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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13 views

Do limits of Waldhausen categories comute with ordinary limits?

Let $(A,\mathcal{W}, \mathcal{C})$ be a Waldhausen category with $A$ an additive category. On one hand, we can define the ordinary limits $lim_A$ of the underlying category $A$. On other hand, we can ...
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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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22 views

Excision sequence for $K_0$ in algebraic $K$-theory

I am looking at Exercise I.2.3 in Weibel's $K$-book. Here is the statement: If $I$ is an ideal of a ring $R$, we form the augmented ring $R\oplus I$ and let $K_0(R,I)$ denote the kernel of $K_0(R\...
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22 views

Recovering components in the projective bundle formula

Projective bundle formula is telling us that for a projective bundle $Y$ over $X$, where the fibers are $\mathbb{P}^n$, we have $G(Y)\cong G(X)^{n+1}$. Here $G$ is the $K$-theory of coherent sheaves. ...
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What are the K groups of $X_n$, the wedge sum of $n$ circles at a single point?

If $X_n$ is the bouquet of n circles, then what is $K_0(X_n)$ and $K_1(X_n)$? I am super confused on how to approach this problem, since I've only seen $K_0$, $K_1$ computed for algebras, but I don't ...
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Help with a problem in K-theory/C- algebras.

For this problem you may assume the fact that $K_1(C_0(D)) = 0$. Let $n > 1$, let $\omega = e^{\frac{2πi}{n}}$, and let $E_n$ be the space obtained from $D$ by identifying $z$ and $\omega z$ for ...
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43 views

What are the K groups of $X_n$?

If $X_n$ is the bouquet of n circles, then what is $K_0(X_n)$ and $K_1(X_n)$? I am super confused on how to approach this problem, since I've only seen $K_0$, $K_1$ computed for algebras, but I don't ...
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17 definitions of algebraic K-theory of a ring. Which should I take?

There are multiple definitions of algebraic K-theory, but I have trouble differentiating between them. Could someone help me out? Let $R$ be a commutative ring. I would like to define $K_n(R)$, and ...
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83 views

Compatibility of tensor product and external tensor product in algebraic K-theory

Let $X$ be a smooth, compact, quasi-projective, complex algebraic variety and let $K(X)=K_0(X)=K_0(\mathrm{Coh}(X))$ be the Grothendieck group of coherent sheaves on $X$. There are several notions of ...
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Properties of injective modules

I am reading A course in Homological Algebra by Hilton and Stammbach. In the first chapter they showed that a $\Lambda$-module is projective iff it is a direct summand of a free module. They then ...
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Show that $K_2(F)$ is a direct summand of $K_2F(t)$

I have a question regarding Example 6.1.2 (page 252) from the book "The K-Book" by Charles Weibel. Here is the statement: Example 6.1.2: Let $F(t)$ be a rational function field in one variable $t$ ...
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Progress on the Bass Finiteness Conjecture?

I am interested in proving the Bass Finiteness Conjecture, and making it my main project... I was wondering what recent developments there have been in the proof of the Bass Finiteness Conjecture... ...
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78 views

$a\otimes(-a) = a \otimes (1-a) + a^{-1}\otimes(1-a^{-1})$???

J. Browkin writes in his article "K-Theory, Cyclotomic Equations, and Clausen's Function" (which appears on Chapter 11 of Lewin's book "Structural properties of polylogarithms") that for any field $F$ ...
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41 views

Identification of $M_n(\tilde{SA})$,where $\tilde{SA} $ is the unitalization of suspension $C^*$-algebra

In the section 11.1 of Rordam's book, there is a remark : If $A$ is a unital $C^*$-algebra, denote $SA$ by the suspension of $A$, $\tilde{SA} $ is the unitalization of $SA$. We can identify $M_n(\...
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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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33 views

Functor between derivators to commute with homotopy colimits?

I am reading the following paper by Non Connective K theory via Universal Invariants. On page 35. One defines $$Hom_{!}(D,D')$$ To be the "homotopy colimit" preservsing functors of two derivators $D$...
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17 views

Admissible layers Q-construction

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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103 views

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 ...
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Additivity in algebraic K-theory — what does it truly mean?

--- Question --- I have seen several definitions of 'additivity' in algebraic K-theory. In all cases, I can more or less see that there is something additive going on. But I have difficulty seeing ...
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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 ...
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The development of $k$-theory for operator algebras

What is the motivation of introducing the $k$-theory for operator algebras? Are there any interesting open problems in this area?
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How to get isomorphism $K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X)$

I'm reading Atiyah's K-theory, on page 44, the Lemma 2.1.1 claims that $$ K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X) $$ I'm confused about how to get this isomorphism. Please ...
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A doubt in the proof of Heller's localization theorem in algebraic K-theory

Currently I am reading through Dr. Weibel's The K Book. There I encountered this exact sequence in the localization theorem of Heller: $$K_{0}(\mathcal{B}) \longrightarrow K_{0}(\mathcal{A}) \...
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91 views

On the ring structure of $K_0$ of the punctured spectrum of a regular local ring

Let $(R, \mathcal m, k)$ be a regular local ring of dimension at least $3$. Let $K_0(X)$ be the Grothendieck group of algebraic vector bundles over the punctured spectrum $X =Spec R \setminus \{\...
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99 views

injective envelope in the category of left exact functors

Let $\mathcal{A}$ be an Abelian category. $\mathcal{L}$ is the category of absolutely pure objects of $\mathcal{A}$ and $\mathcal{L}(\mathcal{A})$ is the category of the exact left functors of $\...
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the isomorphism of $D(B(H))$,where ,$H$ is an infinite dimensional inseparable Hilbert space.

We define $D( B(H))=\cup_n P(M_n( B(H)))/\sim$, where $ P(M_n(B(H))$ is the set of projections in $M_n(B(H))$,$\sim$ is the equivalence relation as follows:suppose $p$ is a projection in $P(M_n(B(H)))$...
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43 views

Equivalence of categories of coherent sheaves

Let $Z$ be a closed subscheme of noetherian scheme $X$ and $U = X-Z$. Let $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $\mathcal{B}$ = $ \lbrace \mathcal{F} \in M(X) : \mathcal{F}...
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a unitary does not in general lift to a unitary

Consider the restriction map $\pi:C(\mathbb{D})\rightarrow C(\mathbb{T})$, where $\Bbb D$ is the closed unit disk and $\Bbb T$ is the unit circle. Suppose $v\in C(\Bbb T)$ is a unitary such that $v(z)=...
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48 views

On equivalence of Bass conjecture of finite generatation of $K_0$ and $G_0$

The following two statements are known as Bass conjecture for $K_0$ (resp. $G_0$) : (1) If $R$ is a finitely generated $\mathbb Z$-algebra ( similarly $X$ is a $\mathbb Z $-scheme of finite type ) , ...
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The group of units of $K_0$ and the Picard group of a finitely generated regular $\mathbb Z$-algebra

Let $A$ be a finitely generated $\mathbb Z$-algebra which is also a regular ring (https://en.wikipedia.org/wiki/Regular_ring). Consider $K_0(A)$ which has a commutative ring structure (the ...
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26 views

the semigroup $D(\Bbb C)$

We define $D(\Bbb C)=\cup_n P(M_n(\Bbb C))/\sim$,where $P(M_n(\Bbb C)) $ is the set of projections in $M_n(\Bbb C)$.$\sim$ is the equivalence relation as follows:suppose $p$ is a projection in $P(M_n(\...
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compute the abelian semigroup $V(A)$

$proj(A)$ is the set of algebraic equivalence classes of idempotents in $A$,we set $V(A)=proj(M_{\infty}(A))$. If $A$ is a $II_{1}$ factor,then $V(A)\cong \Bbb R_{+}\cup\{0\}.$If $A$ is a countably ...
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The Grothendieck group of *quasi*-coherent sheaves on a projective smooth variety $X$.

Let $X$ be a smooth projective $k$-variety (separated, finite type, geometrically integral $k$-scheme). Denote by $\textbf{Qcoh}\ X$ the category of quasi-coherent sheaves on $X$. I have heard it ...
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65 views

unitisation of suspension of a $C^*$ algebra

Let $A$ be a $C^*$ algebra,the suspension of $A$ is defined by $SA=\{f\in C([0,1],A)|f(0)=f(1)=0\}$,then the unitisation of $SA$ is $\tilde{SA}=\{f\in C_0([0,1],A)|f(0)=f(1)\in \Bbb C\}$. How to ...
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Does the Grothendieck group of finitely generated modules form a commutative ring where the multiplication structure is induced from tensor product?

For a commutative ring $R$, let $\mathrm{mod-}R$ denote the category of finitely generated $R$-modules. Let $\mathcal C$ be an abelian, full, isomorphism closed (i.e. $M\cong N$ in $\mathrm{mod-}R$ ...
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Grothendieck group of Abelian categories , with “coefficients” in a ring

For an Abelian category $\mathcal C$ the Grothendieck group $G_0(\mathcal C)$ is defined as $$\dfrac{\bigoplus_{X\in \operatorname{Ob}(\mathcal C)}\mathbb Z[X]}{\langle [A]-[B]+[C] \mid 0\to A \to B \...
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On the two definitions of Milnor's K-theory of a field

I observed two slightly different following definitions of the $n$th group of Milnor's K-theory of a field $k$. The first one is $$K^M_n(k)=(k^\times)^{\otimes n}/G_1=\langle\{a_1\otimes\dots\otimes ...
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Quotient of category of f.g. modules by subcategory

Let $\mathcal A$ be the category of finitely generated modules over $A[t]$ and $\mathcal B$ be its subcategory of modules which is annihilated by some power of $t$. Then I want to show that quotient ...
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135 views

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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46 views

Computing $K$-theory elements in a $C^*$ algebra $A$

Let $A$ be a unital $C^*$ algebra. Let $p,q$ be projections in $M_n(A)$. Then $[p]-[q]$ defines an element in $K_0(A)$. Now consider the matrices, the projections, $$ \left[ \begin{pmatrix} 1-p &...
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98 views

$K$ theory of $C^*$ algebra is different to algebraic $K$ Theory?

Is the $K_0$ group for a $C^*$ algebras $A$ same as that for the $K_0$ group of ring $A$ from algebraic $K$ theory? We assume $A$ is unital (I am not sure if this matters), i.e. what is an example ...
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Certain exact functor on the Grothendieck group of a module category

Let $A$ be a be a finite dimensional, associative, and unital $\mathbb{C}$-algebra. Let $\mathcal{A}$ be the category of finitely generated $A$-modules. Since $A$ is an Artinian ring, there are only ...
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K Theory of $C^*$ algebras I, Higson's notes

Let $B$ be a $C^*$ algebra, and $A(B)$ denote the algebra of bounded continuous functions from $[1,\infty)$ to $B$. Let $I(B)$ be the ideal of functions which vanish at infinity. Let $Q(B)$ be the ...
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On rings whose $K_0$ has nice properties

Let $R$ be a commutative, reduced ring. It can be seen that $K_0(R) \cong \mathbb Z$ as groups if and only if every finitely generated projective module is stably free. My question is, are there ...
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57 views

What are the inverses in the Grothendieck group of a module category?

Let $R$ be a unital commutative ring and $\mathcal{C}$ be the category of finitely generated $R$-modules. The Grothendieck group $K_{0}(\mathcal{C})$ is the free abelian group generated by the ...
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35 views

$K_1(\mathbb{Z})$ is covered by diagonal matrices with only $\pm 1$ entries

Let me give some definitions first: We define the inclusion $GL(n,\mathbb{Z})\to GL(n+1,\mathbb{Z})$ as $A\mapsto \begin{pmatrix}A&0\\0&1\end{pmatrix}$ We call $E(n,\mathbb{Z})$ the subgroup ...
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55 views

The transfer map from $K_2L$ to $K_2F$ for a galois extension $L/F$

In J. Tate's "Relations between $K_2$ and Galois Cohomology" Lemma 3.4 J. Tate talks about the transfer map in k-theory, and references the Milnor's "Introduction to Algebraic K-Theory", but in the ...
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248 views

Torsion Grothendieck group

What is an example of an abelian/exact/triangulated category that has Grothendieck group isomorphic to $\mathbb{Z}/k\mathbb{Z}$ for some $k$? The category of finitely generated abelian groups has ...