definition: A projective variety over a field $k$ is a closed subscheme (over $k$) of $\mathbb P^n_k=\operatorname{Proj}(k[T_0,\ldots,T_n])$.

Now it can be proved (I have done it) that if $I=(f_1,\ldots,f_m)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$, then the projective algebraic set $Z_+(f_1,\ldots,f_m)\subseteq\mathbb P^n(k)$ is in bijection with the set of $k$-rational points of $\operatorname{Proj}\frac{k[T_0,\ldots,T_n]}{I}$. Pratically a scheme of the type $\operatorname{Proj}\frac{k[T_0,\ldots,T_n]}{I}$ corresponds to the naive intuition of an algebraic variety; now I can't find a proof of the following statement:

Every closed subscheme (over $k$) of $\mathbb P^n_k$ is isomorphic to a scheme $\operatorname{Proj}\frac{k[T_0,\ldots,T_n]}{I}$ for certain homogeneous ideal $I\subseteq k[T_0,\ldots,T_n]$.

So firstly I ask any hint, or complete proof or even a reference of the above proposition. Then I have the following question:

Is it true that $\operatorname{Proj}\frac{k[T_0,\ldots,T_n]}{I}$ is irreducible as scheme if and only if $f_1,\ldots,f_m$ are irreducible polynomials? (For affine schemes this is true)

Thanks in advance.


Your last claim is not true "even" in the affine case. For example the affine plan set cut by the ideal $(x,(y-1)^2-x)$ is reducible yet the generators of the ideal are irreducible. You can construct a similar example in the projective plane by intersecting a line and a conic.

As for your main question, look at Hartshorne's Corollary 5.16 in chapter II.


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