How many elements are there in $\Bbb P_\Bbb Z^n(C)$ for $C=\Bbb Z/m\Bbb Z$($m\ge 1$)?

Question

How many elements are there in $\Bbb P_\Bbb Z^n(C)$ for $C=\Bbb Z/m\Bbb Z$($m\ge 1$)?

Here $\Bbb P$ denote the projective space which consists for $C$-lines for $\Bbb Z$-algebra $C$

No idea for counting it in a neat way. Any help, please?

Thank you.

Answer 1

Step 1. The problem boils down to computing $\mathbb{P}^n_{\mathbb{Z}}(\mathbb{Z}/p^i \mathbb{Z})$ for some prime $p$. This seems easy as $spec(\mathbb{Z}/m\mathbb{Z}) = \bigsqcup spec(\mathbb{Z}/p^i\mathbb{Z})$ for various primes.

Step2. $\mathbb{P}^n_{\mathbb{Z}}(\mathbb{Z}/p^i\mathbb{Z}) = \mathbb{P}^n_{\mathbb{Z}/p^i\mathbb{Z}}(\mathbb{Z}/p^i\mathbb{Z})$. This is just saying that for a $Z-$scheme $X$ and $Y$, $Mor_Z(X,Y) = Mor_X(X,X\times_Z Y)$ which is sections of the scheme $X \times_Z Y \rightarrow X$.

Step 3. $|\mathbb{P}^n_{\mathbb{Z}/p^i\mathbb{Z}}(\mathbb{Z}/p^i\mathbb{Z})| = \frac{p^{(i-1)n}(p^{n+1} - 1)}{p-1}$.

Note that a section of a separated morphism is a closed immersion and closed immersions of $\mathbb{P}^n_R$ for any ring $R$ is given by are given by an ideal $I$ of $R[X_0,X_1, . . . ,X_n]$. Moreover if we are looking at sections of the structure morphism then $Proj(R/[X_0, X_1 . . . , X_n]/I) \xleftarrow{\cong} Proj(R[t])$ as $Proj(R[t]) = spec(R)$ and hence the map $R[X_0, X_1, . . . , X_n] \rightarrow R[t]$ is surjective in high enough graded parts. Now any such map is dictated by where $X_i$ goes and since this is a map of graded algebras $X_i \mapsto \alpha_i t$ and it should satisfy that $(\alpha_i^d) = 1$ for some high enough d.

Using this theory in our case $R = \mathbb{Z}/p^i\mathbb{Z}$ we get that $\alpha_i^d \in \mathbb{Z}/p^i\mathbb{Z}$ must generated the unit ideal for some lage enough d. Now this is possible if and only if there exists $\alpha_i$ which is not divisible by $p$. Hence the number of possibilities for the mappings are $p^{i(n+1)} - p^{(i-1)(n+1)}$. Also some of these mappings might give the same point and those are exactly the maps which differ by a unit in $\mathbb{Z}/p^i\mathbb{Z}$ that is $X_i \mapsto a\alpha_i$ for some $a \in (\mathbb{Z}/p^i\mathbb{Z}))^*$ whose cardinality is $p^{i-1}(p-1)$. Dividing the former number by latter number we get the result.

Note that this agrees for $i = 1$ with the known quantity of number of elements in a projective space over a field $\mathbb{F}_p$.