# Affine charts for projective scheme

Am I correct in assuming that any projective scheme $$X \subset \mathbb P^n$$ can be written as $$X \cong \mathrm{Proj} (\Bbbk [x_0, \dotsc, x_n] / I)$$ for some homogeneous ideal $$I = (f_1, \dotsc, f_r)$$ so that $$X$$ can be described concretely by considering the restriction of affine charts of $$\mathbb P^n$$ to $$X$$ and affine charts of $$X$$ can be given by $$\mathrm{Spec} (A_i)$$ where $$A_i = \Bbbk [\tfrac{x_0}{x_i}, \dotsc, \hat{\tfrac{x_i}{x_i}}, \dotsc, \tfrac{x_n}{x_i}] / (f_{1, i}, \dotsc, f_{r,i})$$ where $$[x_0 : \dotsb : x_n]$$ are homogeneous coordinates of $$\mathbb P^n$$ and $$i$$ runs from $$0$$ to $$n$$ and $$f_{k,i}$$ is the dehomogenization of the polynomial $$f_k$$ that appears as a relation in $$I$$?

When comparing schemes and varieties from the point of view of affine coordinate charts is the only difference that for schemes the $$A_i$$'s may contain nilpotent elements?

Regarding the first question: yes, this is true. Given a closed subscheme $$X$$ of $$\mathbb P^n_k$$, on each of the standard affine opens $$U_i = D_+(x_i)$$ it is defined by some ideal $$J_i$$ of $$(k[x_0,\dots ,x_n]_{x_i})_0$$, by affineness. Each $$J_i$$ corresponds to a homogeneous ideal $$I_i$$ of the homogeneous coordinate ring $$k[x_0,\dots ,x_n]$$, and you can check that $$I = \bigcap_i I_i$$ does the job. In fact, this argument works for closed subschemes of any projective scheme $$\operatorname{Proj} S_\bullet$$ (even when $$S_\bullet$$ is not generated in degree 1, with slight modification).
As for the second question, a (not necessarily irreducible) variety can be defined as a reduced, separated scheme of finite type over a field. A projective scheme is always separated and finite type, so the two things which can be relaxed are "reduced" and "field" - in general, your $$A_i$$ could be quotients of $$R[x_{0/i}, \dots ,\hat x_{i/i} ,\dots ,x_{n/i}]$$ for $$R$$ some ring, with $$X\subset\mathbb P^n_R$$. If you insist on working over $$k$$, then nilpotents are the only new thing that you will see in $$A_i$$.
Note also that if $$X$$ is a variety, then it is not necessarily true that $$I$$ is a radical ideal - consider $$\operatorname{Proj} k[x_0,x_1]/(x_0^2,x_0 x_1)$$. So, it's possible that the coordinate rings of the standard open affines have no nilpotents, yet the homogeneous coordinate ring of $$X$$ is nonreduced.
• Sorry, that was more of a remark about nilpotents in $A_i$ vs nilpotents in the homogeneous coordinate ring. I edited to add an actual answer to your second question and clarified my other statement. The Nullstellensatz gives a correspondence between algebraic subsets of $\mathbb A^n_k$ and radical ideals; there isn't such a correspondence between projective algebraic sets and homogeneous radical ideals, as this example shows. – Alex K Apr 28 at 22:59