I'm still studying basic Algebraic Geometry by myself and I got to the point of moving to the projective space. It's clear that there is a need of using homogeneous polynomials to keep the notion of belonging to a projective variety.
The thing that stumps me is defining the algebraic set of an homogeneous ideal. The book I'm using states that an homogeneous ideal is an ideal which contains all the homogeneous parts of the polynomials that it contains, and is equivalent to saying that it's generated by homogeneous polynomials. This looks fine and relatively clear.
Now, when we define the algebraic set, that later we'll call variety, we say that if $I$ is an homogeneous ideal, $V(I) = \{P \in P^n_{K}\ | f(P) = 0$ for all homogeneous $f \in I \}$.
Now, does this mean that it's defined as in the affine case? Because if all the homogeneous components lie in $I$, then for any $f = f_{0} + \ldots + f_{d}$ where $f_{i} =$ the homogeneous component of $f$ of degree $i$, $f_{i} (P) = 0$ and then $f(P) = f_{0}(P) + \ldots + f_{d}(P) = 0$. I think this flows relatively well but then it makes no sense that in the definition given by the book, the condition is only applied to homogeneous polynomials. Am I right or am I missing something huge? I'm pretty insicure about it because I don't know anyone who I can talk with about these things, and clear my doubts, so thank you in advance!
If $P$ is a point in a projective space, then the relation $f(P) = 0$ is only well defined for homogeneous polynomials.
A point is a projective space $\mathbb{P}^n_K$ is defined as a line through origin in $K^{n + 1}$, or the equivalence class of points in $K^{n + 1}\setminus\{0\}$ under equivalence relation defined by $v \sim \lambda v$ for all $\lambda \in K$.
In other words, a point $P$ in the projective space $\mathbb{P}^n_K$ is given by $n + 1$ scalars (called homogeneous coordinates), which is usually denotes by $P = (P_0:P_1:\dots:P_n)$, but these coordinates are not defined uniquely, only up to scaling: $(P_0:P_1:\dots:P_n) = (\lambda P_0: \lambda P_1:\dots:\lambda P_n)$.
Now, if a polynomial $f$ is homogeneous of degree $d$, then $f(\lambda P_0, \dots, \lambda P_n) = \lambda^d f(P_0, \dots, P_n)$, so the relation $f(P) = 0$ is invariant under scaling and this gives us a well-defined notion of $f(P) = 0$ for $P \in \mathbb{P}^n_K$. But if $f$ is not homogeneous, then it is possible that $f(P_0, \dots, P_n) = 0$ and $f(\lambda P_0, \dots, \lambda P_n) \neq 0$, so vanishing of a non-homogeneous polynomial on a point in a projective space is not well defined.