There is a well-known Quillen's localization sequence for (algebraic) K-theory: $\dots\to K_p^Y(X)\to K_p(X)\to K_p(X-Y)\to \dots$, where $Y\to X$ is a closed embedding of schemes. …

Take $A$ to be a simplical, commutative ring. Then $\pi_0(A)$ (defined via Dold-Kan or by the geometric realization) is a discrete commutative ring, and all the other $\pi_i$'s are …

What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?

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 modul …

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 t …

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 se …

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 con …

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. Howeve …

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 …

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 $\hat{G}^\prime(X)$. Th …

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 …

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 polyloga …

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 …

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. …

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 g …