minor corrections

Anonymous

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Anonymous

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- Bertrand Toën,
*Higher and derived stacks: a global overview*, In:*Algebraic Geometry Seattle 2005*, Proceedings of Symposia in Pure Mathematics, Vol. 80.1, AMS 2009 (arXiv:math/0604504, doi:10.1090/pspum/080.1)

I added the notes

- Clark Barwick,
*Applications of derived algebraic geometry to homotopy theory*, lecture notes, mini-course in Salamanca, 2009, pdf.

to some of the relevant pages.

]]>Thanks!

]]>I added links to video recordings of Lurie’s 2014 UOregon Moursund Lectures at spectral algebraic geometry.

]]>Yes, but maybe I didn’t say it well. Shouldn’t one really think of it like this: arithmetic geometry is fundamentally geometry over $\mathbb{Z}$, hence with no condition on the commutative rings whatsoever. Then looking at $\mathbb{Z}_p$ and $\mathbb{Q}$ is “just” a way to decompose this difficult subject into slightly less difficult subjects.

In this vein, since the sphere spectrum $\mathbb{S}$ is the homotopy-theoretic refinement of $\mathbb{Z}$, we are led to saying that “higher arithmetic geometry” (or “derived”, if you insist) is geometry over $\mathbb{S}$, hence with $E_\infty$-rings without any further conditions. Again, you may (and you will) go and decompose this into problems over $\mathbb{Z}_p$ and $\mathbb{Q}$, but as before, this is more a “tool” than an end in itself, the subject as such is really geometry over $\mathbb{S}$. Hence “homotopy arithmetic”.

]]>64 “Complex algebraic geometry” is not a good way to describe characteristic 0. Most characteristic 0 stuff cannot use anything from complex geometry. I think what Urs was getting at is that doing geometry over $\mathbb{Z}_p$ or even $\mathbb{Q}$ is clearly arithmetic, but it lives in characteristic 0.

]]>Oh, I see. Okay.

]]>I actually find the term “E-infinity geometry” good, “objectively” speaking, my only complaint was about the disagreement with the literature.

]]>Okay, the “fundamental” argument is too vague to be useful, admitted. But “spectral geometry” is an unfortunaley ambiguous term for such a profound concept. And about mysuggestion “$E_\infty$-geometry” it was actually you who raised complaints above!

Moreover, if “arithmetic” makes you think “positive characteristic” then that’s pretty good already, I’d say, since it makes clear already that we are not talking about dg-algebras. And “arithmetic” doesn’t suggest anything motivic, does it?

So it seems you should actually be happy with the suggestion of “arithmetic higher geometry”!:-) But of course it was just a thought. If it doesn’t resonate with you or anyone, Iwon’t insist.

]]>Ok, I see what you mean. I’ve always understood arithmetic geometry to refer specifically to characteristic p methods. And I would probably use “complex algebraic geometry” for characteristic zero. That’s why I prefer either the term spectral algebraic geometry or E-infinity geometry.

Also I am not sure that this is the most general or fundamental version of algebraic geometry one could consider (which is what the name “higher arithmetic geometry” might suggest). Perhaps one could take different categories than spectra, for example motivic spectra (which would allow viewing the algebraic K-theory spectrum as an affine scheme, and studying its K-theory or other invariants…).

]]>So also the term “arithmetic geometry” refers to just algebraic geometry, but highlighting the fact that one will not restrict attention to algebraically closed fields, not to characteristic 0 and possibly not even to fields. It’s still “algebraic geometry”, but with the emphasis that the “most fundamental” variant is meant.

Similarly here: derived algebraic geometry over $E_\infty$-rings is just part of derived algebraic geometry, but one needs a word to highlight that one will consider $E_\infty$-algebras *not* in characteristic 0 – where they give just the dg-algebra sector of DAG – but that one really means the “most fundamental” version of the theory over general $E_\infty$-rings with no simplifying assumptions.

arithmetic geometry is about the formal duals of $\mathbb{Z}$-algebras aka commutative rings

But isn’t this a description of algebraic geometry in general?

]]>I mean, arithmetic geometry is about the formal duals of $\mathbb{Z}$-algebras aka commutative rings, whereas here we are dealing with formal duals of $\mathbb{S}$-algebras, aka $E_\infty$-rings.

]]>What is the connection with arithmetic geometry?

]]>Coming back to the discussion about good termionology for geometry modeled on formal duals of $E_\infty$-rings (from earlier this year above):

wouldn’t **higher arithmetic geometry** be a good term? (or maybe “arithmetic higher geometry”)?

I mean on absolute grounds, independent of whether it’s wise to use a different term than already established.

]]>So I forgot about this discussion and accidentally duplicated the page E-infinity geometry at spectral algebraic geometry. Can we delete the latter page somehow?

]]>re#55 Woops, you are right, I managed to trick myself here, first changing the terminology and then forgetting that it was me who changed it. Sorry.

]]>So I think disambiguation pages are a good idea, but still I would prefer to stick to the standard usage of terms (complicial and spectral vs DG and E-infinity) because disambiguation pages are necessary in either case, and there is less risk of confusion anyway between spectral algebraic geometry and spectral theory, than between dg-geometry and the older notion of dg-scheme.

]]>For instance I think that Toen-Vezzosi in their HAG-articles do speak precisely of “dg-geometry”. We have an entry dg-geometry. So E-infinity geometry seems not too bad.

Our entry on dg-geometry is what Toën-Vezzosi call complicial algebraic geometry (from “complex”). In the literature dg-scheme refers to something different, which is just a historical precursor to the correct notion of derived scheme. Kontsevich and Toën have both suggested forgetting this notion entirely, but there are still important papers of Ciocan-Fontanine and Kapranov where they construct eg derived Hilbert scheme and an explicit translation into DAG still hasn’t appeared as far as I know. So in my opinion it is still too early to start using the term dg-geometry without risk of confusion.

It is worthy to mention also that the algebraic geometry over spectra of algebraic topology has been called new brave algebraic geometry by Peter May and was quite an accepted term already a decade ago.

This should be the same as the brave new algebraic geometry considered in HAG II. This is the third HAG context they consider, but it is not the same thing as spectral algebraic geometry. If I recall correctly spectral AG doesn’t fit into the TV framework (it’s not a HAG context).

]]>I added the sentence

Most of those belong either to the geometry as seen either by point spectra of spectral theory (of operators, families of operators, operator algebras, rings, associative algebras, abelian categories etc.), or by spectra in the sense of stable homotopy theory like symmetric spectra, $E_\infty$-spectra, ring spectra…

to spectral geometry and updated discussion in 52 above.

]]>By the way, we also have an entry *brave new algebra*. In case anyone feels that’s worth pointing to for clarification in some entry.

it is safe to call this spectral

algebraicgeometry,

No, it is not ! Not only spctra of Laplacians and Dirac operators but also spectra of operator algebras extend beyond operator algebraic context. There are spectra and their elaborate geometry (structure sheaves, structure stacks, geometric morphisms, cyclic homology, coherent sheaves etc.) not only for operator algebras but also for various classes associative (possibly noncommutative) algebras, abstract noncommutative schemes, abelian categories, exact categories and so on.
They nontrivially extend the notion of prime spectrum of a ring, cf. entry spectral theory. This is an extensive subject which has *preceded* noncommutative geometry a la Connes (say works of P. Gabriel, P.M. Cohn, Golan, Van Oystaeyen and others in 1960-s and 1970-s). Therefore for lots of people, specially in “my” subject, spectral algebraic geometry means something different. Also, what is very important, the study of spectra of operator algebras and other topological algebras and of non-topological algebras and some other kinds of algebras are all quite related and it would be non-natural to separate those. by allowing spectral geometry terminology for one and not for another.

It is worthy to mention also that the algebraic geometry over spectra of algebraic topology has been called new brave algebraic geometry by Peter May and was quite an accepted term already a decade ago. Of course, introducing clashing terminology is allowed to mathematicians who made important work (and in contemporary hyperproduction of new elaborate abstract formalisms almost impossible to avoid), but calling this terminology “safe” is not justified.

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