cohen macaulay modules co torsion pairs and virtually gorenstein algebras abstract we use torsion pairs in stable categories and cotorsion pairs in modules categories to study in general infinitely generated cohen macaulay modules and a generalization of modules of finite projective or injective dimension over an artin algebra we concentrate our investigation to the study of virtually gorenstein algebras which provide a common generalization of gorenstein algebras and algebras of finite representation or cohen macaulay type this class of algebras on the one hand has rich homological structure and satisfies several representation torsion theoretic finiteness conditions and on the other hand it is closed under various operations for instance derived equivalences and stable equivalences of morita type in addition virtual gorensteinness provides a useful tool for the study of the gorenstein symmetry conjecture and modified versions of the telescope conjecture for module or stable categories

on the telescopic homotopy theory of spaces in telescopic homotopy theory a space or spectrum x is approximated by a tower of localizations lnx n 0 taking account of vn periodic homotopy groups for progressively higher n for each n 1 we construct a telescopic kuhn functor n carrying a space to a spectrum with the same vn periodic homotopy groups and we construct a new functor n left adjoint to n using these functors we show that the nth stable monocular homotopy category comprising the nth fibers of stable telescopic towers embeds as a retract of the nth unstable monocular homotopy category in two ways one giving infinite loop spaces and the other giving infinite ln suspension spaces we deduce that ravenel s stable telescope conjectures are equivalent to unstable telescope conjectures in particular we show that the failure of ravenel s nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial johnson wilson e n homology but nontrivial vn periodic homotopy groups showing a fundamental difference between the unstable chromatic and telescopic theories as a stable chromatic application we show that each spectrum is k n equivalent to a suspension spectrum as an unstable chromatic application we determine the e n localizations and k n localizations of infinite loop spaces in terms of e n localizations of spectra under suitable conditions we also determine the e n localizations and k n localizations of arbitrary postnikov h spaces

gauge fields in a dsd and connections of its symmetry algebra the generalized connections of the de sitter algebra so d 1 and anti de sitter algebra so d 1 2 which are differential forms of arbitrary degree with values in any irreducible spin tensor representation of the anti de sitter algebra are studied it is shown that an arbitrary spin gauge field in anti de sitter space massless or partially massless can be described by a single connection a one to one correspondence between the connections of the anti de sitter algebra and the gauge fields is established the gauge symmetry is manifest and auxiliary fields are automatically included in the formalism

morava modules and brown comenetz duality fix a prime number p and let x be a finite spectrum whose n 1 st morava k theory is trivial but whose n th morava k theory is nontrivial n 0 we prove following a method outlined to us by m j hopkins that if 2 p n 2 n 2 the morava module of the brown comenetz dual of the e n localization of x is isomorphic to a suspension of the pontryagin dual of the morava module of x to complete this proof we found it necessary to develop a more canonical construction of certain modified adams spectral sequences this construction should be of independent interest

comodules and landweber exact homology theories abstract we show that if e is a commutative mu algebra spectrum such that e is landweber exact over mu then the category of e e comodules is equivalent to a localization of the category of mu mu comodules this localization depends only on the heights of e at the integer primes p it follows for example that the category of e n e n comodules is equivalent to the category of v n 1 bp v n 1 bp comodules these equivalences give simple proofs and generalizations of the miller ravenel and morava change of rings theorems we also deduce structural results about the category of e e comodules we prove that every e e comodule has a primitive we give a classification of invariant prime ideals in e and we give a version of the landweber filtration theorem

buildings elliptic curves and the k 2 local sphere we investigate a dense subgroup of the second morava stabilizer group given by a certain group of quasi isogenies of a supersingular elliptic curve in characteristic p the group acts on the bruhat tits building for gl2 through its action on the adic tate module this action has finite stabilizers giving a small resolution for the homotopy fixed point spectrum eh 2 hgal by spectra of topological modular forms here e2 is a version of morava e theory and gal gal

tate cohomology and periodic localization of polynomial functors in this paper we show that goodwillie calculus as applied to functors from stable homotopy to itself interacts in striking ways with chromatic aspects of the stable category localized at a fixed prime p let t n be the telescope of a v n self map of a finite s module of type n the periodicity theorem of hopkins and smith implies that the bousfield localization functor associated to t n is independent of choices goodwillie s general theory says that to any homotopy functor f from s modules to s modules there is an associated tower under f p d f such that f p d f is the universal arrow to a d excisive functor our first main theorem says that p d f pd 1f always admits a homotopy section after localization with respect to t n and so also after localization with respect to morava k theory k n thus after periodic localization polynomial functors split as the product of their homogeneous factors this theorem follows from our second main theorem which is equivalent to the following for any finite group g the tate spectrum mathcal t g t n is weakly contractible this strengthens and extends previous theorems of greenlees sadofsky hovey sadofsky and mahowald shick the periodicity theorem is used in an essential way in our proof the connection between the two theorems is via a reformulation of a result of mccarthy on dual calculus

harmonic localization of algebraic k theory spectra the lichtenbaum quillen conjectures hold for the harmonic localization of the k theory spectrum of a nice scheme various consequences of this fact are explored for example the harmonic localization of the k theory of the integers at a regular prime is explicitly identified

generalized tensor idempotents and the telescope conjecture we transpose rickard s construction of idempotent representations to the general context of tensor triangulated categories we connect our construction to the telescope conjecture and prove that the latter is of local nature as an illustration we extend the affine telescope theorem of neeman to arbitrary noetherian schemes we also develop supports for non compact objects

on the structure of the hopf algebroid e n e n this paper is devoted to giving a description of e n e n ravenel and hopkins have studied a certain completion f n of e n and have showed that f n f n contains a subalgebra isomorphic to c s n a the algebra of continuous functions on the n th morava stabilizer group taking values in the ring of integers in an unramified degree n extension of the p adic numbers they also show that this completion is split with respect to this subalgebra i e f n f n c s n a a f n we display an injection of e n e n into c s n s n a which extends to the hopkins ravenel completion and identify the image of their subalgebra as the subalgebra of functions which are invariant under translation by elements of s n using this we give a formula for the coaction of c s n a on e n and another proof of the hopkins ravenel splitting theorem

phantom maps and chromatic phantom maps in the first part we determine conditions on spectra x and y under which either every map from x to y is phantom or no nonzero maps are we also address the question of whether such all or nothing behavior is preserved when x is replaced with v a x for v finite in the second part we introduce chromatic phantom maps a map is n phantom if it is null when restricted to finite spectra of type at least n we define divisibility and finite type conditions which are suitable for studying phantom maps we show that the duality functor wn defined by mahowald and rezk is the analog of brown comenetz duality for chromatic phantom maps and give conditions under which the natural map y w x y is an isomorphism 1 introduction this paper takes place in the stable homotopy category a map x y of spectra is said to be phantom if for every finite spectrum v and every map v x the composite v x y is zero we write ph x y for the group of all phantom maps from x to y the first half of the paper section 2 deals with some results of a general nature the theme is to determine conditions on spectra x and y under which either every map from x to y is phantom or no nonzero maps are we also address the question of whether such all or nothing behavior is preserved when x is replaced with v a x for v finite our main tool is the observation that for y of finite type a map x y is phantom if and only if it is a divisible element of the group x y we also make use of the connection between phantom maps and brown comenetz duality in the second half of the paper section 3 we work in the 7 local stable homotopy category and introduce generalizations of phantom maps called chro matic phantom maps a map x y is n phantom if for every finite spectrum v of type at least n and every map v x the composite v x y is zero the results of this section which require more background of the reader gener alize some of the results of the first half of the paper we define analogs of the divisibility and finite type conditions which are suitable for studying chromatic phantom maps the significance of chromatic phantom maps is demonstrated by the fact that the kernel in homotopy of the important map lnsfi lx n sp is pre cisely the n phantom homotopy classes we show that the duality functor wn defined by mahowald and rezk 11 is the analog of brown comenetz duality our theorem 3 13 gives conditions under which the natural map y w2 xy is

fourier mukai transform in the quantized setting we prove that a coherent dq kernel induces an equivalence between the derived categories of dq modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves

k 1 local topological modular forms we construct the witten orientation of the topological modular forms spectrum tmf in the k 1 local setting by attaching e cells to the bordism theory mo we also identify the ko homology of tmf with the ring of divided congruences