MathOverflow Asked by PowerToThePeople on January 1, 2022
Some people define total cohomology of a space $X$ to be $bigoplus_{i geq 0} H^i(X)$, which would make $H^*(mathbb{C} P^infty)$ a polynomial ring in one generator of degree 2.
However, it seems like thinking of $H^*(mathbb{C} P^infty)$ as a power series ring is more natural for several reasons. For one thing, if cohomology is like the dual of homology, then the dual to an infinite direct sum is a direct product. Many algebraic formulas are also simplified if one allows for the entire power series ring rather than the polynomial ring.
Question: Are there compelling reasons to define total cohomology as $bigoplus_i H^i$ or as $prod_i H^i$?
Addendum:
The question itself is quite concrete, but there are other reasons I am contemplating this, so perhaps I should list them.
If I think of $H^*mathbb{C}P^infty$ as somehow Koszul dual to a circle, this question might be closer to whether one should think of (this kind of) Koszul duality as always happening in a filtered/pro setting. If there are strong views/philosophies on viewing infinite projective space as an instance of Koszul duality, or on whether Koszul duality should always ask for filtration (e.g., adic-near-a-point) structures, do share.
One can think of $mathbb{C}P^infty$ as a space in its own right, or as a filtered diagram of spaces. This changes, for example, what kind of condensed set I think of $mathbb{C}P^infty$ as. Accordingly, the cohomology of the condensed set obtained as an ind-object of $mathbb{C}P^n$ should look more pro-y (and hence look more like a power series), while the cohomology of the condensed set called "what does $mathbb{C}P^infty$ represent as a space" feels more like a polynomial ring.
In my opinion the most natural statement is that $H^*(mathbb{CP}^infty)$ is a graded power series ring. That is we can write $$H^*(mathbb{CP}^infty)=lim_n mathbb{Z}[x]/x^n$$ where the limit is taken in the category of graded rings and $x$ has degree 2. Note that in this particular case (where the ring of coefficients is concentrated in degree 0) it coincides with the graded polynomial ring. This has the advantage that the formula works for all complex-oriented cohomology theories, e.g. for complex K-theory: $$KU^*(mathbb{CP}^infty)=lim_n KU^*[x]/x^n$$ in which case it does not coincide with the graded polynomial ring!
Whether to present graded rings as direct sums or direct products is largely a matter of personal preference, although the direct sum option has always seemed the most natural to me.
One quick comment about one of your addenda: you can always think of homotopy types as the ind-category of finite homotopy types (precisely, this is true at the level of the ∞-category of spaces), so every cohomology ring $E^*X$ has a natural enhancement to a pro-(graded ring) (it's not quite true that $E^*X$ is always the limit of this pro-ring, because of the possible presence of $lim^1$-terms, but let's ignore this for the moment). Under this correspondence $H^*mathbb{CP}^infty=mathbb{Z}[[x]]$ seen as a pro-(graded ring) in the canonical way. This is the start of the connections between homotopy theory and formal geometry, which has been extremely fruitful. This book is an excellent source if you want to learn more about this.
Answered by Denis Nardin on January 1, 2022
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