# Basic question on the connection between Complex-oriented cohomology theories and Formal Group Law

In wikipedia is stated that a complex-orientable cohomology theory is a multiplicative cohomology theory $$E$$ such that the restriction map $$E^2(\mathbb{C}\mathbf{P}^\infty) \to E^2(\mathbb{C}\mathbf{P}^1) = \mathbb{Z}$$ is surjective. An element $$t \in E^2(\mathbb{C}\mathbf{P}^{\infty})$$ that restricts to the canonical generator of the reduced theory $$\widetilde{E}^2(\mathbb{C}\mathbf{P}^1)$$ is called a ''complex orientation''.

Subsequently there is an explanation how $$t$$ gives rise to formal group law on $$E^*(\mathbb{C}\mathbf{P}^\infty) = \varprojlim E^*(\mathbb{C}\mathbf{P}^n)$$:

Let $$m$$ be the multiplication

$$\mathbb{C}\mathbf{P}^\infty \times \mathbb{C}\mathbf{P}^\infty \to \mathbb{C}\mathbf{P}^\infty, ([x], [y]) \mapsto [xy]$$

where $$[x]$$ denotes a line passing through $$x$$ in the underlying vector space $$\mathbb{C}[t]$$ of $$\mathbb{C}\mathbf{P}^\infty$$.

There are several issues in this approach that are not clear at all. What does the notation "$$xy$$" resp. $$[xy]$$ mean? Is it tacitly exploited that $$\mathbb{C}\mathbf{P}^\infty$$ has a natural multiplicative structure?

Moreover why is this structure as suggested in last sentence identical to ring of polynomials $$\mathbb{C}[t]$$?

Here $$\mathbb{CP}^\infty$$ is identified with the projectivization of the polynomial ring $$\mathbb{C}[t]$$ as a vector space over $$\mathbb{C}$$. So, a point of $$\mathbb{CP}^\infty$$ is a 1-dimensional subspace of $$\mathbb{C}[t]$$, or equivalently a nonzero element up to scalar multiples. This then induces a multiplication map on $$\mathbb{CP}^\infty$$: given two points of $$\mathbb{CP}^\infty$$, represented by elements $$x,y\in\mathbb{C}[t]$$, you can multiply the polynomials $$x$$ and $$y$$ to get another nonzero polynomial $$xy$$ which represents another element of $$\mathbb{CP}^\infty$$. Changing $$x$$ or $$y$$ by a scalar multiple will change $$xy$$ by a scalar multiple, so this gives a well-defined map $$\mathbb{CP}^\infty\times\mathbb{CP}^\infty\to\mathbb{CP}^\infty$$.
(The Wikipedia article seems to make the very unfortunate choice of using "$$t$$" to denote both the complex orientation and the variable in the polynomial ring $$\mathbb{C}[t]$$. These are two totally different things which should be given different names.)
• I see, that's very enlightening, thank you. What I not see if this projectivization is in some way compatible with truncations of of $\mathbb{C}[t]$? Indeed $\mathbb{CP}^\infty$ is naturally given as colimit $\bigcup_n \mathbb{CP}^n$, while $\mathbb{C}[t]$ comes which natural truncations to $\mathbb{C}[t]/t^{n+1}$. Does the projectivization of $\mathbb{C}[t]$ in your answer prove any interesting natural relation between the finite proj spaces a $\mathbb{CP}^n$ and the $\mathbb{C}[t]/t^{n+1}$? Jul 8, 2022 at 17:14
• I'm not sure what you're asking. The quotient map $\mathbb{C}[t]\to\mathbb{C}[t]/t^{n+1}$ does not induce a map on the projectivizations, since it maps some nonzero elements to zero. The more natural thing to consider is instead that $\mathbb{CP}^n$ is the projectivization of the vector subspace of $\mathbb{C}[t]$ consisting of polynomials of degree at most $n$. Jul 8, 2022 at 18:37