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Path-Connected Components of Affine Schemes and Algebraic K-Theory

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 Added by Maysam Maysami Sadr
 Publication date 2019
  fields
and research's language is English




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We introduce a functor $mathfrak{M}:mathbf{Alg}timesmathbf{Alg}^mathrm{op}rightarrowmathrm{pro}text{-}mathbf{Alg}$ constructed from representations of $mathrm{Hom}_mathbf{Alg}(A,Botimes ?)$. As applications, the following items are introduced and studied: (i) Analogue of the functor $pi_0$ for algebras and affine schemes. (ii) Cotype of Weibels concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $mathbf{Alg}$ that are enriched over idempotent semigroups. (v) Classifying homotopy pro-algebras for Corti~{n}as-Thoms KK-groups and Weibels homotopy K-groups.



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We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $operatorname{Pic}(X)$ and pointed monoid of regular functions $Gamma(X, mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.
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