Hartshorne Lemma(II 6.1)

Question

Hi I am wondering what Hartshorne means by "an open affine subset on which $f$ is regular". This is part of the first sentence in the proof of lemma 6.1 in chapter two of the book "Algebraic Geometry by Hartshorne". I Tried looking up regular in the index, but this just refers me to regular functions on varieties which as far as I know has absolutely nothing to do with the lemma. Could someone please tell me what he means by this.

Thanks

Answer 1

Let $X$ be an integral scheme. Then, for any open subset $U$, the ring $\mathscr{O}_X (U)$ embeds as a subring of the function field $K (X)$ in a canonical way, and so for every point $x$ of $X$, the local ring $\mathscr{O}_{X, x}$ is canonically a subring of $K (X)$.

Definition. An element $f$ of $K (X)$ is regular at $x$ if it is in $\mathscr{O}_{X, x}$, considered as a subring of $K (X)$, and it is regular on an open subset $U$ if it is in $\mathscr{O}_X (U)$, considered as a subring of $K (X)$.

Answer 2

"$f$ is regular on $U$" just means $f\in\mathcal{O}(U)$.