# Tagged Questions

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### Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory? However, to my taste, the answers there consider the subject from a more modern point of view. When I open a book ...
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### Riemann-Roch without denominators?

The Riemann-Roch Theorem, the Grothendieck-Riemann-Roch Theorem , the Grothendieck-Hirzebruch-Riemann-Roch Theorem , all of them are well explained at Wikipedia .I would like to understand the meaning ...
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### Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
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### K-theory and completion

I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community. I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...
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### an elementary question on K-theory

Sorry for asking such an elementary question. 1) What is Quillen's $K_1$ of a (nice) scheme $X$? If $X=Spec(k)$, I guess one gets $k^\times$, is that correct? What about the case of a curve $C$ ...
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### Faltings-Riemann-Roch Theorem

I found the famous Faltings book Lectures on arithmetic Riemann-Roch theorem". In the book, very analytic techniques such as Garding inequality or heat kernel are explained. I have no idea where ...
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### Codimension zero embeddings and diffeomorphism groups

Let $V$ be a smooth manifold obtained by attaching the open collar'' $[0,1)\times \partial N$ to a compact smooth manifold $N$ along the boundary. Let $\mathrm{Emb}(N, V)$ be the space of embeddings ...
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### Comparison of products in Quillen and Waldhausen K-theory

I'm relatively new to algebraic K-theory and stumbled upon the following question. I would be very glad If someone could provide a reference to an answer or a short argument. We are given an exact ...
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### Can any suspension spectrum be realized as Waldhausen K-theory?

If we consider the category of finite, pointed sets and declare cofibrations to be inclusions and weak equivalences to be bijections, we get a Waldhausen category whose $K$-theory spectrum is the ...
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### Do GE rings have matrix completion?

If $R$ is a ring, $E_n(R)$ is the subgroup of the group $GL_n(R)$ generated by matrices obtained from the multiplicative identity matrix by replacing an off-diagonal entry by some $r \in R$. The ...
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### Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first: Given an integral ...
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### Additivity theorem for algebraic L-theory?

There is an additivity theorem for algebraic K-theory. My question is is there an additivity theorem for algebraic L-theory?
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### Algebraic K-theory of odd-dimensional spheres

Let $A(X)$ denote the Waldhausen's algebraic K-theory of a space $X$, and let $n$ be odd. Are the rational homotopy groups of $A(S^n)$ known? Is the group $\pi_{2k}(A(S^n))$ finite for all ...
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### Path components of a monoidal category form a monoid?

In Grayson's 'Higher Algebraic K-theory II', leading up to the categorical generalisation of the plus construction, he considers $\pi_0(S) = \pi_0(BS)$, where $S$ is a (small, symmetric) monoidal ...
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### Motivic cohomology and cohomology of Milnor K-theory sheaf

Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}_p$). ...