Some reflections on exactness and coexactness

Jeffrey D. Carlson

22nd December 2015

Abstract We try to make sense of the notions of exactness and coexactness in a generic pointed category and then in the category of connected commutative graded algebras over a field. It is found that under the most natural definitions, these concepts agree entirely.

1. Summary and conclusions Exactness and coexactness as defined by Moore and Smith [MS68, p. 762] are dual concepts, defined entirely categorically and originally discussed in the context of commutative or co- commutative connected Hopf algebras. They do not always agree: for example, they differ in the category of groups. The category-theoretic notions of exactness and coexactness discussed in your letter [Smi15] are completely equivalent to one another, in all pointed categories, and disagree with those in your earlier paper, being generically weaker. The two versions of exactness agree in a pointed category if and only if the first map in every image factorization A Ñ Imp f q B is epic, and dually the two versions of coexactness agree im f coim g if and only if the second map in every coimage factorization B Coimpgq Ñ B is monic. These conditions both hold, for example, in the category of connected CGAs, and dually in the category of connected cocommutative graded coalgebras, and consequently all four concepts of exactness and coexactness agree there. In your letter [Smi15], you make two conflicting claims in the footnotes. Footnote 3 claims that in the category of connected CGAs over a field, the categorical image imp f q “ kerpcoker f q agrees with the set-theoretic image, while Footnote 4 claims that we should not reason with the categorical image, because it is not very meaningful, and we should instead focus on the set-theoretic image. The first of these claims is definitely wrong, and the above observations seem to make a strong argument that the category-theoretic image is indeed the right object to look at. Your analysis of the alternate version of exactness when “image” is taken to be the set- theoretic image is correct, but as you point out, the resulting map is nearly useless, making the choice to ignore the better-functioning categorical image questionable. The putative disagree- ment between exactness and coexactness in the context of commutative graded algebras and the dismissal of exactness of CGAs as a fruitless concept seems to be solely the result of this insistence on using the set-theoretic image rather than the more “correct” category-theoretic one, which insistence it seems may be spurious. Consequently, exact sequences in the category of CGAs, have over the last fifty years or so needlessly (but accurately) described as “coexact.” In fact, I made no real use of the commutativity hypothesis in dealing with connected graded k-algebras, so I removed it. The only properties of connected graded rings A we seem to use are

1 2

• the Noether first isomorphism theorem,

• the fact the kernel of a homomorphism is a two-sided ideal,

• the construction X ÞÝÑ pXq :“ AXA that generates an ideal from a subset X Ď A,

• the fact that graded connected k-algebra maps preserve the additive decomposition A “

k ‘ ně1 An, and • theÀ fact that monomorphisms are injective.

As a complement, I looked briefly at these notions in the category of “cohomological Hopf algebras” from your paper [MS68]. I am not confident in these results and await your feedback.

2. Categorical generalities Let pC , ˚q be a pointed category, meaning ˚ is a zero object in C . For any two objects X, Y P C , denote again by ˚ the unique trivial morphism X Ñ ˚ Ñ Y. Given an arrow f : X Ñ Y, recall that, if they exist,

• the kernel kerp f q is the final arrow k : K Ñ X such that f ˝ k “ ˚;

• the cokernel cokerp f q, if it exists, is the initial arrow c : Y Ñ C such that c ˝ f “ ˚;

• the image imp f q is kerpcoker f q, meaning the final arrow i with cokerp f q ˝ i “ ˚;

• the coimage coimp f q is the cokerpker f q, meaning the initial arrow c1 with c1 ˝ kerp f q “ ˚.

The kernel kerp f q can be seen as the equalizer of f and ˚, and the cokernel cokerp f q as the coequalizer of f and ˚, and consequently kernels and images are regular monomorphisms and cokernels and coimages regular epimorphisms. Because cokerp f q ˝ f “ ˚, one has always a image factorization

f im f X ÝÝÑim Imp f q Y of f and dually since f ˝ kerp f q “ ˚ a coimage factorization

coim f f X Coimp f q ÝÝÝÑcoim Y.

Moreover, ˚ “ cokerp f q ˝ f “ cokerp f q ˝ fcoim ˝ coimp f q, and since coimp f q is epic, also cokerp f q ˝ fcoim “ ˚, meaning fcoim factors through kerpcoker f q “ im f . Thus there arises a compound factorization: fim

( im f X / / Coimp f q f / Imp f q / / Y. coim f 7

fcoim

From it one sees that if f is epi, then so is fim, and that if f is mono, then so is fcoim. In the former case, we say f induces an epimorphism to its image and in the latter that it induces a monomorphism from its coimage. If f is an isomorphism, of course, then f does both. In the pointed categories of greatest interest to us, monomorphisms will turn out to be injections and fcoim to be monomorphic, but not all monomorphisms will be kernels. Thus the subobject Coimp f q ãÑ Y will correspond more closely to the intutitive set-theoretic image, while the regular image object Imp f q will morally be the smallest “kernel object” generated by Coimp f q. 3

Consider a sequence f g X ÝÑ Y ÝÑ Z (1) in C and assume the kernel kerpgq: Kerpgq Ñ Y exists. If the composition g ˝ f “ ˚ is trivial, then by the definition of kernel, f factors through kerpgq, say as

f¯ ker g X ÝÑ Kerpgq ÝÝÝÑ Y.

One says [MS68, p. 762] this sequence is exact at Y if the factor map f¯ is an epimorphism. Dually, if we assume the cokernel coker f : X Ñ Cokerp f q exists, then by the definition of cokernel, g factors through coker f , say as

coker f g¯ X ÝÝÝÝÑ Cokerp f q ÝÑ Y.

One says the sequence is coexact at Y if this map g¯ is an monomorphism. In the situation above, note that

g ˝ imp f q “ g¯ ˝ cokerp f q ˝ imp f q “ ˚ since imp f q “ kerpcoker f q, and consequently imp f q: Imp f q Ñ Y factors through kerpgq, yield- ing a natural arrow e : Imp f q Ñ Kerpgq as in the diagram

Imp f q (2) > fim  ! im f ! X e Y. >

¯ f  > ker g Kerpgq

Here the right triangle commutes by definition, and the compositions along the top and bottom are f . The left triangle then commutes because

kerpgq ˝ f¯ “ f “ imp f q ˝ fim “ kerpgq ˝ e ˝ fim and kerpgq is mono. Define the sequence (1) to be exact2 if e is an isomorphism, as in [Smi15]. Dually, since coimpgq ˝ f “ ˚, we find coimpgq factors through cokerp f q via some epimorphism c. We analogously say (1) is coexact2 if c is an isomorphism, again as in [Smi15]. The combined diagram is Imp f q Coimpgq (3) > < O fim  < O gcoim im f coim g " X e Y c Z. > < ker g coker f ¯ g¯ f  > " " Kerpgq χ / Cokerp f q

The arrows f¯ and g¯ at bottom are those figuring in the original definitions of exactness and coexactness, and the arrows e and c are those figuring in the new definitions.

We note a curious thing about the revised definitions: they are identical to one another. 4

First proof. Fix an object Y in a pointed category C and recall [Mac78, p. 189] that

coker % ker forms a Galois connection (an adjunction of preorders) between the “over” category C {Y whose objects are arrows to Y in C and the “under” category Y{C whose objects are arrows from Y. If e is an isomorphism, so that imp f q – kerpgq in C {Y, then the cokernels

cokerp f q “ coker kerpcoker f q “ cokerpim f q – cokerpker gq “ coimpgq will be isomorphic in Y{C , and` the isomorphism˘ will be precisely c. Dually, if c is an isomor- phism, then writing cokerp f q – coimpgq we see the kernels

imp f q “ kerpcoker f q – kerpcoim gq “ ker cokerpker gq “ kerpgq will be isomorphic, the isomorphism given by e. ` ˘

Here is another, less sophisticated proof.

Second proof. Exactness2 occurs just if kerpgq factors through imp f q. Indeed, right-multiplying the equation imp f q “ kerpgq ˝ e by a hypothetical inverse e´1 shows kerpgq would factor through imp f q. Conversely, the existence of a factorization kerpgq “ imp f q ˝ e1 would imply e1 is monic and that imp f q “ kerpgq ˝ e “ imp f q ˝ e1 ˝ e. Left-cancelling the monic imp f q, this would mean e1 ˝ e “ id. Right-multiplying by e1 to get e1 ˝ e ˝ e1 “ e1 and then left-cancelling e1 since it is monic, we see e1 “ e´1. On the other hand, kerpgq factors through imp f q “ kerpcoker f q if and only if

χ “ cokerp f q ˝ kerpgq “ ˚ if and only cokerp f q factors through coimpgq “ cokerpker gq. But by the dual of the argument in the previous paragraph, cokerp f q factors through coimpgq precisely if c admits an inverse.

To compare the notions of exactness and exactness2, note first that e is the kernel of χ “ cokerp f q ˝ kerpgq. Indeed, if cokerp f q ˝ kerpgq ˝ h “ ˚, then kerpgq ˝ h must through im f “ kerpcoker f q, say as kerpgq ˝ h “ imp f q ˝ j “ kerpgq ˝ e ˝ j. Since kerpgq is monic, we left-cancel it and find h “ e ˝ j. Second, recall that any epimorphic equalizer is an isomorphism. Indeed, suppose e is the equalizer of f , g : X Ñ Y. Then since f ˝ idX “ g ˝ idX, it follows idX factors through e, say as 1 1 1 1 1 1 idX “ e ˝ e . Left-multiply by e to get e “ e ˝ e ˝ e and right-cancel e , since it is epic, to find e1 “ e´1. Combining these two facts, we see e will be an isomorphism precisely if it is epic. Thus exactness implies exactness2, and the converse holds when f induces an epimorphism to its image. To see this condition is necessary to the converse, consider the case g “ cokerp f q: then kerpgq “ kerpcoker f q “ imp f q, so exactness2 holds, and exactness holds as well if and only if fim is an epimorphism. Dually c will be iso just if it is monic, so coexactness implies coexactness2, and the converse holds exactly when g induces a monomorphism from its coimage. We summarize.

Proposition 2.1. Let C be a pointed category. Then exactness2 and coexactness2 are equivalent in C . Exactness implies exactness2 and coexactness implies coexactness2. f g Consider a sequence ¨ Ñ ¨ Ñ ¨ with trivial composition ˚ “ g ˝ f . For fixed f , exactness will be equivalent to exactness2 for all g if and only if f induces an epimorphism onto its image, and for fixed g, coexactness will be equivalent to coexactness2 for all f if and only if g induces an monomorphism from its coimage. 5

It is possible for these conditions to not hold, as we will show for the category of groups in Section 3. This proposition allows one to prove a claim included in your letter and your paper both, the reasoning for which was not entirely clear to me on first reading.

Corollary 2.2. Let C be a pointed category in which f : Coimp f q Ñ Imp f q is an isomorphism for every arrow f . Then exactness, exactness2, coexactness2, and coexactness are equivalent in C .

To obtain the hypothesis that fim is epic, Mac Lane [Mac78, Lemma VIII.1, p. 189] uses the following theorem.

Proposition 2.3. Let C be a pointed category with all equalizers and all monomorphisms m normal in the equivalent senses that m “ impmq or that m be the kernel of some arrow. Then every arrow of C induces an epimorphism to its image. Dually, if C has all coequalizers and all epimorphisms are normal (hence are cokernels, their own coimages), then every arrow of C induces a monomorphism from its coimage.

These sufficient conditions are emphatically too strong in the category of connected CGAs, where proper inclusions of subobjects are frequently epimorphic and most monomorphisms ˚ are not kernels, but the category k-Hopf0 of commutative connected Hopf algebras has the former property and the category of cocommutative connected Hopf algebras has the latter [MS68, p. 756].

3. The category of groups Consider the pointed category pGrp, 1q of groups and group homomorphisms, with zero object the trivial group. Recall the straightforward result that monomorphisms in Grp are precisely injections and the less trivial result that epimorphisms are precisely surjections ([Mac78, Exer. 5, p. 21]1). The kernel of a group homomorphism φ : G Ñ H is exactly the inclusion in H of the group- theoretic kernel Kerpφq, so the coimage is the surjection G Ñ G{ Kerpφq. The second map gcoim in the factorization G Ñ G{ Kerpφq Ñ H is always injective, being essentially the inclusion of the set-theoretic image by the first isomorphism theorem, and hence a monomorphism, so by Proposition 2.1, coexactness, coexactness2, and exactness2 are identical in Grp. We may thus consider exactness and exactness2 only. The cokernel of φ : G Ñ H is the initial surjection out of H annihilating φpGq. This exists and consists in modding out from H the normal closure

φpGq :“ hφpGqh´1 hPH @@ DD A ď E of φpGq. The kernel of cokerpφq is the inclusion impφq: φpGq ãÑ H, whose domain is the set-theoretic image of φ if and only if φpGq is normal in H. Now, in the factorization @@ DD

φ impφq G ÝÑim φpGq ãÝÑ H, @@ DD the first map φim is epic if and only if φpGq is already normal in H, and consequently we expect exactness and exactness2 might differ. Indeed, consider a sequence φ ψ G ÝÑ H ÝÑ K (4)

1 or http://ncatlab.org/toddtrimble/published/epimorphisms+in+the+category+of+groups 6 of group homomorphisms, for which the diagram (2) specializes to

φpGq = φim  ! im φ @@ DD ! G H. <

φ¯ !  < ker ψ Kerpgq

Exactness occurs just if φ¯pGq “ φpGq “ Kerpψq, while exactness2, or equivalently coexactness, occurs if φpGq “ Kerpψq, a strictly weaker condition unless φpGq is normal in H. For example, if we take ψ “ cokerpφq for any inclusion φ : G Ñ H of a non-normal subgroup, then the @@ DD sequence (4) will be coexact but not exact.

4. The category of connected graded k-algebras

Let k be a field. Then the category pk-Alg0, kq of all connected graded (associative) k-algebras is pointed category, with zero object k itself. It contains as a full subcategory the connected graded-commutative algebras k-CGA0, which in turn contains the category of connected graded, commutative k-algebras (which is to say commutative graded rings with equipped with k- „ algebra homomorphism k Ñ A0 ãÑ A) via the “even regrading” i given on objects by

ipAq2n :“ An,

ipAqodd :“ 0.

Write A for the augmentation ideal of positive-degree elements in an object A P k-Alg0. One has a canonical k-module decomposition r A “ k ‘ A, 2 and moreover, by a degree argument, a map f : ArÑ B is trivial if and only if the set-theoretic image fA :“ f pAq of the augmentation ideal is 0.3 We analyze kernels, cokernels, images, and coimages in this pointed category. r r κ f In k-Alg0, the kernel of f : A Ñ B always exists. A composition K Ñ A Ñ B is trivial if and only if p f ˝ κqpKq “ 0, or in other words if we have a containment ´1 r κpKq Ď f p0q 4 ´1 of sets. This in turn happens just if κ factorsr through the subring k ` f p0q in A. It follows the kernel is the inclusion kerp f q: k ` f ´1p0q ãÝÑ A.

f c In k-Alg0, the cokernel of f : A Ñ B always exists. A composition A Ñ B Ñ C is trivial if and only if pc ˝ f qpAq “ 0, or in other words if we have a containment

´1 r fA Ď c p0q 2 It seems we can go further. All we need is basic ring theory and the existence of this splitting of k ÝÑ A, so r ε one could define a category k´Algε of augmented k-algebras k ãÑ A Ñ k and ring homomorphisms preserving the augmentations ε, and the arguments would seem to still go through without grading or commutativity assumptions on A. 3 This affected omission of parentheses serves only to avoid irritating nestings later for which I would have to resize the outer parentheses. 4 Here f ´1p0q is the ideal usually called the kernel of A. 7 of sets. Since c´1p0q is an ideal, this is equivalent to the whole ideal p fAq being contained in c´1p0q. This in turn happens just if c descends to a well-defined map out of B {{ A :“ B{p fAq. It follows the cokernel is the quotient map r r cokerp f q: B B {{ A.

In k-Alg0, the coimage of f : A Ñ B exists because cokernels do: it is the quotient map

´1 coimp f q: A A k ` f ´1p0q “ A{ f p0q.

That is to say, the target of the categorical coimageLL is isomorphic to the set-theoretic image, but ´1 only rarely to the category-theoretic one. The induced map fcoim : A{ f p0q Ñ B is an injection and thus a monomorphism because the ideal-theoretic kernel of f is definitionally f ´1p0q. In k-Alg0, the image of f : A Ñ B exists because kernels do: it is the inclusion

imp f q: k ` pcoker f q´1p0q “ k ` p fAq ãÝÑ B.

Note that this is quite distinct in general from the set-theoretic image.r Again, it is a sort of “normal closure” of the coimage object.

Now consider a sequence f g A ÝÑ B ÝÑ C in k-Alg0. The resulting diagram is

k ` p fA p q >  _ fim im f r

A B. >

f¯   . ker g k ` g´1p0q

To determine what exactness and coexactness are under the original definition, we need to establish what monic and epic maps in k-Alg0 are. If a map f : A Ñ B is a monomorphism, then its kernel is trivial, as can be seen by left-cancelling f from the trivial composition ˚ “ f ˝ pker f q: k ` f ´1p0q ÝÑ B, which means f ´1p0q “ 0 and f is an injection. On the other hand, if f is an injection, it is clearly also left-cancellable.5 Recalling that coimage objects inject into codomains, we see every map in k-Alg0 induces a monomorphism from its codomain. If a map f : A ÝÑ B in k-Alg0 is an epimorphism, its cokernel is trivial, as can be seen by right-cancelling f from the trivial composition pcoker f q ˝ f “ ˚: A Ñ B Ñ B {{ A. By definition, then, the ideal generated in B by fA is all of B. On the other hand, if f verifies this condition, it’s clearly right-cancellable. Indeed, if fA generates B and two homomorphisms from B agree on the subring f pAq, then they agreer on all ofrB. Now one can see that every map induces an epimorphism to its image: in the imager factorizationr A Ñ k ` p fAq Ñ B of f , the first map

5 Set This all follows purely categorically from the fact the forgetful functor to isr faithful and admits a left adjoint, namely the “tensor algebra on a basis” construction. Explicitly, if f : A Ñ B in k-Alg0 is injective and we are given g1, g2 : Z Ñ A such that f ˝ g1 “ f ˝ g2 then for all a P A we have f pg1paqq “ f pg2paqq, so that g1paq “ g2paq. Since this happens for all a, one has g1 “ g2. For the other direction, if f fails to be injective, one has f paq “ 0 for some nonzero homogeneous a. Construct the free associative unital algebra Tpxq “ krxs on one variable x with deg x “ deg a. (In the case of the subcategory k-CGA0, replace this with an exterior algebra if deg a is odd) and consider the unique map g : Tpxq Ñ A sending x ÞÑ a. Then f ˝ g “ ˚ is the trivial map although g is not, so f cannot be a monomorphism. 8

fim is an epimorphism, because the image fA of the augmentation ideal of A generates the augmentation ideal p fAq of the image k ` p fAq. If follows from Proposition 2.1 that the four concepts of exactness, coexactness, exactness2,r and coexactness2 agree in k-Alg0. r r Proposition 4.1. In the category k-Alg0 of connected graded k-algebras or the subcategory k-CGA0 of connected, graded-commutative k-algebras, the four concepts of exactness, coexactness, exactness2, and coexactness2 are identical.

Alternate, low-tech proof. We can also easily verify these claims manually. Indeed, exactness2 is the condition k ` p fAq “ Imp f q “ Kerpgq “ k ` g´1p0q, and coexactness that 2 r B{p fAq “ B {{ A “ Cokerp f q – Coimpgq “ B{g´1p0q, and these both mean p fAqr “ g´1p0q. For exactness and coexactness being the same as well, note our f : A Ñ B induces an epimorphism A Ñ Kerpgq “ k ` g´1p0q just if p fAq “ g´1p0q just if g : B Ñ C induces a monomorphismr B {{ A “ Cokerp f q Ñ C. r 5. The category of commutative Hopf algebras ˚ We consider the pointed category pk-Hopf0 , kq of connected commutative graded Hopf k-algebras over a field k, making auxiliary use of the category pk-Hopf0, kq of all connected graded Hopf k-algebras. ˚ Recall [MS68, Prop. 2.2], all monomorphisms are normal in k-Hopf0 , so by Mac Lane’s Propo- sition 2.3, every map induces an epimorphism to its image, and it follows from Proposition 2.1 ˚ again that exactness, exactness2 and coexactness2 are identical in k-Hopf0 . The difference be- tween exactness and coexactness in this category, then, is mediated solely by whether the map fcoim : Coim f ÝÑ C is monic. To determine when this might happen, we consider the relation with the algebra coimage CoimpUf q ÝÑ C, where U : k-Hopf0 ÝÑ k-Alg0 is the functor that forgets the comultiplication, which we omit in notation when it is clear what underlying object is at issue. It is known [Swe69, p. 134] that U admits a right adjoint, namely the “cofree coalgebra on a vector space” applied to the underlying vector space of an algebra, given the unique consistent algebra structure. This construction does not require a grading, but produces a graded Hopf algebra if given a graded algebra. In particular, U is a left adjoint, so it preserves colimits, and in particular cokernels and epimorphisms. Write Kerp f q ãÑ B for the Hopf kernel and KerpUf q ãÑ B for the algebra kernel. Applying U to the first inclusion, one obtains a factorization Kerp f q ãÑ KerpUf q ãÑ B, where the first map is a monomorphism since the composition is. It follows KH is the maximal Hopf subalgebra of B contained in the subalgebra K “ k ` f ´1p0q. Taking cokernels, one finds a factorization

B Coimp f q CoimpUf q of coimpUf q in k-Alg0, where the second map is an epi because the composition is. In fact, since both these maps are cokernels in k-Alg0 and such maps are quotient maps, this second epi is a quotient map too. Coexactness will be equivalent to the other conditions, by Proposition 2.1, just if the map fcoim is monomorphic. This map factors as

Coimp f q ÝÑ CoimpUf q C, References 9

where the second map pUf qcoim is injective by the ring -theoretic considerations of the preceding section. It follows that the composition is a monomorphism (injective) just if the first map, a quotient map, is really an isomorphism. Back at the level of kernels, since U preserves cokernels, this means Kerp f q “ KerpUf q “ k ` f ´1p0q is a Hopf algebra, or that f ´1p0q itself is a Hopf ideal. Thus we see that it is actually coexactness which is the unnecessarily restrictive condition in the setting of connected, commutative graded Hopf algebras.

˚ Proposition 5.1. In the category pk-Hopf0 , kq of connected commutative graded Hopf k-algebras, ex- f g actness, exactness2, and coexactness2 are equivalent. Consider a sequence ¨ Ñ ¨ Ñ ¨ with trivial com- position ˚ “ g ˝ f . For fixed g, coexactness will be equivalent to coexactness2 for all f if and only if k ` g´1p0q is a Hopf algebra.

References [Mac78] S. Mac Lane, Categories for the working mathematician, ser. Grad. Texts in Math. Springer, 1978, vol. 5. [MS68] J. C. Moore and L. Smith, “Hopf algebras and multiplicative fibrations, I,” Amer. J. Math., pp. 752–780, 1968. [Smi15] L. Smith, personal communication, Nov. 2015. [Swe69] M. E. Sweedler, Hopf algebras. New York: Benjamin, 1969.