Hopfological algebra for infinite dimensional Hopf algebras
Marco A. Farinati

TL;DR
This paper extends Hopfological techniques to infinite dimensional co-Frobenius Hopf algebras, introducing a new homology theory and analyzing its properties, including examples and K-theoretic implications.
Contribution
It defines a homology functor for co-Frobenius Hopf algebras, generalizing classical homology, and explores its applications and K-theory in this broader context.
Findings
The homology functor is homological for co-Frobenius Hopf algebras.
The homology theory differs from cyclic homology but has an SBI-sequence.
K_0 of the category can be computed using a localization sequence.
Abstract
We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the "Homology functor" (we prove it is homological) for any co-Frobenius algebra, with coefficients in -comodules, that recover usual homology of a complex when . Another easy example of co-Frobenius Hopf algebra gives the category of "mixed complexes" and we see (by computing an example) that this homology theory differs from cyclic homology, although there exists a long exact sequence analogous to the SBI-sequence. Finally, because we have a tensor triangulated category, its is a ring, and we prove a "last part of a localization exact sequence" for …
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Hopfological algebra for infinite dimensional Hopf algebras
Marco A. Farinati Partially supported by UBACyT 2018-2021, 256BA. Research member of CONICET - I.M.A.S. - Depto. de Matemática, F.C.E.y N. Universidad de Buenos Aires, Ciudad Universitaria Pabellón I (1428) C.A.B.A. Argentina, *E-mail address:*[email protected]
Abstract
We consider ”Hopfological” techniques as in [K] but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the ”Homology functor” (we prove it is homological) for any co-Frobenius algebra, with coefficients in -comodules, that recover usual homology of a complex when . Another easy example of co-Frobenius Hopf algebra gives the category of ”mixed complexes” and we see (by computing an example) that this homology theory differs from cyclic homology, although there exists a long exact sequence analogous to the SBI-sequence. Finally, because we have a tensor triangulated category, its is a ring, and we prove a ”last part of a localization exact sequence” for that allows us to compute -or describe- of some family of examples, giving light of what kind of rings can be categorified using this techniques.
2010 Mathematics Subject Classification: 16T05 16E35, 18G99, 18D99, 19A49, 81R50
Keywords: Co-Frobenius Hopf Algebras, Tensor Triangulated Categories, Homology Theories, , Categorification.
Introduction
This paper has mainly 3 contributions:
(1) The ”Hopfological algebra” can be developed not only for finite dimensional Hopf algebras but also for infinite dimensional ones, provided they are co-Frobenius. The language of comodules is better addapted than the language of modules.
(2) The formula ”” can written in Hopf-co-Frobenius language.
(3) Some -theoretical results allow us to compute of the stable categories associated to co-Frobenius Hopf algebras of the form , with cosemisimple and finite dimensional.
The paper is organized as follows: In Section 1 we exhibit point (1) above, in Section 2 we make point (2). In Section 3 we develop some tools to understand the triangulated structure. In Section 4 we exhibit the first examples. Section 5 deals with . Section 6 illustrate the first step on how to develop -in the setting of co-Frobenius Hopf algebras- the direction taken in [Qi1] for finite dimensional Hopf algebras.
Acknowledgments
I wish to thank Juan Cuadra for answering questions and pointing useful references on co-Frobenius coalgebras. I also thanks Gastón A. García for helping me with coradical problems. This work was partially supported by UBACyT 2018-2021, 256BA.
1 Integrals, Co-Frobenius and triangulated structure
will be a field, a Hopf algebra over , all comodules are right comodules. The category of comodules is denoted and the subcategory of finite dimensional comodules is denoted .
1.1 Integrals
Definition 1.1**.**
(Hochschild, 1965; G. I. Kac, 1961; Larson-Sweedler, 1969). A (left) integral is a linear map such that
[TABLE]
that is, .
It is well-known that the dimension of the space of (left) integral is . In case admits a non-zero (left) integral , will be called co-Frobenius. The following is well-known, we refer to [ACE] and [AC] and references therein for the proofs:
Theorem 1.2**.**
If is co-Frobenius then, in the category of (say right) -comodules
there exists enough projectives; 2. 2.
every finite dimensional comodule is a quotient of a finite dimensional projective, and embeds into a finite dimensional injective; 3. 3.
being projective is the same as being injective.
If and are two objects in a category , denote set of morphisms and the subset of consisting on morphisms that factors through an injective object of . Denote
[TABLE]
The category whose objects are -comodules and morphism is called the stable category and it is denoted . Similarly is the stable category associated to . By the above theorem, is embedded fully faithfully in . With these preliminaries, one can prove the following main construction:
Theorem 1.3**.**
If is a co-Frobenius Hopf algebra then has a natural structure of triangulated category, is a triangulated subcategory.
Proof.
We apply directly Happel’s Theorem 2.6 of [Ha]. The only thing to do is to notice that (and ) are Frobenius exact categories. Using Happel’s notation, let be an additive category embedded as a full and extension-closed subcategory in some abelian category , and the set of short exact sequences in with terms in . For as, since both and are already abelian, we have and the notion of -projective and -injective is the same as usual projectives and injectives. Maybe we just remark that an object in is injective in if and only if it is injective in and similarly for projectives (see Lemma 1.4 as an illustration).
An exact category is called a Frobenius category if has enough -projectives and enough -injectives and if moreover the -projectives coincide with the -injectives. In our case, or are clearly Frobenius categories if is a co-Frobenius Hopf algebra. Theorem 2.6 in [Ha] just state that the stable category is triangulated. ∎
Lemma 1.4**.**
If then is projective in if and only if it is projective in .
Proof.
If is projective in then then it has the lifting property for all comodules, in particular for the finite dimensional ones. Assume is projective in and consider a diagram of comodules
[TABLE]
where and are not necessarily finite dimensional. Since is finite dimensional, one can consider , clearly is a finite dimensional comodule, with generators say . Since is surjective, one may found () with and there exists a finite dimensional subcomodule containing all ’s, hence we have a diagram
[TABLE]
Now all comodules are finite dimensional, and because is projective within finite dimensional comodules, there exists a lifting of , hence, a lifting of the original . ∎
For clarity of the exposition we recall the definition of suspension, desuspension and triangles in . For this particular case of comodules over a co-Frobenius Hopf algebra, the general definitions can be more explicitly realized. Moreover, for as in Section 3, concrete and functorial constructions can be done in . The reader familiar with Happel’s results may go directly to Section 2.
1.2 Suspension and desuspension functors
In [K], when is finite dimensional and is an integral in (not in ), the author embeds an -module via and define as . For us, and this definition makes no sense, but (even without using the integral) one can always embed an -comodule into by means of its structural map. The structure map is -colinear provided we use the (co)free -comodule structure on (and not the diagonal one).
Definition 1.5**.**
For a right comodule with structure , define
[TABLE]
If is finite dimensional this definition also makes sense in . If is co-Frobenius and is finite dimensional, is not finite dimensional, however, there exists a finite dimensional injective and a monomorphism , so, one can define in and we know in . Moreover, for co-Frobenius Hopf algebras, one can give functorial embeddings in that works in (see Corollary 2.16).
Remark 1.6**.**
If the notation is confusing because is Hopf and one also has the diagonal action, one may consider another injecting embedding:
[TABLE]
[TABLE]
This map is clearly an embedding, and it is -colinear if one uses the diagonal action on . Both embeddings are ok because with diagonal action and with structure coming only from are isomorphic (see Lemma 2.14).
Similar (or dually) to [K] one can define desuspension. Consider the map
[TABLE]
Recall that co-Frobenius implies is bijective (see for instance [DNR]) and it is easy to prove that is a right integral:
[TABLE]
[TABLE]
We use because is right colinear:
[TABLE]
Hence, is a right -comodule and we have, for any , a short exact sequence
[TABLE]
Definition 1.7**.**
The desuspension functor is
Remark 1.8**.**
When considering , we know every finite dimensional comodule has a finite dimensional projective cover , so we can consider and this is isomorphic to in the stable category. Also (see Corollary 2.16), one can define in and in in a functorial way.
As an illustration of the need of stabilization for having a triangulated category one see that, for any a comodule, we have a short exact sequence in
[TABLE]
In particular, considering instead of , there is a short exact sequence
[TABLE]
But there is also a short exact sequence
[TABLE]
So ” computes using another injective embedding”. Usually in but in . Similar argument for , hence these are mutually inverse functors in the stable category, but not in .
1.3 Triangles
One of the axioms of triangulated categories is that any map is a part of a triangle . Triangles are defined via the mapping cone construction. For , is defined in the following way:
Choose an injective embedding and define by the square
[TABLE]
One can see that this definition does not depend -in the stable category- on the choice of the injective embedding . Notice also a well defined map given by the universal property of the push-out:
[TABLE]
Triangles in are (by definition) all sequences isomorphic (in ) to some sequence of the form . Next two Lemmas emphasize the strong relation between the exact structure of (resp ) and the triangulated structure of (resp )
Lemma 1.9**.**
*If
\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}
is a short exact sequence in then the sequence is isomorphic to in the stable category.*
Proof.
We assume . Consider the diagram
[TABLE]
Let be an embedding into an injective object, for simplicity we assume . We define the map
[TABLE]
[TABLE]
It has the property that, for any ,
[TABLE]
So, it induces a well defined map
[TABLE]
[TABLE]
Now from the injectivity of we know there exists a map fitting into the diagram
[TABLE]
So, define the map
[TABLE]
[TABLE]
It has the property
[TABLE]
so, it induces a well defined map
[TABLE]
One composition is the identity:
[TABLE]
[TABLE]
The other composition is
[TABLE]
[TABLE]
so, the Kernel is
[TABLE]
that is an injective comodule, so, these morphisms are mutually inverses in . ∎
The second Lemma is a useful one, maybe it is folklore but it is not usually written:
Lemma 1.10**.**
*If \textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TX} is a triangle in the stable category then there exists a short exact sequence in such that the sequence
\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z} is isomorphic to
\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{\prime}} in the stable category.*
Proof.
One of the axioms of triangulated categories says that
\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TX} is a triangle if and only if
\textstyle{T^{-1}Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z} is so. Hence,
\textstyle{T^{-1}Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z} is isomorphic to a distinguished triangle, that is, there is an isomorphism (in the stable category) of t-uples
[TABLE]
In particular, there is a commutative diagram in the stable category
[TABLE]
and clearly -or equivalently
[TABLE]
is a short exact sequence in . Notice that if and are finite dimensional, one can find a finite dimensional injective hull and hence the short exact sequence also belongs to . ∎
2 Integrals and coinvariants
If is a coalgebra and a right -comodule then is a left module via
[TABLE]
where, as usual, if is a right -comodule, we denote its structural map and we use Seedler-type notation . In particular, for and , being left integral means . Moreover, multiplication by in has the following standard and main property:
[TABLE]
That is, . We list some examples, keeping in mind the above formula.
Examples
If is co-semisimple (e.g. with an affine reductive group) then the inclusion split as -comodules. One can check that an -colinear splitting is an integral. In the cosemisimple case, the inclusion is actually an equality (this will be clear in Subsection 2.1). Nevertheless, the integral may not be so explicitly described. An easy and explicit example is: 2. 2.
If is a finite group and , then is an integral. For any :
[TABLE]
Actually, every finite dimensional Hopf algebra is (Frobenius and) co-Frobenius. Notice that is co-semisimple if and only if is semisimple, if and only if the characteristic of the ground field does not divide the order of . 3. 3.
Let be a group (possibly infinite, e.g. ) and , define
[TABLE]
A right -comodule is the same as -graded vector space . The action of gives the projection into . 4. 4.
Tensor product of co-Frobenius algebras is co-Frobenius, the integral can be computed using tensor products of integrals. 5. 5.
Let be a Hopf algebra and its coradical. Notice that does not need to be a Hopf subalgebra in general. Nevertheless, one of the main results in [ACE] is that is co-Frobenius if and only if the coradical filtration is finite. A particular case is illustrated in the following: 6. 6.
Let be a cosemisimple Hopf algebra and let be a finite dimensional Yetter-Drinfel’d module such that its Nichols algebra is finite dimensional. Then is co-Frobenius. The integral is essentially given by the ”volume form”, or ”Fermionic integration” in (see Remark 3.5).
- (a)
The simplest example is: generated by and with relations and . Comultiplication given by
[TABLE]
[TABLE]
The antipode is
[TABLE]
We have . An element of may be uniquely written as
[TABLE]
A left integral is given by
[TABLE]
This particular example motivates all definitions of this paper. The second simplest example of this kind is the following: 2. (b)
generated by and with relations , , , and comultiplication given by
[TABLE]
[TABLE]
[TABLE]
If we write an element as
[TABLE]
then a left integral is given by . We will compute some invariants of the (stable) comodule category associated to this .
One of the main goals of this paper is to translate into Hopf-co-Frobenius language the notion of homology ””. The definition is very natural:
2.1 Hopf homology for algebras with a non-zero integral
Definition 2.1**.**
Given a co-Frobenius Hopf algebra and , denote
[TABLE]
For , we define
[TABLE]
and
[TABLE]
Example 2.2**.**
If and is co-Frobenius with , then , hence and the functor is non trivial.
Example 2.3**.**
For , .
Example 2.4**.**
The condition “” is stable under arbitrary direct sums and direct summands, so for any injective module .
As a corollary, if is an -colinear map such that if factors through an injective:
[TABLE]
then the induced map
[TABLE]
is necessarily zero. So, the functor is actually defined in the stable category
[TABLE]
Remark 2.5**.**
For all , the functors are defined in the stable category.
Corollary 2.6**.**
If is co-semisimple then every -comodule is injective, hence for all comodule . In other words, for all comodule .
Remark 2.7**.**
From the point of view of invariant theory, is the most convenient situation, but from the point of view of homological algebra, is most interesting.
Lemma 2.8**.**
Let be a Hopf algebra with nonzero integral and denote the Hom space in the stable category of comodules, then there exists an epimorphism
[TABLE]
Proof.
Notice that
[TABLE]
[TABLE]
is an isomorphism. We will show that this map fits into a commutative square
[TABLE]
Assume factors through an injective object
[TABLE]
then one may consider the unit map and the diagram
[TABLE]
Since is a monomorphism and is injective, one may find a dashed morphism making a commutative diagram, so, it is enough to consider the case .
[TABLE]
Now if is such that , then
[TABLE]
so . This proves that the induced map
[TABLE]
is both well-defined, and clearly surjective. ∎
Remark 2.9**.**
One may wonder if the epimorphism of the above Lemma is in fact an isomorphism. This will be the case (see Theorem 2.12). For finite dimensional Hopf algebras it is due to You Qi [Qi2], where he proves actually for finite dimensional Frobenius algebras, in particular for finite dimensional Hopf algebras. It is not clear for the author how to adapt Qi’s arguments to our case, maybe one can find a simpler proof, but we provide a proof with some homological machinery first.
Remark 2.10**.**
Lemmas 1.9 and 1.10 gives an alternative proof that the composition of two consecutive morphisms in a triangle is zero (in the stable category), and so, every functor defined in the stable category sends triangles to complexes. For the particular case of , without knowing that it is representable or not, we have the expected result:
Theorem 2.11**.**
If is a triangle in the stable category then there is a long exact sequence of vector spaces
[TABLE]
Proof.
We will prove that
[TABLE]
is exact in when is a short exact sequence. The general result follows from Lemma 1.10 and the shifting axiom of triangles. So assume is a short exact sequence in , then multiplication by the integral gives as a commutative diagram (of vector spaces) with exact rows
[TABLE]
So, even forgetting that is injective, the snake Lemma gives in particular that
[TABLE]
is exact. ∎
Now, the above Theorem together with Lemma 2.8 gives the following:
Theorem 2.12**.**
Let be a Hopf algebra with nonzero integral and denote the Hom space in the stable category of comodules, then the natural map
[TABLE]
is an isomorphism.
Proof.
First recall the following version of the ”5”-lemma: given a commutative diagram with exact rows:
[TABLE]
if and are monomorphisms and is an epimorphism, then is a monomorphism.
Consider the class of -comodules such that the map is an isomorphism. Because short exact sequences in gives both long exact sequences for and , given a short exact sequence of comodules
[TABLE]
where nd are in , then we have a diagram
[TABLE]
Every vertical map is an epimorphism (Lemma 2.8) and both and are monomorphism because they are isomorphisms (), so is monomorphism, hence, an isomorphism.
We conclude that the theorem is true for any finite dimensional comodule , provided it is true on simple comodules.
If and is not injective then and . (If is injective, the theorem is noninteresting, but still true).
If is simple and then , so trivially .
Now let be a possibly infinite dimensional comodule and such that for some . Consider a finite dimensional subcomodule containing and . Then, the class of in is zero. But because is finite dimensional we know and so there exists a factorization
[TABLE]
with injective. So, is zero in . ∎
2.2 Multiplicative structure
Because is Hopf, the categories and are tensor categories, and the tensor structure descends to the stable category, as one can see after these standard facts:
Lemma 2.13**.**
If is a coalgebra and a vector space, the right comodule with structure map is an injective comodule. 2. 2.
Every injective comodule is a direct summand of one as above. The category of comodules has enough injectives.
Proof.
It follows from the adjunction formula
[TABLE]
[TABLE]
and that every vector space is an injective object in -Vect. 2. 2.
If is a comodule, the structure morphism
[TABLE]
gives an embedding into an injective object: -colinearity is by coassociativity and injectivity is because of counitarity. If is injective, then the monomorphism splits, hence, is a direct summand of where is the underlying vector space of .
∎
Lemma 2.14**.**
Let be a Hopf algebra, . Denote the underlying vector space of .
* (with diagonal coaction) is isomorphic to (with ).* 2. 2.
Also 3. 3.
If is injective then and are both injectives.
Proof.
We only exhibit the maps:
[TABLE]
[TABLE]
with inverse
[TABLE]
The composition is
[TABLE]
The other composition is similar. The surprising part is that these maps are -colinear. For instance:
[TABLE] 2. 2.
The maps are similar: consider
[TABLE]
[TABLE]
with inverse
[TABLE]
The composition is
[TABLE]
The other composition is similar. The colinearity follows the same lines. 3. 3.
If is injective then it is isomorphic to a direct summand of for some vector space (e.g. ), and so is isomorphic to a direct summand of
[TABLE]
and is a direct summand of
[TABLE]
In any case, a direct summand of a comodule of the form for some vector space .
∎
There are several corollaries:
Corollary 2.15**.**
The tensor product is well defined in the stable category. In particular, is an associative ring.
Let be the injective hull of in . It is well-known that is co-Frobenius if and only if is finite dimensional (see Theorem 2.1 in [AC]). Also, for co-Frobenius Hopf algebras, there exists a finite dimensional projective comodule with a surjective map .
Corollary 2.16**.**
Define . The map () is a functorial injective embedding, if then as well. Also, gives a functorial projective surjection , if then as well.
Proof.
The injective part is clear. Let us prove the existence of a surjective map with finite dimensional:
Since is surjective, there exists such that , and there exists a finite dimensional subcomodule containing . In particular, . Because is co-Frobenius, there exists a finite dimensional injective hull of , let’s call it . Looking at the diagram
[TABLE]
Because is injective there exist the dashed arrow. Because is the injective hull and is injective, the map is injective and is a direct summand of . Eventually changing by we get a finite dimensional direct summand of such that the restriction of is non-zero, hence, a surjection with projective and finite dimensional. ∎
Corollary 2.17**.**
For any , in , there are isomorphisms in the stable category
[TABLE]
and similarly for . Hence, and are tensor triangulated categories.
Proof.
Let and be embeddings into injective comodules, then is injective and one can compute via
[TABLE]
But and are injectives too, and we have the following short exact sequences with injective objects in the middle:
[TABLE]
∎
Notice that the morphisms are not canonical in the category of comodules, but they are canonically determined in the stable category
Corollary 2.18**.**
For any there is an isomorphism in the stable category
[TABLE]
Künneth map
Let and be two comodules. It is clear that and also one can easily check that
[TABLE]
and
[TABLE]
So, there is a canonical map
[TABLE]
Moreover, using Corollary 2.18 on can define maps
[TABLE]
(If a number is negative, we use the convention .) In this way, one can assembly all those maps and get, for any fixed , a map that we call ”Künneth map”
[TABLE]
For , from concrete computations (see Corollary 4.10) we know this map cannot be an isomorphism in general. It would be interesting to know their general properties. In any case, is a graded algebra.
3 Small injective embeddings
for
During this section we assume
is a co-semisimple Hopf algebra, 2. 2.
is such that , the Nichols algebra associated to the braided vector space , is finite dimensional.
Let us recall briefly the conditions above and set notations and conventions. First, is the category whose objects are left -modules and right -comodules with the compatibility
[TABLE]
where , and , . Morphisms are -linear and colinear maps. For any Hopf algebra , the category is braided with
[TABLE]
[TABLE]
Recall that if is a braided vector space (e.g. ) then both (the tensor algebra) and (the tensor coalgebra) are braided Hopf algebras. has free product and braided-shuffle coproduct, while has deconcatenation coproduct and braided-shuffle product. The Nichols algebra is, by definition, the image of the unique (bi)algebra map that is the identity on :
[TABLE]
It happens to be, degree by degree, the image of the quantum symmetrizer map associated to the braiding. We refer to Andruskievitch’s notes [A] for a gentle introduction and full discussion on Nichols algebras. The reader may keep in mind the easy example when the braiding is -flip. The braided bialgebra is actually a braided Hopf algebra, and the bicross product is a usual Hopf algebra. Since there is a lot of structures around we recall them:
- •
is a coalgebra, we denote ,
- •
, we denote the structure ,
- •
is a coalgebra, the comultiplication is given by the following diagram (recall the underlying vector space of is ):
[TABLE]
In Sweedler-type notation:
[TABLE]
- •
In particular . Denoting , we have that is a right -subcomodule of . With this structure we consider as an object in . To emphasize the difference with we call it .
Example 3.1**.**
Let , . Assume . In order to compute we proceed as follows:
[TABLE]
[TABLE]
[TABLE]
The main fact of this section is the following:
Proposition 3.2**.**
* is an injective object.*
Proof.
Since is cosemisimple, the inclusion splits as -comodule. Choose a splitting . This is actually right integral for , that is, it satisfies
[TABLE]
and additionally .
Now we define a splitting of the inclusion via
[TABLE]
We need to see that it is -colinear. Recall the -structure in is given by the identification , so
[TABLE]
We check -colinearity:
[TABLE]
∎
Remark 3.3**.**
The proof is independent of the fact of being finite dimensional, but we are interested in the case so that is co-Frobenius.
As a corollary we have
Corollary 3.4**.**
For any , the map defined by
[TABLE]
is a functorial injective embedding. In particular, if is finite dimensional then () is a finite dimensional embedding working in . From the short exact sequence
[TABLE]
we have . Recall is graded (with the tensor grading) and (its maximal degree) has dimension 1. The Kernel of is . From we have the short exact sequence
[TABLE]
hence .
Remark 3.5**.**
is *not * isomorphic to in general, but it is 1-dimensional. So, in order to compute one should ”twist by the inverse of the quantum determinant”:
If is a generator of the 1-dimensional vector space then is not in general an -subcomodule of , but is so, hence
[TABLE]
for a unique group-like element , that we call ”quantum determinant”. From the surjective map
[TABLE]
we get a surjective map into the trivial comodule :
[TABLE]
If we call , it is a projective -comodule that surjects into and from it one has functorial projective surjections for any comodule :
[TABLE]
and functorial , since from:
[TABLE]
we get is a functor in (resp. in if is finite dimensional) that gives the desuspension functor in (resp. in ).
Before going into rings, we look at some examples.
4 First examples
4.1 The example
Let be the -algebra generated by and with relations
[TABLE]
It is a Hopf algebra if one defines the comultiplication by
[TABLE]
(to be compared with Example 3.1). That is, is group-like and is 1--primitive. Notice that is not a Hopf algebra in the usual sense (unless characteristic=2), but it is a super Hopf algebra. Nevertheless, is a Hopf algebra in the usual sense. Maybe all computations in this example are folklore, but for clearness we include them.
For an element
[TABLE]
define
[TABLE]
The main fact about the category , noticed by Bodo Pareigis [P], is
A right -comodule is the same as a d.g. structure on
Notice that evaluation at gives a map , so any -comodule is a -comodule (i.e. a -graded object), but the presence of keep track of a square-zero differential. We just write the correspondence: if with and then, for , the right comodule structure is
[TABLE]
and every right -comodule is of this form.
It is a pleasant exercise to check that the standard differential on the tensor products of complexes with the usual Koszul agree with the standard -comodule structure on the tensor product of -comodules.
Notice that
[TABLE]
[TABLE]
means that is a right -subcomodule of . As d.g. vector space is the complex
[TABLE]
where , .
Smaller injective embeddings for
In this case we have , considered as -comodule via
[TABLE]
[TABLE]
The general argument developed in the previous section gives us that, for any , the map given by is an embedding of into an injective object. In particular and we have proven the following:
Corollary 4.1**.**
In the stable category of for ,
[TABLE]
We leave as an exercise the following:
Corollary 4.2**.**
Identifying d.g. and , the comodule identifies with the mapping cone of the identity of . Moreover, the ”stable -comodule mapping cone” of a colinear map identifies with the classical mapping cone of a map between complexes.
Homology
For a d.g. vector space viewed as -comodule, the coinvariants are
[TABLE]
But ” ” means that and , so
[TABLE]
On the other side, the action of the integral on an element gives
[TABLE]
[TABLE]
That is
[TABLE]
Hence
[TABLE]
4.2 The example and -complexes
Fix , . Let be the algebra generated by and with the relations
[TABLE]
where is an -primitive root of unity. This algebra is Hopf with comultiplication
[TABLE]
To have an -comodule is the same as a -graded vector space together with a degree -1 map satisfying . The tensor structure for is given by the usual total grading in , and the differential on homogeneous elements is
[TABLE]
For an homogeneous element of degree , the coaction is given by
[TABLE]
[TABLE]
where as usual , and .
If is an -complex, there are several ways to associate an ”homology” in degree . For each , since , one may consider . The general machinery of co-Frobenius algebras and stable categories, however, choose one particular . Since and we have
[TABLE]
The other (homological) degrees are not the of the degree-shiftings of the -complex. As an illustration we compute in terms of the -complex data:
Proposition 4.3**.**
**
Remark 4.4**.**
From Corollary 3.4 and the isomorphism (notice is generated by the class of and +lower degree terms) it follows that
[TABLE]
Proof.
Recall , the structure is given by
[TABLE]
[TABLE]
The degree zero part of , if , is
[TABLE]
A typical element is of the form
[TABLE]
where . The differential has the form
[TABLE]
[TABLE]
[TABLE]
This expression is equal to zero if and only if
[TABLE]
From the second set of equalities we see that the only parameter is , because is, up to scalar, . The equation means . We conclude
[TABLE]
We leave to the reader to check that, under this bijection, corresponds to . ∎
4.3 The example and Mixed complexes
Denote
[TABLE]
It is not a Hopf algebra in the usual sense, but it is a Hopf algebra in the (signed) graded sense. The algebra
[TABLE]
is a Hopf algebra with comultiplication
[TABLE]
Notice that produce a differential of degree -1, while produce a differential of degree +1. This Hopf algebra is isomorphic to where and . Writing multiplicatively , the action is given by
[TABLE]
and the coaction is determined by
[TABLE]
Lemma 4.5**.**
* identifies with objects where is a -graded vector space, and are square zero differentials with , , and . In other words, are mixed complexes.*
The proof is straightforward, we only indicate the correspondence: for a mixed complex , the corresponding right comodule structure
[TABLE]
for an homogeneous , is given by
[TABLE]
It is clear that . Also, an easy computation shows (see Example 6(b) of Section 2 for the expression of the integral)
[TABLE]
So,
[TABLE]
Remark 4.6**.**
In this (stable) category, the suspension functor is not the shifting degree in general. However, we have the following lemma:
Lemma 4.7**.**
* and are -subcomodules of . as objects in . For denote and . The following assertions follows:*
- •
* is an injective object in .*
- •
,
- •
**
- •
**
- •
**
Proof.
The first item follows from the obvious isomorphism
[TABLE]
Observe that and . Now from the short exact sequence
[TABLE]
we get
[TABLE]
Since is injective, we conclude . Similarly for .
In order to compute cohomology we consider first
[TABLE]
We visualize it using the following diagram
[TABLE]
So,
[TABLE]
[TABLE]
[TABLE]
but , so .
We also must compute :
[TABLE]
[TABLE]
So, under the isomorphism , the subspace corresponds to . We conclude .
Now from the second item we get
[TABLE]
The parts with instead of is completely analogous. ∎
Corollary 4.8**.**
For any mixed complex there are long exact sequences
[TABLE]
and
[TABLE]
Proof.
We consider the short exact sequences in :
[TABLE]
and
[TABLE]
Recall and . These short exact sequences in gives triangles in the stable category; their log exact sequences together with the previous Lemma gives the result. ∎
Corollary 4.9**.**
* ;
.*
Another corollary is the following computation:
Corollary 4.10**.**
Considering as trivial mixed complex concentrated in degree zero,
[TABLE]
Proof.
Specializing the long exact sequence
[TABLE]
at and gives
[TABLE]
If we have
[TABLE]
Inductively, for
[TABLE]
because do not have 0-degree component if . It remains to compute . and .
Clearly . For , since (notice has degree zero), the formula for is
[TABLE]
We have and in , so . ∎
Remark 4.11**.**
Notice the asymmetry in the gradings, but , as we can see from the general argument above, or compute directly:
[TABLE]
The degree zero component is , but (also ), so and
[TABLE]
5 (De)Categorification: computation of
5.1 of exact and triangulated categories
The main result of this section is Theorem 5.1, that gives a general presentation of . We recall the main constructions:
If is an exact category such that the isomorphism classes of objects is a set, then is defined as the free abelian group on the set of isomorphism classes of objects module the relations
[TABLE]
whenever there is a short exact sequence
[TABLE]
We remark that and, for , , so, every element in can be written in the form for some objects in . For triangulated categories, is defined similarly, taking the free abelian group on isomorphism classes of objects modulo the relations
[TABLE]
whenever there is a triangle
[TABLE]
By we understand the K-theory of the category of finite dimensional -comodules, that is an exact category with usual short exact sequences. We denote the K-theory of the stable category as triangulated category.
Almost by definition, if denotes the full subcategory of injective objects in , there is a short exact exact sequence of categories
[TABLE]
One could expect a general result in K-theory concluding a long exact sequence ending with
[TABLE]
This is actually the case for short exact sequences of exact categories where the left hand side is also a Serre subcategory. Recall a Serre (sub)category is closed under quotients, subobjects and extensions. In our case, is an exact category but is not a Serre subcategory in general. Also, is not in general an exact category, it is triangulated, but the other two are not triangulated.
For exact sequences of Waldhausen categories there is also a long exact sequence in -theory, both and are Waldhausen categories, but it is not clear that is so, in any case one should prove it. Instead, one can prove directly the following:
Theorem 5.1**.**
The natural functors and induce an exact sequence
[TABLE]
In particular, the ring can be presented as the quotient of by the ideal generated by injective objects.
Proof.
The functor is the identity on objects, so the induced map is surjective. By definition of , the composition is zero, so the composition is zero as well. Let us denote
[TABLE]
We have a surjective ring homomorphism
[TABLE]
To see injectivity of this map we argue as follows: Assume in goes to zero in . From the short exact sequence
[TABLE]
in with injective, in , hence in . So, we have
[TABLE]
Eventually changing by , we can assume that, modulo , the element is equal to for some object . Now if is zero in , then there exists integers and triangles in the stable category
[TABLE]
such that
[TABLE]
But, using that direct sum of triangles is a triangle, for the positive ’s we get
[TABLE]
and similarly for the negative ’s. From this, we may assume that there are two triangles , such that
[TABLE]
But because is a triangle, then so is , hence in , and for a triangle, we have is also a triangle and
[TABLE]
So, we can conclude that there exists a single triangle such that
[TABLE]
But we know (Lemma 1.10) that any triangle in the stable category is isomorphic, in the stable category, to a short exact sequence
[TABLE]
Recall also that in if and only if there exist injectives and such that in . But Modulo , clearly . So, we finally get that
[TABLE]
Hence, in is zero Mod . ∎
Remark 5.2**.**
It could be interesting to know if this is the last part of a long exact sequence for higher K-groups.
5.2 and the coradical
Let be a Hopf algebra and its coradical. Since is a subcoalgebra, every -comodule is an -comodule. Consider the category and ; is a non-empty full subcategory closed under taking subobjects, quotient objects, and finite products in . Also is an abelian category and the inclusion functor is exact, so Quillen’s theorem gives.
Theorem 5.3**.**
([Q], Theorem 4. (Devissage)) Let and be as above. Suppose that every object of has a finite filtration such that is in for each . Then the inclusion induces an isomorphism
If is a nonzero comodule, then its socle is a nonzero subcomodule that is actually an -comodule (see Exercise 3.1.2. of [DNR], page 117, its solution on page 140). If in addition is finite dimensional, considering and induction in the dimension of one can easily define a finite filtration
[TABLE]
such that , hence . Now Quillen’s theorem implies the following:
Corollary 5.4**.**
As abelian groups, . If is a Hopf subalgebra then this isomorphism is also a ring isomorphism.
5.3 Smash products
Let where is cosemisimple and a finite dimensional braided Hopf algebra in . For an element , denote the associated graded with respect to the ”socle filtration”. Recall that the assignment implements the isomorphism . If denote the set of (isomorphism classes of) simple objects in , then , for ,
[TABLE]
for uniques (and finite non zero) multiplicity integers . We define
[TABLE]
In particular is a finite dimensional -comodule, so it makes sense
[TABLE]
In the case with a group, the isomorphism classes of simple comodules can be parametrized by and , so we identify . The main result of this section is the following:
Theorem 5.5**.**
Let be a group and . Assume , with finite dimensional . The assignment induces an isomorphism of rings
[TABLE]
Proof.
From Theorem 5.1 it follows that . But from Corollary 5.4 we know
[TABLE]
[TABLE]
For we also know . We need to identify inside .
Recall that is injective and . Now let be a finite dimensional indecomposable injective -comodule. Because is indecomposable and injective, is an indecomposable -comodule (here, injectivity of is essential), hence simple and
[TABLE]
for some . Clearly is an injective indecomposable -comodule and . Since is injective, there exist a dashed morphism in the diagram:
[TABLE]
This map restricted to the socle is injective, so the map is injective and we have . But is injective, so the same argument in the opposite direction gives and so . In other words,
[TABLE]
for some . We can conclude that if is finite dimensional injective (non necessarily indecomposable) -comodule, then there exists integers with
[TABLE]
That is, is the ideal generated by . ∎
5.4 Examples
The first two examples are well-known:
:
[TABLE] 2. 2.
(Khovanov) :
[TABLE]
If is is a prime then . 3. 3.
where , , then
[TABLE]
hence and so
[TABLE] 4. 4.
If , for , a list of nonzero scalars is given, then define as the algebra generated by with relations
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
It is a Hopf algebra with comultiplication given by
[TABLE]
[TABLE]
Then is a Hopf algebra of the form . The algebra has monomial basis , so, writing ,
[TABLE]
Hence,
[TABLE]
Remark 5.6**.**
It would be interesting to compute for some non pointed cosemisimple , for instance, with non abelian reductive affine group.
6 -comodule algebras and the category
An -comodule algebra is a -algebra together with an -comodule structure such that the multiplication map
[TABLE]
and the unit
[TABLE]
are -colinear. Usual examples are:
- •
: comodule algebra = -graded algebra.
- •
finite, : comodule algebra = algebra with a -action by ring homomorphisms.
- •
affine group, : comodule algebra = algebra with a rational -action.
- •
, comodule algebra =algebra with a -action acting by derivations.
For our purpose, the motivating example is . In this case, an -comodule algebra = d.g. algebra.
Also, if is any Hopf algebra and is any algebra, then viewed as trivial -comodule is an -comodule algebra.
The main fact for our interest is the following:
[TABLE]
where -module structure in is the one coming from and the -comodule structure is the diagonal one. Moreover, if is finitely generated as -module and is finite dimensional, then is finitely generated as -module. In this way, the subcategory of consisting on -finitely generated modules, denoted by , is naturally a module over the category . Following [K], we consider the restriction functor
[TABLE]
and define to be acyclic (or -acyclic to emphasize the role of ) if is injective as -comodule. In other words, if in . A map is called quasi-isomorphimsm (qis) if becomes an isomorphism in . Denote the class of objects in that are injective as -comodules.
Example 6.1**.**
Let be an arbitrary object and an injective -comodule. In virtue of Lemma 2.14, .
If , denote the set of maps that factors through an object in . The stable category - or the -derived category- , denoted by and also by , is defined as the category with same objects as but morphism
[TABLE]
The subcategory of whose objects are in (i.e. are finitely generated as -modules) is denoted by .
Recall that if is the injective hull of , is a finite dimensional injective -comodule (because is co-Frobenius), and for any , then
[TABLE]
is an embedding of into an acyclic object in . If is a (finite dimensional) projective cover of , then
[TABLE]
is an epimorphism from an -acyclic object in into . If is finitely generated as -module, then so is and . The definition of , of and of the mapping cone of objects and maps in (resp. in ) actually gives objects in (resp. in ). One can easily see that all constructions and proof’s of Happel’s Theorem 2.6 in [Ha], when starting with objects in in (resp. in ) always stay in (resp. in ). So and are triangulated categories, and by Example 6.1, they are modules over and respectively.
Remark 6.2**.**
is a module over the ring .
Example 6.3**.**
If then and .
Example 6.4**.**
if and is an ordinary algebra viewed as trivial -comodule algebra then , the (unbounded) derived category of .
Example 6.5**.**
If is a semisimple Hopf algebra and is a co-Frobenius Hopf algebra, we view as trivial -comodule algebra, then
[TABLE]
Since is semisimple, is co-semisimple and is co-Frobenius. In this case we have . Also if as in Section 5.3 then is a quotient of . Assuming algebraically closed, every simple corepresentation of the tensor product is given by the tensor product of a simple -comodule and a simple -comodule, hence .
Enriched Hom
If , there are several Hom spaces that one can consider. We begin with the discussion for d.g. -modules:
If and are d.g. A-modules, then one may consider
- •
Chain maps: = maps preserving degree and commuting with the differential.
- •
Chain maps up to homotopy: , where if for some degree +1 -linear map .
- •
The HOM complex: where = -linear maps of degree . If is concentrated in degree zero (i.e. is a trivial -comodule) then
- •
Morphisms in the derived category: .
In general is different from . Assume for simplicity is an ordinary alegbra (i.e. d.g. algebra concentrated in degree zero), if and have infinite nonzero degrees, then
[TABLE]
For instance, if and , then
[TABLE]
Nevertheles, the set of chain maps agree with and the set of chain maps up to homotopy is the same as .
For general co-Frobenius Hopf alegbras (i.e. not necesariily finite dimensional ones) one has the same ”problems” but also analogous solutions. First of all, if is a (not finite dimensional) Hopf algebra, an -comodule algebra and , then is not an -comodule in general. For instance, if and , then is a not rational -module, so it is not an -comodule. In this way, if one consider
[TABLE]
it is not expectable to have an object in .
It is not clear to the author how to get an object in analogous to (maybe , where runs over all -finitely generated subobjects?). To have an object would provide the notion of map up to homotopy just by taking . Nevertheles, we have the following
Proposition 6.6**.**
* is a (left) -module and the definition of can be naturally extenbded to -modules.*
Proof.
The first statement is probably well-known, for completenes we exhibit the proof: First recall that if is a finite dimensional Hopf algebra and are -modules (e.g. if is finite dimensional and ) then the standard action of an element in a map , acting on an alement is given by
[TABLE]
If and then the above formula is
[TABLE]
[TABLE]
and the last term in the equality make sense for , independently on the dimension of , so one *defines * the -action of on via
[TABLE]
In other words,
[TABLE]
One can proof by standard diagramatic methods that this is an action, and is -colinear (if and only if it is -linear) if and only if
[TABLE]
Concering the second statement, if is an -module, one may define
[TABLE]
If is a right -comodule then it is clear that , so one can extend the definition of on simply by
[TABLE]
If then = linear and -linear maps =, and a definition of ”chain maps up to homotopy” is available definig
[TABLE]
This recover the definition given in [Qi1] for finite dimensional Hopf algebras and when and are -graded vector spaces, but we emphasizes that this definition makes sense in full generality for a co-Frobenius algebra (whose coradical is not necesarily finite over ). ∎
A warning on the notation in [Qi1], we call what he calls in the ungraded case. He defines only in the graded case but using the degree shifting, and not the triangulated structure, so in [Qi1] is different from our .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[ACE] N. Andruskiewitsch, J. Cuadra, P. Etingof: On two finiteness conditions for Hopf algebras with nonzero integral , Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) Vol. XIV (2015), 1-40.
- 3[AC] N. Andruskiewitsch, J. Cuadra: On the structure of (co-Frobenius) Hopf algebras Journal of Noncommutative Geometry. Volume 7, Issue 1, pp. 83–104 (2013).
- 4[AD] N. Andruskiewitsch, S. Dascalescu: Co-Frobenius Hopf algebras and the coradical filtration . Math. Z. 243 (2003), 145-154 .
- 5[DNR] S. Dascalescu, C. Nastasescu, S. Raianu: Hopf algebras. An introduction . P. and App. Math., Marcel Dekker. 235. New York, NY: Marcel Dekker. ix, 401 p. (2001).
- 6[Ha] D. Happel, Triangulated categories in the representation of finite dimensional algebras , London Mathematical Society Lecture Note Series, 1988.
- 7[K] M. Khovanov: Hopfological algebra and categorification at a root of unity: the first step , J. Knot Theory and its Ramifications 25, No. 3, 26 p. (2016).
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