefficient change.
Representation theory: $Ext^{1} (L, M)$ between simple modules builds the quiver/Ext-algebra describing how irreducibles glue; foundational for highest-weight categories and Koszul duality.
Commutative algebra / singularity theory: the Koszul complex computes $Tor$ to detect regular sequences and depth; $Ext$ depth measures (Auslander–Buchsbaum formula) classify Cohen–Macaulay and Gorenstein rings.
String theory / mirror symmetry (physics): D-branes are objects in derived categories of coherent sheaves; $Ext$ groups count open-string states (massless spectrum) between branes.

Historical Development

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 $Ext (A, B)$ to compute homology of spaces with twisted coefficients; the Universal Coefficient Theorem named.
1943–1945 — Eilenberg & Mac Lane: systematic study of $Ext$ and the bifunctor "Tor" (torsion product); cohomology of groups via $Ext_{Z G}$ .
1947 — Hopf, Eckmann: independent group (co)homology constructions clarified as derived functors.
1948 — D.K. Harrison / Mac Lane: extension-theoretic interpretation of $Ext^{1}$ 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; $δ$ -functors; the Grothendieck spectral sequence.
1960s — Yoneda: $Ext^{n}$ realized as equivalence classes of $n$ -fold extensions, with composition product (Yoneda product), making $Ext^{∙}$ 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.

Subtleties & Limits

Boundaries, generalizations, and hard cases

Infinite global dimension: for general (non-regular, non-hereditary) rings the global dimension is infinite, so $Ext^{n}$ and $Tor_{n}$ 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 $Z G$ with $G$ infinite, explicit resolutions exist (bar resolution) but computation of $Ext_{Z G}^{*} (Z, Z)$ (= 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 $\hat{H}^{n}$ , $n \in Z$ — a refinement requiring complete resolutions.
Derived $\otimes$ and Tor amplitude: the derived tensor product $\otimes^{L}$ packages all Tor; understanding Tor-amplitude and perfect complexes underlies modern (derived) algebraic geometry and is an active research interface.