In exercise, Prof. Weibel asks to show that $\text{Tot}^{\oplus}(D)$ is not acyclic if we follow his own errata sheet for his book An Introduction to Homological Algebra 1995 edition ($D$ is the unbounded double complex $D_{p,q}=\mathbb{Z}/4\mathbb{Z}$ with $p,q\in \mathbb{Z}$ with horizontal and vertical differentials equal to multiplication by $2$).

I fail to see why. I only can prove it is acyclic (by looking at a cycle which either is a boundary either should have infinite odd components ...). Where am I wrong ?

  • 1
    $\begingroup$ How do you prove it is acyclic? The element $\overline{2}$ in $D_{0,0}\subset\text{Tot}^{\oplus}(D)_0$ is a cycle but not a boundary. $\endgroup$ – Hanno Jan 3 '16 at 21:24
  • 1
    $\begingroup$ Maybe you are interested in the fact that there is also a mixture of the sum- and the product-totalization by taking semi-infinite products. See Example II.C.3.2 in my thesis - but please don't find mistakes! ;) $\endgroup$ – Hanno Jan 3 '16 at 21:28
  • $\begingroup$ @Hanno this time you are right and I am wrong. I missed this one ! Thank you very much for your time. Can you make your comment an answer then I can accept it? $\endgroup$ – brunoh Jan 3 '16 at 21:33
  • $\begingroup$ @Hanno thank you very much for your quite interesting suggestion! I will learn a lot & quickly from your thesis. $\endgroup$ – brunoh Jan 3 '16 at 21:38

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.