Modules, comodules and cotensor products
over Frobenius algebras
Lowell Abrams
Department of Mathematics
Rutgers University
New Brunswick, NJ 08903
labrams@math.rutgers.edu
Abstract
We characterize noncommutative Frobenius algebras A in terms
of the existence of a coproduct which is a map of left Ae-modules.
We show that the category of right comodules over A, relative to this
coproduct, is isomorphic to the category of right modules. This iso-
morphism enables a reformulation of the cotensor product of Eilenberg
and Moore as a functor of modules rather than comodules.
We prove that the cotensor product M2N of a right A-module
M and a left A-module N is isomorphic to the vector space of homo-
morphisms from a particular right Ae-module D to M N, viewed
as a right Ae-module. Some of the properties of D are investigated,
and some sample calculations are given. Finally, we show that when
A is commutative or semisimple, the cotensor product M2N and its
derived functors are given by the Hochschild cohomology over A of
M N.
Keywords: Frobenius algebra, comodule, cotensor product, Hochschild
cohomology
1 Introduction
Eilenberg and Moore originally introduced the cotensor product M2N
and its derived functors Cotor(M; N) on comodules M; N as tools for the
calculation of the homology of the fiber space in a fibration [5]. This paper
investigates these functors in the context where the coalgebra is a Frobenius
algebra (defined in section 2).
The Frobenius case is not far removed from that of Eilenberg and Moore,
whose coalgebra is the set of normalized singular chains in some space X; in
the presence of sufficient flatness, all the relevant constructions yield exact*
*ly
the same data upon passing to homology [ibid.]. When the space X under
1
consideration is compact and oriented, its homology is in fact a Frobenius
algebra.
Nevertheless, our approach diverges from that of Eilenberg and Moore in
an important way. The results presented here rest on a new characterization
of Frobenius algebras as algebras possessing a coassociative comultiplication
ffi: A ! A A, with counit, which is a map of regular bimodules. (This is
formulated slightly differently as theorem 2.1 below.) This comultiplica-
tion is decidedly different from the one used by Eilenberg and Moore. The
relationship between the two coproducts will be discussed elsewhere.
The Frobenius algebra coproduct, and in particular the element ffi(1A ),
has already begun to find its place in a variety of contexts. In two dimen-
sional topological quantum field theory, it gives rise to the handle operator
[1]. In quantum cohomology it provides a generalization of the classical
Euler class [2]. It also plays an important role in the study of quantum
Yang-Baxter equations and serves as a separability idempotent [3]. Here,
we will consider left and right submodules of A A generated by ffi(1A ).
These will be discussed more later in this section.
The bimodule property of the Frobenius algebra coproduct implies an-
other important property of Frobenius algebras, appearing as theorem 3.3:
The category of right modules over a Frobenius algebra A is isomorphic to
the category of right comodules over A. This result makes it possible to
view Eilenberg and Moore's functors on comodules as functors on modules.
Now, using the Snake Lemma, one can show that the cotensor product is
left exact in both variables. (This also follows from theorem 4.6, of course.)
This suggests that M2N should be expressible as a module of homorphisms
from some module D to M N. In fact, this is the case, as stated in theorem
4.6. The concern is to develop a satisfactory understanding of the module
D.
Specifically, D denotes the right Ae-submodule of A A generated by
ffi(1A ). This is not the same as the left Ae-submodule ffi(A) of AA generated
by ffi(1A ). The latter module is a very natural object to consider, since ffi
itself is a left Ae-module map, but the importance of D in this context is
somewhat surprising. Under certain conditions, delineated in 4.3 and 4.3.1,
D and ffi(A) are in fact the same up to a canonical involution. But in other
cases, such as those presented below as examples 4.1 and 4.2, quite the
contrary is true. For instance, in example 4.2, ffi(A) is four dimensional,
whereas D is eight dimensional.
There are two important corollaries to the main results discussed above.
One (4.6.1 below) is that the right derived functors of the cotensor product
M2N, i.e. Cotor*(M; N), are in fact the modules Ext*(D; M N). The
2
other (4.6.2 below) is that when A is commutative, the cotensor product
M2N and its derived functors are given by the Hochschild cohomology
over A of M N.
The author extends heartfelt thanks to Chuck Weibel who, in addition
to being free with helpful advice, is a living index to [6].
Notation and Conventions
All algebras A considered here are assumed to be finite dimensional as
a vector space over their coefficient field K, and to possess a multiplicative
identity element 1A . We let : A A ! A denote the multiplication map.
The symbols An will always denote An , i.e. the tensor product of n copies
of A, and never the Cartesian product. For any object X, we will use "X"
or "." to denote the identity map X ! X, and the symbols . . will be
abbreviated "..".
2 Noncommutative Frobenius Algebras
An algebra A is defined to be a Frobenius algebra if it possesses a left
A-module isomorphism L: A ! A* with its vector space dual. Here, A is
viewed as the left regular module over itself, and A* is a left A-module by
the action a . i(b) := i(ab) for any a; b 2 A and i 2 A*. It is easy to show
that the existence of the isomorphism of left modules implies the existence
of an isomorphism R of right modules, where the right module structures
are defined analogously.
There are many equivalent definitions of Frobenius algebras; see [4] for
more information. For our purposes, the new characterization of Frobenius
algebras presented below is very useful.
Theorem 2.1 An algebra is a Frobenius algebra if and only if it has a
coassociative comultiplication, with counit, which is a map of left regular
Ae-modules.
Here, Ae denotes the ring A Aop, and A has the left Ae-action defined
by (b b0) . a := bab0.
In many respects, the proof of this result follows the proof of an analogous
result for the commutative case, found in [1]. For the sake of space, we
merely indicate how this proof differs from the one given there.
Proof. Assume A denotes a Frobenius algebra with left-module isomor-
phism L: A ! A*. Let T : A A ! A denote the composition O T , where
T : AA ! AA denotes the canonical involution. Define the comultiplica-
tion map ffiL: A ! A A to be the composition (-1L -1L) O *TO L. With
3
the appropriate adjustments, the discussion in [1] shows that the following
diagram commutes:
A A __________A-
| |
| |
|.ffiL |ffiL
| |
|? . |?
A A A ______A- A
In words, ffiL is a map of left modules.
Using the right-module isomorphism R : A ! A*, it is an analogous
exercise to define ffiR and show that this comultiplication map is a map of
right modules. Let ffl: A ! K denote R (1A ). Note that R (1A ) = L(1A ),
and thus that ffl serves as a counit for both ffiR and ffiL.
Now consider the following diagram:
A
@ ffiR
@
@@R
ffiR. -
A2 ______A3- ______A2.
@ | | @ ..
ffi @ffi ||.AffiL ||ffiL@
R @@RL|? |? @@R
A4 ______A3-.A_____A2-.ffl.
This diagram commutes because of the properties of ffiR , ffiL and ffl mentioned
just above. It follows that ffiR O is the same as the composition of maps
from the far left down and along the bottom row to the lower righthand
corner. A corresponding diagram shows that ffiR O is also the same as that
composition, i.e. ffiR O = ffiL O . Since A has an identity element, we see
that ffiR = ffiL. Define ffi := ffiR = ffiL. We have just shown that this map
ffi: A ! A A is a map of bimodules, i.e. is an Ae-module map, and has a
counit.
The remainder of the proof follows as in [1].
Throughout the sequel, ffi and ffl will denote the comultiplication and
counit respectively. Let ffi(A) denote the image of ffi.
Corollary 2.1.1 The map ffi is an injection of left Ae-modules.
Proof. By theorem 2.1, ffi is a map of left Ae-modules. Since ffi has a
counit, it is certainly injective.
4
3 Modules and Comodules
We let 1A : K ! A denote the map sending 1K to 1A . Since X and
X K are canonically isomorphic, for any map f: X ! X we will abuse
notation and write f 1A : X ! X A instead of f 1A : X K ! X A.
When discussing compositions of maps, the term "switch" will always refer
to reversing the order of noninteracting maps.
Suppose M is a right A-module with structure map m: M A ! M.
Define the map 5m : M ! M A to be the composition (m .) O (. ffi) O
(. 1A ):
M .1A-!M A .ffi-!M A2 m.-!M A
Lemma 3.1 The map 5m endows M with the structure of a right A-
comodule.
Proof. It is necessary to show that the following diagram commutes:
5m
M ________-M A
| |
|5m |.ffi (1)
| |
|? 5m |?
M A ______-M A2
Expanding each of the occurrences of 5m in accordance with the definition
of that map yields the outer edge of this diagram:
.1A .ffi m.
M _________M- A ______-M A2 ______M- A
| | | |
|.1A |.ffi |..ffi |.ffi
| | | |
|? .ffi |? .ffi. |? m.. |?
M A ______M- A2 ______M- A3 ______M- A2
| |6 |6 |6
|.ffi |.. |.A. |m..
| | | |
|? ..1A. | .Affi. | m... |
M A2 ______M- A3 ______M- A4 ______M- A3
Q Q jj3
m. Q Q j j.ffi.
Qs j
M A ______-M.1A2A.
From left to right and top down, the squares inside this large diagram com-
mute for the following reasons: Vacuity, coassociativity of ffi, switch, prop-
erty of the multiplicative identity, ffi being a module map, m being a module
5
map. The hexagon on the bottom is commutative because it only involves
a switch.
It follows that the outer edge forms a commutative square, i.e. diagram
(1) is commutative.
Suppose now that M is a right A-comodule, with comodule structure
map 5: M ! MA. Define the map m5 : MA ! M to be the composition
(. ffl) O (. ) O (5 .):
M A 5.-!M A2 .-! M A .ffl-!M
Lemma 3.2 The map m5 endows M with the structure of a right A-
module.
Proof. It is necessary to show that the following diagram commutes:
m5 .
M A2 ______-M A
| |
|. |m5 (2)
| |
|? m5 |?
M A ________-M
Expanding each occurrence of m5 in accordance with the definition of
that map yields the outer edge of the following diagram:
5.. .. .ffl.
M A2| ______M- A3 ______-M A2 ______-M A
|| Q Q 5.. Q 5... Q 5.. Q Q 5.
| Q Q QQ Q Q Q
|| Qs .ffiQAQs. .A. QQs ..ffQQsl.
|| M A3 ______-M A4 ______-M A3 ______M- A2
| | Q .. jj3 |
|. | Q .ffij. |.
| | Q j |
| | Q Qs j |?
| | _____________________-.
| |.. M A2 M A
| | j3
| | .. j |
| | j |.ffl
| | j |
|? 5. |? . j .ffl |?
M A ______-M A2 _____________________-M A _________M-
The subdiagrams of this diagram are commutative for the following reasons:
In the top row of squares, the leftmost square expresses the comodule prop-
erty of 5. The other two squares simply involve switches, as does the large
6
square on the far left. The square in the center (between the second and
third rows of maps) uses the module property of ffi. The square to its right
uses the counit property of ffl. The large pentagon on the bottom expresses
the associativity of . The triangle in the lower right hand corner is vacuous.
It follows that the outer edge forms a commutative square, i.e. diagram
(2) is commutative.
Lemmas 3.1 and 3.2 show that there are canonical maps between the
category of modules over A and the category of comodules over A. In fact,
these provide an isomorphism.
Theorem 3.3 The category of right modules over a Frobenius algebra A is
isomorphic to the category of right comodules over A.
Proof. First we will show that the constructions m 7! 5m and 5 7! m5
are mutual inverses. Then we will show that every module map is a comodule
map for the corresponding comodule structures, and vice-versa.
Suppose m: M A ! M is a right module structure map. Consider the
following diagram:
.1A. .ffi. m..
M A ______-M A2 ______M- A3 ______-M A2
| | | | Q .
|.ffi |..ffi |...ffi |..ffQi
| | | | Q Q
|? .1A.. |? .ffiA. |? m... |? Qs
M A2 ______M- A3 ______M- A4 ______-M A3 M A|
| j | | j j3 |
|... j .A. | .. | j |
| j j .. | | j .ffl. |
|? j+ |? |? j |
_____________________- ______- |
M A2 M A3 m.. M A2 |.ffl
.ffi. |
Q Q |
Q |
..fflfQfl |
Qs m |?
M A _________M-
The composition of maps across the top and down the right is nothing other
than the definition of the map m5m : M A ! M. Since the composition
of maps down the left and across the bottom is m itself (by the counit
property), the identity m5m m will follow if the diagram is commutative.
This is in fact the case, because the subdiagrams are commutative for the
following reasons: With the exception of those that will now be mentioned
explicitely, the subdiagrams are commutative simply because they involve
switches. The triangle on the lower left uses the multiplicative unit property.
7
The square to its right expresses the module property of ffi. The square on
the far upper right is commutative because it is essentially the outer edge
of the following diagram:
.ffiA2
||?@@R
A3 A A
. @@R ||?ffiffl.
A2
This latter diagram is commutative because the square on the left expresses
the module property of ffi, and the square on the right express the counit
property of ffl.
It follows that m5m m. Suppose, on the other hand, that 5: M !
M A is a comodule structure. We now show that 5m5 5. Consider
the following diagram:
.1A .ffi 5.. .A
M _________M- A ______-M A2 ______M- A3 ______M- A2
Q Q Q j3 j3 |
5Q Q Q5Q j j j j |.ffl.
QQs QQs j j .Affi jj .ffi ||?
M A ______-M..A21A ______-M. A _______-M..A
From left to right, the subdiagrams are commutative for the following rea-
sons: Switch, switch, the module property of ffi, the counit property of ffl.
Because the composition of maps across the top and down the right of this
diagram is simply the definition of 5m5 , and the composition of maps down
the left and across the bottom is just 5 (by the unit property of 1A ), we
see that 5m5 5.
Suppose that M and N are right A modules with module structure maps
m and n respectively. In order to verify that a map f: M ! N of right
modules is also a map of right comodules (for the corresponding comodule
structures), consider the following diagram:
.1A .ffi m.
M ________-M A ______M- A2 ______-M A
| | | |
|f |f. |f.. |f.
| | | |
|? .1B |? .ffi |? n. |?
N _________N- A ______-N A2 ______-N A
Two of the subdiagrams simply involve switches. The third is commutative
because f is a map of modules. Thus, the outer edges form a commutative
8
diagram as well. But this diagram asserts that f is a map of comodules,
where the comodule structure maps are 5m and 5n.
If f: M ! N is assumed to be a map of right comodules, where the
comodule structure maps are 5 and 50, then by reasoning analogous to
that of the previous paragraph, the following diagram shows that f is a
map of right modules:
5. . .ffl
M A ______-M A2 ______M- A ________-M
| | | |
|f. |f.. |f. |f
| | | |
|? 50. |? . |? .ffl |?
N A ______-N A2 ______-N A _________N-
This completes the proof.
With appropriate changes, all the results and proofs in this section apply
to left modules and left comodules as well.
4 Cotensor Product
Suppose that M is a right A-module with module structure map m, and
that N is a left A-module with module structure map n. By theorem 3.3,
M is a right comodule with structure map 5m and N is a left comodule
with structure map 5n. Let OE denote the map
OE := 5m N - M 5n : M N -! M A N:
The cotensor product [5] M2N of M and N is defined to be the kernel
of OE.
Viewing A as both the right and left regular modules over itself (i.e. the
module structure maps are both ), we can form A2A. Note that 5 is
just the map ffi, by the module property of ffi.
Proposition 4.1 The cotensor product A2A is exactly ffi(A).
Proof. By the definition of OE, it suffices to show that the two maps
(5 A) O ffi and (A 5 ) O ffi are the same. But these two maps are
just (ffi A) O ffi and (A ffi) O ffi, respectively. These are the same, by t*
*he
coassociativity of ffi.
Definition 4.2 Let D denote the right Ae-submodule of A A generated
by ffi(1A ). Note that D and ffi(A) (see corollary 2.1.1 above) are different
objects.
9
Proposition 4.3 If ffi(1A ) is symmetric, i.e. T O ffi(1A ) = ffi(1A ), then D*
* and
T O ffi(A) are isomorphic as right Ae-modules.
Proof. For any a; b; x; y 2 A, we have
[T (x y)]. (a b) = ya bx = T [(b a) . (x y)]:
In our case, the hypothesis on ffi(1A ) and the module property of ffi therefore
imply that
ffi(1A ) . (a b) = T [(b a) . ffi(1A=)]T O ffi(ba) :(3)
Because A contains 1A , this equality shows that D and T Offi(A) are identical
sets. Define the right Ae-action on T O ffi(A) to be the action inherited from
the right Ae-action on A A. Equation (3) guarantees that this action is
well defined and that the correspondence between D and T Offi(A) is actually
an isomorphism of modules.
Corollary 4.3.1
1. If A is commutative then D and ffi(A) are isomorphic as right and left
Ae-modules.
2. If A is semisimple then D and T O ffi(A) are isomorphic as right Ae-
modules.
Proof. If A is commutative then ffi is a cocommutative map, by defi-
nition. (See the proof of theorem 2.1.) Thus the hypothesis of proposition
4.3 is automatically satisfied, and also T O ffi = ffi. Of course, in the case *
*of a
commutative algebra there is no distinction between left and right regular
actions.
By Wedderburn's first structure theorem, to prove the result in the case
when A is semisimple it suffices to assume that A is a matrix ring. In
that case, A has a Frobenius algebra structure given by the map L(a) :=
TrO (a -). Let eijdenote the elementary matrix with the entry 1 in the
i; j position. Of course, the set of elementary matrices in A forms a basis.
Let e*ijdenote the element of A* such that e*ij(ekl) = ffiikffijl. It is easy t*
*o see
that -1L(e*ij) = eji. Applying proposition 5 from [1], we have
X X
ffi(1A ) = eij -1L(e*ij) = eij eji:
i;j i;j
10
Clearly, the hypothesis of proposition 4.3 is satisfied.
When A is neither commutative nor semisimple, proposition 4.3 does
not necessarily apply. In fact, D and ffi(A) may differ quite strongly.
Example 4.4 Let A denote the exterior algebra on two generators, x and
y. Then
ffi(1A ) = 1A xy + xy 1A - x y + y x;
and ffi(A) has the basis
{ffi(1A ); x xy + xy x; y xy + xy y; xy xy} ;
whereas D has the basis
{ffi(1A ); x xy - xy x; y xy - xy y; xy xy} :
Example 4.5 Let A denote the algebra with generators x and y satisfying
the following relations: x2 = 0; y2 = 0; yx = xy + x. A has the basis
{1A ; x; y; xy}, and is a Frobenius algebra with map L: A ! A* given by
1A 7! (xy)* xy 7! 1*A
x 7! y* y 7! x* + (xy)*
We have
ffi(1A ) = 1A xy + xy 1A + x (y - 1) + y x;
and ffi(A) is four-dimensional with the basis
8 9
>>< ffi(1A ); >>
x xy + xy x; = ;
>>:y xy + xy y + x (y - 1); >>;
xy xy
whereas, after some manipulation, D can be seen to have the basis
8 9
>>>x x; x xy; xy x; xy xy; >>
>< y x - x y + xy 1A ; >>=
1A xy + 2x y - x 1A ; :
>>> >>
>: 2y xy - xy 1A ; >>;
y xy - xy y
11
In particular, D is eight dimensional. These basis elements, in the order
shown, are the elements obtained from right action on ffi(1A ) by the following
elements of Ae, respectively:
1_
1 2(x 1A - 1A x);
_2(x 1A + 1A x) - xy y - 2y xy;
xy y + 2y xy;
y xy; 1A y - y 1A ;
1A 1A - 1A y + y 1a;
y 1A + y y; y y
This example shows very clearly that D and ffi(A) are fundamentally differ-
ent.
Given a right module M and a left module N as above, endow M N
with the right Ae-module structure. Let Hom Ae(D ; M N) denote the
vector space of right Ae-module maps.
Theorem 4.6 There is a vector space isomorphism
M2N ~= Hom Ae (D ; M N) :
Proof. Note first that an element f 2 Hom Ae(D ; M N) is deter-
mined by its value on ffi(1A ), the generator of D.
The following diagram is commutative, since f is a map of modules:
K _____________________A4-ffi(1____________________________-A3
| A)ffi(1A)(..)OT23- (..)OT234 |
| |
|f.. |f.
| |
| |
|? (m..)OT23- (..n)OT234 |?
M N A2 ________________________M- N A
By proposition 4.1, the composition of maps across the top of the diagram is
0. Since the composition of maps from the upper left, down and across the
bottom is T23O OE O f [ffi(1A,)]it follows that f [ffi(1A2)]M2N. Thus, there
is a well defined injective map oe: Hom Ae(D ; M N) ! M2N sending
f 7! f [ffi(1A.)]Since each element e 2 M N defines a unique Ae-module
map o(e): ffi(1A ) 7! e, restriction of o to M2N provides an inverse to oe.
Allowing for abuse of notation, define the cotensor product functor 2A
by 2: M N 7! M2N, and let Cotor iA(M; N) denote its right derived
functors. Let Hi(A; -) denote the Hochschild cohomology functors.
12
Corollary 4.6.1 Over a Frobenius algebra A, the Cotor functor is given by
Cotor*A(M; N) ~=Ext*Ae(D; M N) :
Proof. In light of theorem 4.6, this is purely a matter of definitions.
Corollary 4.6.2 If A is a commutative or semisimple Frobenius algebra,
then cotensor product and its derived functors are Hochschild cohomology,
i.e.
Cotor*A(M; N) ~=H*(A; M N) :
Proof. By corollary 4.3.1 we have ffi(1A ) = T O ffi(1A ) and D ~=T O ffi(A).
Since, by corollary 2.1.1, ffi is an injective map of left Ae-modules (deter-
mined by its value on ffi(1A )), so is T O ffi. Thus D and A are isomorphic as
Ae-modules. It follows from theorem 4.6 that M2N ~=Hom Ae(A ; M N) .
But this is exactly H0(A; MN) [6, pg. 301]. Since H*(A; -) ~=Ext*Ae(A; -)
[ibid. pg. 303], this corollary follows from 4.6.1.
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3823-3836.
[4]Charles Curtis and Irving Reiner. Representation Theory of Finite Groups
and Associative Algebras. Interscience Publishers, New York, 1962.
[5]Samuel Eilenberg and John C. Moore. Homology and fibrations I: Coalgebras,
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[6]Charles A. Weibel. An Introduction to Homological Algebra. Cambridge studies
in advanced mathematics, 38. Cambridge University Press, Cambridge, 1994.
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