In Chapter 1 of John McCleary's book A User's Guide to Spectral Sequences the condition assumed for the filtrations is that they are bounded below. In pg. 4 it is said that $H^*$ (if good enough) can be recovered from the $E_0^*$ by taking the direct sum but if the filtration is not bounded above this is not true:

$$ H^*=F^0H^*\supset F^1H^*\supset\cdots\supset F^nH^*\supset F^{n+1}H^*=F^{n+2}H^*=\cdots\supset0$$

and you will never recover the dimensions in $F^{n+k}H^*$ taking the sum of the $E_0^p$.

There is also the same problem in Example 1.K. You can see the page in this question Example 1.K in A User's Guide to Spectral Sequences

Is it part of the informality of the chapter and one should ask for the filtrations to be bounded above and below to recover the full space? Then, in general, a spectral sequence of vectorial spaces (hence more complex structures) does not recover $H^*$ and the limit is not unique unless something is known about the filtrtion that gives $E_0^{*,*}(H^*,F^*)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.