- And how Topological cyclic homology is related and more computable.
Algebraic K theory
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Prismatic cohomology
The category \(\mathsf{Prism}\) doesn’t have a final object, so has interesting cohomology. Relates to Algebraic K theory of \({\mathbb{Z}}_p\)?
Algebraic K theory is hard, using topological Hochschild homology somehow makes computations easier.
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2021-06-06
Algebraic K theory : finitely generated projective \(R{\hbox{-}}\)modules mod equivalence with \(\oplus\), then take group completion to get \(K_0(R)\).
- Topics
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Milnor Conjecture
References: Algebraic K theory | Motives
- Giant Math Term Index
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Algebraic K Theory for schemes
Most of the important results about Algebraic K theory arise from the study of the spectrum \({\mathsf{K}}(X)\), rather than that of the disembodied abelian groups \({\mathsf{K}}^n(X)\). For example, if the scheme \(X\) is covered by two open sets \(U\) and \(V\) , one wants a Mayer-Vietoris sequence \begin{align*} \cdots \rightarrow K ^ { n } ( X ) \rightarrow K ^ { n } ( U ) \oplus K ^ { n } ( V ) \rightarrow K ^ { n } ( U \cap V ) \rightarrow K ^ { n + 1 } ( X ) \rightarrow \cdots \end{align*}