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.

will bewhen 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 Dec 25 '17 at 15:44betterin 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 Dec 25 '17 at 22:22