0

There are many ways to define algebraic varieties, but over $\mathbb{C}$ most of them coincide. However, there are some things about which I am not sure and which I would like to verify. I know there are many similar questions around, but I am interested in this specific formulation. References (or sketches of proofs) are very welcome.

First question: is it the same to define an affine variety as

  1. A closed, irreducible subset of some $\mathbb{C}^n$ with respect to the Zariski topology
  2. $\mathrm{Spm}(\mathbb{C}[X_1,\dots ,X_n]/I)$, where $I$ is a prime ideal and $\mathrm{Spm}$ denotes the set of maximal ideals endowed with the Zariski topology
  3. $\mathrm{Spec}(\mathbb{C}[X_1,\dots ,X_n]/I)$, where $I$ is a prime ideal and $\mathrm{Spec}$ denotes the set of prime ideals endowed with the Zariski topology.

More in detail: do all three definitions give rise to equivalent categories, when I define a variety as a separated locally ringed space over $\mathbb{C}$ that can be covered by finitely many affine varieties? (And is this even the correct definition?)

The second question is: if I am only working with closed points, is the following definition of regular function completely general and correct?

Let $X$ be a variety of dimension $n$. A function $f: X \to \mathbb{C}$ is regular at a point $x \in X$ if there is an affine open neighbourhood $U$ of $x$ and polynomials $g,h \in \mathbb{C}[X_1,\dots,X_n]$ such that $h$ is nowhere zero on $U$ and $f=\frac{g}{h}$ on $U$. The function $f$ is a regular function if it is regular at each point of $X$. We denote by $\mathbb{C}[X]$ the $\mathbb{C}$-algebra of regular functions on $X$. Equivalently, $\mathbb{C}[X] := \mathcal{O}_X(X)$.

57Jimmy
  • 6,266
  • $2$ and $3$ are not even homeomorphic. Indeed, $2$ has only closed points, while $3$ has non-closed points corresponding to prime but not maximal ideals. – Alekos Robotis May 04 '20 at 17:42
  • @AlekosRobotis Agreed, I chose the wrong formulation, which I am going to correct. They are certainly not isomorphic. But there is a bijection between the topologies, so the question on them giving rise to equivalent categories is still legitimate. – 57Jimmy May 04 '20 at 20:37
  • @KReiser It does answer the first part (very nice answer btw!), thanks for the link. I would still like a comment on the correctness of the definition of regular function. How's the official policy in these cases, should I close this and ask a new question? – 57Jimmy May 04 '20 at 20:52
  • I'm not aware of an official policy, so here's what I think. You're really asking two different questions in this post - "what's the difference between all these definitions of variety?" and "what's the right definition of regular function?". If the first one has been resolved by the link, maybe turning this post in to just the second question ("is this the appropriate definition of a regular function in each of these settings?") would be best. – KReiser May 04 '20 at 21:11
  • @KReiser I didn't know how to change this appropriately, so I closed this and asked a new question here. Thanks for the suggestion – 57Jimmy May 04 '20 at 23:04

0 Answers0