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As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:

$\bullet$ In the approach by Toen-Vaquié we fix a nice symmetric monoidal category $C$, also called a relative context. An affine scheme is defined to be an algebra object in $C$, and an arbitrary scheme is a certain presheaf on affine schemes. We optain the category $\mathrm{Sch}(C)$ of schemes relative to $C$.

$\bullet$ In Durov's theory a generalized ring is an algebraic monad which is commutative in a certain sense. Then affine schemes are defined to be the spectra of generalized rings and arbitrary schemes are optained by gluing. This results in the category $\mathrm{genSch}$.

$\bullet$ In his book Categories of commutative algebras Yves Diers considers Zariski categories, which seem to axiomatize familiar properties of categories of commutative algebras. If $\mathcal{A}$ is such a Zariski category, then one can develope commutative algebra internal to $\mathcal{A}$, construct affine schemes and then by gluing also schemes as usual. We optain the category $\mathrm{Sch}(\mathcal{A})$.

$\bullet$ In derived algebraic geometry one replaces the category of rings with the category of simplicial rings (but I don't really know enough about that, yet).

My question is: What are the connections between these 'generalized algebraic geometries'?

Fortunately there is a map of $\mathbb{F}_1$-land which draws connections between all these various approaches to schemes over $\mathbb{F}_1$. For example monoid schemes à la Deitmar/Kato are in the intersection of Toen-Vaquiè and Durov. Note, however, that the theories mentioned above are far more general.

Specifically, one might ask the following questions: Is the category of generalized rings a Zariski category and is Durov's theory (say, with the unary localization theory) a special case of the one by Yves Diers? What is the relationship between Toen-Vaquié schemes relative to the symmetric monoidal category of simplicial rings and derived schemes? If $C$ is a relative context, is then the category of algebra objects in $C$ a Zariski category and do the corresponding schemes coincide? Probably not because Diers never mentions monoids as an example, but perhaps it's the other way round? Of course, many more questions are out there ...

Probably I'm not the first one with this question, therefore I've also put the "reference-request" tag. It would be great if there is some paper like "Mapping AG-land".

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    $\begingroup$ Can you shed light what is the motivation and expected outcomes of these activities ? $\endgroup$
    – Alexander Chervov
    Commented Feb 25, 2012 at 17:51
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    $\begingroup$ @Alexander: Fair question. I'm aware that each of these theories is motivated by specific problems. Although the basic idea is always the same, they seem to be so isolated. It would be great if there is some unification, or at least overlappings. Currently I would like to use some spectrum construction for a symm. mon. category and thereby I lost track of all the possible approaches mentioned above (even worse, I invented one by my own and recently studied it with Alexandru Chirvasitu). I hope that an answer to this question here will eventually clear up my confusion. $\endgroup$
    – Martin Brandenburg
    Commented Feb 25, 2012 at 22:42
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    $\begingroup$ @Martin is there any problem in classical alg.geom which is expected to be solved using new approaches ? $\endgroup$
    – Alexander Chervov
    Commented Feb 26, 2012 at 15:26
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    $\begingroup$ Saša, understanding the interrelations in the world we kind of know is worthy by itself. $\endgroup$
    – Zoran Skoda
    Commented Feb 1, 2013 at 16:36
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    $\begingroup$ I forgot blue schemes (ncatlab.org/nlab/show/blueprint). They are connected to schemes à la Toen-Vaquié: arxiv.org/abs/1212.3261 $\endgroup$
    – Martin Brandenburg
    Commented Jun 6, 2013 at 10:38

2 Answers 2

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First, here are some things about the four generalizations you mention:

Monoids don't fall into Diers' framework: By his Proposition 1.4.1 the terminal object in his framework is strict, i.e. any morphism $1 \to A$ is an isomorphism, which is definitely not the case for monoids. I also wouldn't expect Diers' examples to be instances of Toen/Vaquie's framework in general, Diers' example 1.3.16, the category of pairs (ring, module over it), might be a counterexample. I don't know about Durov's setting.

Durov's geometry is in no obvious way an instance of Toen/Vaquie's framework. If you want to force it into that framework, this might be an idea to go after: Monads are monoids in the monoidal category of endofunctors. Commutative monads, however, are not commutative monoids in that category; indeed it doesn't even make sense to say that since the category is not symmetric monoidal. So first you have to find a symmetric monoidal ambient category in which Durov's generalized rings live. Seeing monads as Lawvere algebraic theories or as (things presented by) sketches might do the job - a commutative theory is probably exactly a sketch with an isomorphism from its tensor square. Another idea could be to consider a category of monads where morphisms are natural monad transformations together with distributive laws...

Derived algebraic geometry on the other hand is an instance of the homotopical version of Toen/Vaquie's framework, also contained in that article - see also below.

Second let me point out that there are many more generalizations of algebraic geometry than those four:

° Rings with extra structure can count as generalization, if one can endow any usual ring with such an extra structure, e.g.

  • Not Borger's geometry with lambda-rings: Not any ring can be endowed with the trivial lambda-ring structure - see his comment

  • Berkovich's analytic geometry: Any ring can be endowed with the trivial metric

° One can replace rings by first order structures in several ways:

  • Several Russian authors do this in somewhat similar ways, a recent reference is this one by Daniyarova, Myasnikov, Remeslennikov which has many references to other work in this direction; see also this one by B. Plotkin.

  • First order structures can be described by sketches and there is an outline of algebraic geometry along this line in this text by R. Guitart.

° There are hyperrings (used for algebraic geometry by Connes/Consani) and fuzzy rings (by Walter Wenzel and Andreas Dress, e.g. this), which are certain second order structures.

° There are two generalizations of rings used by Shai Haran to compactify the integers, F-rings and the one given in his "Non-additive prolegomena".

° There is another generalization made by Shai Haran in his article on "dyslectic geometry" in which rings are endowed with gradings over general monoids (see here). Something quite similar seems to be going on in James Dolan's generalization of algebraic geometry (unpublished, but see here, there also was a series of videos of talks somewhere)

° derived algebraic geometry doesn't necessarily have to be based on simplicial rings; dg-algebras and $E_\infty$-ring spectra are equally important inputs, and there are many others, captured in

  • Toen-Vezzosi's HAG-contexts (these are homotopically additive)

  • Lurie's structured spaces from DAG 5 which capture about everything based on the idea of glueing together homotopical algebraic structure.

° note that Toen/Vaquie in their relative geometry do not stop at commutative monoids in some monoidal category but also give a homotopical version - this is something like a non-additive version of HAG-contexts and covers e.g. geometry over the "spectrum with one element" whose input are simplicial monoids.

° replacing rings by groupoid objects in rings together with an appropriate notion of equivalence gives you stack theory. Of course you can go on to higher stacks.

° of course there are the several approaches to non-commutative algebraic geometry - see Mahanta's survey for some of them, many related to the next point

° Chirvasitu/Johnson-Freyd's 2-schemes

° Takagi's generalized schemes

° Deitmar's congruence schemes

° Lorscheid's blue schemes

° I am sure I forgot several things...

Third, since I am at it, let me note that there are also generalizations of algebraic geometry which do not exactly build upon a generalized notion of ring, e.g.

° Hrushovski/Zilber's Zariski Geometries (see here) capturing the essential structure which is used in the applications of model theory to arithmetic geometry

° Rosenberg's noncommutative geometry. It doesn't have to be non-commutative, and also not additive, as Z. Skoda pointed out here

° in particular: schemes as dg-categories (Kontsevich, Rosenberg, Tabuada)

° Balmer's triangulated geometry

...

To summarize: Carefully mapping AG-land will keep you busy for a while. I have gathered quite some material for a rudimentary map (or maybe a low resolution satellite photo) accompanied by a few selected closer snapshots, but I won't start writing it before the second half of the year...

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  • $\begingroup$ One correction, which I don't think affects your overall point: It is not true that every ring admits a lambda-ring structure. For example, finite fields and rings of integers in nontrivial number fields don't admit lambda-structures. In fact, for finitely generated Z-algebras, admitting a lambda-structure is a very restrictive condition. In particular, unlike in equivariant algebraic geometry with respect to actions of groups or Lie algebras, there is no "trivial" lambda-structure. This is one way in which lambda-algebraic geometry is more interesting. $\endgroup$
    – JBorger
    Commented Feb 29, 2012 at 2:40
  • $\begingroup$ Thank you, Peter, for this very interesting answer. I was aware that there are many other "theories of algebraic geometry" (Balmer, Rosenberg, Berkovich), but I didn't know that there are so many! My question was only refering to those where the category of rings is replaced by a similar category and then the usual Zariski spectrum construction is generalized to this context. Your answer shows that there are even more ways to generalize algebraic geometry. $\endgroup$
    – Martin Brandenburg
    Commented Mar 2, 2012 at 8:27
  • $\begingroup$ But my main objective was to know if there is a precise connection between Durov and Toen-Vaquie (I'm not so much interested in Diers' work and DAG). It would be great if you can expand your thoughts you have sketeched above. What about the following idea: The category of generalized rings is symmetric monoidal, as well as complete and cocomplete. What happens if we apply Toen-Vaquie to this context? $\endgroup$
    – Martin Brandenburg
    Commented Mar 2, 2012 at 8:31
  • $\begingroup$ Your idea is good! The tensor product of generalized rings is just their coproduct and every generalized ring is a commutative monoid with respect to that via its codiagonal - this embeds generalized rings into commutative monoids in generalized rings and every commutative monoid there arises this way. Problem solved. See Theorem 2.1(2) in epub.ub.uni-muenchen.de/7094/1/7094.pdf $\endgroup$
    – Peter Arndt
    Commented Mar 3, 2012 at 9:33
  • $\begingroup$ My ideas were along the same line, but unecessarily complicated: I was thinking e.g. of the category of finite product sketches - it has a commutative tensor product such that the $S \otimes T$-algebras are exactly the $T$-algebras in $S$-algebras. The latter notion involves some compatibility between the two $S$- and the $T$-structures, which, if one takes $S=T$, might become just the notion of commutativity. Now sketches are just presentations of theories and not yet the right thing, but I think the same kind of tensor product exists for algebraic theories and then one gets back the above.. $\endgroup$
    – Peter Arndt
    Commented Mar 3, 2012 at 9:53
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In 2015 appeared one more very interesting work by Connes and Consani; their Absolute algebra and Segal’s $Γ$-rings (au dessous de $\overline{\operatorname{Spec}(\mathbb Z)}$) also contains the view of interconnections between previous approaches to the subject.

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