Timeline for $K_0$ of a non-separated scheme

4 events

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Mar 8, 2010 at 21:43	comment	added	Clark Barwick		Yes, I'm using $n\geq 2$, but only in (B) at the line you quote. When $n=1$, everything is the same, except the naive K-theory happens to agree with the "true" K-theory. You are, of course, correct that $K_0$ is older notation. Nowadays $K_0$ usually refers to the thing I describe in (A), and $G_0$ refers to the object you were interested in. (Of course you can use any notation you like, as long as you're clear about what you mean, just as you were here! The only risk is that it's sometimes tricky to keep track of which theorems apply to which objects...)
Mar 7, 2010 at 18:34	vote	accept	Ariyan Javanpeykar
Mar 7, 2010 at 8:05	comment	added	Ariyan Javanpeykar		Great! Just a couple of questions. You said, "the inclusion $\mathbf{A}^n \rightarrow X$ induces an equivalence between...". Are you supposing $n\geq 2$ here? What happens when $n=1$? I'm not good with vector bundles so details are much appreciated here. My question was originally just for $K_0(X)$. (I'm guessing this is old notation only used in the 60's.) Thanks for the enlightment on $K^0(X)$, i.e., the Grothendieck group of vector bundles on $X$. This clearly shows that the Cartan morphism need not be an isomorphism when $X$ is not separated.
Mar 6, 2010 at 20:56	history	answered	Clark Barwick	CC BY-SA 2.5