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EXAMPLE 8. Commutative Algebras. Let C be the category of commutative K-
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.
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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
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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

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