The most beautiful mathematical structure: Schemes

Commutative Rings

A commutative ring (with unit) is an algebraic structure with compatible operations addition and multiplication such that both operations are commutative, i.e. order of the elements added/multiplied does not matter (and contains a multiplicative identity).

Prime Spectrum: Spec(R)

Consider the ring of integers: …, -3, -2, -1, 0, 1, 2,…
Among all the integers, there are some integers that stand out. They are prime numbers.
If we consider a general commutative ring, there are subsets that stand out. They are prime ideals. A prime ideal P of a ring R is a proper subset that is closed under addition and multiplication by all elements in R and also that satisfies this following property:
If xy is in P, then at least x or y is in P. (Thinking of containment as division, you can read off it as “if a prime divides a product, then it at least divide one of the multiplicands”)
The significance of prime ideals is that one can consider “the quotient of the commutative ring R by a prime ideal P” and this quotient would carry the property that the product of two nonzero elements cannot be zero. (which might be hard to imagine if you haven’t even worked in say mod10 in high school)
Finally, the prime spectrum of a commutative ring R is the set of its prime ideals and is denoted by Spec(R).
Working with the ring of integers, we never think of prime ideals as sets but merely think about their generators, namely prime elements. In general, prime ideals are not necessarily generated by single elements so that we really have to think them as a whole.

Zariski Topology

Spec(R) is regarded with as a topological space when endowed with the Zariski topology.
If we take an element x of R and consider all the prime ideals that contain this element, we get a subset of R that is designated to be closed (hence, its complement is open). Such open subsets (that will simply be denoted by D_x) are the basic open subsets of Spec(R).

Sheaf of regular functions

Please revisit the writing [1] for a definition of a sheaf.
Now, that we have our topological space Spec(R), we can consider the sheaves on it. We will simply consider the sheaf of regular functions.
Since the localization of a commutative ring might be too much to include in this writing, from now on we will just work with the polynomial ring C[t]. This will give us the affine line (once we define the associated affine scheme to it).
Before defining a sheaf on Spec(C[t]), let us ask ourselves what this spectrum is. The ring C[t] is like the ring of integers in the sense that every prime ideal is generated by a (distinguished) prime element. Distinguished prime elements of C[t] are simply monic polynomials of degree 1 (search the Fundamental Theorem of Algebra). They are polynomials of the form t-c (where c is a complex number). Hence, such an element is determined by c. That said, Spec(R) (as a set) can be identified with the set of complex numbers. For the description of the Zariski topology on Spec(R) and its sheaf of regular functions, please see [1].

Affine Schemes

An affine scheme (arising from some commutative ring R) is the topological space Spec(R) together with its sheaf of functions.

Schemes

A schemes is a topological space with a sheaf of rings on it such that it admits a covering by affine schemes.
A proper way to give an example of a scheme that is not affine (although locally is by definition), one should wait for a Proj construction (a process of creating schemes out of graded commutative rings)

Author: Canberk İrimağzı

Author’s note: The author learned algebraic geometry from Qing Liu’s lovely book “Algebraic Geometry and Arithmetic Curves”. It only makes sense to include his book in the references.

References:
[1]
[2] Liu, Qing. Algebraic Geometry and Arithmetic Curves. https://books.google.com/books/about/Algebraic_Geometry_and_Arithmetic_Curves.html?id=ePpzBAAAQBAJ&printsec=frontcover&source=kp_read_button