The Rational Function Field of an Affine Variety


Definition: Let $K$ be a field and let $V subseteq mathbb{A}^n(K)$ be a nonempty affine variety. The Rational Function Field of $V$ is defined as $K(V) = left { frac{f}{g} : f, g in Gamma(V), g neq 0 right }$. Elements of $K(V)$ are called Rational Functions.

The rational function field of $V$ is also sometimes called the Quotient Field of $V$ or the Field of Fractions of $V$.

Definition: Let $K$ be a field, $V subseteq mathbb{A}^n(K)$ a nonempty affine variety, and let $mathbf{p} in V$. A rational function $displaystyle{f in k(V)}$ is Defined at $mathbf{p}$ if there exists polynomials $a, b in Gamma(V)$ with $displaystyle{f = frac{a}{b}}$ and such that $b(mathbf{p}) neq 0$.
Definition: Let $K$ be a field, $V subseteq mathbb{A}^n(K)$ a nonempty affine variety, and let $mathbf{p} in V$. The Local Ring of $V$ at $mathbf{p}$ is defined to be the subring $O_{mathbf{p}}(V)$ of $K(V)$ of rational functions on $V$ that are defined at $mathbf{p}$.

In the following proposition we prove that $O_{mathbf{p}}(V)$ is indeed a subring of $K(V)$.

Proposition 1: Let $K$ be a field, $V subseteq mathbb{A}^n(K)$ a nonempty affine variety, and let $mathbf{p} in V$. Then the local ring of $V$ at $mathbf{p}$, $O_{mathbf{p}}(V)$ is a subring of $K(V)$ that contains $Gamma(V)$.

We use the subring test which can be found on the Subrings and Ring Extensions page to prove that $O_{mathbf{p}}(V)$ is a subring of $K(V)$.

(1)

begin{align} quad Gamma(V) subseteq O_{mathbf{p}}(V) subseteq K(V) end{align}

  • So all that remains to show is that $O_{mathbf{p}} (V)$ is indeed a subring of $K(V)$.
  • Let $f, g in O_{mathbf{p}}(V)$ with $displaystyle{f = frac{a}{b}}$, $displaystyle{g = frac{c}{d}}$ where $a, b, c, d in Gamma(V)$ and $b(mathbf{p}) neq 0$, $d(mathbf{p}) neq 0$. Then:

(2)

begin{align} quad f + g = frac{a}{b} + frac{c}{d} = frac{ad + bc}{bd} end{align}

(3)

begin{align} quad fg = frac{a}{b} cdot frac{c}{d} = frac{ac}{bd} end{align}

  • Observe that since $b(mathbf{p}), d(mathbf{p}) neq 0$ and $Gamma(V)$ is an integral domain we have that $(bd)(mathbf{p}) neq 0$ and so $f + g$ and $fg$ are defined at $mathbf{p}$ which shows that $O_{mathbf{p}}(V)$ is closed under addition and closed under multiplication.
  • Let $f in O_{mathbf{p}}(V)$. Then there exists $a, b in Gamma(V)$ such that $displaystyle{f = frac{a}{b}}$ and $b(mathbf{p}) neq 0$. Since $Gamma(V)$ is a ring, $-a in Gamma(V)$ and so $displaystyle{-f = frac{-a}{b}}$. So $-f in O_{mathbf{p}}(V)$.
  • Lastly we clearly see that $1$ is defined at $mathbf{p}$ so $1 in O_{mathbf{p}}(V)$.
  • Therefore $O_{mathbf{p}}(V)$ is a subring of $K(V)$. $blacksquare$
Theorem 2: Let $K$ be a field and let $V subseteq mathbb{A}^n(K)$ be a nonempty affine variety. Then $Gamma(V) = bigcap_{mathbf{p} in V} O_{mathbf{p}}(V)$.



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