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Date Created: 09/08/15
Ruth Vanderpool STATEMENT Department of Mathematics University of Oregon My research has focused on understanding whether the generalized notion of derived categories suggested in 2 can be applied to categories like the category of p complete abelian groups Given an abelian category such as left modules over a unital ring R a chain complex of left R modules M is a sequence of left R modules Ml connected by morphisms di Ml H MFl called differentials with ll1 0 di O The objects of the derived category consist of chain complexes M where M is projective for all i The de ning characteristic of the derived category is that morphisms between objects are equivalences if the map induced by homology is an isomorphism Some of the more basic examples occur when R is a eld or the integers If R is a eld7 then the objects of the derived category are isomorphic to chain complexes with di 0 for all i and each Mi is a vector space over R In the case that R Z also known as abelian groups7 the derived category consists of coproducts of chain complexes of the form OHZgZHO and OHZHO The category Ab ofp complete abelian groups is a subcategory of Z modules An abelian group A is in Ab if it is isomorphic to the inverse limit of ApiA where the inverse limit is taken with respect to the natural maps Api1A a ApiA Objects in Ab are called p complete One example of a p complete group is the p adic integers denoted Z The construction for the derived category of left modules sketched above does not apply here as even though all p complete abelian groups are Z modules7 not all Z modules are in Ab The motivation for this work comes from searching for an analogue ofthe derived category for the category known as Morava modules The Morava module of a spectrum X is the input to the Adams Novikov spectral sequence that computes 7134me It is possible that similar techniques used in studying the derived category of modules over the Steenrod Algebra used in 57 97 87 and 7 could be used to study LKWX if we had a derived category of Morava modules Like Ab7 the objects in the category of Morava modules are complete with respect to the ideal of a complete local ring The construction used to build a derived category for left Z modules suggests a starting point to nd the derived category for Ab The process begins by creating a category with objects that are chain complexes Morphisms between chain complexes are sequences of abelian group morphisms that commute with the appropriate differentials Two such morphisms fhgar X a Y are called homotopic if there exists a collection of maps 3139 Xi H Yi1 with fi 7 gi d l o 3139 sFl o if Let ICAb denote the category whose objects are chain complexes and morphisms are chain homotopy classes of maps The extra equivalences in ICAb create a triangulated structure with a suspension functor E ICAb a ICAb that shifts the degree of a given chain complex up by one The triangulated structure is given by triangles which are chain homotopy equivalent to X L Y a Conef a 2X 1 2 for some 1 where Conef XFl 63 Y with differentials dm y ide idyy 7 W The general properties any homology functor h should satisfy are 1 that triangles are sent to long exact sequences and 2 that h respect coproducts A reasonable guess at such a homology functor in the category lCAb is HomKMbg Yi 7 where Y is the chain complex with Z in the ith degree However this fails to satisfy the second condition If we require the functor h to generalize the usual homology H7 on chain complexes of abelian groups we have the following negative result Theorem 1 There exists no homology functor h lCAb H Ab satisfying hX HX in the case that X is a nitely generated free Z module There are functors that satisfy condition 1 and extend H7 but any such h ICAb a Ab cannot also satisfy condition The structure of coproducts in Ab allows for a nontrivial map from a product of Z s with indexing set I to a coproduct of Z s with the same indexing set A result from 3 shows certain products of free Z modules in Ab are isomorphic to coproducts of free modules with respect to an indexing set of strictly larger cardinality The map from a product to a coproduct thus becomes a map between coproducts We can use this to show any functor satisfying conditions 1 and 2 that also extends H7 will lead to a contradiction Recall that the derived category for R modules is a subcategory of chain complexes where each degree is a projective module By considering an analogous subcategory of lCAb where each degree is a coproduct of Z s denoted by lCAb there could be hope of nding a homology functor that can still extend Unfortunately this coproduct structure is what led to the problem that Theorem 1 acknowledges thus there does not exist a homology functor from lCAb that extends One might hope the problems above would be circumvented if instead of considering a functor h ICAb a Ab we consider one from lCAb to Ab Although Ab is a subcategory of Ab the structure of the coproducts from Ab does distinguish this case from that of the h in Theorem 1 However Theorem 2 There exists no homology functor h lCAb H Ab with HX when X is a nitely generated free Z module The category Ab is not quite an abelian category In particular we show there are monomorphisms that are not kernels and this leads to the result above The question about the existence of any nontrivial functor h lCAb H Ab satisfying conditions 1 and 2 remains open Theorem 1 has consequences for stable homotopy categories SHC s de ned in The derived categories discussed thus far are speci c examples of SHC s The de nition given in 2 is axiomatic and slightly more generalized requiring similar structures such as the triangulated structure Rather than using homological functors we consider functors of the form HomTG 7 where T is a triangulated category and G comes from a distinguished set g C T referred to as generators Conditions limit which generators can be chosen but if an SHC D is algebraic the set of generators must be small in the sense that the natural EB HomTG W a HomTG H W I I is a isomorphism Given a generating set that is small HomTG 7 satis es conditions 1 and Theo rem 1 implies the following Corollary 3 Let be an algebraic SHC containing a chain complex of the form 0 gt Z1 a 0 that is a subcategory of lCAb If h D gt Ab is a homology functor then it does not extend Hi 7 on nite complexes This result raises some natural questions Question 4 Do there exist small objects in lCAb that would lead to nontriuial homology functorf2 If so can we use these to construct an algebraic 3H0 Some of the obvious candidates do not work Both chain complexes of the form k OHZPLZPHO and 0gtZpgt0 for all k can be shown to not be small A complete answer has yet to be found for the rst question but current work suggests a negative result Much of the work done in 2 concerns algebraic SHC s but we could relax the require ments on the generating sets and ask Question 5 Is there a class of generators in lCAb which would lead to a nontriuial 3H0 If so what other nice properties might this SHC be lackingf2 REFERENCES 1 M J Hopkins M Mahowald H Sadofsky Constructions of elements in Picard groups Amer Math Soc 15889126 1994 M Hovey J H Palmieri and N P Stickland Axiomatic Stable Homotopy Theory Mem Amer Math Soc 6102128 1997 Irving Kaplansky In nite Abelian Groups University of Michigan Press Michigan 1954 J May Derived Categories from a topological point of view httpwwwmathuchicag0edumayMISCDerivedCatspdf 2006 H R Margolis Spectra and the Steenrod algebra modules over the Steenrod algebra and the stable homotopy category Elsevier Science Pub Co New York 1983 A Neeman Triangulated Categories Ann of Math Stud vol 148 Princeton University Press Princeton NJ 2001 J H Palmieri Quillen strati cation for the Steenrod algebra Ann of Math 2 149 1999 no 2 4217449 J H Palmeiri Nilpotence for modules over the mod 2 Steenrod algebra l HDulce Math J 82 1996 no 1 1957208 209726 J H Palmieri H Sadofsky Selfmaps of spectra a theorem of J Smith and Margolis7 killing construction Math Z 215 1994 no 3 4777490 E EE EEEE E
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