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I do not know very much (in fact, almost nothing) about algebraic geometry but have some knowledge in commutative algebra (e.g. taken some graduate-level courses in the theory of rings, modules, and Galois theory).

I have recently worked through Sheaves in Geometry and Logic because I became interested in topos theory on logical and foundational grounds. I was interested in some of their examples of the Zariski site, but not having any knowledge in Algebraic Geometry, I felt that I missed the full power of topos theory as applied in this way (and it also took me far longer to fill in the details of the examples than it presumably would if I had been familiar with the subject). Also, it's a branch of math which I haven't gotten around to learning that sounds really interesting for its own sake.

What would be a good text for learning algebraic geometry which fully leverages category theory, sheaves, and topos theory from the beginning?

This question is similar to some others on this site, but other questions focus on either (1) looking the applications of topos theory to algebraic geometry, from the perspective of someone who already knows some of both, (2) learning algebraic geometry only as it applies to understanding specific examples in topos theory, or (3) learning enough topos theory to apply it to algebraic geometry. None of these match my particular situation.

I don't mind difficult, dense, or abstract texts. But I would rather not go on a wild nCatLab/Wikipedia chase to find introductory theorems in algebraic geometry that are casually referenced in the book, so I do require that it be reasonably introductory for someone with my knowledge.

Thanks!

Mark Saving
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  • I often see Ravi Vakil's book The Rising Sea recommended as a very categorical/sheafy book on algebraic geometry. I can't vouch for it myself, but it's free online so you can at least check it out. – Malice Vidrine Jul 22 '21 at 19:27
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    However, Vakil says "if I use the word “topoi”, you can shoot me" at the beginning of the book, and his book intersects with Sheaves mainly through (double-)starred sections. At the same time, having myself read Sheaves and wishing, like you, to build understanding of AG on that, I still use The Rising Sea as the main AG textbook. Knowing topoi helps here and there, but I'm not sure a lot more can be derived from it. – Арсений Кряжев Jul 24 '21 at 16:50

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