Rambling Thoughts on Geometry

I am not sure that everyone in the class has been exposed to “classical” algebraic varieties.  Without some background here,  I would think it would be hard to understand why we are doing anything we are doing in class.  So I hope that we can use this space to write up rambling thoughts on the underlying geometry here.

2 Responses to “Rambling Thoughts on Geometry”

1. stevengubkin Says:

1. Points as maximal ideals (later to be expanded to prime ideals).

We are used to thinking about points in the affine plane as $\mathbb{C}^2$ as points of the underlying set of $\mathbb{C}^2$, or more categorically, as morphisms of sets $1 \rightarrow \mathbb{C}^2$. This is geometry. We are interested in algebraic geometry, so we want to reformulate our geometry in terms of algebra (in the hopes that on the algebra side things will generalize in new and exciting ways). So instead of functions of sets, we want to regard points as homomorphisms of finitely generated $\mathbb{C}$ algebras. The natural way to do that is to associate to each point $(a,b)$ the evaluation map $Ev_{(a,b)}: \mathbb{C}[x,y] \rightarrow \mathbb{C}$. Then it is easily seen that each point of $\mathbb{C}^2$ corresponds to a homomorphism, and conversely each homomorphism $\mathbb{C}[x,y] \rightarrow \mathbb{C}$ must be evaluation at some point ($x$ and $y$ have to go somewhere). If we want to be fancy we can write $Hom(1, \mathbb{C}^2) \cong Hom (\mathbb{C}[x,y], \mathbb{C})$. Since these homomorphisms are also determined by their maximal ideals (namely, $(x-a,y-b)$), we have a third equivalent way of looking at points of the affine plane.

So now we have an understanding of how to look at point of the affine plane algebraically: They are $\mathbb{C}$-algebra homomorphisms $\mathbb{C}[x,y] \rightarrow \mathbb{C}$. We might even call $\mathbb{C}[x,y]$ the coordinate ring of the affine plane, because homomorphisms out of it correspond to points of the affine plane. Can we find coordinate rings of other geometric objects? For example, is there a ring $\Gamma(C)$ such that the homomorphisms $\Gamma(C) \rightarrow \mathbb{C}$ correspond to points of the unit circle $C = \{(x,y) \in \mathbb{C}^2| x^2 + y^2 = 1\}$? Yes we can! The ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ does the trick: A homomorphism $\phi: \mathbb{C}[x,y]/(x^2+y^2-1) \rightarrow \mathbb{C}$ is still determined by where it sends $x$ and $y$, but this homomorphism is now subject the the requirement that $\phi(x)^2 + \phi(y)^2 = 1$ (remember these are $\mathbb{C}$ algebra homomorphisms, so they fix 1), i.e. that it corresponds to a point on the unit circle. In general the coordinate ring of an algebraic curve $C$ is $\mathbb{C}[x,y]/I(C)$, where $I(C)$ is the ideal of all polynomials vanishing at every point of $C$. This is the proper way to translate between varieties and algebras!

We want this assignment of coordinate rings to be functorial if it is good for anything, but we have not even defined morphisms between varieties yet. We define away this problem by choosing our maps between varieties to be the ones that correspond to algebra homomorphisms. Essentially, we are looking at the slice category $( \mathbb{C}-Alg \downarrow \mathbb{C} )$. I leave it to you to work out the details. (Hint: The maps are called polynomial maps)

2. stevengubkin Says:

To come soon: Why intersection theory requires schemes (intersection multiplicity is governed by the dimension of coordinate rings – which may have nilpotent elements!)

To come soon after that: A picture which explains why $^a\phi: Spec(A_f) \rightarrow Spec(A)$ is a homeomorphism onto it’s image, which is $D(f)$