# What is the big picture of algebraic geometry? [closed]

I am trying to understand a big-picture for Algebraic Geometry:

Given a category of commutative rings $\mathrm{CRing}$, we can create objects that locally look like these objects called schemes. This is done by taking the topos of sheaves of $\mathrm{CRing}$ with respect to a Zariski (or even etale) topology. The subcategory which has objects covered by representables (objects mapped from $\mathrm{CRing}$ via yoneda embedding and sheafification) via open maps.

So this overall constructs some "locally isomorphic to rings" object. There are a few issues one needs to consider which can be fixed by restricting to finitely presentable rings (does this make the schemes (quasi-)compact?).

So in analogy, if you want to study $\mathrm{CRing}$ you can study modules instead which are "generalised functions".

So we can take our previous schemes and generalise them even more by thinking about sheaves over them with respect to another topology, this time we have all sorts of choices (fppf, zar, h-, etale etc.).

Now these form a "big zariski topos" so we have an internal language, thus we can describe rings and modules internally. Categories of these become sheaves of rings and (quasi-)coherent sheaves respectively.

The category of (quasi-)coherent sheaves over some scheme can have its simplicial objects studied. Traditionally this is done via the Dold-Kan correspondence and gives the derived category construction. This homotopy category is a big package describing (co)homology theories. Other cohomology theories that arise in AG can be studied by changing the topology chosen earlier.

Now this big picture has some quite broad strokes and many different aspects which one can generalise. Why stick with with rings when you can use $E_\infty$-rings, why consider sheaves when you can consider $\infty$-stacks etc.

So my question is: How accurate is this picture?

I know I am being a bit cheeky by calling all of this algebraic geometry when there is so much more, (for example classifying varieties birationally etc).

Oh and just for note: cohomologies I have described above are worth studying, not because of all this silly category theory but because they describe and classify important and useful invariants of the geometric objects in question.

## closed as off-topic by Andy Putman, David White, Mark Sapir, R. van Dobben de Bruyn, Harry GindiJul 23 '18 at 0:11

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andy Putman, David White
If this question can be reworded to fit the rules in the help center, please edit the question.

• This question is far too vague to answer (though I suspect that very few working algebraic geometers will recognize your account as describing how they think about their subject). I have voted to close. – Andy Putman Jul 22 '18 at 21:42
• Not sure what kind of answer you're expecting here. Nothing you've said is blatantly wrong. Does that suffice as an answer? – zzz Jul 22 '18 at 21:47
• Mathematics is what mathematicians do. Algebraic geometry is what algebraic geometers do. – Jason Starr Jul 22 '18 at 23:00
• A correction -- $Spec(R)$ is quasicompact for any commutative ring $R$. So cutting down to finitely-presentable rings is much more drastic than imposing a quasicompactness condition. Of course, the point is that it doesn't matter because a reasonable functor should be determined by its values on finitely-presentable objects. Also, I think there's something to be said about how scheme locally look like the opposite of a ring rather than a ring itself, though I'm not sure exactly what. That being said, personally I sympathize roughly with the perspective here, though I am no algebraic geometer. – Tim Campion Jul 22 '18 at 23:04