Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and arithmetic curves" from Qing Liu. I have had difficulties to prove the equivalence of many definitions. For example Hindry and Silverman defines an affine variety to be an irreducible algebraic subset of $\mathbb{A}^n$ with respect to Zariski topology. On the other hand, Liu defines an affine variety to be the affine scheme associated to a finitely generated algebra over a field.
[Math] Learning schemes
algebraic-geometryschemesself-learning
Related Solutions
Yes to both questions, but I'm not sure what you mean by wanting a global proof. Reducedness is a local property! The proof will consist of picking a point $x \in X$ and an affine chart $U = \text{Spec}(R)$ containing $x$, then checking reducedness on that chart.
In particular, in the affine case, if $R$ is reduced, then all its localizations are reduced. On the other hand, if $R$ is nonreduced, let $f \in R$ be nilpotent. The annihilator $\text{Ann}(f)$ is contained in some associated prime ideal $P$ (this is a defining property of associated prime ideals), hence $f/1 \in R_P$ is nonzero, so $R_P$ is also nonreduced.
First, you should convince yourself that the question whether a given morphism $\operatorname{Spec} \mathcal{O}_{X,x} \to Y$ extends to an open neighborhood of $x$ comes down to the following question in commutative algebra:
Let $A$ and $B$ be two rings and let $\mathfrak{p}$ be a prime ideal of $A$. Does a given ring homomorphism $B \to A_\mathfrak{p}$ factor as $B \to A_f \to A_\mathfrak{p}$ for some $f \in A \setminus \mathfrak{p}$?
Note that there is no reason to hope that such a factorization always exists. Indeed, the answer to the above question might be "no".
But what if $B$ is a finitely generated $R$-algebra with $R$ a Noetherian ring, $A$ is an $R$-algebra and the given morphism is a morphism of $R$-algebras? This is the case you have to deal with in solving your exercise (I'll assume you are able to work out why - if you have trouble, just drop a comment). Then the above question is guaranteed to have a positive answer. Let's prove this. Write $B = R[T_1,\dotsc,T_n] / (g_1,\dotsc,g_m)$, denote the composite of the canonical projection $R[T_1,\dotsc,T_n] \twoheadrightarrow B$ with the given morphism $B \to A_\mathfrak{p}$ by $\varphi$ and write
$$\varphi(T_i) = \frac{a_i}{f_i}, \quad i=1,\dotsc,n \, ,$$ $$\varphi(g_j) = \frac{a_j'}{h_j}, \quad j=1,\dotsc,m \, .$$
Now, for each $j \in \lbrace 1,\dotsc,m \rbrace$, choose some $f_j' \in A \setminus \mathfrak{p}$ such that $f_j' a_j' = 0$ in $A$ (which exists since $\varphi(g_j) = 0 \in A_\mathfrak{p}$). I now claim that $f = \prod_{i=1}^n f_i \prod_{j=1}^m f_j'$ has the desired property, i.e. that the given homomorphism factors through $A_f \to A_\mathfrak{p}$. To see why, we first make use of the fact that, by construction, $f$ is a common denominator of all the $\varphi(T_i)$ and note that
$$ T_i \mapsto \frac{a_i \prod_{k \neq i} f_k \prod_{j=1}^m f_j'}{f}, \quad i=1,\dotsc,n$$
defines a morphism of $R$-algebras $R[T_1,\dotsc,T_n] \to A_f$ whose composite with $A_f \to A_\mathfrak{p}$ agrees with $\varphi$. In addition, we have, by construction, $a_j' f = 0$ for all $j \in \lbrace 1,\dotsc,m \rbrace$; thus, the morphism just defined maps each $g_j$ to $0$ and, as a consequence, factors over $B$.
Best Answer
I have found Kenji Ueno's book Algebraic Geometry 1: From Algebraic Varieties to Schemes to be quite satisfying in introducing the basic theory of schemes. Well, to be fair, this is only the first in a series of three books on the subject by the same author. So this first volume basically just develops the definitions of an affine scheme first and then of a scheme in general by "pasting" together affine schemes. It does not go into cohomology and more advanced stuff, which is the subject of the other two books.
However, what I really like is that he motivates very carefully the passage from the definition of an affine algebraic variety as an irreducible algebraic set in an affine space $\mathbb{A}_{k}^{n}$ to the definition of an affine variety using schemes, which is where you are having some trouble. What he does is that he starts by doing some algebraic geometry in the classic sense, that is, over an algebraically closed field $k$, in the first chapter of the book.
Then the author proves that there is a correspondence between the points in an algebraic set $V$ and the maximal ideals of its associated coordinate ring $k[V]$, where a point $(a_1, \dots , a_n) \in V$ corresponds to the maximal ideal of $k[V]$ determined by the ideal $(x_1 - a_1 , \dots , x_n - a_n) \in k[x_1, \dots, x_n]$, that is, a correspondence between the points in $V$ and the "points" in the maximal spectrum $\text{max-Spec}(k[V])$ of the coordinate ring $k[V \, ]$.
Then Ueno goes on to define an affine algebraic variety as a pair $(V, k[V \, ] )$ where $V$ is an an algebraic set. But he then makes the argument that one can go a little bit further and consider the pair $( \text{max-Spec}(R), R )$ where $R$ is a $k$ algebra. But here Ueno arguments that if the original intention was to study the sets of solutions of polynomial equations, then where is the geometry and where are the equations hidden if an algebraic variety is defined as this pair $( \text{max-Spec}(R), R )$?
The interesting thing is that if the $k$ algebra $R$ is finitely generated over $k$ then
$$ R \simeq \frac{ k[x_1 , \dots , x_n] }{I}$$
so that as a consequence
$$ \text{max-Spec}(R) = V(I)$$
so that again you'll have some equations (this is all done and explained in detail in the book). So then the author (re)defines an algebraic variety over an algebraically closed field $k$ (remember that he is doing everything in the classic sense) as a pair $( \text{max-Spec}(R), R )$, where $R$ is a finitely generated $k$ algebra.
And then at the end of the first chapter the author motivates the need for a more general theory, for example having in mind the needs of number theory, because since everything was done in the context of an algebraically closed field, then the arguments don't work for the fields (and rings) of interest in number theory. In particular, it is noted how an extension of the definitions to include these cases would need to take into account not only the set of maximal ideals, but the set of all prime ideals.
Then chapter two develops first some properties of this set of prime ideals, or prime spectrum of a ring, making it into a topological space with the Zariski topology... and then defines the necessary things in order to be able to define an affine scheme and a scheme (I mean, the concepts of a sheaf of rings, a ringed space, etc).
It is not a short story of course, but again I prefer this type of approach at first, than having to deal with an unmotivated (and difficult) definition that strives for great generality but I have no idea of where it comes from and what is its purpose.
Note that the book that Arturo recommended is great also but it assumes you already know some algebraic geometry and its level is higher than Ueno's book.
You should take a look at it and see if you like it, the book has a fair amount of examples and some exercises interspersed within the text also. You'll have to study from other sources as well but I believe that this book does a pretty good job at motivating the abstract definitions.
I hope this helps at least a little.