Timeline for Why does K-theory need schemes to be Noetherian?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2019 at 18:39 | vote | accept | Li Guanyu | ||
| Nov 19, 2019 at 9:20 | history | edited | Denis Nardin |
edited tags
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| Nov 19, 2019 at 6:37 | comment | added | Denis Nardin | @Qfwfq Regarding your second question, this answer by Denis-Charles Cisinski explains the situation well. | |
| Nov 19, 2019 at 6:36 | comment | added | Denis Nardin | @Qfwfq I've never seen $K^0$, but I'm not an algebraic geometer and I never use cohomological grading. I've always seen $K_i(X):=K_i(\mathrm{Perf}(X))$ and $G_i(X):=K_i(\mathrm{PCoh}(X))$, which is the notation in Thomason-Trobaugh. Sometimes people explicitly decorate Bass K-theory with a superscript B, but most of the time just mention they're using that in the text and add no additional decoration (this changes only the negative K-groups). But I am a homotopy theorist, the conventions in different areas might as well be different | |
| Nov 19, 2019 at 6:03 | history | became hot network question | |||
| Nov 19, 2019 at 1:48 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Gave title consistent capitalisation
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| Nov 18, 2019 at 23:32 | comment | added | Qfwfq | Yes, $K_0(Coh)$ is a module over the ring $K_0(Vect)$. Since we're here I'll ask: do these groups (together with the corresponding pieces with nonzero index) have anything to do with the homology and cohomology groups associated to the same homotopical object (such as ring spectrum...)? | |
| Nov 18, 2019 at 23:24 | comment | added | Li Guanyu | @Qfwfq For algebraic-geometors, B) is good. When $X$ is not regular, $K_i$ is not $K^i$. Your B) is actually my notations, where we found ways to distinguish $K_i(\mathrm{Coh}(X))$ and $K_i(\mathrm{Vect}(X))$. But, both can be either covariant and contravariant. I guess the reason we want $K_i(\mathrm{Vect}(X))$ to be cohomological was $K_0(\mathrm{Vect}(X))$ has a natrual action on $K_0(\mathrm{Coh}(X))$, which is an analogy of cohomology rings acting on homology groups. | |
| Nov 18, 2019 at 23:04 | comment | added | Qfwfq | Not wanting to open a little question about notation, I'll ask here. I've seen people using the following notations A) $K_0(R):=K_0(Vect(Spec(R)))$, $K^0(X):=K_0(Vect(X))$ (upper index cause it's contravariant in $X$, I imagine), and $K_0(X):=K_0(Coh(X))$ (lower index cause it's covariant in $X$, I imagine). And B) $K^0(X):=K_0(Vect(X))$, and $G_0:=K_0(Coh(X))$. Which of A), B) is prevalent? Or the notation of the OP? | |
| Nov 18, 2019 at 22:27 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
typo in the title
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| Nov 18, 2019 at 22:21 | answer | added | Denis Nardin | timeline score: 21 | |
| Nov 18, 2019 at 22:03 | comment | added | Denis Nardin | You don't need the scheme to be Noetherian. Usually quasi compact quasi separated is enough to get a well-behaved theory. But you need to use different definitions to work in full generality (e.g. your scheme might not have enough vector bundles for the definition you are using to be meaningful). Have you tried to have a look at Thomason-Trobaugh? | |
| Nov 18, 2019 at 21:58 | history | asked | Li Guanyu | CC BY-SA 4.0 |