0

I am trying to understand the nullstelensatz theorem and I have a question.

First of all I understand that an ideal $I$ of a ring corresponds to an algebraic variety while maximal ideals (which are of the form $\langle x-a_i \rangle $ ) correspond to points in the variety, however I would like to know what information about the ideals and the coordinate ring of a variety can we learn by studying the variety?

For example, the variety may be connected, compact and have a singular point, what does this tell us about it's coordinate ring and it's corresponding ideal?

Also what do the operations of the coordinate ring correspond to? What does, say multiply polynomials on the coordinate ring mean for the corresponding points in the variety, if I have for example two polynomials $p,q$ in the coordinate ring, $p$ and $q$ correspond to a set of points in a variety, what points do $p+q$ and $pq$ correspond to?

Mason_dns
  • 157
  • 1
    For example, $pq$ corresponds to the union of the set of zeroes of $p$ and the zeroes of $q$. Can't say much about $p+q$, but if you consider the zeroes of the ideal generated by the polynomials $p$ and $q$ (so every such element is a sum of multiples of $p$ and $q$), then this set is the intersection of the set of zeroes of $p$ and the set of the zeroes of $q$. – Evans Gambit Aug 19 '21 at 06:08
  • 2
    If you're talking about an affine variety, all the information about the variety is contained in the coordinate ring. So this question is probably biting off a bit more than you can chew at your current level of knowledge, but on the other hand you've got a lot of stuff to look forward to seeing as you keep studying algebraic geometry. – KReiser Aug 19 '21 at 07:57
  • 1
    to answer your example: I'm assuming you are talking about variety over complex numbers: being compact means that your variety is not affine! so instead of coordinate ring, you should consider coordinate rings on every open subset, and standard theorem in complex geometry says that the only global regular functions are constant ones. being connected means that your cordinate rings do not have idempotents: elements like $f$ such that $f^2=f$. having a singular point means that for example, the maximal ideal associated to that point does not generate by a regular sequence – ali Aug 19 '21 at 09:32
  • Related – Viktor Vaughn Aug 19 '21 at 18:59

1 Answers1

1

Question: "For example, the variety may be connected, compact and have a singular point, what does this tell us about it's coordinate ring and it's corresponding ideal?"

Answer: If $k$ is the field of complex numbers and $f(x,y)\in k[x,y]$ is a polynomial, it follows the quotient ring $A(C):=k[x,y]/(f(x,y))$ is the "coordinate ring" of the curve $C:=Z(f)$ defined by the polynomial $f$. By definition: $Z(f)\subseteq k^2$ is the set of zeros of the polynomial $f(x,y)$.

Example: Let $f(x,y):=y-x^2$. It follows its zero set is the (complex) parabola. As a set $C$ is the set of points $(z,z^2)\in k^2$ where $z\in k$ is an arbitrary complex number.

Properties such as singularity, non-singularity, reducibility, dimension, etc are reflected in the coordinate ring $A(C)$. If $C$ is non-singular it follows $A(C)$ is a regular ring. The dimension $dim(C)$ is reflected in the ring $A(C)$: We may define the "ring theoretic dimension" $krdim(A(C))$ of $A(C)$ and it turns out $krdim(A(C))=dim(C)$. The curve $C$ is irredicble iff the ideal $I(C):=(f) \subseteq k[x,y]$ is a prime ideal. The curve $C$ is a disjoint union of "curves"

$$C:=C_1 \cup \cdots \cup C_n$$

iff there is a direct sum decomposition of rings $A(C) \cong A_1 \oplus \cdots \oplus A_n$. The tangent space $T_x(C)$ of $C$ at a (closed) regular point $x$ may be defined as

$$T_x(C):=(\mathfrak{m}_x/\mathfrak{m}_x^2)^*,$$

where $\mathfrak{m}_x \subseteq A(C)$ is the maximal ideal corresponding to $x\in C$. You may always define the vector space $(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, and it follows

$$D1.\text{ }dim(C) \leq dim_k((\mathfrak{m}_x/\mathfrak{m}_x^2)^*)$$

for all closed points $x$. We say $x$ is a regular point iff $D1$ is an equality. Hence you use the ring $A(C)$ to define "regularity".

Question: "Also what do the operations of the coordinate ring correspond to? What does, say multiply polynomials on the coordinate ring mean for the corresponding points in the variety, if I have for example two polynomials p,q in the coordinate ring, p and q correspond to a set of points in a variety, what points do p+q and pq correspond to?"

Answer: You should view $A(C)$ as the "ring of globally defined functions on $C$". Two (equivalence classes of) polynomials $f,g\in A(C)$ define functions (when you are working over an algebraically closed field $k$)

$$f,g:C \rightarrow k.$$

When you add or multiply $f,g$ you add and multiply these functions. The "nullstellensatz" implies there is a 1-1 correspondence between maximal ideals $\mathfrak{m} \subseteq A(C)$ and points $(a,b)\in k^2$ with $f(a,b)=0$. Hence there is a 1-1 correspondence between (closed) points $x\in C$ and maximal ideals $\mathfrak{m}_x\subseteq A(C)$.

hm2020
  • 10,015