G A the

algebras (associative, with identity), A the category of K-modules, U : C

usual underlying. The free commutative algebra functor F U is given by the symmet-

ric algebra construction (symmetrized tensor algebra)

X —X X —X —X

XF = K • X • • ···.

S(2) S(3)

G AT , where T = F U .

We certainly have C

If Λ is a commutative algebra, then

ker G Right Λ-Modules

Λ-Mod = Ab(C , Λ)

([Cartan & Eilenberg (1956), Mac Lane (1963)]) is an equivalence of categories. As in

G Λ must be of the form Λ•M G Λ as a K-module, where

Example 6, a Λ-module Y

G Λ, and have multiplication (»1 , m1 )(»2 , m2 ) = (»1 »2 , »1 m2 +

M is the kernel of Y

m1 »2 ). However since Y is commutative, the bimodule M must be symmetric. We view

M indi¬erently as a right module, left module, or symmetric bimodule, over Λ.

H 1 (Λ, Y )Λ classi¬es K-split commutative algebra extensions of Λ by Y . Such an

G Λ as a K-module, with multiplication

extension must have the form Λ • M

(»1 , m1 )(»2 , m2 ) = (»1 »2 , »1 m2 + m1 »2 + (»1 — »2 )f )

f

G M is a factor set satisfying, in general, whatever identities are needed

where Λ — Λ

in order to make Λ • M into a commutative algebra (such as the symmetry (»1 — »2 )f ) =

(»2 — »1 )f ).

G Λ via projection

Now take Λ = K[x]/(x2 = 0), M = Λ as a Λ-module, Y = Λ•M

as the module, and to heighten the drama let K be a ¬eld. Let f be the factor set

(1 — 1)f = (1 — x)f = (x — 1)f = 0, (x — x)f = 1.

51

TRIPLES, ALGEBRAS AND COHOMOLOGY

G Λ constructed by means of f represents a nonzero element

The extension E = Λ•M

in H 1 (Λ, Y )Λ . (Otherwise there would be an isomorphism

GY

E cc

c

c

c

c in (C , Λ)

c1

Λ

G Λ is split by its zero section » ’ (», 0), which is an algebra map, E GΛ

Since Y

would also be split by an algebra map. Thanks to the choice of f this is impossible.

Always, in triple cohomology, an extension represents the zero cohomology class ⇐’ it

is inessential, i.e., split in the category of algebras.) Thus we know that H 1 (Λ, Y )Λ = 0.

G Λ, or equivalently its kernel M , is injective in the category of Λ-

However, Y

modules. Indeed, as a Λ-module, M HomK (M, K) (use the obvious 1-1 correspondence

the K-base 1, x of M and the dual base), which proves injectivity, as K is a ¬eld, over

which everything is injective.6 We conclude that triple cohomology need not vanish on

injective coe¬cients.

This example shows that algebra cohomology cannot both classify extensions and be

a derived functor on the module category in the sense of [Cartan & Eilenberg (1956)] or

[Mac Lane (1963)].

Barr also knows an example of a commutative H 2 which fails to vanish on injective

coe¬cients.7 There seems to be no reason why the same thing cannot happen in any

dimension.

If C is tripled over sets, and the ground ring K is not a ¬eld, such examples are even

G Z/2Z, here K = Z, and the kernel is actually

easier to come by. Consider Z/4Z

a vector space over Λ = Z/2Z. Even the relative homological algebra in the module

category does not seem to o¬er much hope (see [Eilenberg & Moore (1965b)], [Heller

(1958)], or [Mac Lane (1963), Chapter IX]).

There is a close relationship between the theories of triples and of sites, or Grothendieck

topologies, which it is beyond the scope of this paper to explore. Using this insight,

one observes that it is possible to write the triple cohomology H n (X, Y ) as a derived

G Ab, where

functor Rn H 0 (X, Y ) in the category of functors (presheaves) (ImG)—

G: AT G AT is the free T-algebra cotriple. This result is analogous to Theorem (3.1)

of [Artin (1962)]. I am indebted to S. U. Chase for showing me this.

EXAMPLE 9. Additive Categories. In additive categories the notion of module simpli-

¬es. Indeed, if B is additive, X ∈ B, we have

ker GB

X-Mod = Ab(B, X)

6

Editors™ note: the proof of injectivity is a little terse. The point is that when R is a K-algebra, then

for any R-projective P and K-injective Q, the R-module Hom(P, Q) is R-injective.

7

Editors™ note: Subsequently, this example was published: M. Barr, A note on commutative algebra

cohomology. Bull. Amer. Math. Soc. 74 (1968), 310“313.

52 JONATHAN MOCK BECK

G X ’ M = ker(Y G X).

is an equivalence of categories. ker is the functor Y

X • M , q.e.d. (We are

Because of the zero section of the module, we must have Y

assuming that additive categories have a • and kernels, i.e., ¬nite projective limits.)

Thus in an additive category every object “is” a module over every other object, in

a unique manner. A typical cohomology theory arising in the additive context is the

classical Extn (A, C) of (right) Λ-modules. The two variables in Ext give the illusion of

Λ

being on the same footing, in contrast with group cohomology, say, where one variable is

a group and the other is a module. But in view of the above proposition, C is equally an

A-module, so there is no real contrast between Ext and the group case.

Phrased di¬erently, there is no need to pass to the comma category (B, X) in order to

obtain enough abelian group objects. Thus the only cohomology theory we are concerned

with is of the type H n (X, Y ), where X, Y ∈ B, which is the same as AbB. This arises as

follows. Let

FG UG

A B A (F U)

G B be U F , the functor part of the standard cotriple

be an adjoint pair and let G: B

in B arising from adjointness. (A is not assumed additive, hence G need not be additive,

nor need 0G = 0 ∈ B.) Form the standard resolution

‚0 ‚1 ‚n

0o Xo XG o ··· o XGn o XGn+1 o ···

where X is in dimension ’1 and ‚n = Σ(’1)i X i , 0 ¤ i ¤ n, i = Gi Gn’i , using

additivity of B to add up the face operators in advance. Applying the functor ( , Y ): B —

G Ab, one gets a cochain complex

d1 dn

G (XG2 , Y )

G (XG, Y ) G ··· G (XGn , Y ) G (XGn+1 , Y ) G ···

0

where dn = (‚n , Y ). H n (X, Y ), relative to F U of course, is the n-th cohomology group

of this complex.

U is tripleable, H 0 (X, Y ) is the hom functor and H 1 (X, Y ) classi¬es U -exact

If F

GY GE GX G 0. U -exactness is de¬ned in Theorem 2 above,

sequences 0

and these facts are contained in the interpretation of the cohomology given in §3, provided

one can identify short U -exact sequences with U -split principal homogeneous objects.

GY GE GX G 0 is U -exact, then Y operates on E by additivity,

But if 0

G X, and EU XU — Y U .

moreover simply-transitively and compatibly with E

G X operated on by Y , a U -exact sequence is de¬ned by

Conversely, given such an E

GY GE GX G0

0 XX ”f

XX ”

””—¦

X

(Y,0) X( ””

Y •E

Now, doing the obvious, let E n (X, Y ) be the set of Yoneda equivalence classes of U -

GY G Yn’1

exact sequences (n-dimensional extensions relative to U ) of the form 0

53

TRIPLES, ALGEBRAS AND COHOMOLOGY

G · · · Y0 GX G 0 [Mitchell (1965)]. The standard resolution XGn+1 , ‚n (n ≥ 0)

is F -free (as well as U -exact). Thus, given an n-dimensional extension, we can construct

a map of complexes over X in the usual manner:

‚n+1

‚0 ‚1 ‚n

0o X o XG o XG2 o ··· o XGn o XGn+1 o XGn+2 o ···

a

0o Xo Y0 o Y1 o ··· o o Yo 0o ···

Yn’1

The component a is an n-cocycle. This de¬nes the map needed in the following additive

extension of Theorems 5, 6 of §3:

THEOREM 7. The natural map

G H n (X, Y )

E n (X, Y )

is an isomorphism if F U is tripleable.

Several proofs of this result are possible, we will omit all of them. One proof involves

breaking up long U -exact sequences into composites of short ones (this requires kernels

in B), and then using the fact that short ones are classi¬ed by H 1 , or more precisely,

G Y in A. In an additive category it is possible to

determined by 1-cocycles XT

characterize H n (X, Y ) by a fairly obvious set of axioms, as a functor of X. Another proof

of Theorem 7 then proceeds by verifying these axioms for E n (X, Y ).

Note the following gap. In the general, nonadditive case, Barr™s examples referred

to above show that H n (X, Y ), X ∈ B, Y ∈ AbB, does not classify mixed additive-

nonadditive “extensions” of the form

GY G Yn’1 G ··· G Y0 G 0,

0 U -exact

Y0

G X,

E U -split principal homogeneous object.

(Such have been considered, from the point of view of cohomology classi¬cation, in [Barr

(1965), Barr & Rinehart (1966), Gerstenhaber (1964)].) Otherwise H n (X, Y ) would vanish

when Y is injective. In general, does H n (X, Y ) classify any concept of n-dimensional

extensions of X by Y ? Should these “extensions” perhaps be required to have a simplicial

structure?

As examples of the situation envisaged in Theorem 7, we cite the following:

(a) The category of right Λ-modules tripleable over the category of sets A. We have F : A

G MΛ , U : MΛ G A, where AF = A·Λ (A-fold coproduct of Λ™s) and XU is the usual

underlying set of X; F U . U -exactness is the ordinary abelian-category exactness, the

standard complex XG— is a Λ-free resolution of X, hence (as Theorem 7 also shows)

G Extn (X, Y ), n≥0.

H n (X, Y ) Λ

54 JONATHAN MOCK BECK

•

G “ is a ring map, we get a tripleable adjoint pair ’ —Λ “: MΛ G M“ ,

(b) If Λ

G MΛ ([Cartan & Eilenberg (1956)], p. 29). As cohomology we obtain

M• : M “

Hochschild™s relative Ext [Hochschild (1956)]:

G Extn (X, Y ), n≥0.

H n (X, Y ) •

We have been emphasizing cohomology. But one can take coe¬cients in functors other

than hom functors, for example, the tensor product with a ¬xed Λ-module. Thus TorΛ ,

Tor• can be introduced into our theory, as well as a general homology theory of algebras

(which we pass over in silence).

(c) If A is a graded abelian group and C is the category of chain complexes, adjoint

functors (tripleable)

GC GA

F U

A

are de¬ned by: A = (An ) =’ AF = (AFn ) with AFn = An • An+1 and boundary

G AFn’1 by shifting (an , an+1 ) =’ (0, an ); U forgets the boundary

operator AFn

operator. Then the cohomology theory H n (X, Y ) classi¬es sequences of chain complexes

GY G Yn’1 G ··· G Y0 GX G 0 which are split exact in A, i.e.,

0

ignoring the boundary operator.

If negative degrees are permitted (the reader can make the categories precise), this

example coalesces with the obvious graded variant of (b), as AF = A — D where D is the

graded ring generated by an element ‚ ∈ D’1 with ‚‚ = 0 in D’2 .

(d) We refer to [Eilenberg & Moore (1965a)] for the examples of co- and contra-modules

over a coalgebra (as well as an account of the situation when both categories, base and

algebras, are additive).

(e) A = graded, connected, commutative K-coalgebras, C = graded, connected, bicom-

mutative Hopf algebras. Then C = AbA [Milnor & Moore (1965)]. The graded tensor al-

G A [Moore (1961)], and tripleably. H n (X, Y )

gebra is left adjoint to the underlying C

classi¬es sequences of Hopf algebras which on the coalgebra level decompose into short

GA—B G B (cartesian products in A).

exact sequences of the form A

Example (e) also goes through in the ungraded case.

There is a graded dual (e— ) in which one takes A = commutative algebras, C as above

G A is cotripleable. The sequences

is then abelian cogroups in A and the underlying C

classi¬ed by the cohomology are split as algebras, into tensor products.

G K-

I do not know whether the ungraded dual (e— ) works; the underlying C

algebras may lack a right adjoint.

Obviously (with notation as in (e)), Hopf algebras which are only cocommutative are

group objects in A. Examples 3, 4, 5 on groups and monoids can all be reworked in this

context, replacing the category of sets by that of commutative coalgebras; and dually.

In additive categories where a notion of exactness is de¬ned, or in abelian categories,

Yoneda™s theory of long extensions supplies a cohomology theory Extn (X, Y ) without the

55

TRIPLES, ALGEBRAS AND COHOMOLOGY

intervention of projective or injective resolutions of any kind. The triple theory cannot