Consider a complex-oriented multiplicative generalized cohomology theory $h^{*}(X)$. It is complex-oriented, if by the definition the following two conditions hold:

1) There exists an element $t\in h^{2}(\mathbb{C}P^{\infty})$, restricting (the abuse of notation) to the canonical generator $t$ of $h^{2}(\mathbb{C}P^{1})$. (Canonical means that we have $\widetilde{h}^{2}(S^{2}) \cong \widetilde{h}^{2}(\mathbb{C}P^{1}) \cong h^{0} $ from the suspension map $\Sigma^{2}:\ \lbrace x, y \rbrace \rightarrow S^{2}$, hence $h^{2}(\mathbb{C}P^{1})$ is free $h^{*}$ - module of rank $1$).

2) $t$ goes to $-t$ under the involution of $\mathbb{C}P^{1}$.

Using the Atiyah-Hirzebruch spectral sequence it isn't difficult to show that $$h^{*}(\mathbb{C}P^{n}) \cong h^{*}[t]/(\ t^{n+1}).$$ I want to deduce $$h^{*}(\mathbb{C}P^{\infty}) \cong h^{*}[[ t]].$$

The article below at p.3 tells me to use the Milnor exact sequence to make it: http://www.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf

$$0 → lim^{1}\ h^{∗−1}(\mathbb{C}P^{n}) → h^{∗}(\mathbb{C}P^{∞}) → lim\ h^{∗}(\mathbb{C}P^{n}) → 0.$$

$lim^{1} A_{i}$ is defined for an inverse system of groups $\lbrace A_{i},\ f_{i}\rbrace$ as follows. Let $\Delta:\ \prod\limits_{i=1}^{\infty} A_{i}\rightarrow \prod\limits_{i=1}^{\infty}A_{i}$ be defined by $\Delta(a_{i}) = a_{i} - f_{i+1}(a_{i+1}).$ Then $$lim^{1} A_{i} = \prod\limits_{i=1}^{\infty} A_{i}\Big/ \lbrace x_{i}\rbrace\sim *\\ \text{iff}\qquad \exists\lbrace a_{i}\rbrace:\ \lbrace x_{i}\rbrace = \Delta(\lbrace a_{i}\rbrace)$$

I haven't found any refences to the expalanation of the existence of that sequence. My questions are:

1) What is the Milnor exact sequence?

2) Why does the first part of the sequence vanish?

3) Are there any examples unveiling the meaning of $lim^{1}$?

  • $\begingroup$ This is just the derived functor of $\lim$ (and corresponding exact sequence); it can be found in any reasonably complete text in homological algebra. $\endgroup$ – Alex Degtyarev Nov 13 '14 at 20:55

Milnor's paper is called On axiomatic homology theory and is published in Pacific J. Math. Volume 12, Number 1 (1962), 337-341. It is also reprinted in Frank Adams' Algebraic Topology: A Student's Guide. It is wonderfully lucid (as with all of Milnor's writings) and should tell you all you need to know.

(The answer to your second question is given in the document you linked, after the proof of Lemma 1.4 on page 3.)


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