车讯:同期推新概念车 斯柯达柯迪亚克4月上市

Question

I'm reading the paper On Localization of Tight Closure in Line-$S_4$ Quartics where the authors used the following strange notion in Definition 2.1:

Let $F$ be an algebraically closed field of characteristic 2 and $\alpha\in F$, consider a map $\phi_{\alpha}:F\cup \{\infty\}\rightarrow F\cup \{\infty\}:t\rightarrow t^4+\alpha t^{-4}$

Since this comes from a paper, I guess I just missed a few prerequisites, but I didn't found anything useful by googling "infinity in positive characteristic". If we treat $F$ as an affine scheme, then this set $F\cup \{\infty\}$ reminds me of the Riemann sphere more generally this can be viewed as closed points of the projective scheme $\mathbb{P}^1_F$, but in this way, I find it hard to consider what $\phi_{\alpha}(0)=0^4+\alpha 0^{-4}$ is. So I wonder if there is a standard definition for the set $F\cup \{\infty\}$? Does it have an algebraic structure derived from $F$?

Answer 1

Every rational map on $F$ can be extended to a morphism on $\mathbb{P}^1_F$. If $\frac{P(t)}{Q(t)}$ is a ratio of two polynomials without common divisors, then we can homogenize both $P$ and $Q$ to the forms $\hat{P}(t, s)$ and $\hat{Q}(t, s)$ of the same degree and consider the map gvien by $(t:s) \mapsto (\hat{P}(t, s):\hat{Q}(t, s))$ in homogeneous coordinates on $\mathbb{P}^1_F$. One can check that it maps any root of $Q$ to $\infty$ and $\infty$ to $0$ if $\deg P < \deg Q$, to $\infty$ if $\deg P > \deg Q$, and to the ratio of leading coefficients $\frac{p_n}{q_n}$ if $\deg P = \deg Q$.

In your case $0$ and $\infty$ are mapped to $\infty$ when $\alpha \neq 0$.