Timeline for What can be said about $A$ and $B$ given the exact sequence $0 \to R^p \to A \to R^r \to R^q \to B \to 0$?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 7, 2016 at 18:12 | comment | added | Steven Landsburg | If the projective dimension of $B$ is at most $2$, then $A$ is projective. If the projective dimension of $B$ is $3$, then the projective dimension of $A$ is at most $1$. If the projective dimension of $B$ is greater than $3$, then the projective dimension of $A$ is exactly two less than the projective dimension of $B$. | |
| Jan 7, 2016 at 18:07 | answer | added | Vladimir Dotsenko | timeline score: 1 | |
| Jan 7, 2016 at 17:49 | comment | added | Chan | In the first instance any field, finite field, or PID. Information on more general rings would be useful also though. | |
| Jan 7, 2016 at 17:48 | comment | added | Qiaochu Yuan | It follows that $[A] = [B]$ in the quotient of the Grothendieck group $K_0(R)$ by the subgroup generated by $[R]$. | |
| Jan 7, 2016 at 17:46 | comment | added | Vladimir Dotsenko | So how general would be the class of rings you care about? | |
| Jan 7, 2016 at 17:43 | comment | added | Chan | Note I have removed the condition $r \geq q$ (it was a mistake!). | |
| Jan 7, 2016 at 17:41 | history | edited | Chan | CC BY-SA 3.0 |
deleted 10 characters in body; edited title
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| Jan 7, 2016 at 17:26 | comment | added | Chan | For example, I want to know what can be said about relative homology groups $H_1(X,A)$ and $H_0(X,A)$ when it is known that $H_1(A) = 0$, $H_1(X) \cong R^p$, and the connectivity of $A$ and $X$ is $r$ and $q$, respectively. This is just an analogy, but should suffice. Computed means compute $H_1(X,A)$ and $H_0(X,A)$ in the usual sense (or at least determine rank). | |
| Jan 7, 2016 at 17:15 | comment | added | Vladimir Dotsenko | Can you be a bit more specific about why you need this? When $R$ is a vector space, for instance, all the answers are essentially trivial. When $R$ is a PID (e.g. $\mathbb{Z}$), it is also quite easy to give very precise answer. But it all depends on the generality you care for. Also, it would help if you explain what "computed" means in question 1). | |
| Jan 7, 2016 at 16:57 | review | First posts | |||
| Jan 7, 2016 at 17:30 | |||||
| Jan 7, 2016 at 16:53 | history | asked | Chan | CC BY-SA 3.0 |