I need some help with some bisimplicial stuff... Suppose I have a bisimplicial A-module, $M$. I know I can express $diag(M)$ as a coequalizer of simplicial A-modules, so I can correspondingly express $Tot(CC_{\ast}M)$ as the coequalizer of complexes of $A$-modules (here $CC_{\ast}M$ is the bicomplex associated to $M$. Now suppose I use put the Postnikov/standard filtrations of the rows of $CC_{\ast}M$ together to get one on $CC_{\ast}M$.

Is knowing that $Tot(CC_{\ast}M$ is a coequalizer as above enough to conclude that $Tot$ in this case commutes with $ker(\tau_{\leq n}CC_{\ast}M \rightarrow \tau_{\leq n-1}CC_{\ast}M)$?

I should mention I only care about this in $D(A)$.

Also, if anyone knows a better way to prove the tot/ker compatibility above, I would love something much cleaner.

Please disregard the above brain fart, I just realized my mistake!

@JanDeMeyer notice that there's also math.harvard.edu/~lurie/papers/tamagawa.pdf and this is 394 pages...

tamagawa-abridged is Weil’s Conjecture for Function Fields I and is 336 pages

And for reference, here's the book: press.princeton.edu/titles/14199.html

Which is also a part 1. The second part will prove the product formula, whereas the first part proves the product formula implies the conjecture.

Thank you both for your help already!

So let me get this straight... It seems like the original paper math.harvard.edu/~lurie/papers/tamagawa.pdf was found to be too dense by the authors, so they decided to expand the proofs and split the new version into two parts, of which the second part is entirely devoted to proving the trace formula, but it's an open question how the published book compares to "tamagawa-abridged"

I might just start reading the original and see how that goes