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        Questions tagged [kt.k-theory-and-homology]

        Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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        3
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        1answer
        191 views

        Description of higher chow groups

        In the literature there are several descriptions of motivic cohomology groups, some of them rather explicit, but I don't always understand why they are equivalent. The simplest example I have in mind ...
        5
        votes
        1answer
        140 views

        Equivalence between categories of coherent sheaf of codimension p

        Let $X$ be a noetherian and separated scheme and $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $M^{P}(X) = \lbrace \mathcal{F} \in M(X) \hspace{2mm} : Codim(sup(\mathcal{F}), X) \...
        7
        votes
        1answer
        201 views

        Reference request: mod 2 cohomology of periodic KO theory

        The mod 2 cohomology of the connective ko spectrum is known to be the module $\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$, where $\mathcal{A}$ denotes the Steenrod algebra, and $...
        4
        votes
        1answer
        94 views

        Kuenneth short exact sequence for K-homology

        Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one? Using general spectra stuff, one gets a ...
        4
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        0answers
        175 views

        Different algebra-structures on $\operatorname{THH}(\mathbb F_p)$?

        By definition, we have a ring map $\mathbb F_p\to\operatorname{THH}(\mathbb F_p)$. Post-compose with the canonical map $\mathbb Z_p\to\mathbb F_p$, we get a ring map $\mathbb Z_p\to\operatorname{THH}(\...
        7
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        0answers
        247 views

        Reference request: complex K-theory as a commutative ring spectrum

        Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum? For real $K$-theory I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...
        6
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        1answer
        115 views

        Coarse index of Dirac operator on $\mathbb{R}$

        Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index $$\text{Ind}(...
        9
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        0answers
        154 views

        Geometric motivation behind the Fredholm module definition

        If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
        3
        votes
        1answer
        174 views

        A question about the group $[HZ,KU]$

        I don't know if the following question is obvious, but can't figure it out. I want to ask if it is known what $[HZ,KU]$ is? Here $KU$ is the complex $K$-theory.
        4
        votes
        1answer
        120 views

        Mapping cone and derived tensor product

        This question is in some sense a continuation to this question: Derived Nakayama for complete modules For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...
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        0answers
        455 views

        Algebraic K-theory of schemes and cohomology

        Are there examples of: two smooth projective schemes over a field having homotopy equivalent algebraic K-theory spectra and having different rational Voevodsky motives; two smooth projective schemes ...
        10
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        1answer
        267 views

        Where does the $\hat A$ class get its name?

        In K-theory we have the Todd class and the $\hat A$ class. The Todd class is named after the Cambridge geometer John Arthur Todd. Where does the name $\hat A$ come from? Does the A stand for Atiyah?...
        3
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        0answers
        41 views

        Minimum rank of inverse complex vector bundles

        When considering vector bundles (real or complex) over a compact manifold, i know about the existence of inverse bundles. That is, if $\xi$ is a vector bundle over $M$, then there is a bundle $\nu$ ...
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        2answers
        272 views

        Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

        I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg ...
        9
        votes
        0answers
        191 views

        The term “absolute geometry”

        My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...

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