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I have a very vague question, but also a fairly specific wish. Namely, I'm wondering what the similarities and differences are between the theory of ordinary schemes on the one hand, and the theory of derived schemes on the other.

I know we have an adjunction between schemes and derived schemes, induced by truncation/inclusion between rings and simplicial rings. I'm not sure how much mileage one gets out of this.

Ideally, one would have some mechanism or theory, stating how to transport results from the classical case to the derived case. For example. maybe one can make a precise logical statement of the form: if my proposition is not too complex in some sense, then replacing identities with homotopies will give me a proposition in the derived setting.

I've once heard that the cotangent complex can help you with this. Any reference for this would be greatly appreciated.

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    $\begingroup$ One of the traditional motivations is intersection theory. If you consider two ordinary affine schemes $SpecR$, $SpecS$ over a base $SpecA$, then their fibered product as schemes is given by $Spec(R\otimes_A S)$. However if we consider them as constant derived schemes, then their fibered product as such is the affine derived scheme $Spec(R \otimes^L _A S)$ (derived tensor product of simplicial rings). The significance of this is that the derived fibered product remembers all the $Tor^A_n(R,S)$. So by Serre's intersection formula, derived intersections remember all intersection multiplicities. $\endgroup$
    – JJJ
    Commented Mar 31, 2019 at 2:52
  • $\begingroup$ Right. I guess right now I'm more looking for similarities than differences. For example, from the perspective of derived schemes, these derived fiber products still behave like intersection, right? E.g., I believe that the non-empty fibers of open immersions are given by fields. Basically, my question is: does one need to prove all of the expected behavior one statement at a time, or is there some general machinery for this? $\endgroup$
    – Maanroof
    Commented Mar 31, 2019 at 3:08

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