# 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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### Bott Periodictity for Real and Complex Algebras

In Rordam's Introduction to K-Theory for $C^*$ Algebras they prove \begin{equation} K_{n+2}(A) \cong K_{n}(A) \end{equation} using isomorphism in terms of the suspensions of $A$. I have also heard ...
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### Intuition behind 4-fold periodicity of $L$-theory

The quadratic and the symmetric L-groups are 4-fold periodic. What is the simple argument to obtain the intuition behind the 4-fold periodicity of $L$-theory? (For example, why not have the Bott ...
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### A reference question on the Dedekind zeta function and other L-functions.

On page 501 in the following ICM1983 conference proceeding https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1983.1/ICM1983.1.ocr.pdf you will find a conjecture on the L-function $L(X,s)$ of an ...
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### Note or textbook about algebraic k-theory

I'm reading the Weibel's K-book. It's difficult for me. I have read some $K_0$ with the hlep of Bass' book. But Bass' book is so old. So, are there any more detailed books or notes to recommend. I ...
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### Does isomorphism at the level of residue fields imply isomorphism of etale $K$-theory?

I am wondering whether something like the following argument is correct or not? If a morphism of regular schemes $f:X\rightarrow Y$ induces an isomorphism on residue fields and also induces a ...
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### Bott inverted algebraic $K$-theory vs algebraic $K$-theory.

Bott inverted mod $l$ algebraic $K$-theory coincides with mod $l$ etale $K$-theory under mild conditions. The Bott element is an element in $K_2$ that is inverted with respect to the graded product ...
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### Restriction of $G$-equivariant sheaves to fixed locus

Let $G$ be a group acting on a Noetherian scheme $X$ over $\mathbb{C}$ and $\mathcal{E}$ be a $G$-equivariant coherent sheaf on $X$ where the support of $\mathcal{E}$ contains the fixed locus $X^G$. I ...
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### When existence of transfer map of algebraic $K$-theory implies rational injection.

Given a map of smooth projective varieties $f:X\rightarrow Y$ over fields, there is a projection formula in algebraic $K$-theory given by $f_*(\alpha.f^*(\beta))=\beta.f_*(\alpha)$. I was wondering ...
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### $K$-theory of formal power series.

I was wondering whether there is a calculation of algebraic $K$-groups of the formal power series $\mathbb{F}_p[[x]]$?
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### $K$-theory of union of smooth curves.

Is there a method that one can calculate algebraic $K$ theory of a number of smooth curves with some singularity points corresponding to the intersection points of two different curves, given we know ...
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### $\mathcal{K}$-homology sheaf on affine space.

The $n$-th $\mathcal{K}$-homology sheaf on $X$ is defined as the sheafification of the presheaf $U\mapsto K_n(U)$. Here $K_n(U)$ is the $n$-th algebraic $K$-group. I was just wondering whether it is ...
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### Simply connected and rationally connected varieties in char $p$.

Consider a smooth projective variety in char $p$ with trivial etale fundamental group. You can assume it is also rationally connected. I believe there are a lot of examples for these types of ...
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### $G_1$ of a nodal projective curve.

Let $C$ be the projective nodal curve over the field $k$ i.e. the projective line with two points identified. I'm having trouble calculating the $G$-theory of $C$, especially $G_1$. Writing the ...
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### K Theory: Book Recommendations

Good people! So I've been hoping to get into K Theory for a while now, and the book that I have been trying to use (and failing) has been Charles Weibel's book by that very title. The book itself isn'...
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### Elementary computations for hermitian and non-hermitian algebraic K-theory

As the title states, I am looking for some examples of non trivial albeit elementary calculations of algebraic hermitian and non-hermitian $K$-theory groups. Concretely, I would be interested in ...
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### A problem on $K_0$

Let $i: R \hookrightarrow R[[t]]$. If $R$ is left regular, then $K_0i$ is an isomorphism ? It's well known that this holds for $R[t]$.
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### Relating $G_0(R/fR)$ and $\oplus_{P\in V(f)} G_0(R/P)$?

For a Commutative Noetherian ring $R$, let $G_0(R)$ denote the Grothendieck group of the abelian category of finitely generated $R$-modules i.e. it is the abelian group generated by the isomorphism ...
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### When the classes of two finitely generated modules are equal in the Grothendieck group

Let $R$ be a Commutative Noetherian ring. Let $G_0(R)$ denote the Grothendieck group of the abelian category of finitely generated $R$-modules i.e. it is the abelian group generated by the isomorphism ...
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### Is $St(n,R) \rightarrow St(n+1,R)$ injective?

For every (not necessarily commutative) ring R and every $n\geq 3$, one can define the n-th (unstable) Steinberg group St(n,R) as in https://ncatlab.org/nlab/show/Steinberg+group. Is the canonical map ...
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### On transfer and base change map on Grothendieck groups induced from injective ring homomorphism

For a commutative Noetherian ring $R$, let $G_0(R)$ denote the Grothendieck group (an abelian group) of the abelian category of finitely generated $R$-modules (Note that I'm Not talking about $K_0(R)$ ...
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### The generator of $K_0(C(\partial(]0,1[^2)))$ and $K_1(C(\partial(]0,1[^2)))$

Let $C=[0,1]^2 \subseteq \mathbb{C}$ and $\partial C$ the boundary of $C$. I'm looking for the $K_0(C(\partial C))$ and $K_1(C(\partial C))$ and its generator.
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### Proving $K_1(\mathcal{T})=0$ (is trivial)

Let $\mathcal{T}$ the toeplitz algebra and we define the short exact sequence? where $C(\mathbb{T})=\{z\in \mathbb{C}/ |z|\leq 1\}$:  0 \rightarrow \mathcal{K} \rightarrow \mathcal{T }\rightarrow C(...
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### Grothendieck group of local affine- surfaces with rational singularities

Let $(R, \mathfrak m)$ be an excellent, normal, local domain of dimension $2$ containing an algebraically closed field $k=R/\mathfrak m$. Let $\pi: Y \to X=\operatorname {Spec}(R)$ be a resolution ...
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### When two Algebraic vector bundles on a Noetherian quasi-affine scheme are equal in $K_0$ of the scheme

Let $X$ be a (connected) Noetherian scheme and $K_0(X)$ denote the Grothendieck group of the category of Algebraic vector bundles (coherent sheaves that are locally free and of constant rank ( as $X$ ...
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### Is there a categorical way to see that $K_1(A)\twoheadrightarrow K_1(A/I)$ for a nilpotent ideal $I\subset A$?

Let $A$ ba a (not necessarily commutative) unital ring. The $K_1$ group of $A$ is defined as $K_1(A)=\pi_1((\mathrm{Proj}(A)^\simeq)^\mathrm{gp})$. Here $\mathrm{Proj}^\simeq$ is the core of the ...
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### Geometric Resolutions of C* -Algebras

I am reading "K-theory FOR OPERATOR ALGEBRAS" Bruce Blackadar, Proposition 23.5.1. Let B be a separable $C^{*}$-algebra. Then there is a separable commutative $C^{∗}$-algebra F, whose spectrum ...
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I have a question which seems to be extremely trivial but for some reason I don't get it and am very confused about it. In the paper by Quillen, "Higher Algebraic K-theory I" page 94 top. Quillen ...
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### Grothendieck group of equivalent categories

Let $\Gamma : \mathcal A \to \mathcal B$ be an equivalence of categories where $\mathcal A$ is an exact category, $\mathcal B$ is an additive full subcategory of the category of $R$-modules for some ...
### Projective modules of rank $n$ over a ring with $\operatorname{Pic}(R) = 0$
Let $R$ a ring with trivial Picard group, so every rank $1$ projective module is free. What does that tell me about the structure of projective modules of rank $n$? For starters, if $P$ is any ...