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Operads and Gamma-Homology of Commutative Rings

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Operads and Gamma-Homology of Commutative Rings See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/2745568 Operads And Gamma-Homology Of Commutative Rings Article in Mathematical Proceedings of the Cambridge Philosophical Society · June 1998 DOI: 10.1017/S0305004102005534 · Source: CiteSeer CITATIONS READS 42 21 2 authors, including: Sarah Whitehouse The University of Sheffield 33 PUBLICATIONS 241 CITATIONS SEE PROFILE All content following this page was uploaded by Sarah Whitehouse on 30 July 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS Alan Robinson and Sarah Whitehouse May Introduction In this pap er we construct and investigate the natural homology theory for coherently homotopy commutative dg algebras usually known as E algebras We call the theory homology for historical reasons see for instance Since discrete commutative rings are E rings we obtain by sp ecialization a new homology theory for commutative rings This sp ecial case is far from trivial It has the following application in stable homotopy theory which was our original motivation and which will b e treated in a sequel to this pap er The obstructions to an E multiplicative structure on a sp ectrum lie under mild hyp otheses in the cohomology of the corresp onding dual Steenro d algebra just as the obstructions to an A structure lie in the Ho chschild cohomology of that algebra The homology of a discrete commutative algebra B can b e understo o d as a renement of Harrison homology which was originally dened as the homology of the quotient of the Ho chschild complex by the sub complex generated by nontrivial shue pro ducts It is b etter dened as the homology of a related complex which n one obtains by tensoring each term B with a certain integral representation V of n the symmetric group and passing to covariants Harrison theory works very n n well in characteristic zero but not otherwise A more satisfactory theory necessarily involves the higher homology of the symmetric groups not only the covariants H Our homology theory is constructed to do just that It is furthermore completely dierent except in characteristic zero from AndreQuillen homology which is related to a completely dierent class of problems Polynomial algebras are acyclic for AndreQuillen theory by its construction but they are not generally free E algebras and their homology is generally nonzero There are two further signicant generalizations First there is a cyclic variant of the homology of any E dg algebra This arises very naturally from our construction in x The cyclic theory like standard cohomology is connected with an obstructiontheoretic problem A full account will app ear elsewhere Second the domain of denition can b e widened from the ab elian situation of dg algebras to the case of sp ectra in stable homotopy theory so that one can dene the homology of an E ring sp ectrum This is analogous to extending Ho chschild homology to top ological Ho chschild homology which includes Mac Lane homology as a sp ecial The second author was supp orted by a TMR grant from the Europ ean Union held at the Lab oratoire dAnalyse Geometrie et Applications UMR au CNRS Universite ParisNord The rst author was supp orted by the EU Homotopy Theory Network Typ eset by A ST X M E ALAN ROBINSON AND SARAH WHITEHOUSE case The generalization to sp ectra is not dicult but we have p ostp oned the details to another sequel We mention that E homology was invented indep endently by the rst author and by Waldhausen during the s and outlined in various lectures including a plenary lecture by the rst author to the Adams Memorial meeting in Some details subsequently app eared in the second authors thesis and a preprint Meanwhile Kriz and later Basterra were developing by metho ds very dierent from ours an E cohomology theory for ring sp ectra which is extremely likely to b e equivalent to ours The pap er is organised as follows Section contains material on op erads In Section we intro duce complexes over op erads and dene the realization of such a complex A cyclic variant of the construction is also given Section covers the most imp ortant case of realization namely the cotangent complex of an E algebra This section also contains the denitions of homology and cyclic homology and a transitivity theorem The pro of of a key acyclicity lemma is deferred to App endix A In Section we further justify our constructions by showing that their A analogues lead to Ho chschild and cyclic homology of asso ciative algebras Section is devoted to the sp ecial case of homology of discrete commutative algebras It is shown that for pairs of Q algebras homology coincides with AndreQuillen homology and an example is given to show the theories are dierent in general The nal section describ es a pro duct in cohomology of a discrete commutative algebra Operads cyclic operads and cofibrancy We work in the category of chain complexes dg mo dules over a commutative ground ring K We might equally well of course have chosen simplicial mo dules Our principal denitions use Getzler and Kapranovs theory of cyclic op erads but we require Markls nonunital version which is describ ed in Op erads Let S denote the category of nite sets S and isomorphisms of sets S the sub category of nonempty sets and S the category of based nite sets and isomorphisms To avoid foundational diculties we assume without further mention where necessary that these have b een replaced by equivalent small sub categories Our constructions do not dep end up on the choice of sub category One can for instance take just one set f ng in S for each n and similarly f ng for each n in S so that b oth categories b ecome disjoint unions of symmetric groups An op erad C has ob jects chain complexes C indexed by S all nite sets S isomorphisms C C induced by isomorphisms S T S T of sets and comp osition maps C C C t S T S t T t for all nite sets S and T and all elements t T where S t T is the deleted t sum S t T n ftg these data must satisfy standard conditions of functoriality and asso ciativity of comp osition The induced isomorphisms give a left action of the symmetric group of automorphisms of S on C One thinks of C as a parameter S S S space of op erations in the sense of universal algebra with inputs lab elled by S and a single output the induced isomorphisms corresp ond to p ermutation of inputs and the comp osition to substitution of the output of C for the input lab elled t in t S OPERADS AND HOMOLOGY OF COMMUTATIVE RINGS C The op erad C is said to b e E if for each S S the complex C is contractible T S and free It is obviously sucient to check this for S f ng for all S n The standard example of an E op erad is D in which D is the nerve of the S category SS of isomorphisms of nite sets over S Comp osition in D is induced by the deleted sum functor in S Cyclic op erads A cyclic op erad can b e dened as an op erad with extra structure a action on C which makes comp osition symmetric by n f ng putting the output variable on the same fo oting as the n inputs see Then it is clearly desirable to change the notation and denote by C what was S t previously denoted C Confusion can arise so we stress that from now on we shall S use the cyclic convention and include the output in the lab elling set It seems b est to dene cyclic op erads directly A cyclic operad is a functor E from the category S of nonempty nite sets to the category of chain complexes together with comp osition op erations E E E st S T S t T st for all nite sets S T with at least two elements and all choices of s S t T Here S t T denotes the deleted sum S n fsg t T n ftg and is required to st st b e a natural transformation of functors from S S to chain complexes having the asso ciativity prop erty st t u t u st for s S t t T t t u U and the symmetry prop erty st ts where E E is induced by the isomorphism of sets and E S t T T t S S st ts E E E interchanges factors and intro duces the usual sign T T S Some further notation will b e needed The comp osition E E E st S T V is asso ciated with a partition of V into two subsets S n fsg and T n ftg Conversely let V P t Q b e any partition of V into two nonempty sets We can dene the asso ciated comp osition by writing P and Q for the disjoint unions P t fg and Q t fg and taking 1 2 E E E V P Q E cyclic op erads We call E an E cyclic operad if for all S S the complex E is contractible and free It suces to check this for S f ng S S for all n The op erad D dened as in is an E cyclic op erad the comp osition again b eing induced by the deleted sum functor Cobrant op erads We adopt the notation intro duced in for adding new p oints to a set S S and S are to denote S t fg S t fg and S t f g resp ectively For each partition V S t T of V we have a comp osition map 1 2 E In a cobrant op erad provided S and T each have more E E V T S than one element we want this map to b e the inclusion of a face of E so we V require it to b e an equivariant cobration and we require faces to intersect only in faces of faces This leads to the following ALAN ROBINSON AND SARAH WHITEHOUSE Denition Let E denote the co equalizer of the maps V 1 21 M M 2 1 2 12 1 E E E E E
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