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New stubs noncommutative projective geometry and Michael Artin.
One question: you write $\mathbb{N}$-graded algebra, shouldn’t it be $\mathbb{Z}$-graded?
One remark: So what about Kontsevich-style noncommutative projective geometry: characterizing the projective noncommutative schemes by their $A_\infty$-categories of would-be quasi-coherent sheaves? This is what Jim Dolan was indicating on the nCafe.
I have edited graded algebra a little and moved the proof that gradings on $R$ are equivalent to $\mathbb{G}_m$-actions on $Spec R$ from projective space to here.
$\mathbb{N}$-graded is right. Quasi-coherent sheaves over projective schemes correspond to ℕ-graded modules (modulo a subcategory).
Thanks, my bad. A $\mathbb{G}_m$-action on $Spec R$ corresponds to a $\mathbb{Z}$-grading, but if we think of $Spec R$ as being $(Spec S) - \{0\}$ then this comes from an $\mathbb{N}$-grading on $S$.
I was not thinking about grading in the fundamental sense, just by experience what I know about projective geometry. Thanks for making arguments why they hold :)
I have archived now obsolete discussion about an early state in the development of the entry noncommutative algebraic geometry which was fopr a while part of that same entry and I left a link there to this archived version (btw. it is a pity Lieven did not become a contributor in $n$Lab yet):
Lieven Le Bruyn to the Lab-elves : what follows is part of the original text by Zoran Skoda based on his association with Alex Rosenberg. I’ve tried to include the major developments above with precise references. I don’t understand the Van Oystaeyen-Willaert vs. Rosenberg part nor the Le Bruyn (that is,me) vs. Laudal part. But, feel free to edit in any way you feel like.
Zoran Škoda: my (very unfinished and criticised before any level of completion beyond quick initial start) entry is not at all based on any association, but rather on much more general viewpoint on noncommutative geometry including not only ring-theoretic aspect but also the categorical aspect on the line GABRIEL-Manin-Rosenberg in abelian and Kapranov-Bondal-Orlov-van den Bergh-Kontsevich in triangulated, dg and A-infinity setup. My emphasis was more on concepts, and categorical viewpoint, than on listing all references (history entry should be separately and welcome). For example, in the addition above casually is mixed graded and nongraded setup: nlab regulars should be warned about points of entry and motivation of affine and proijective geometries. On the other hand, I am not competent enough to write extensively on the very developed, but for category theorists rather special case of slightly noncommutative projective geometry of line Artin-Schelter-Tate-Zhang-Smith though intend to do part of this in specialized entries (one of the reasons why I listed ALL papers in Grothendieck Festschrift was that interest in Artin’s work; last week I went to integrable systems conference partly because of interest in elliptic algebras which belong to the class from this paper; Le Bruyn should notice that I was long time ago passworded member of his web chat group on his style of ncg which to my regret was not very active). I placed some of the stuff from below into the text above to make it all-inclusive without disposing any material except the repetitions of facts. Much more is missing here, for example algebraic K-theory for nc rings and cyclic homology etc.
I have made also some additions to noncommutative algebraic geometry, including some new text passages and additions and corrections to links.
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