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Some Reflections on Exactness and Coexactness

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Some Reflections on Exactness and Coexactness 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 ` n¥1 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 f¯ g¯ > " " Kerpgq c / 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 c “ 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.
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