Algebraic K-theory of the infinite place

In this note, we show that the algebraic K-theory of generalized archimedean valuation rings occurring in Durov's compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces. We also show that the "residue field at infinity" is badly behaved from a K-theoretic point of view.Comment: Final version. To appear in Journal of Homotopy and Related Structure

f-cohomology and motives over number rings

This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives $M_\eta$ over F--one via ramification conditions on $\ell$-adic realizations, another one via the K-theory of proper regular models--both agree with motivic cohomology of $\eta_{!*} M_\eta[1]$. Here $\eta_{!*}$ is constructed by a limiting process in terms of intermediate extension functors $j_{!*}$ defined in analogy to perverse sheaves.Comment: numbering has been updated to agree with the published versio

Dynamic Sonographic Tissue Perfusion Measurement

The amount of blood passing through a tissue is a fundamental parameter since metabolism and its adaptation in disease is reflected by changes of perfusion. To evaluate the functional state of a tissue or an organ it is therefore helpful to know its perfusion intensity. Inflammation for example is highlighted by an increase of perfusion whereas chronic diseases are often accompanied by atrophy of tissue and reduction of organ perfusion. We developed and present here an overview of a simple but sensitive method to quantify tissue perfusion by means of simple color Doppler sonography. This dynamic tissue perfusion measurement (DTPM) uses color hue data to calculate the mean perfusion velocity and color pixel area to calculate the perfused part of a certain region of interest. All data are referred to full heart cycles thus reflecting all changes during a heart beat. With this approach a substantial step forward is made compared to traditional resistance index (RI) or contrast enhanced ultrasound (CEUS) sonographic techniques of blood flow evaluation. This paper describes DTPM basics and shows applications in a variety of fields

Symmetric operads in abstract symmetric spectra

This paper sets up the foundations for derived algebraic geometry, Goerss--Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la To\"en--Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith's stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and E-infinity ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of E-infinity rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent.Comment: 34 pages. Comments and questions are very welcome. v2: Identical to the journal version except for formatting and styl