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### Seeking examples or proof: injectivity of Cartan homomorphism for commutative rings?

This question is motivated by some issue raised by David Speyer in this question.
Let $R$ be a ring. Let $K_0(R)$ and $G_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules ...

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### How to determine kernels of maps between algebraic K_1-groups

Suppose we have a ring homomorphism $\varphi: R \to S$, say an injection (e.g. coming from an injection $H \to G$ of finite groups and $R=\mathbb{Z}_p[H],S=\mathbb{Z}_p[G])$, what can be said about ...

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### Why is Riemann-Roch for stacks so hard?

First some indication that it really is a difficult problem: Both Vistoli and Gillet in their classics on intersection theory on stacks remark that their should be a Riemann-Roch theorem for proper ...

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### Is a field uniquely determined by its multiplicative group/how much knows K_1 about fields?

As the title says I would like to know if $K_1(k)=k^*$ uniquely determines a field $k$.
For finite fields this is clearly the case, but I suspect it is not ture in general. However I guess cooking up ...

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### Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...

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### What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?

More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...

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### Is there a category-theoretic definition of the arithmetic Grothendieck group

Let $X$ be a regular scheme which is flat over $\mathbf{Z}$. The arithmetic Grothendieck group $\hat{K}(X)$ is defined to be the quotient of $\hat{G}(X)$ by $\hat{G}^\prime(X)$. This is actually quite ...

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### Symplectic Steinberg group

I have several questions about Steinberg group and K2 for symplectic group:
Can I extend the definition of Steinberg symbols to symplectic case? Will they generate the center of Steinberg group?
...

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546 views

### Does every projective A/I-module come from A?

Let $A$ be a Noetherian commutative ring and $I$ an ideal in $A$. It is pretty much trivial to see that every free $A/I$-Module is obtained from a free $A$-module by tensoring over $A$ with $A/I$: ...

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### On finite endomorphisms of $\mathbf{P}^r$

This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot ...

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### Can we decompose Diff(MxN)?

If you have two manifolds $M^m$ and $N^n$, how does one / can one decompose the diffeomorphisms $\text{Diff}(M\times N)$ in terms of $\text{Diff}(M)$ and $\text{Diff}(N)$? Is there anything we can say ...

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**1**answer

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### $K_0$ of a non-separated scheme

This question is on "computing" the Grothendieck group of the projective $n$-space with $m$ origins ($m\geq 1$). For any (noetherian) scheme $X$, let $K_0(X)$ be the Grothendieck group of coherent ...

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### Link: Serre's intersection formula <-> Bloch-Quillen Thm / When only intersecting divisors, is there 'shorter' approach of proof known?

In very short:
When proving the equivalence of intersection theory constructed through (Milnor) K-sheaves and their product vs. defining the product via Serre's local multiplicity formula + moving, I ...

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731 views

### Is every Adams ring morphism a lambda-ring morphism?

A lambda-ring $R$ is called "special" if it satisfies the $\lambda^i\left(xy\right)=...$ and $\lambda^i\left(\lambda^j\left(x\right)\right)=...$ relations, or, equivalently, if the map ...

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### Quillen's Morphism Inverting Functors

In "Higher algebraic K-theory I" Quillen defines a morphism inverting functor to be a functor from a category C to the category Sets which maps "arrows" in C to isomorphisms in Sets.
Proposition 1:
...

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### Isolated hypersurface singularities, Chow groups and D-branes

Say a ring $R$ is an isolated hypersurface singularity if $R = k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}/(W)$, where $k$ is a field and $W \in k[x_1, \ldots, x_n]$ is such that the ideal $(\partial_1 ...

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### Is Higher K-functor the derived functor of K0?

It might be a stupid question. I wonder whether the derived functor of functor K0 is Quillen Higher K-functor?
If not, is there any relationship between derived functor of K0(or satellites of ...

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### Is there a simple relationship between K-theory and Galois theory?

I can (barely) understand the definition of the higher algebraic K-groups a la the plus construction right now (I have some past familiarity with K-theory for C*-algebras and can recall the rudiments ...

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### Does Milnor K-Theory arise from Waldhausen K-Theory

Quillens higher K-groups of rings can be realized as πnK(C) - the Waldhausen K-Theory of a suitable Waldhausen category C. Is this also true for Milnor K-Theory of Rings? Is there a functor F from ...

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### (Co-) Homology associated to Waldhausen K-Theory

Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...

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### Maps between K-groups induced by rings homomorphism

Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free ...

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### Differences between reflexives and projectives modules

Let R be a normal noetherian domain.
What is the difference between a finitely generated reflexive module and a finitely generated projective module?
Can anybody recommend any references or make ...

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### The localisation long exact sequence in K-theory over an arbitrary base

If I work over a field k,write D for the formal disk k[[t]] and Dx for the formal punctured disk k((t)), then there is an associated long exact sequence in algebraic K-theory
... Kn+1(Dx) --> Kn(k) ...

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### Why is Milnor K-theory not ad hoc?

When Milnor introduced in "Algebraic K-Theory and Quadratic Forms" the Milnor K-groups he said that his definition is motivated by Matsumoto's presentation of algebraic $K_2(k)$ for a field $k$ but is ...

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### Stable Homology of arithmetic groups.

Suppose that F/Q is a number field.
Using automorphic forms, Borel computed the (R-) stable cohomology of SL_n(O_F), and as a ...

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### K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

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### Motivation/interpretation for Quillen's Q-construction?

This question has been on my mind for a while. As I understand it, the Q-construction was the first definition for higher algebraic K-theory. Some details can be found here.
...

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### algebraic K-theory and tensor products

Algebraic K-theory defines a functor K taking commutative rings to E_\infty ring spectra. I'm interested in which pushouts (tensor/smash products) K preserves. For example, if R is a regular ...

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### Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...