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.

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    $\begingroup$ 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. $\endgroup$ – Andy Putman Jul 22 '18 at 21:42
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    $\begingroup$ Not sure what kind of answer you're expecting here. Nothing you've said is blatantly wrong. Does that suffice as an answer? $\endgroup$ – zzz Jul 22 '18 at 21:47
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    $\begingroup$ Mathematics is what mathematicians do. Algebraic geometry is what algebraic geometers do. $\endgroup$ – Jason Starr Jul 22 '18 at 23:00
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    $\begingroup$ 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. $\endgroup$ – Tim Campion Jul 22 '18 at 23:04

I am leaving this as an answer rather than a comment, only because I do not have enough reputation to leave comments; I will delete this answer in a few hours, once enough time has passed that it seems likely that the original poster has seen it.

SleepyGraduate, what you have sketched are some ideas and constructions which are used by people who work on a mixture of homotopy theory and algebraic geometry; it sounds to me like the picture of algebraic geometry that a graduate student might get from going to seminar talks in algebraic topology but never studying algebraic geometry from or with algebraic geometers. It is not at all an accurate picture of the entire subject of algebraic geometry, which is quite vast; if there is any unifying theme to all of it, it is probably "the study of the geometric objects which can be described in terms of vanishing of polynomials," but that description doesn't do justice to the breadth of the field. While simplicial and cohomological methods can be very useful in algebraic geometry, there is much, much, much, much more to the subject than the basic setup for (higher) stacks and their module categories and associated cohomology theories, which seems to be the focus of what you wrote about.

Here is a nice resource on Mathoverflow for suggestions for algebraic geometry textbooks. From the lists of textbooks and topics covered, you can see something of the breadth of the subject: Best Algebraic Geometry text book? (other than Hartshorne)


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