# Spectral Sequence involving "Triple Tor"

Can someone help me with the first 4 lines of Page 111 of Local Algebra by Serre?

I would like to know which spectral sequence is being used.

Initially I thought it is the Grothendieck spectral sequence, but I don't think that is the case, and I don't know what a "Triple Tor" is. Any help would be appreciated. Thank you.

## 1 Answer

Basically he's computing the homology of the complex $M \otimes^{\mathbb{L}}N \otimes^{\mathbb{L}} k$ (where tensors are taken over $A$). Associativity of the tensor product tells us that $$(M\otimes^{\mathbb{L}}N) \otimes^{\mathbb{L}} k = M \otimes^{\mathbb{L}}(N \otimes^{\mathbb{L}} k)$$

And so we get two different spectral sequences... $$\operatorname{Tor}_p(\operatorname{Tor}_q(M, N), k) \Rightarrow \operatorname{Tor}_{p+q}(M, N, k)$$ $$\operatorname{Tor}_p(M, \operatorname{Tor}_q(N,k)) \Rightarrow \operatorname{Tor}_{p+q}(M, N, k)$$

which you can think of in many different ways. One comes from the spectral sequence associated to a double complex. That is, in general if we have complexes $A_*$ and $B_*$ and we want to compute $H_*(A_* \otimes^{\mathbb{L}} B_*)$ (this is sometimes called "hypertor" but that makes it sound scarier than it is...) then we can choose Cartan-Eilenberg resolutions $P \rightarrow A$ and $Q \rightarrow B$, and consider the spectral sequence of the double complex $P \otimes Q$. This gives: $$H_*(H_*(A) \otimes^{\mathbb{L}} H_*(B)) \Rightarrow H_*(A \otimes^{\mathbb{L}} B)$$ which specializes to the aforementioned spectral sequences in your example.

• @Dylan Thanks, yes you are right, I finally got it. Apr 7, 2013 at 11:24
• Do you know if we have a spectral sequence involving Ext. Something similar to above where all the Tor is replaced by Ext? Thanks May 4, 2013 at 5:12
• I mean if we have something like "Triple Ext"? May 5, 2013 at 6:08
• @messi: If you can say what you mean by "triple Hom" then that should tell you what to do with "triple Ext" May 15, 2013 at 17:58