the additivity of traces in triangulated categories abstract we explain a fundamental additivity theorem for euler characteristics and generalized trace maps in triangulated categories the proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation the refinement consists of several new axioms relating products and distinguished triangles the axioms hold in the examples and shed light on generalized homology and cohomology theories

a note on k theory and triangulated categories we provide an example of two closed model categories having equivalent homotopy categories but different waldhausen k theories we also show that there cannot exist a functor from small triangulated categories to spaces which recovers quillen s k theory for exact categories and which satisfies localization

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

classifying thick subcategories of the stable category of cohen macaulay modules abstract various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics in this paper as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p group due to benson carlson and rickard we consider classifying thick subcategories of the stable category of cohen macaulay modules over a gorenstein local ring the main result of this paper yields a complete classification of the thick subcategories of the stable category of cohen macaulay modules over a local hypersurface in terms of specialization closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus

intersection products for tensor triangular chow groups abstract we show that under favorable circumstances one can construct an intersection product on the chow groups of a tensor triangulated category t as defined in 5 which generalizes the usual intersection product on a non singular algebraic variety our construction depends on the choice of an algebraic model for t a tensor frobenius pair which has to satisfy a k theoretic regularity condition analogous to the gersten conjecture from algebraic geometry in this situation we are able to prove an analogue of the bloch formula and use it to define an intersection product similar to grayson s construction from 14 we then recover the usual intersection product on a non singular algebraic variety assuming a k theoretic compatibility condition

the spectrum of prime ideals in tensor triangulated categories we define the spectrum of a tensor triangulated category k as the set of so called prime ideals endowed with a suitable topology in this very generality the spectrum is the universal space in which one can define supports for objects of k this construction is functorial with respect to all tensor triangulated functors several elementary properties of schemes hold for such spaces e g the existence of generic points or some quasi compactness locally trivial morphisms are proved to be nilpotent we establish in complete generality a classification of thick ideal subcategories in terms of arbitrary unions of closed subsets with quasi compact complements thomason s theorem for schemes mutatis mutandis we also equip this spectrum with a sheaf of rings turning it into a locally ringed space we compute examples and show that our spectrum unifies the schemes of algebraic geometry and the support varieties of modular representation theory

on support varieties and tensor products for finite dimensional algebras abstract it has been asked whether there is a version of the tensor product property for support varieties over finite dimensional algebras defined in terms of hochschild cohomology we show that in general no such version can exist in particular we show that for certain quantum complete intersections there are modules and bimodules for which the variety of the tensor product is not even contained in the variety of the one sided module

detecting cohomology for lie superalgebras abstract in this paper we use invariant theory to develop the notion of cohomological detection for type i classical lie superalgebras in particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras these results are used later to affirmatively answer questions which were originally posed in boe et al 2010 5 and bagci et al 2008 2 about realizing support varieties for lie superalgebras via rank varieties constructed for the smaller detecting subalgebras

supports and filtrations in algebraic geometry and modular representation theory working with tensor triangulated categories we prove two theorems relative to supports and then discuss their incarnations in algebraic geometry and in modular representation theory first we show that an indecomposable object has connected support then we consider the filtrations by dimension or codimension of the support and prove that the associated subquotients decompose as sums of local terms

the exponential homomorphism for the second syntomic cohomology abstract let ok be a complete discrete valuation ring with perfect residue field f of characteristic p 3 and m a free filtered dieudonne module the second relative syntomic cohomology h2 ok f s m 2 with coefficients in m is an object related to arithmetic geometry such as the albanese kernel in this article we study h2 ok f s m 2 by constructing the exponential homomorphism we determine the structure of h2 ok f s m 2 under the assumption that m is of hodge witt type and that the absolute ramification index of ok is prime to p

katoʼs residue homomorphisms and reciprocity laws on arithmetic surfaces abstract we explicitly study kato s residue homomorphisms in milnor k theory which are closely related to contou carrere symbols as applications we establish several reciprocity laws for certain locally defined maps on k groups that are associated to arithmetic surfaces

the spectrum of the equivariant stable homotopy category of a finite group we study the spectrum of prime ideals in the tensor triangulated category of compact equivariant spectra over a finite group we completely describe this spectrum as a set for all finite groups we also make significant progress in determining its topology and obtain a complete answer for groups of square free order for general finite groups we describe the topology up to an unresolved indeterminacy which we reduce to the case of p groups we then translate the remaining unresolved question into a new chromatic blue shift phenomenon for tate cohomology finally we draw conclusions on the classification of thick tensor ideals

support varieties an axiomatic approach we provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category where the tensor product is not necessarily symmetric this is illustrated by examples taken in particular from the representation theory of finite dimensional algebras

reconstructing projective schemes from serre subcategories abstract given a positively graded commutative coherent ring a j 0 a j finitely generated as an a 0 algebra a bijection between the tensor serre subcategories of qgra and the set of all subsets y proj a of the form y i y i with quasi compact open complement proj a y i for all i is established to construct this correspondence properties of the ziegler and zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way also there is constructed an isomorphism of ringed spaces proj a o proj a spec qgr a o qgr a where spec qgr a o qgr a is a ringed space associated to the lattice l serre qgr a of tensor serre subcategories of qgra

negative k theory of derived categories we define negative k groups for exact categories and for derived categories in the framework of frobenius pairs generalizing definitions of bass karoubi carter pedersen weibel and thomason we prove localization and vanishing theorems for these groups devissage for noetherian abelian categories additivity and resolution hold we show that the first negative k group of an abelian category vanishes and that in general negative k groups of a noetherian abelian category vanish our methods yield an explicit non connective delooping of the k theory of exact categories and chain complexes generalizing constructions of wagoner and pedersen weibel extending a theorem of auslander and sherman we discuss the k theory homotopy fiber of e e and its implications for negative k groups in the appendix we replace waldhausen s cylinder functor by a slightly weaker form of non functorial factorization which is still sufficient to prove his approximation and fibration theorems

on ramification in transcendental extensions of local fields abstract let l k be an extension of complete discrete valuation fields and assume that the residue field of k is perfect and of positive characteristic the residue field of l is not assumed to be perfect in this paper we prove a formula for the swan conductor of the image of a character h 1 k q z in h 1 l q z for sufficiently ramified further we define generalizations l k ab and l k as of the classical function and prove a formula for l k ab t for sufficiently large t r

p adic deformation of algebraic cycle classes we study the p adic deformation properties of algebraic cycle classes modulo rational equivalence we show that the crystalline chern character of a vector bundle on the closed fibre lies in a certain part of the hodge filtration if and only if rationally the class of the vector bundle lifts to a formal pro class in k theory on the p adic scheme

the bousfield lattice of a triangulated category and stratification for a tensor triangulated category which is well generated in the sense of neeman it is shown that the collection of bousfield classes forms a set this set has a natural structure of a complete lattice which is then studied using the notions of stratification and support

robust estimation of sinusoidal signals with colored noise using decentralized processing a technique is developed for the estimation of the number of signals and their central frequencies using decentralized processing when it is known a priori that the observations consist of a finite number of sinusoidal signals corrupted by an additive colored random noise process with unknown correlations such a noise sequence may be caused by jamming from a hostile agent the authors decentralized processing scheme is one in which each sensor estimates the frequencies and their covariance matrix and sends the results to the fusion center at the fusion center since the estimates from the sensors have a mixture density that is possibly not gaussian a robust technique is utilized to combine the estimates even when the numbers of frequencies transmitted by the various sensors are identical determining corresponding frequencies from each sensor is not a straightforward task also outliers caused by line splitting or by spurious frequencies are hard to detect these problems can be resolved by two methods the so called refitting method and the ranking method algorithms for both are presented in detail

on the derived category of a graded commutative noetherian ring abstract for any graded commutative noetherian ring where the grading group is abelian and where commutativity is allowed to hold in a quite general sense we establish an inclusion preserving bijection between on the one hand the twist closed localizing subcategories of the derived category and on the other hand subsets of the homogeneous spectrum of prime ideals of the ring we provide an application to weighted projective schemes

monoidal infinity category of complexes from tannakian viewpoint in this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infty categories of quasi coherent complexes it can be viewed as a derived analogue of tannaka duality as a consequence we deduce that an algebraic stack satisfying a certain condition can be recovered from the symmetric monoidal stable infty category of quasi coherent complexes with tensor operation

the balmer spectrum of the equivariant homotopy category of a finite abelian group for a finite abelian group a we determine the balmer spectrum of mathrm sp a omega the compact objects in genuine a spectra this generalizes the case a mathbb z p mathbb z due to balmer and sanders invent math 208 1 283 326 2017 by establishing a corrected version of their hbox log p conjecture for abelian groups we also work out the consequences for the chromatic type of fixed points and establish a generalization of kuhn s blue shift theorem for tate constructions kuhn in invent math 157 2 345 370 2004

approximately optimum detection of deterministic signals in gaussian and compound poisson noise an approximate but explicit likelihood ratio is derived for detecting deterministic signals in gaussian and compound poisson noise the approximation in the derivation is based on the assumption that the localized noise elements rarely overlap each other the derived log likelihood ratio consists of two distinct parts one is the conventional correlation detector for detecting deterministic signals in gaussian noise the other is a nonlinear processor which compensates for the degradation of the correlation detector caused by the localized noise and is activated only by the presence of the localized noise as such it involves covariance operators of both the gaussian and the localized noise and is obtained by using the simultaneous diagonalization and orthogonalization of quadratic forms in function space involving eigenfunctions of certain composite operators

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 graded dual numbers arcs and non crossing partitions of the integers abstract we give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity using this model we describe the lattice of thick subcategories of the bounded derived category and of the perfect complexes in terms of non crossing partitions we also make some comments on the symmetries of these lattices exceptional collections and the analogous problem for the unbounded derived category

picard groups in triangular geometry and applications to modular representation theory for a tensor triangulated p category k with spectrum spc k we construct an injective group homomorphism h 1 spc k m 1 p pic k 1 p where pic k is the group of invertible objects of k in modular representation theory we prove that this homomorphism induces a rational isomorphism between the picard group of the projective support variety and the group of endotrivial representations

separability and triangulated categories abstract we prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one this applies in particular to etale algebras more generally we do this for exact separable monads

integration of complementary detection localization systems an example the integration or fusion of two or more detection localization systems often results in performance improvement especially if they are complementary to each other on the other hand the cost of integration in additional signal processing and overhead may be considerable as a figure of merit for this tradeoff the authors propose the db gain to cost ratio and illustrate its evaluation by using a simple example the example consists of a pair of detection localization systems which have complementary features one has higher resolution that the other but is more susceptible to random disturbances the author models the first as a system which detects a signal in multiplicative and additive random disturbances and additive white noise thus one main result is the derivation of the maximum likelihood detector under these disturbances in the second system the author assumes the random disturbances to be negligible hence the maximum likelihood detector is the matched filter input data to both systems are two dimensional arrays of real numbers corresponding to the same physical area under observation where the second data array has a coarser grid than the first thus yielding poorer resolution the detection probabilities in terms of which the db gains are computed are evaluated by monte carlo simulation

presheaves of triangulated categories and reconstruction of schemes to any triangulated category with tensor product k otimes we associate a topological space rm spc k otimes by means of thick subcategories of k a la hopkins neeman thomason moreover to each open subset u of this space spc k otimes we associate a triangulated category cal k u producing what could be thought of as a presheaf of triangulated categories applying this to the derived category k otimes rm d rm perf x otimes l of perfect complexes on a noetherian scheme x the topological space rm spc k otimes turns out to be the underlying topological space of x moreover for each open u subset x the category cal k u is naturally equivalent to rm d rm perf u as an application we give a method to reconstruct any reduced noetherian scheme x from its derived category of perfect complexes rm d rm perf x considering the latter as a tensor triangulated category with otimes l