Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutative setting and which parts don't?
Edit: I also welcome any comments/references regarding any reasonable notions whatsoever of "non-commutative algebraic geometry".
I think it is helpful to remember that there are basic differences between the commutative and non-commutative settings, which can't be eliminated just by technical devices.
At a basic level, commuting operators on a finite-dimensional vector space can be simultaneously diagonalized [added: technically, I should say upper-triangularized, but not let me not worry about this distinction here], but this is not true of non-commuting operators. This already suggests that one can't in any naive way define the spectrum of a non-commutative ring. (Remember that all rings are morally rings of operators, and that the spectrum of a commutative ring has the same meaning as the [added: simultaneous] spectrum of a collection of commuting operators.)
At a higher level, suppose that $M$ and $N$ are finitely generated modules over a commutative ring $A$ such that $M\otimes_A N = 0$, then $Tor_i^A(M,N) = 0$ for all $i$. If $A$ is non-commutative, this is no longer true in general. This reflects the fact that $M$ and $N$ no longer have well-defined supports on some concrete spectrum of $A$. This is why localization is not possible (at least in any naive sense) in general in the non-commutative setting. It is the same phenomenon as the uncertainty principle in quantum mechanics, and manifests itself in the same way: objects cannot be localized at points in the non-commutative setting.
These are genuine complexities that have to be confronted in any study of non-commutative geometry. They are the same ones faced by beginning students when they first discover that in general matrices don't commute. I would say that they are real, fascinating, and difficult, and people have put, and are currently putting, a lot of effort into understanding them. But it is a far cry from just generalizing the statements in Hartshorne.
There are 2 major problems in extending Hartshorne chp 2 to the noncommutative case.
The first is a good definition of a topological space and a structure sheaf on it so that one recovers the ring back as its global sections.
The second is the failure of functoriality.
Several proposals have been made for a noncommutative structure sheaf. As topological space one has taken either the set of all twosided prime ideals or the space of prime torsion functors. On the prime ideals some sheaves of rings have been defined by symmetric or bimodule localizations such that one recovers the ring back as the global sections (as least in the Noetherian case). On the prime torsion space this fails.
However, a ringmorphism A-->B does not in general induce a map on the twosided prime ideals (notable exceptions are Procesi's central extensions, explaining why functionality is no problem in the commutative case).
It is mainly failure of functoriality that has brought some people to DEFINE Mod(A) as the 'sheaves of coherent modules on a non-existant space associated to A'. Clearly, any ringmorphism A-->B defines a functor Mod(B)-->Mod(A). As long as one is interested in homological/geometric properties things can be extended to the noncommutative world. Some might argue that in this proposal one is doing category theory rather than geometry, failing a topological space and structure sheaves.
I guess there is some consensus these days that there is NO SINGLE 'noncommutative geometry' suitable for all noncommutative algebras. That is, depending on the class of algebras under investigation one might consider other spaces/sheaves. For example, for PI-rings (roughly rings finite over their center) one can go a long way defining everything over the central scheme or by considering moduli spaces of finite dimensional representations. On the other hand, for filtered rings with associated graded a commutative ring, it might be more fruitful to take sheaves of micro-localizations. Etc. etc.
Attempts to get as close as possible to Hartshorne's were made in the 70ties, so if you would like to use some of it, a good starting point might be the books (many in the Dekker monograph series or in Springer's LNM) by Golan (prime torsion theories), Van Oystaeyen et at (prime ideals) or Procesi (PI-algebras and GIT).
The short answer is that if you tried doing this, you would be getting into lots of (mathematical) trouble. There were a number of papers and preprints by Alexander Rosenberg devoted to this problem, with the titles like "Noncommutative affine schemes", "Noncommutative local algebra", "Noncommutative globalization", "Noncommutative schemes", etc., some of them apparently still unpublished (for reasons not known to me). This culminated in the paper "Noncommutative smooth spaces", by Kontsevich and Rosenberg, which available from the arXiv. Hardly any traces of Hartshorne's Chapter II exposition remained in this final approach.
One of the big problems you run into is localization. Not all rings can be localized. You'll almost certainly need to restrict to rings satisfying the Ore Conditions. However, lots of natural rings do satisfy them, for instance "almost commutative" rings do. These are filtered rings whose associated graded ring is commutative. Among the rings of this form is the ring of linear differential operators on an open subset of a variety, and localization works out, so you get a sheaf, and you can look at modules, etc, they're called D-modules.
But anyway, the first few things to do are to decide on a class of rings where you can localize (or if you can't, you REALLY need to rework things from the bottom), and then you need to decide if you're looking at left, right or bimodules, including whether you're going to look at prime left/right/two-sided ideals, etc.
CAVEAT: I am a nonexpert in noncommutative AG, I just know a few places where the standard things in commutative AG break down a bit.
The problem is that the category of (non necessarily commutative) rings does not have all the nice properties of the category of commutative rings. First and most obvious obstacle comes when ones tries to define the spectrum, since there are three different valid options for prime ideals (left, right or two sided); by picking two sided ideals one could define the Zariski topology just fine, but normally noncommutative rings don't have enough two-sided prime ideals to make the noncommutative spectrum interesting. Among other things it is in general not possible to reconstruct the ring out of the spectrum. As someone told me once "it doesn't matter how you define a point in a noncommutative space, there are never enough of them".
More subtle problems arise at the level of localization, on the hand one need to impose the Ore conditions, but even for rings that satisfy them one still has the problem that for noncommutative rings localization functors do not commute with each other. A possible detour around this problem was taken in the late seventies and early eighties (in what could be called the origin of noncommutative algebraic geometry) by Fred Van Oystaeyen, involving mainly replacing the naive notion of prime spectrum by more subtle ones (torsion spectrum, localization spectrum). A more recent summary of those viewpoints and their developments is in the book by Van Oystaeyen Algebraic Geometry for Associative Algebras.
Edit: After Kevin's clarification, a nice survey on the history and different approaches to noncommutative algebraic geometry can be found at the entry noncommutative algebraic geometry in nLab.
I can say more for this topic. As some people might argued, one approach to noncommutative geometry of noncommutative algebra is to consider module category over this algebra. Say, one define Mod(A)(noncommutative affine scheme)as category of quasi coherent sheaves over non-existence space and then once we have algebra morphism A to B, we have functor Mod(B)--->Mod(A). But I DO NOT think we failed to define topological space and structure sheaf in this approach. In fact, in a much more general settings, say consider abelian/grothendieck categories as category of quasi coherent sheaves on non-existence space(noncommutative scheme). Rosenberg defined spectrum of ableian category and it is indeed underlying topological space of noncommutative scheme. And,we can associate sheaf of rings on this topological space,then we can reconstruct Mod(A)(category defined as noncommutative scheme),and in particular, if A is commutative ring and Mod(A) as Qcoh(SpecA,O), we can reconstruct commutative scheme
Actually, Rosenberg has a paper stick to language of module and ring theory describing spectrum of a noncommutative ringThe left spectrum, the Levitzki radical, and noncommutative schemes. In this paper he introduced the left spectrum for a noncommutative ring which is a topological space and defined the structure presheaf on this spectrum. Moreover in the chapter I of his bookNoncommutaive algebraic geometry and representation of quantized algebra, based on the left spectrum(LspecR)of noncommutative ring R and sheaf of module on this left spectrum, he succeeded in reconstructing the module over this ring R as a global section of this sheaf (which is quasi coherent presheaf on (LSpecR, O)). Besides, in his paper above, he also compared his definition of left spectrum to F. van Oystaeyen's scheme.
I wonder whether this might be an answer for the question proposed by Professor Lebruyn.
More comments: In fact,in 1962, Gabriel introduced the injective spectrum for abelian category in his dissertations Des Catégories Abéliennes and a nice revision paper for this topic is injective spectrum of noncommutative spaces. H,Krause and C.Weibel also gave a definition of spectrum of locally coherent category,More recently, Rosenberg has several papers in Max Plank preprint series which introduced vairous spectrum for abelian category,triangulated category and exact category(some of them are defined in his book)in a more categorical flavor. such as spectra of noncommutative space,Underlying space of noncommutative schemeIn particular, P.Balmer has defined spectrum for tensor triangulated category
One other approach to noncommutative algebraic geometry is summed up by the philosophy
"a noncommutative variety should be thought of entirely in terms of the corresponding derived category of coherent sheaves"
This way we can study noncommutative geometry by studying the bounded derived category of the category of finitely generated modules over an algebra. In this way we can introduce interesting classes of algebras corresponding to interesting classes of commutative spaces. For example an algebra $A$ is called d Calabi-Yau if the d-th shift in $D^b(A)$ is a Serre functor, this then corresponds to open CY-varieties.
This can then be extended to DG-algebras and $A_{\infty}$ algebras.
A very nice paper covering these algebras, and thus to some extent this point of view of noncommutative algebraic geometry is the paper Calabi-Yau algebras by Ginzburg.
In fact, I revised some of problems in Chapter II Scheme theory in Hartshorne using Kontsevich-Rosenberg's machine. I have to notice, what you should deal with is probably module category over noncommutative ring. In noncommutative algebraic geometry, this is just category of quasi coherent sheaves on noncommutative affine schemes. But I did not restrict myself to noncommutative ring case. I try to do it in general noncommutative scheme, say a Grothendieck category or an abelian category. Noticed that, one can take grothendieck category as category of quasi coherent sheaves on quasi compact and quasi separated "would be scheme". So one should consider category of grothendieck category as category of "space" and morphism between spaces as iso class of inverse image functor. Rosenberg developed algebraic geometry in this 2-category. He introduced various spectrum for various destination. I should mention, spectrums for abelian category in his sense coincides with prime spectrum of a commutative ring when you take module category over commutative ring. In fact, one can define Zariski topology on this 2-categories using a family of conservative(faithful)exact localization functors(Serre subcategory in dual language). Then one can introduced the associated topology on the spectrum of abelian category. Then one can continue to introduce the "fiber" at each point of spectrum as stack of local category.(as fibered category) This is called geometric realization of an abelian category or Grothendieck category. Then one can take category of quasi coherent sheaves on this fibered category. At last, we get reconstruction theorem for noncommtative scheme. If we take the original category as quasi coherent sheaves of quasi compact(or not in general)quasi separated commutative scheme. Then we get reconstruction theorem for commutative scheme which means commutative algebraic geometry can be fully embedded into noncommutative algebraic geometry.
Because of this "Justify" theorem, we can develop various notions correspondent to commutative algebraic geometry. One can define noncommutative affine scheme (can be seen as category with projective cogenerator, then by Gabriel cheating theorem, equivalent to a module category). One can also define affine morphism, open/closed immersion/coimmersion(for the motivation of representation theory) picard group, vector bundles
One can also define differential operators in abelian category, monoidal category(for motivation of representation theory of quantum group and math phy), in particular, Noncommutative D-modules on noncommutative space, in particular quantum D-module on quantized flag variety. (I think this is related to the problem mentioned by siegels).
As is well known to all, Beilinson Bernstein's framework aiming to representation theory lives in triangulated category. Actually, there is indeed whole abelian picture developed mainly by Rosenberg and Lunts-Rosenberg-Tanisaki later.
In fact, for most(I think it should be all) problems in Hartshorne (facts in commutative algebraic geometry)has correspondence version in noncommutative algebraic geometry (in particular, what you mentioned, noncommutative ring)
There is indeed noncommutative flat descent theory in Konstevich-Rosenberg's work. I think the more accurate name should be categorical flat descent theory(Beck's theorem)
One more comment: What I mentioned above is ONE framework they developed.(Mainly for representation theory). There is ANOTHER framework introduced by them base on presheave view point(proposed by Gabriel-Grothendieck). They develop algebraic geometry in this view point which is NOT equivalent to the CATEGORICAL GEOMETRY I mentioned above in general. They coincides in affine case and then go to completely different direction. The mainly motivation for this view point is from Konstevich, he wanted to consider noncommutative grassmannian which might be helpful in understanding M-theory in Physics. Along this direction, they define noncommutative algebraic space, stack (DM and Artin) and so on.
last comment: One guy mentioned above, the category of rings did not have good property as commutative ones. But I guess this is not a very big deal. Rosenberg define so called right exact category(say category of rings, category of affine schemes, category of vector bundles). He developed whole homological algebra in this settings and Universal algebraic k theory, algebraic cycles. chow group and so on
I am sorry I have to go back to work instead of typing here. There are various noncommutatve algebraic geometry. If you are dealing with projective scheme, you might be interested in work of Artin's School on NC projective geometry
Several other comments: we have the notion of locally noetherian category whose objects is generated by noetherian object. For example, category of quasi coherent sheaves on noetherian commutative scheme is a locally noetherian category. We can play game in this setting. Then we can get whole commutative algebraic geometry of noetherian scheme
Rosenberg and Kontsevich will publish three volumes books soon. I have got the second draft of these books which Rosenberg has already taught us. There is a historical observations in the preface of the first volume of these books. I posted here:
Historical observations on noncommutative algebraic geometry I
[Historical observations on noncommutative algebraic geometry II ] 2
Historical observations on noncommutative algebraic geometry III
Historical observations on noncommutative algebraic geometry IV
Historical observations on noncommutative algebraic geometry V
For what I do (i.e. noncommutative Iwasawa theory) it would be interesting if there exists or someone finds a nice notion of algebraic geometry for groups rings.
As mentioned above, the lack of localization will be a major issue. It is interesting to note that many homological constructions will work. For example, major work was done by Van den bergh, Zhang, Yekutieli and others in constrtuing dualizing complexes over some noncommutative rings, thus allowing some of the features of the Grothendieck duality to be recovered. See for example https://www.jstor.org/stable/2161576
