A question in Proposition $6.3$ of Chapter $-2$ of Daniel Perrin's Algebraic Geometry ( Page $32$)
问题内容
This question is from Proposition $6.3$ of the textbook Algebraic geometry by Daniel Perrin( Page 32).
Here $k$ is a commutative field.Let $V$ be a projective algebraic set and consider a homogeneous element $f \in \Gamma_h(V)= k[X_0,...,X_n]/I_p(V)$ of degree>0. $I_p(V)$ is the ideal of projective algebraic set.Set $D^+(f)=${$x\in V| f(x)\neq 0$}.
Proposition $6.3 :$ With the above notations, every non empty open set of $V$ is a finite union of open sets of the form $D^+(f)$.
Proof: I am not able to understand the following line: If $U$ be an non -empty open set of $V$,then how did author wrote $V-U=V_P(I)$, where $I$ is a homogeneous ideal of $R$?
Can you please tell me how can we write this?
回答 (1)
I suppose that $R=\Gamma_h(V)$.
If $U\subset V$ is a non-empty open, then $Z=V\setminus U$ is a proper Zariski closed subset of $V$. Just let $$ I=\{f\in R\,|\,f_{|Z}=0\}. $$