# Questions tagged [algebraic-k-theory]

Algebraic K-theory is a tool from homological algebra that defines a sequence of functors from rings to abelian groups. It has many applications in algebraic geometry. See also (topological-k-theory).

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### K theory of integers [on hold]

How can i calculate $K_2$ of $\mathbb{Z}$ i.e. $K_2 (\mathbb{Z})$
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### What does it mean to say $K$-theory satisfy Mayer–Vietoris sequence?

I saw the statement "a $K$-theory is expected to satisfy Mayer–Vietoris property and Bott Periodicity" somewhere and I am trying to understand what it means. What does it mean to say a $K$-theory ...
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### Kernel of the Adams operation on the algebraic K-theory

Adams operation $\psi^n$ acts on the algebraic $K$-theory groups of a commutative ring $R$. Its known that eigenvalues of $\psi^n$ on $\mathbb{Q}\otimes K_i(R)$ can only be non-negative powers of $n$. ...
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### Is there an 'easy' way to calculate $K_0(\mathbb{Z}[C_p])$?

For $C_2$ the cyclic group of order 2, I want to calculate $\tilde{K}_{0}(\mathbb{Z}[C_2])$. Now so far, I know by a theorem of Rim that $\tilde{K}_{0}(\mathbb{Z}[C_2])$ is isomorphic to the ideal ...
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### Showing an isomorphism of $R[G]$-modules using the regular representation

Let $R$ be a commutative ring, and $G$ a finite group. Now suppose $M,N$ are finitely generated $R[G]$-modules such that $M\cong_R N$ (let's say they have $R$-rank$=n$). To show $M\cong_{R[G]}N$, is ...
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### Example of Waldhausen category with quotient maps not closed under composition

Let $\mathscr{C}$ be a Waldhausen category, i.e. a category pointed by $0$, with cofibrations and weak equivalences. Recall a map $B \to B/A$ in $\mathscr{C}$ is called a quotient map if it is the ...
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