Algebraic Geometry & Commutative Algebra

Let's start with some setup. A ring $(R, +, *)$ is an algebraic object so that $(R,+)$ is an abelian group:

  1. Closure: $a + b \in R$ for every $a,b \in R$.
  2. Associative: $a + (b + c) = (a + b) + c$
  3. Identity: There exists $0_R$ such that, for every $x \in R$, $0_R + x = x = x + 0_R$.
  4. Inverses: Given $x \in R$, there exists a $y \in R$ such that $x + y = 0_R = y + x$. We usually write $y = -x$.
That the group is abelian means that $x + y = y + x$, for every $x,y \in R$. We also need the multiplication to be compatible with addition, in the sense that, the distributive laws hold:
  1. $a * (b + c) = a*b + a*c$
  2. $(a + b) * c = a*c + b*c$
The multiplication, in a ring, is associative $a*(b*c) = (a*b)*c$.

Notes: A couple of notes.

  1. It is not necessary for a ring to contain a multiplicative identity, $1_R$. However, we will not pursue such rings here and often the definition of a ring is given, with the assumption that a $1_R$ exists.
  2. Nor is it necessary for the multiplication of a ring to be commutative; that is $ab \neq ba$ in a noncommutative ring. An example of a noncommutative ring is the set of $n \times n$ matrices, over a field $k$, $\operatorname{Mat}_{n \times n}(k)$.

Often times, rings are too large to study outright. The following definition and the forming of quotients helps with this problem.

Defn. An ideal $I$ of a ring $R$ is a subset of $R$ such that:

  1. $I$ is closed under addition.
  2. $rf$ is an element of $I$, for every element of $r \in R$ and $f \in I$.

Addtional Note: Noncommutative rings are interesting to study. There are some structure theorems, such as the Artin-Wedderburn theorem, that are helpful. On the other hand, noncommutative rings display some bizarre properties. A simple ring is one whose two-sided ideals are the trivial ones. However, there exist simple rings which have nontrivial left ideals.

While Hilbert was pursuing a problem in invariant theory, he proved the following two theorems.

Nullstellensatz: Let $R$ be a polynomial ring, in $n$ variables, over an algebraically closed field $k$, and $J \subset R$ an ideal. Then \[ I(V(J)) = \sqrt{J} \] where $V(J)$ is the set of points which vanish on all of the functions in $J$.

Basissatz: Every ideal in the polynomial ring, in $n$ variables over a field, is finitely generated.

These two theorems give a connection between the algebraic structure of rings, and the geometry of varieties. A variety is determined by the vanishing of a finite number of polynomials. These polynomials generate an ideal, and the nullstellensatz tells us that this ideal is determined up to radical. One can start looking at an arbitrary ring instead of just being restricted to polynomial rings. This leads one to look at the spectrum of the ring, to which one can apply the Zariski topology. This is the beginning of abstracting the ideas in classical algebraic geometry.

There are a number of nice numerical invariants, for a commutative Noetherian ring. Two primary examples are:

  1. The dimension of the ring.
  2. The depth of the ring.
When the dimension and the depth are equal, we say that a ring is Cohen-Macaulay . A Gorenstein ring is a Cohen-Macaulay ring which is Type 1. These are very nice properties to have in a ring. However, there are still some behaviors that they exhibit we would like to get under control. Grothendieck put forward the idea of an excellent ring. These are the rings that are believed to have a resolution of singularities. These are but some of the notions that are used in the modern study of commutative algebra.

The above is mostly motivated from the algebraic point-of-view. What of the geometric issues? The above tells us that we can take some variety $X$ and consider the ideal $I(X)$. We can then take consideration of the ring of regular function $k[X] = k[x_1, \ldots, x_n]/I(X)$. Taking quotients of regular functions, one forms the rational functions $k(X)$. One can then consider the transcendence degree of $k(X)$. Even though this is a rather course description of a transcendental extension, it conincides with the dimension of the variety $X$.

The tools, one uses in algebraic geometry, are dependent on the type of problems that one is working on. I have a large interest in elliptic curves, which has lead me to consider diophantine problems, and the tools of arithmetic algebraic geometry. This tends to lead one quickly down the road of sheaves (local descriptions and gluing) and schemes. These tools are very technical and abstract; too the point that some mathematicians readily express their fear regarding them. One of the things mathematics can teach us is to not be afraid of language. It also reinforces the fact that one must be able to communicate more than the definitions, for language to be effective.

As a graduate student, I had the opportunity to take courses in tight closure, local cohomology, and algebraic geometry. I hope to get some notes up that were put together for an underground seminar in commutative algebra.