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. References [1]Lowell Abrams. Two-dimensional topological quantum field theories and Frobenius algebras. J. Knot Theory and its Ramifications, 5 (1996) 569-587. [2]Lowell Abrams. The quantum Euler class and the quantum cohomology of the Grassmannians. preprint q-alg/9712025. [3]K. I. Beidar, Y. Fong, and A. Stolin. On Frobenius algebras and the quantum Yang-Baxter equation. Trans. Amer. Math. Soc., 349 number 9 (1997) 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, cotensor product and its derived functors. Comment. Math. Helv., 40 (1966) 199-236. [6]Charles A. Weibel. An Introduction to Homological Algebra. Cambridge studies in advanced mathematics, 38. Cambridge University Press, Cambridge, 1994. 13