Let $K$ be an algebraically closed field.

Define an algebraic variety to be a pair $(X,\mathscr{O}_X$) for a topological space $X$ together with a sheaf $\mathscr{O}_X$ that is a subsheaf of the sheaf of germs of function on $X$ with values in $K$ subject to the following axioms:

A1) There exists a finite open cover $\{V_i\}_{i=1}^{n}$ of $X$ such that each $V_i$, equipped with the induced structure from $X$, is isomorphic to a locally closed subspace $U_i$ of an affine space, equipped with the sheaf $\mathscr{O}_{U_i}$.

A2) The diagonal $\Delta$ of $X \times X$ is closed in $X \times X$.

Define an algebraic variety to be affine if it is isomorphic to a closed subvariety of an affine space.

Define an open subset $U$ of an algebraic variety $V$ to be affine if, equipped with the induced structure of an algebraic variety from $V$, $U$ is an affine variety.

Define For any algebraic variety $V$, and any regular function $f \colon V \to K$, let $V_f$ be the set of points $x \in V$ for which $f(x) \neq 0$.

Proposition 2: $V_f$ is open in $V$ and $V_f$ is affine.

Proposition 3: Let $V$ be a closed subvariety of $K^r$, $F$ be a closed subset of $V$ and let $U = V - F$. Then the open subsets $V_p$ form a base for the topology of $U$ when $P$ runs over the set of polynomials vanishing on $F$.

Remark The terminology and concepts here are a bit outdated from modern algebraic geometry. If you need more definitions / terminology in order to answer my question, let me know and I will provide the details. All of this is from Serre's FAC, most all of this is in Chapter II.

Theorem 1. The open affine subsets of an algebraic variety $X$ form an open base for the topology of $X$.

Proof. The question being local, we can assume that $X$ is locally closed subspace of an affine space $K^r$; in this case, the theorem follows immediatley from Propositions 2 and 3.

Question / Thoughts: First, I am not sure I understand the form of the argument. Would you agree the following is more or less the structure of how Serre is proving the theorem?

  1. Let $V$ be an algebraic variety such that the open cover in axiom A1) has one element. This means all of $V$ is isomorphic to some locally closed subspace of an affine space (which WLOG we can choose to be $K^r$?)
  2. In this case, we can apply Proposition 2 and Proposition 3 to yield the result, i.e. open affine subsets of $V$ form a basis for the topology of $V$.
  3. The result holding in the case where the open cover has one element implies that the result holds in the case where the open cover in axiom A1) has any finite number of elements.

If this is how Serre is arguing, please help me see why 2) and 3) in the list above are true. How can I use proposition 2) and proposition 3) to yield the result in the special case? And how can I argue that I can generalize the result to when $V$ has more than one chart?

  • $\begingroup$ Serre is saying since by definition a variety has a cover by sets which are (isomorphic to) locally closed subspaces, it suffices to check on these local patches. Hence we might as well just assume the variety in question itself is affine, since once we find a basis for all affines, the topology carries over when we glue them together to create a non-affine variety. $\endgroup$ – Ryan Keleti Feb 23 at 23:57
  • $\begingroup$ I am asking for the recipe to “carry the topology over when we glue” and how to prove it in the affine case. $\endgroup$ – Prince M Feb 24 at 3:16
  • $\begingroup$ Also, the charts are not affine, they are locally closed which is quasi affine. $\endgroup$ – Prince M Feb 24 at 3:18
  • $\begingroup$ My bad, replace affine with quasi-affine. I think what you're looking for is "gluing data" (for example see here math.ucdavis.edu/~osserman/classes/248A-F13/…). $\endgroup$ – Ryan Keleti Feb 24 at 3:47
  • $\begingroup$ The first question is purely topological it says if you have a basis on the open subspace for each element of an open cover you can lift it to a basis on the space, we just need to check that the lifted element is also of the for $V_f$, for the second questions we need a clever way to express X as V - F so we can apply proposition 3. $\endgroup$ – Prince M Feb 24 at 4:08

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