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I am reading John McCleary's A User's Guide to Spectral Sequence and was quite confused about one result: On page 15 of the version I was reading, it says that if $E^{\star,\star}_2$ is the bigraded vector space in Example 1.E, then $P(E^{\star,\star}_2,t)=(1+t^{11})(1+t^4+t^8+t^{12})(1+t^3)$. I am quite confused on how to obtain this result from Example 1.E. It seems to me that $P(E^{\star,\star}_t)$ has a term $t^{11+12+3}=t^{26}$, which by definition of $P(E^{\star,\star}_2,t)$ implies that $\text{dim}_k(\bigoplus _{p+q=26}E^{p,q})=1$. Why is that? I am not sure if I have understood Example 1.E wrongly. Any explanation will be greatly appreciated.

Update: Thank you @Neil Strickland for reminding me. The conditions of Example 1.E are: Suppose $E^{\star,\star}_2$ is given as an algebra by

$E^{\star,\star}_2\cong\mathbb{Q}[x,y,z]/(x^2=y^4=z^2=0)$,

where the bidegree of each generator is given by $\text{bideg}x=(7,1)$, $\text{bideg}y=(3,0)$ and $\text{bideg}z=(0,2)$. Furthermore, suppose $d_2(x)=y^3$ amd $d_3(z)=y$. In this case, the spectral sequence collapses at $E_4$ and, though $x$ and $y$ do not survive to $E_{\infty}$, the product $xy$ does.

Zuriel
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    You'll have a much better chance of getting a useful answer if you actually tell us what Example 1.E is. – Neil Strickland Apr 09 '12 at 21:23
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    Also, I had found McCleary's book quite unenlightening, but maybe it's just me. – Igor Rivin Apr 09 '12 at 21:38
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    My guess is that the numbers in example 1.E were changed to (7,1), (3,0) and (0,2) from (10,1), (4,0) and (0,3), perhaps to fit the page better, but the discussion a page later was not changed. – Ben Williams Apr 10 '12 at 00:25
  • @Ben Williams, this change is not mentioned in the book. Does this mean that the case is that this part of the book contains some flaws, not I understood it wrongly? – Zuriel Apr 10 '12 at 02:47
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    That's my guess. I think you understood it fine, and it's an erratum in the book. – Ben Williams Apr 10 '12 at 04:56

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I think the Poincar\'{e} polynomial should be $P(E_2^{*,*},t)=(1+t^{8})(1+t^3+t^6+t^9)(1+t^2)$ instead of the one you mentioned so this is an erratum in the book. Its clear from the algebra structure that there can not be any term of degree 26.

George
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