# Weibel IHA exercise 1.2.6 : Example of total complex

1.2.6 is below;

Give examples of (1) a second quadrant double complex C with exact columns such that $Tot^{\prod}(C)$ is acyclic but $Tot^{\oplus}(C)$ is not; (2) a second quadrant double complex C with exact row such that $Tot^{\oplus}(C)$ is acyclic but $Tot^{\prod}(C)$ is not; and (3) a double complex (in entire plane) for which every row and every column is exact, yet neither $Tot^{\prod}(C)$ nor $Tot^{\oplus}(C)$ is acyclic.

I think (1) and (2) is wrong question, since direct sum and direct product are isomorphic when they deal with finite sum (or product); But second quadrant double complex $C$ deals with finite product for $Tot_{1}, Tot_{2}, ...$ and so on, so it is not possible that $Tot^{\prod}(C)$ is acyclic but $Tot^{\oplus}(C)$ is not or vice versa.

Am I correct? if not, is there any examples for this problem?

• Since you're in the second quadrant (horizontal index negative, vertical index positive), every term of $Tot$ will have infinitely many components, no matter if you use $\prod$ or $\oplus$. Recall that for $\oplus$ you must have finitely many terms non-zero in an infinite sum, whereas this is not a condition for $\prod$. This gives you the difference between the two. – Jānis Lazovskis May 10 '16 at 4:26
• @jiv Oh I see... thank you :) – user124697 May 10 '16 at 13:56