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So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through the whole subject, can anyone briefly tell me what to expect from the subject? More precisely, I'd like to know how much I can expect my intuition from classical (scheme-theoretic) algebraic geometry to transfer over to this setting.

For example, can I expect standard properties of morphisms (finite-type-ness, finiteness, flatness, etc.) to transfer over to the derived setting? Can one define reasonable analogues of group schemes in this new setting (what are Hopf algebras in $\mathbb{E}_{\infty}$ rings, what provides the cofibered product)? If so, can we construct quotients etc.?

Just for reference, the way I understand the basic object of derived algebraic geometry (the derived scheme) is as follows. Let $A$ be an $\mathbb{E}_{\infty}$ ring, that is, a commutative monoid in the category of spectra. As far as my (very limited) understanding goes, the spectral affine scheme $\mathbf{Spec}(A)$ associated with $A$ may be identified with the spectrally ringed $\infty$-topos $(Shv(\operatorname{Spec}(\pi_0A)), \mathcal{O}_A)$, where $\mathcal{O}_A(U_f)=A[\frac{1}{f}]$. Then a spectral scheme is a spectrally ringed $\infty$-topos that is locally affine.

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    $\begingroup$ I have added a top-level tag (ag.algebraic-geometry) - I hope it is the correct one. I guess that (big-picture) tag might also be suitable for this question, but all five spots are already taken. $\endgroup$
    – Martin Sleziak
    Commented Dec 25, 2017 at 9:07
  • $\begingroup$ @MartinSleziak Good suggestion! I removed schemes and added the big-picture tag $\endgroup$
    – Exit path
    Commented Dec 25, 2017 at 9:10
  • $\begingroup$ I feel like this question is a bit pre-emptive, since so much of SAG is still being developed. The best you can get is an answer advertising how great it will be when it's sorted out. What references are there on spectral algebraic geometry other than Lurie's newest book project? Speaking of which, searching in math.harvard.edu/~lurie/papers/SAG-rootfile.pdf only turned up 3 hits for "Hopf algebra" and none related to your question. I'll bet there are a lot of hits for "flat", but I'm not going to comb through them all. $\endgroup$
    – David White
    Commented Dec 25, 2017 at 15:44
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    $\begingroup$ I heard people say that what algebraic geometers really need is actually derived (higher or not) geometry (the one done, say over $\mathbb{C}$, via functors on commutative dg-algebras), while the "brave new" variant (aka spectral algebraic geometry) is more suited for the needs of homotopy theorists. I don't know if it's true. But your learning plan should take into account where you want to arrive. There a MO question about motivation for DAG: mathoverflow.net/questions/226082/… $\endgroup$
    – Qfwfq
    Commented Dec 25, 2017 at 20:25
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    $\begingroup$ Some things will carry true (mainly the formal things like what's the fiber product of affines), some things will be awful (there's no good notion of closed subscheme in SAG that incorporates all the examples we care about), some things work better in SAG than in classical algebraic geometry(for one, everything is "flat" in some weak sense, and you can expect much more stuff to be "affine" in a weak sense, and the cotangent complex of a good stack is always perfect!). It is a different if strongly related subject.DAG with simplicial rings is much more similar to classical algebraic geometry. $\endgroup$
    – Denis Nardin
    Commented Dec 25, 2017 at 22:22

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