arakelov geometry

8/2/2019 Arakelov Geometry

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arXiv:0704

.2030v1

[math.AG

]16Apr2007

New Approach to Arakelov Geometry

Nikolai Durov

February 1, 2008

Introduction

The principal aim of this work is to provide an alternative algebraic frame-work for Arakelov geometry, and to demonstrate its usefulness by presenting

several simple applications. This framework, called theory of generalizedrings and schemes, appears to be useful beyond the scope of Arakelov ge-ometry, providing a uniform description of classical scheme-theoretical alge-braic geometry (schemes over SpecZ), Arakelov geometry (schemes overSpecZ and SpecZ), tropical geometry (schemes over Spec T and SpecN)and the geometry over the so-called field with one element (schemes overSpecF1). Therefore, we develop this theory a bit further than it is strictlynecessary for Arakelov geometry.

The approach to Arakelov geometry developed in this work is completelyalgebraic, in the sense that it doesnt require the combination of scheme-theoretical algebraic geometry and complex differential geometry, tradition-

ally used in Arakelov geometry since the works of Arakelov himself.However, we show that our models X/SpecZ of algebraic varieties X/Q

define both a model X/ SpecZ in the usual sense and a (possibly singular)Banach (co)metric on (the smooth locus of) the complex analytic varietyX(C). This metric cannot be chosen arbitrarily; however, some classicalmetrics like the FubiniStudy metric on Pn do arise in this way. It is in-teresting to note that good models from the algebraic point of view (e.g.finitely presented) usually give rise to not very nice metrics on X(C), andconversely, nice smooth metrics like FubiniStudy correspond to models withbad algebraic properties (e.g. not finitely presented).

Our algebraic approach has some obvious advantages over the classicalone. For example, we never need to require X to be smooth or proper, andwe can deal with singular metrics.

In order to achieve this goal we construct a theory of generalized rings,commutative or not, which include classical rings (always supposed to be

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2 Introduction

associative with unity), then define spectraof such (commutative) generalizedrings, and construct generalized schemes by patching together spectra ofgeneralized rings. Of course, these generalized schemes are generalized ringedspaces, i.e. topological spaces, endowed with a sheaf of generalized rings.

Then the compactified SpecZ, denoted by

SpecZ, is constructed as a(pro-)generalized scheme, and our models X/SpecZ are (pro-)generalizedschemes as well.

All the generalized notions we discuss are indeed generalizations ofcorresponding classical notions. More precisely, classical objects (e.g.commutative rings) always constitute a full subcategory of the category ofcorresponding generalized objects (e.g. commutative generalized rings). Inthis way we can always treat for example a classical scheme as a generalizedscheme, since no new morphisms between classical schemes arise in the largercategory of generalized schemes.

In particular, the category of (commutative) generalized rings contains

all classical (commutative) rings like Z, Q, R, C, Zp, ..., as well as somenew objects, such as Z (the archimedian valuation ring ofR, similar to p-adic integers Zp Qp), Z() (the non-completed localization at , or thearchimedian valuation ring ofQ), Z (the integral closure ofZ in C).Furthermore, once these archimedian valuation rings are constructed, wecan define some other generalized rings, such as F1 := Z Z, or the fieldwith one element F1. Tropical numbers T and other semirings are alsogeneralized rings, thus fitting nicely into this picture as well.

In this way we obtain not only a compact model SpecZ ofQ (calledalso compactification of SpecZ), and models X

SpecZ of algebraic

varieties X/Q, but a geometry over the field with one element as well. Forexample, SpecZ itself is a pro-generalized scheme over F1 and F1.

In other words, we obtain rigorous definitions both of the archimedianlocal ring Z and of the field with one element F1. They have beendiscussed in mathematical folklore for quite a long time, but usually only ina very informal fashion.

We would like to say a few words here about some applications of thetheory of generalized rings and schemes presented in this work. Apart fromdefining generalized rings, their spectra, and generalized schemes, we discusssome basic properties of generalized schemes, essentially transferring some

results of EGA I and II to our case. For example, we discuss projective(generalized) schemes and morphisms, study line and vector bundles, definePicard groups and so on.

Afterwards we do some homological (actually homotopic) algebra overgeneralized rings and schemes, define perfect simplicial objects and cofibra-


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Introduction 3

tions (which replace perfect complexes in this theory), define K0 of per-fect simplicial objects and vector bundles, briefly discuss higher algebraicK-theory (Waldhausens construction seems to be well-adapted to our sit-uation), and construct Chow rings and Chern classes using the -filtrationon K0, in the way essentially known since Grothendiecks proof of RiemannRoch theorem.

We apply the above notions to Arakelov geometry as well. For example,we compute Picard group, Chow ring and Chern classes of vector bundles overSpecZ, and construct the moduli space of such vector bundles. In particular,we obtain the notion of(arithmetic) degree degE logQ+ of a vector bundleE over SpecZ; it induces an isomorphism deg : Pic(SpecZ) logQ+. Wealso prove that any affine or projective algebraic variety X over Q admitsa finitely presented model X over SpecZ, and show (under some naturalconditions) that rational points P X(Q) extend to uniquely determinedsections P : SpecZ

X. We show that when X is a closed subvariety

of the projective space PnQ, and its model X is chosen accordingly (e.g. Xis the scheme-theoretical closure of X in PnSpecZ), then the (arithmetic)

degree of the pullback P(OX(1)) of the ample line bundle ofX equals thelogarithmic height of point P X(Q) Pn(Q).

There are several reasons to believe that our algebraic Arakelov geom-etry can be related to its more classical variants, based on Kahler metrics,differential forms and Green currents, as developed first by Arakelov himself,and then in the series of works of H. Gillet, C. Soule and their collobora-tors. The simplest reason is that our algebraic models give rise to some(co)metrics, and classical metrics like the FubiniStudy do appear in this way.

More sophisticated arguments involve comparison with the non-archimedianvariant of classical Arakelov geometry, developed in [BGS] and [GS]. Thisnon-archimedian Arakelov geometry is quite similar to (classical) archime-dian Arakelov geometry, and at the same time admits a natural interpretationin terms of models of algebraic varieties over discrete valuation rings. An-alytic torsion corresponds in this picture to torsion in the special fiber (i.e.lack of flatness).

Therefore, one might hope to transfer eventually the results of these twoworks to archimedian context, using our theory of generalized schemes, thusestablishing a direct connection between our algebraic and classical ana-

lytic variant of Arakelov geometry.Acknowledgements. First of all, I would like to thank my scientificadvisor, Gerd Faltings, for his constant attention to this work, as well asfor teaching me a lot of arithmetic geometry during my graduate studies inBonn. I would also like to thank Alexandr Smirnov for some very interesting


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4 Introduction

discussions and remarks, some of which have considerably influenced thiswork. Id like to thank Christophe Soule for bringing very interesting works[GS] and [BGS] on non-archimedian Arakelov geometry to my attention.

This work has been written during my graduate studies in Bonn, as a partof the IMPRS (International Max Planck Research School for Moduli Spacesand their applications), a joint program of the Bonn University and the Max-Planck-Institut fur Mathematik. Id like to thank all people involved fromthese organizations for their help and support. Special thanks go to Chris-tian Kaiser, the coordinator of the IMPRS, whose help was indispensablethroughout all my stay in Bonn.


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Introduction 5

Contents

Overview 9

0. 1. M ot i v at i on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90.2. Z

-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

0.3. -categories, algebras and monads . . . . . . . . . . . . . . . . 150.4. Algebraic monads as non-commutative generalized rings . . . . 170.5. Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 280.6. Localization and generalized schemes . . . . . . . . . . . . . . 370.7. Applications to Arakelov geometry . . . . . . . . . . . . . . . . 450.8. Homological and homotopic algebra . . . . . . . . . . . . . . . 500.9. Homotopic algebra over topoi . . . . . . . . . . . . . . . . . . . 570.10. Perfect cofibrations and intersection theory . . . . . . . . . . 61

1 Motivation: Looking for a compactification of SpecZ 65

1.1. Original motivation . . . . . . . . . . . . . . . . . . . . . . . . 651.5. Vector bundles over SpecZ . . . . . . . . . . . . . . . . . . . . 701.6. Usual description of Arakelov varieties . . . . . . . . . . . . . . 71

2 Z-Lattices and flat Z-modules 732.1. Lattices stable under multiplication . . . . . . . . . . . . . . . 732.3. Maximal compact submonoids of End(E) . . . . . . . . . . . . 752.4. Category ofZ-lattices . . . . . . . . . . . . . . . . . . . . . . 782.7. Torsion-free Z-modules . . . . . . . . . . . . . . . . . . . . . 842.11. Category of torsion-free algebras and modules . . . . . . . . . 922.12. Arakelov affine line . . . . . . . . . . . . . . . . . . . . . . . . 952.13. Spectra of flat Z-algebras . . . . . . . . . . . . . . . . . . . 1022.14. Abstract Z-modules . . . . . . . . . . . . . . . . . . . . . . 109

3 Generalities on monads 119

3.1. AU - c at e gor i e s . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.2. Categories of functors . . . . . . . . . . . . . . . . . . . . . . . 1243 . 3 . M o n a d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.4. Examples of monads . . . . . . . . . . . . . . . . . . . . . . . . 1373.5. Inner functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4 Algebraic monads and algebraic systems 1554.1. Algebraic endofunctors on Sets . . . . . . . . . . . . . . . . . . 1554.3. Algebraic monads . . . . . . . . . . . . . . . . . . . . . . . . . 1614.4. Algebraic submonads and strict quotients . . . . . . . . . . . . 1684.5. Free algebraic monads . . . . . . . . . . . . . . . . . . . . . . . 173


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6 Introduction

4.6. Modules over an algebraic monad . . . . . . . . . . . . . . . . 1814.7. Categories of algebraic modules . . . . . . . . . . . . . . . . . . 1904.8. Addition. Hypoadditivity and hyperadditivity . . . . . . . . . 1974.9. Algebraic monads over a topos . . . . . . . . . . . . . . . . . . 203

5 Commutative monads 215

5.1. Definition of commutativity . . . . . . . . . . . . . . . . . . . . 2155.2. Topos case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.3. Modules over generalized rings . . . . . . . . . . . . . . . . . . 2285.4. Flatness and unarity . . . . . . . . . . . . . . . . . . . . . . . . 2395.5. Alternating monads and exterior powers . . . . . . . . . . . . . 2445.6. Matrices with invertible determinant . . . . . . . . . . . . . . . 2535.7. Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

6 Localization, spectra and schemes 269

6.1. Unary localization . . . . . . . . . . . . . . . . . . . . . . . . . 2696.2. Prime spectrum of a generalized ring . . . . . . . . . . . . . . . 2826.3. Localization theories . . . . . . . . . . . . . . . . . . . . . . . . 2856.4. Weak topology and quasicoherent sheaves . . . . . . . . . . . . 2966.5. Generalized schemes . . . . . . . . . . . . . . . . . . . . . . . . 3026.6. Projective generalized schemes and morphisms . . . . . . . . . 313

7 Arakelov geometry 329

7.1. Construction of SpecZ . . . . . . . . . . . . . . . . . . . . . . 3297.2. Models over

SpecZ, Z and Z() . . . . . . . . . . . . . . . . 350

7.3. Z

-models and metrics . . . . . . . . . . . . . . . . . . . . . . 3567.4. Heights of rational points . . . . . . . . . . . . . . . . . . . . . 357

8 Homological and homotopical algebra 361

8.1. Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . 3688.2. Simplicial and cosimplicial objects . . . . . . . . . . . . . . . . 3768.3. Simplicial categories . . . . . . . . . . . . . . . . . . . . . . . . 3818.4. Simplicial model categories . . . . . . . . . . . . . . . . . . . . 3838.5. Chain complexes and simplicial objects over abelian categories 3888.6. Simplicial -modules . . . . . . . . . . . . . . . . . . . . . . . 3938.7. Derived tensor product . . . . . . . . . . . . . . . . . . . . . . 398

9 Homotopic algebra over topoi 405

9.1. Generalities on stacks . . . . . . . . . . . . . . . . . . . . . . . 4059.2. KripkeJoyal semantics . . . . . . . . . . . . . . . . . . . . . . 4119. 3. M ode l st ac k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419


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Introduction 7

9.4. Homotopies in a model stack . . . . . . . . . . . . . . . . . . . 4309.5. Pseudomodel stack structure on simplicial sheaves . . . . . . . 4419.6. Pseudomodel stacks . . . . . . . . . . . . . . . . . . . . . . . . 4539.7. Pseudomodel structure on sO-MODE and sO-Mod . . . . . . 4599.8. Derived local tensor products . . . . . . . . . . . . . . . . . . . 4649.9. Derived symmetric powers . . . . . . . . . . . . . . . . . . . . 468

10 Perfect cofibrations and Chow rings 487

10.1. Simplicial dimension theory . . . . . . . . . . . . . . . . . . . 48710.2. Finitary closures and perfect cofibrations . . . . . . . . . . . . 49610.3. K0 of perfect morphisms and objects . . . . . . . . . . . . . . 50610.4. Projective modules over Z . . . . . . . . . . . . . . . . . . . 52410.5. Vector bundles over SpecZ . . . . . . . . . . . . . . . . . . . 53210.6. Chow rings, Chern classes and intersection theory . . . . . . . 54410.7. Vector bundles over

SpecZ: further properties . . . . . . . . . 557


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Overview 9

Overview

We would like to start with a brief overview of the rest of this work, discussingchapter after chapter. Purely technical definitions and statements will beomitted or just briefly mentioned, while those notions and results, which weconsider crucial for the understanding of the remainder of this work, will beexplained at some length.

0.1. (Motivation.) Chapter 1 is purely motivational. Here we discuss propersmooth models both of functional and number fields, and indicate why non-proper models (e.g. the affine line A1k as a model of k(t), or SpecZ as amodel ofQ) are not sufficient for some interesting applications. We also in-troduce some notations. For example, SpecZ denotes the compactificationof SpecZ. Its closed points must correspond to all valuations ofQ, archime-dian or not, i.e. we expect SpecZ = SpecZ{} as a set, where denotesa new archimedian point. We denote by Z Q := R and Z() Qthe completed and non-completed local rings of SpecZ at , analogous toclassical notations Zp Qp and Z(p) Q.0.1.1. It is important to notice here that SpecZ, Z and Z() are not definedin this chapter. Instead, they are used in an informal way to describe theproperties we would expect these objects to have. In this way we are evenable to explain the classical approach to Arakelov geometry, which insists ondefining an Arakelov model X of a smooth projective algebraic variety X/Qas a flat proper model X SpecZ together with a metric on X(C) subjectto certain restrictions (e.g. being a Kahler metric).

0.1.2. Another thing discussed in this chapter is that the problem of con-

structing models over SpecZ can be essentially reduced to the problem ofconstructing Z()-models of algebraic varieties X/Q, or Z-models of alge-braic varieties X/R. In other words, we need a notion of a Z-structure onan algebraic variety X over R; ifX = Spec A is affine, this is the same thingas a Z-structure on an R-algebra A. So we see that a proper understand-ing of compactified models of algebraic varieties over Q must include anunderstanding ofZ-structures on R-algebras and vector spaces.

0.2. (Z-structures.) Chapter 2 is dedicated to a detailed study of Z-

structures on real vector spaces and algebras. We start from the simplestcases and extend our definitions step by step, arriving at the end to thecorrect definition of Z-Mod, the category ofZ-modules. In this waywe learn what the Z-modules are, without still having a definition ofZitself.


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10 Overview

The main method employed here to obtain correct definitions is thecomparison with the p-adic case.

0.2.1. (Z-lattices: classical description.) The first step is to describe Z-structures on a finite-dimensional real vector space E, i.e. Z-lattices A

E.

The classical solution is this. In the p-adic case a Zp-lattice A in afinite-dimensional Qp-vector space E defines a maximal compact subgroupKA := AutZp(A)

= GL(n,Zp) in locally compact group G := AutQp(E) =GL(n,Qp), all maximal compact subgroups of G arise in this way, andKA = KA iff A

and A are similar, i.e. A = A for some Qp .Therefore, it is reasonable to expect similarity classes ofZ-lattices inside

real vector space Eto be in one-to-one correspondence with maximal compactsubgroups K of locally compact group G := AutR(E) = GL(n,R). Suchmaximal compact subgroups are exactly the orthogonal subgroups KQ =O(n,R), defined by positive definite quadratic forms Q on E, and KQ = KQ

iff Q and Q are proportional, i.e. Q = Q for some > 0.In this way the classical answer is that a Z-structure on a finite di-mensional real space E is just a positive definite quadratic form on E, andsimilarly, a Z-structure on a finite dimensional complex vector space is apositive definite hermitian form. This point of view, if developed further, ex-plains why classical Arakelov geometry insists on equipping (complex pointsof) all varieties and vector bundles involved with hermitian metrics.

0.2.2. (Z-structures on finite R-algebras.) Now suppose that E is a finiteR-algebra. We would like to describe Z-structures on this algebra, i.e. Z-lattices A E, compatible with the multiplication and unit of E. In the

p-adic case this would actually mean 1 A and A A A, but if we wantto obtain correct definitions in the archimedian case, we must re-writethese conditions for A in terms of corresponding maximal compact subgroupKA G.

And here a certain problem arises. These compatibility conditions cannotbe easily expressed in terms of maximal compact subgroups of automorphismgroups even in the p-adic case. However, if we consider maximal compactsubmonoids of endomorphism monoids instead, this problem disappears.

0.2.3. (Maximal compact submonoids: p-adic case.) Thus we are induced todescribe Z-lattices A in a finite-dimensional real space E in terms of max-

imal compact submonoids MA of locally compact monoid M := EndR(E) =M(n,R). When we study the corresponding p-adic problem, we see that allmaximal compact submonoids of EndQp(E) are of form MA := { : (A) A} = EndZp(A) = M(n,Zp) for a Zp-lattice A E, and that MA = MA iffA and A are similar, i.e. in the p-adic case maximal compact submonoids of


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0.2. Z-structures 11

End(E) are classified by similarity classes of lattices A E, exactly in thesame way as maximal compact subgroups of Aut(E).

0.2.4. (Maximal compact submonoids: archimedian case.) However, in thearchimedian case there are much more maximal compact submonoids MA

inside M := EndR(E) than positive definite quadratic forms. Namely, wecan take any symmetric compact convex body A E (a convex body isalways required to be absorbent, i.e. E = R A), and put MA := { EndR(E) : (A) A}. Any such MA is a maximal compact submonoid inM = EndR(E), all maximal compact submonoids in M arise in this way, andMA = MA iff A

= A for some R.0.2.5. (Z-lattices.) This leads us to define a Z-lattice A in a finite-dimensional real space E as a symmetric compact convex body A. It iswell-known that any such A is the unit ball with respect to some Banachnorm

on E; in other words, aZ-structure on E is essentially a Banach

norm on E.Next, we can define the category Z-Lat ofZ-lattices as follows. Ob-

jects ofZ-Lat are couples A = (AZ, AR), where AR is a finite-dimensionalreal vector space, and AZ AR is a symmetric compact convex body in AR.Morphisms f : A B are couples (fZ, fR), where fR : AR BR is an R-linear map, and fZ : AZ BZ is its restriction.

Of course, this definition is again motivated by the p-adic case. Notice,however, that in the p-adic case we might define the category ofZp-latticeswithout any reference to ambient Qp-vector spaces, while we are still not ableto describe Z-lattices without reference to a real vector space.

Another interesting observation is that Z-Lat is essentially the categoryof finite-dimensional Banach vector spaces, with R-linear maps of norm 1as morphisms. While this description establishes a connection to Banachnorms, we dont insist on using it too much, since the p-adic case suggeststhat we should concentrate our attention on AZ, not on ambient space AR.

0.2.6. (Z-structures on finite R-algebras.) Now we are able to define aZ-structure A on a finite R-algebra E, i.e. a Z-lattice A E, compatiblewith the multiplication and unit of E. The key idea here is to express thiscompatibility in terms of corresponding maximal compact submonoid MA End(E) first in the p-adic case, and then to transfer the conditions on MA

thus obtained to the archimedian case.The final result is that we must consider symmetric compact convex bod-ies A E, such that 1 A and A A A. In the language of Banachnorms this means 1 = 1 and xy x y, i.e. we recover the notionof a finite-dimensional real Banach algebra.


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12 Overview

0.2.7. (From lattices to torsion-free modules.) Next step is to embed Z-Latinto a larger category of torsion-free Z-modules Z-Fl.Mod. Comparingto the p-adic case, we see that Z-Fl.Mod might be constructed as the cat-egory Ind(Z-Lat) of ind-objects over Z-Lat (cf. SGA 4 I). This cate-gory consists of formal inductive limits lim

M of Z

-lattices, taken

along filtered ordered sets or small categories, with morphisms given byHom(lim M, lim N) = lim lim Hom(M, N).

0.2.8. (Direct description ofZ-Fl.Mod.) However, the category Z-Fl.Modof torsion-free Z admits a more direct description, similar to 0.2.5. Namely,one can define it as the category of couples (AZ, AR), where AR is a realvector space (not required more to be finite-dimensional), and AZ AR isa symmetric convex body (not required to be compact, but still required tobe absorbent: AR = R AZ). Morphisms are defined in the same way as forZ-lattices, and it is immediate from this construction that Z-Lat is a full

subcategory ofZ-Fl.Mod.0.2.9. (Inductive and projective limits in Z-Fl.Mod.) We show that ar-bitrary inductive and projective limits exist in Z-Fl.Mod. For example,product A B equals (AZ BZ, AR BR), and the direct sum (i.e. co-product) A B can be computed as (conv(AZ BZ), AR BR), whereconv(S) denotes the convex hull of a subset S in a real vector space.

0.2.10. (Tensor structure on Z-Fl.Mod.) We define an ACU -structure onZ-Fl.Mod and Z-Lat, having the following property. An algebra (A,,)in Z-Lat is the same thing as a couple (AZ, AR), consisting of a finitedimensional R-algebra AR, and a Z

-lattice AZ

AR, compatible with the

algebra structure of AR in the sense of 0.2.6.

This tensor structure is extremely natural in several other respects. Forexample, when translated into the language of (semi)norms on real vec-tor spaces, it corresponds to Grothendiecks projective tensor product of(semi)norms.

The unit object for this tensor structure is Z := ([1, 1],R). One shouldthink of this Z as Z, considered as a left module over itself, not as thering Z.

0.2.11. (Underlying set of a torsion-free Z-module.) Once we have a unit

object Z, we can define the forgetful functor : Z-Fl.Mod Sets,A Hom(Z, A). It is natural to say that (A) is the underlying setof A. Direct computation shows that (A) = AZ for A = (AZ, AR),thus suggesting that we ought to concentrate our attention on AZ, not onauxiliary vector space AR.


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0.2. Z-structures 13

0.2.12. (Free Z-modules. Octahedral combinations.) Since arbitrary directsums exist in Z-Fl.Mod, we can construct free Z-modules Z

(S) , by taking

the direct sum of S copies ofZ. It is immediate that LZ : S Z(S)is a left adjoint to = Z : Z-Fl.Mod Sets. One can describe this

Z

(S)

= ((S),R(S)

) explicitly. Its vector space component is simply R-vector space R(S) freely generated by S. Its standard basis elements will bedenoted by {s}, s S. Then the symmetric convex subset (S) R(S)consists of all octahedral (linear) combinations of these basis vectors:

(S) = conv{s} : s S (0.2.12.1)

=

s

s{s} :

s

|s| 1, almost all s = 0

(0.2.12.2)

In other words, (S) is the standard octahedron in R(S).

0.2.13. (Notation: finite sets and basis elements.) We would like to mentionhere some notation, used throughout this work. If M is a free object,generated by a set S (e.g. M = R(S) for some ring R), we denote by {s} thebasis element of M corresponding to s S. Thus s {s} is the naturalembedding S M.

Another notation: we denote by n the standard finite set {1, 2, . . . , n},where n 0 is any integer. For example, 0 = , and 2 = {1, 2}. Further-more, whenever we have a functor defined on the category of sets, we write(n) instead of (n), for any n 0. In this way R(n) = Rn is the standardn-dimensional real vector space, with standard basis {k}, 1 k n, and(n) Rn is the standard n-dimensional octahedron.0.2.14. (From Z-Fl.Mod to Z-Mod.) Our next step is to recover the cat-egory Z-Mod of all Z-modules, starting from the category Z-Fl.Mod oftorsion-freeZ-modules. We use adjoint functors LZ : Sets Z-Fl.Mod : for this. Namely, we observe that this pair of adjoint functors defines amonad structure (, , ) on endofunctor := LZ : Sets Sets.However, functor happens not to be monadic, i.e. the arising comparisonfunctor I : Z-Fl.Mod Sets from Z-Fl.Mod into the category of -algebras in Sets (which we prefer to call -modules) is not an equivalenceof categories, but just a fully faithful functor.

The p-adic analogy suggests to define Z-Mod := Sets. Then category

Z-Fl.Mod can be identified with a full subcategory ofZ-Mod with the aidof functor I, and the forgetful functor Z : Z-Mod Sets is now monadicby construction, similarly to the forgetful functor on any category defined byan algebraic system (e.g. category R-Mod of modules over an associativering R).


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14 Overview

0.2.15. (Explicit description ofZ-modules.) We can obtain a more explicitdescription of Z-modules, i.e. of objects M = (M, ) ObZ-Mod =Ob Sets. Indeed, by definition M = (M, ) consists of a set M, togetherwith a Z-structure, i.e. a map : (M) M, subject to certain con-ditions. Since

(M) consists of formal octahedral combinations x x{x},

x |x| 1, of elements ofM, such a map should be thought of as a way ofevaluating such formal octahedral combinations. Thus we write

x xx in-

stead of(

x{x}). We use this notation for finite sums as well, e.g. x + yactually means ({x} + {y}), for any x, y M, , R, || + || 1.Notice, however, that the + sign in expression x + y is completely formal:in general we dont get any addition operation on M.

As to the conditions for , they are exactly all the usual relations satisfiedby octahedral combinations of elements of real vector spaces. For example,(x + y) = x + y.

In this way we see that a Z-module is nothing else than a set M, together

with a way of evaluating octahedral combinations of its elements, in such away that all usual relations hold in M.

0.2.16. (Torsion modules.) Category Z-Mod is strictly larger than the cat-egory Z-Fl.Mod of torsion-free Z-modules. In fact, it contains objects likeF := Coker(m Z), where m = ((1, 1),R) in Z-Fl.Mod, and thetwo morphisms m Z are the natural inclusion and the zero morphism.Notice that this cokernel is zero in Z-Fl.Mod, but not in Z-Mod: in fact,this F is a three-element set with a certain Z-module structure. Thisgives an example of a non-trivial torsion Z-module.

0.2.17. (Arakelov affine line.) Among other things discussed in Chapter 2,

we would like to mention the study of Arakelov affine line Spec Z[T], car-ried both from the point of view of (co)metrics and from that of (generalized)schemes.

When we compute the naturally arising cometric on A1R or A1C, coming

from this Z-structure Z[T] on R[T], it turns out to be identically zerooutside unit disk { : || < 1}, i.e. all points outside this disk are at infinitedistance from each other. Inside the disk we obtain a continuous piecewisesmooth cometric, very similar to Poincare model of hyperbolic plane in theunit disk.

On the other hand, we can use the -structure defined on Z-Fl.Mod(and actually on all ofZ-Mod) to define Z-algebras A, modules over them,ideals and prime ideals inside them and so on, thus obtaining a definition ofprime spectrum Spec A. For example, topological space SpecZ looks likethe spectrum of a DVR. When we study the Arakelov affine line SpecZ[T]from this point of view, we observe some unexpected phenomena, e.g. Krull


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0.3. -categories, algebras and monads 15

dimension of this topological space turns out to be infinite.

0.3. (-categories, algebras and monads.) Chapter 3 collects some generaldefinitions and constructions, related to -categories, algebras and mon-ads. It is quite technical, but nevertheless quite important for the next

two chapters. Most results collected here are well-known and can be foundin [MacLane]; however, we want to fix the terminology, and discuss the gen-eralization to the topos case.

0.3.1. (AU -categories and -actions.) We discuss AU -categories, i.e.categories A, equipped with a tensor product functor : A A Aand a unit object 1 Ob A, together with some associativity constraintX,Y,Z : (XY)Z = X(YZ) and unit constraints 1X = X = X1,subject to certain axioms (e.g. the pentagon axiom). Roughly speaking, theseaxioms ensure that multiple tensor products X1 X2 Xn are well-defined and have all the usual properties of multiple tensor products, apart

from commutativity.After that we discuss external (left) -actions : A B B of an

AU -category A on a category B. Here we have to impose some externalassociativity and unit constraints (XY)M = X(YM) and 1M =M, subject to similar relations.

Of course, any AU -category A = (A, ) admits a natural -action onitself, given by := .0.3.2. (Algebras and modules.) Whenever we have an AU -category A, wecan consider algebras A = (A,,), A Ob A, : A A A, : 1 A,always supposed to be associative with unity (but not commutative in

fact, commutativity doesnt make sense without a commutativity constrainton A). Thus we obtain a category of algebras in A, denoted by Alg(A).Next, if : A B B is an external -action, and A is an algebra

in A, we denote by A-Mod or BA the category ofA-modules inB, consistingof couples (M, ), M Ob B, : AM M, subject to classical relations ( 1M) = 1M and (1A ) = ( 1M).0.3.3. (Monads over a category C.) Now ifC is an arbitrary category, thecategory of endofunctors A := Endof(C) = Funct(C, C) admits a natural AU-structure, given by composition of functors: FG := FG. Furthermore,there is a natural -action ofA on C, defined by FX := F(X).

Then a monad = (, , ) over a category C is simply an algebra in thiscategory of endofunctors A, i.e. : C C is an endofunctor and : 2 , : IdC are natural transformations, subject to associativity and unitrelations: ( ) = ( ) and ( ) = id = ( ), orequivalently, X (X ) = X (X) : 3(X) (X), and X (X) =


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16 Overview

id(X) = X (X ), for any X Ob C.Similarly, the category C of -modules (in C) is defined as the category

of -modules in C with respect to the external -action of A on C justdiscussed. In this way C consists of couples M = (M, ), with M Ob C, : (M)

M, such that

() =

M :

2(M)

M, and

M = idM.

0.3.4. (Monads and adjoint functors.) Whenever we have a monad overa category C, we get a forgetful functor : C C, (M, ) M, whichadmits a left adjoint L : C C, S ((S), S), such that L = .Conversely, given any two adjoint functors L : C D : , we obtain a canon-ical monad structure on := L : C C, together with a comparisonfunctor I : D C, such that = I. Functor is said to be monadicif I is an equivalence of categories.

0.3.5. (Examples: R-modules, Z-modules. . . ) For example, if C = Sets,D = R-Mod is the category of left modules over a ring R (always supposed

to be associative with unity), then the forgetful functor : R-Mod Setsturns out to be monadic. The corresponding monad R : Sets Setstransforms a set S into the underlying set of free R-module R(S) generatedby S, i.e. into the set of all formal R-linear combinations of basis elements{s}, s S. In this way an R-module M is just a set M together with amethod : R(M) M of evaluating formal R-linear combinations of itselements, subject to some natural conditions.

Similarly, Z-Mod was defined to be Sets for a certain monad over Sets, so the set (S) of formal octahedral combinations of elementsof S should be thought of as the set of all formal Z-linear combinations ofelements ofS, or as the underlying set of free Z-module Z

(S) . This was the

way weve defined in the first place.This analogy between R-Mod = SetsR and Z-Mod = Sets suggests

that our category of generalized rings, which is expected to contain all classi-cal rings R as well as exotic objects like Z, might be constructed as a certainfull subcategory of the category of monads over Sets. Then we might put-Mod := Sets for any monad from this full subcategory, thus obtaininga reasonable definition of -modules without any special considerations.

0.3.6. (Topos case: inner endofunctors and inner monads.) Apart fromthe things just discussed, Chapter 3 contains some technical definitions andstatements, related to inner endofunctors and inner monads over a topos E.These notions are used later to transfer the definitions of generalized ringsand modules over them from the category of sets into arbitrary topoi, e.g.categories of sheaves of sets over a topological space X. This is necessaryto obtain a monadic interpretation of sheaves of generalized rings and ofmodules over them. Without such notions we wouldnt be able to discuss


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0.4. Algebraic monads as non-commutative generalized rings 17

generalized ringed spaces and in particular generalized schemes. However,we suggest to the reader to skip these technical topos-related pages duringthe first reading.

0.4. (Algebraic monads as non-commutative generalized rings.) Chapter 4

is dedicated to the study of algebraic endofunctors and monads over Sets(and algebraic inner endofunctors and monads over a topos Eas well). Thisnotion of algebraicity is actually the first condition we need to impose onmonads over Sets in order to define the category of generalized rings. Infact, algebraic monads are non-commutative generalized rings. For example,any monad R defined by a classical ring R (as usual, associative with unity)is algebraic, as well as monad , used to define Z-Mod.

0.4.1. (Algebraic monads vs. algebraic systems.) Another important remarkis that an algebraic monad overSets is essentially the same thing as an alge-braic system. We discuss this correspondence in more detail below. In some

sense algebraic systems are something like presentations (by a system of gen-erators and relations) of algebraic monads. Thus different (but equivalent)algebraic systems may correspond to isomorphic algebraic monads, and al-gebraic monads provide an invariant way of describing algebraic systems. Inthis way the study of algebraic monads might be thought of as the study ofalgebraic systems from a categorical point of view, i.e. a categorical approachto universal algebra.

0.4.2. (Algebraic endofunctors over Sets.) We say that an endofunctor : Sets Sets is algebraic if it commutes with filtered inductive limits:(lim

S) = lim

(S). Since any set is a filtered inductive limit of its finite

subsets, we obtain (S) = lim(n

S)N/S (n), where N = {0, 1, . . . , n, . . .}n0denotes the category ofstandard finite sets, considered as a full subcategoryof Sets. Therefore, any algebraic endofunctor : Sets Sets is completelydetermined by its restriction |N : N Sets. In fact, this restriction functor |N induces an equivalence between the category of algebraic endofunc-tors Aalg A = Endof(Sets) and Funct(N, Sets) = SetsN. This means thatan algebraic endofunctor is essentially the same thing as a countable col-lection of sets {(n)}n0, together with maps () : (n) (m), defined

for any : n m, such that ( ) = () () and (idn) = id(n).0.4.3. (Algebraic monads.) On the other hand, if two endofunctors and

commute with filtered inductive limits, the same is true for their composition = , i.e. Aalg = SetsN is a full-subcategory ofA = Endof(Sets).Therefore, we can define an algebraic monad = (, , ) as an algebra inAalg. Of course, an algebraic monad is just a monad, such that its underlyingendofunctor commutes with filtered inductive limits.


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18 Overview

0.4.4. (Why algebraic?) Notice that monads R defined by an associativering R are algebraic, i.e. R(S) =

finite I S R(I), just because any element

of R(S) = R(S), i.e. any formal R-linear combination of elements of S,

involves only finitely many elements of S, hence comes from R(I) for somefinite subset I

S. Similarly,

is algebraic, because any octahedral

combination sS s{s} (S) has only finitely many s = 0.We can express this by saying that we consider only operations de-

pending on finitely many arguments. For example, we might remove therequirement s = 0 only for finitely many s S and consider infiniteoctahedral combinations

s{s}, with the only requirement

s |s| 1.

In this way we obtain a larger monad , which coincides with on finite sets, but is different on larger sets. A -structure on a set Mis essentially a way of computing such infinite octahedral combinations ofelements of M. This is definitely not an algebraic operation, and is notan algebraic monad.

Therefore, word algebraic means here something like expressible interms of operations involving only finitely many arguments.

0.4.5. (Algebraic monads and operations.) Now let be an algebraic end-ofunctor (over Sets), M be a set, : (M) M be any map of sets. Forexample, we might take an algebraic monad and a -module M.

Let t (n) for some integer n. Take any n-tuple x = (x1, . . . , xn) Mnof elements of M. It can be considered as a map x : n M, k xk, hencewe get a map (x) : (n) (M). Now we can apply (x) to t, thusobtaining an element of M:

t(x1, . . . , xn) = [t]M(x1, . . . , xn) := ( (x))(t) (0.4.5.1)In this way any t (n) defines an n-ary operation [t]M : Mn M on M.

Thats why we say that (n) is the set of n-ary operations of , and call itselements n-ary operations of . Furthermore, we say that [t]M : M

n Mis the value of operation t on M. Of course, when n = 0, we speak aboutconstants and their values (any constant c (0) has a value [c]M M),and when n = 1, 2, 3, . . . we obtain unary, binary, ternary, . . . operations.

One can show that giving a map : (M) M is actually equivalent togiving a family of evaluation maps (n) : (n)Mn M, (t, x1, . . . , xn) [t]M(x1, . . . , xn), n 0, satisfying natural compatibility relations, which canbe written as

[(t)]M(x1, . . . , xn) = [t]M(x(1), . . . , x(m)), t (m), : m n(0.4.5.2)

Here is a shorthand for ().


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0.4.6. (Elementary description of algebraic monads.) The above descriptionis applicable to maps n : ((n)) (n). We see that such a map iscompletely determined by a sequence of evaluation or substitution maps(k)n : (k) (n)k (n), (t, t1, . . . , tk) t(t1, . . . , tk) = [t](n)(t1, . . . , tk).

In this way we obtain an elementary description of an algebraic monad ,consisting of a sequence of sets {(n)}n0, transition maps () : (m) (n), defined for all : m n, evaluation maps (k)n : (k)(n)k (n),and a unit element e := 1(1) (1).

In fact, this collection completely determines the algebraic monad , andconversely, if we start from such a collection, satisfying some natural com-patibility conditions (certain associativity and unit axioms), we obtaina uniquely determined algebraic monad .

Of course, -modules M = (M, ) also admit such an elementarydescription, consisting of a set M, and a collection of evaluation maps(n) : (n)

Mn

M, subject to certain compatibility, associativity and

unit conditions (e.g. [e]M = idM).Therefore, we might have defined algebraic monads and modules over

them in such an elementary fashion. We didnt do this just because arisingdefinitions and especially relations seem to be quite complicated and not tooenlightening, unless one knows that they come from the definition of algebraicmonads, i.e. they are the algebra relations in Aalg, written in explicit form.0.4.7. (Unit and basis elements.) Notice that the unit element e = 1(1) (1) actually completely determines the unit : IdSets , since X (x) =((ix))(e) for any x X, where ix : 1 X is the map 1 x.

Recall that we denote by

{k

}n or simply by

{k

}the basis element

n(k) (n), 1 k n. For example, e = {1}1. These basis elementshave some natural properties, e.g. {k}m = {(k)}n, for any : m n,1 k m. More interesting properties come from the unit axioms for and-modules. For example, ( ) = id implies (and actually is equivalentto)

t({1}, {2}, . . . , {n}) = [t](n)({1}n, . . . , {n}n) = t, t (n). (0.4.7.1)

The other unit axiom ( ) = id is actually equivalent to e(t) = t forall n 0, t (n).

Unit axiom for a -module M is equivalent to [

{k

}n]M = prk : M

n

M,

or just to [e]M = idM.

0.4.8. (Special notation for unary operations.) Ifu (1) is a unary opera-tion, and t (n) is an n-ary operation, we usually write ut or u t insteadof [u](n)(t) (n), and similarly ux := [u]M(x) for any -module M and


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20 Overview

any x M. This notation is unambiguous because of associativity relations(ut)(x1, . . . , xn) = u t(x1, . . . , xn) for any u (1), t (n), xi M or(m).

In this way we obtain on set || := (1) a monoid structure with iden-tity e, and a monoid action of

|

|on the underlying set of any -module M.

0.4.9. (Special notation for binary operations.) Of course, when we have abinary operation, denoted by a sign like +, , ..., usually written in infixform, we write x + y, x y etc. instead of [+](x, y), [](x, y) etc.

Since t = t({1}, . . . , {n}) for any t (n), we can write [+] = {1}+{2} (2) when we want to point out the corresponding element of (2).

0.4.10. (Free modules.) Notice that L(n) = ((n), n) is the free -moduleof rank n, i.e. Hom(L(n), M) = Mn for any -module M. Of course, themap Hom(L(n), M) Mn is given by evaluating f : L(n) M on basiselements

{k

}n. We also denote L(n) simply by (n), when no confusion

can arise.

0.4.11. (Set of unary operations.) For any algebraic monad the set || :=(1) has two natural structures: that of a monoid, and that of a -module(free of rank one). If = R for a classical associative ring R, then || isthe underlying set of R, its monoid structure is given by the multiplicationof R, and its module structure is the natural left R-module structure on R.

In this way || plays the role of the underlying set of algebraic monad .Notice that it is always a monoid under multiplication, but in general itdoesnt have any addition, i.e. multiplication is in some sense more funda-mental than addition.

Another interesting observation is that, while the addition of a classicalring R indeed corresponds to a binary operation [+] = (1, 1) R(2) =R2, the multiplication of R doesnt correspond to any element of R(n).

Instead, it is built in the composition maps (k)n , i.e. it is part of the more

fundamental structure of algebraic monad.

0.4.12. (Matrix description. Comparison to Shai Harans approach.) SinceHom((n), (m)) = (m)n, we define the set ofm n-matrices over byM(m, n; ) := (m)n. Composition of morphisms defines maps M(n, k; )M(k, m; ) M(n, m; ), i.e. we obtain a well-defined product of matrices.Putting here m = 1, we get maps (n)

k

(k) (n), which are nothingelse than the structural maps (k)n of , up to a permutation of the twoarguments. This matrix language is often quite convenient. For example,the associativity relations for maps

(k)n are essentially equivalent to asso-

ciativity of matrix products, and the unit relations actually mean that the


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identity matrix In := ({1}n, . . . , {n}n) M(n, n; ) is indeed left and rightidentity with respect to matrix multiplication.

One might actually try to describe generalized rings in terms of collec-tions of matrix sets {M(m, n; )}m,n0, together with composition (i.e.matrix multiplication) maps as above, and direct sum maps M(m, n; )

M(m, n; ) M(m + m, n + n;), (u, v) uv. Pursuing this path oneessentially recovers Shai Harans notion of an F-algebra (or rather its non-commutative counterpart, since we dont say anything about tensor productsat this point), defined in [ShaiHaran]. Furthermore, this notion ofF-algebrais more general than our notion of generalized ring, since Shai Haran neverrequires M(m, n; ) = M(1, n; )m.

In fact, once we impose this condition (which is quite natural if we wantto have (m) (n) = (m + n) for directs sums, i.e. coproducts of freemodules) on (non-commutative) F-algebras, we recover our notion of gener-alized ring. However, Shai Haran doesnt impose such restrictions. Actually,

he doesnt define a module over an F-algebra A as a set Q with some ad-ditional structure (in fact, any algebraic structure on a set is described bysome algebraic monad, so our theory of generalized rings is the largest al-gebraic theory of ring-like objects, which admit a notion of module overthem), but considers infinite collections {M(m, n; Q)}m,n0 instead, thoughtof as m n-matrices with entries in Q. This more complicated notion ofmodule roughly corresponds to our -bimodules in Aalg, i.e. to algebraicendofunctors, equipped with compatible left and right actions of .

It is not clear whether one might transfer more sophisticated constructionsof our work to Shai Harans more general case, since these constructionsinvolve forgetful functors : -Mod

Sets, i.e. expect -modules to be sets

with additional structure, and heavily rely on formulas like (m) (n) =(m + n) for categorial coproducts.

Unfortunately, we cannot say much more about this right now, since wehavent found any publications of Shai Haran on this topic apart from hisoriginal preprint [ShaiHaran], which deals only with the basic definitions ofhis theory.

0.4.13. (Initial and final algebraic monads.) Notice that the category ofalgebraic monads has an initial object F, given by the only monad structureon IdSets , as well as a final object 1, given by the constant functor with

value 1, equipped with its only monad structure.This initial algebraic monad F has the property F(n) = n, i.e. a freeF-module of rank n consists only of basis elements {k}n, 1 k n.Furthermore, any set admits a unique F-module structure, i.e. F-Mod =Sets.


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22 Overview

As to 1-Mod, the only sets which admit a 1-module structure are theone-element sets, hence 1-Mod is equivalent to the final category 1.

0.4.14. (Projective limits of algebraic monads. Submonads.) Projectivelimits of algebraic monads can be computed componentwise: (lim

)(n) =

lim (n), for any n 0. For example, ( )(n) = (n) (n). Animmediate consequence is that an algebraic monad homomorphism f : is a monomorphism iff all maps fn :

(n) (n) are injective. (Noticethat injectivity of f1 = |f| : || || does not suffice.) Therefore, is an(algebraic) submonad of iff (n) (n) for all n 0, e = e, and thecomposition maps

(k)n of are restrictions of those of .

We can easily compute intersections of algebraic submonads inside analgebraic monad. For example, we can intersect Z R and Q R (herewe denote by Z the algebraic monad previously denoted by , and identifyclassical associative rings R with corresponding algebraic monads R), thusobtaining the non-completed local ring at

:

Z() := Z Q (0.4.14.1)

0.4.15. (Classical associative rings as algebraic monads.) In fact, the functorR R, transforming a classical ring into corresponding algebraic monad,is fully faithful, and we have R-Mod = R-Mod. Therefore, we can safelyidentify R with R and treat classical associative rings as algebraic monads,i.e. (non-commutative) generalized rings.

0.4.16. (Underlying set of an algebraic monad.) We denote by thetotal or graded underlying set of an algebraic monad , given by

:= n0

(n) (0.4.16.1)

This is a N0-graded set, i.e. a set together with a fixed decomposition intoa disjoint union indexed by N0, or equivalently, a set together with adegree map r : N0. In our case we say that r is the arity map of .0.4.17. (Free algebraic monads.) The underlying set functor fromthe category of algebraic monads into the category Sets/N0 ofN0-graded setsadmits a left adjoint, called the free algebraic monad functor and denoted byS

S

or SF

S. IfS is a finite set

{f1, . . . , f n

}, consisting of elements

fi of degrees or arities ri = r(fi), we also write Ff[r1]1 , . . . , f [rn]n or justf[r1]1 , . . . , f [rn]n .

This = FS is something like an algebra of polynomials over F innon-commuting indeterminates from S. Notice, however, that the structure


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of depends on the choice of arities of indeterminates from S: if all of themare unary, the result is indeed very much like an algebra of polynomialsin non-commuting variables, while non-unary free algebraic monads oftenexhibit more complicated behaviour.

Existence of free algebraic monads = FS

is shown as follows. Weexplicitly define (X) as the set of all terms, constructed from free variables{x}, x X, with the aid of formal operations f S. In other words, (X)is the set of all terms over X, defined by structural induction as follows:

Any {x}, x X, is a term. If f S is a formal generator of arity r, and t1, . . . , tr are terms, then

f t1 . . . tr is also a term.

Of course, this is just the formal construction of free algebraic systems, bor-rowed from mathematical logic. When we dont want to be too formal, we

write f(t1, . . . , tn) instead of f t1 . . . tn, and t1 t2 instead of t1 t2, if is abinary operation, traditionally written in infix form.

0.4.18. (Generators of an algebraic monad.) Given any (graded) subsetS , we can always find the smallest algebraic submonad ,containing S (i.e. such that S), for example by taking the intersectionof all such algebraic submonads. Another description: is the image of thenatural homomorphism FS from the free algebraic monad generatedby S into , induced by the embedding S . Therefore, (n) consistsof all operations which can be obtained by applying operations from S to thebasis elements {k}n finitely many times.

If = , i.e. ifFS is surjective, we say that S generates .0.4.19. (Relations and strict quotients of algebraic monads.) Notice thatepimorphisms of algebraic monads neednt be surjective, as illustrated byepimorphism Z Q. However, an algebraic monad homomorphism f : is a strict epimorphism (i.e. coincides with the cokernel of its kernel pair) iffall components fn : (n) (n) are surjective. Therefore, strict quotients of are in one-to-one correspondence with compatible (algebraic)equivalence relations R . Any such equivalence relation is completelydetermined by R , i.e. by the collection of equivalence relationsR(n) (n) (n). Compatibility of such a family of equivalence relationswith the algebraic monad structure of essentially means compatibility withall maps () : (n) (m) and (k)n : (k) (n)k (n).

Given any system of equations E , we can construct thesmallest compatible algebraic equivalence relation R = E contain-ing E, e.g. by taking the intersection of all such equivalence relations. The


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24 Overview

corresponding strict quotient /E is universal among all homomor-phisms

f , such that all relations from f(E) are satisfied in .0.4.20. (Presentations of algebraic monads.) This is applicable in particularto subsets E

F

S

F

S

, where S

r

N0 is any N0-graded set.

Corresponding strict quotient FS/E will be denoted by FS| E orS| E, and called the free algebraic monad generated by S modulo relations

fromE. When S or E are finite sets, we can replace S or E in S| E by anexplicit list of generators (with arities) or relations. Furthermore, when wewrite relations from E, we often replace standard free variables {1}, {2},. . . , by lowercase letters like x, y, . . . , thus writing for example F1 = 0[0],[1] | 0 = 0, (x) = x.

Whenever = S| E, we say that (S, E) is a presentation of algebraicmonad . Clearly, any algebraic monad admits a presentation: we justhave to choose any system of generators S (e.g. itself), and takeany system of equations E generating the kernel R FS FS of thecanonical surjection FS (e.g. E = R). If both Sand E can be chosento be finite, we say that is finitely presented (absolutely, i.e. over F).

0.4.21. (Inductive limits of algebraic monads.) Presentations of algebraicmonads are very handy when we need to compute inductive limits of alge-braic monads. For example, the coproduct of two algebraic monads = S| E and = S | E can be computed as S, S| E, E. Since thecokernel of a couple of morphisms can be computed as a strict quotient mod-ulo a suitable compatible algebraic equivalence relation, and filtered induc-tive limits of algebraic monads can be computed componentwise, similarly to

the projective limits, we can conclude that arbitrary inductive and projectivelimits exist in the category of algebraic monads.

0.4.22. (Endomorphism monad of a set.) Let X be any set. We de-note by END(X) its endomorphism monad, constructed as follows. Weput (END(X))(n) := HomSets(X

n, X), e := idX (END(X))(1), and de-fine

(k)n : Hom(Xk, X) Hom(Xn, X)k = Hom(Xk, X) Hom(Xn, Xk)

Hom(Xn, X) to be the composition of maps: (f, g) g f. It is easy tocheck that this indeed defines an algebraic monad END(X), which acts on Xin a canonical way. Furthermore, giving an action of an algebraic monad on set X turns out to be the same thing as giving algebraic monad homo-

morphism END(X), since a -action on X is essentially a rule thattransforms formal operations t (n) into their values [t]X : Xn X.We transfer some classical terminology to our case. For example, a -

module X is said to be exact or faithful if corresponding homomorphism END(X) is injective.


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0.4.23. (Presentations of algebraic monads and algebraic systems.) Let =S| E be a presentation of an algebraic monad . Clearly, algebraic monadhomomorphisms are in one-to-one correspondence to graded maps : S , such that the images under of the relations from E holdin . Applying this to = END(X), we see that a -module structure ona set X is the same thing as assignment of a map [f]X : X

r X to eachgenerator f S of arity r = r(f), in such a way that all relations from Ehold. Since Sis a set of operations with some arities, and Eis a set of relationsbetween expressions involving operations from S and free variables, we seethat (S, E) is an algebraic system, and a -module is just an algebra forthis algebraic system (S, E). In other words, algebras under an algebraicsystem (S, E) are exactly the S| E-modules. Since any algebraic monadadmits a presentation, the converse is also true: the category of modules overan algebraic monad can be described as the category of algebras for somealgebraic system.

In this way we see that algebraic systems are nothing else than presenta-tions of algebraic monads, and algebraic monads are just a convenient cat-egorical way of describing algebraic systems, i.e. theory of algebraic monadsis a category-theoretic exposition of universal algebra.

The reader might think that algebraic monads are not so useful, becausethey are just algebraic systems in another guise. If this be the case, and thereader is still not convinced of the convenience of using algebraic monads, wesuggest to do following two exercises:

Describe the intersection of two algebraic submonads of an algebraicmonads in terms of their presentations.

Find a presentation of END(X), and prove the correspondence betweenhomomorphisms END(X) and -module structures on X, interms of presentations of these algebraic monads.

0.4.24. (Examples: Z and F1.) For example, we can write

Z = 0[0], [1], +[2] | 0 + x = x, (x + y) + z = x + (y + z),x + y = y + x, x + (x) = 0 (0.4.24.1)

Strictly speaking, we should write + + {1}{2}{3} = +{1} + {2}{3} insteadof (x + y) + z = x + (y + z),...In any case the above equality actually means that the category Z-Modconsists of sets X, endowed by a constant 0X , a unary operation X : X X, and a binary operation +X : X

2 X, satisfying the above relations, i.e.abelian groups.


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26 Overview

On the other hand, this algebraic monad Z = Z can be described ex-plicitly, since Z(n) = Z(n) = Zn is the free Z-module of rank n.

Of course, we can remove the commutativity relation from the abovepresentation ofZ, and add relations x + 0 = x and (x) + x = 0 instead,thus obtaining an algebraic monad G, such that G-Mod is the category ofgroups. Sets G(n) are free groups in n generators {1}, . . . , {n}, and we havea natural homomorphism G Z. Well see later that algebraic monad Z iscommutative, while G is not.

Another example: the field with one element F1 can be defined by

F1 = F0[0] , (0.4.24.2)i.e. it is the free algebraic monad generated by one constant. We see thatF1-Mod is the category of sets X with one pointed element 0X X, andF1-homomorphisms are just maps of pointed sets, i.e. maps f : X Y, suchthat f(0X ) = 0Y.

On the other hand, we can describe sets F1(n) explicitly: any such setconsists of n + 1 elements, namely, n basis elements {k}n, and a constant 0.

We hope that these examples illustrate how algebraic systems and alge-braic monads are related to each other.

0.4.25. (Modules over algebraic monads.) After discussing the propertiesof algebraic monads themselves, and their relation to algebraic systems, weconsider some properties of the category -Mod of modules over an alge-braic monad . These are actually properties of algebras under a fixedalgebraic system; we prefer to prove these properties again, using the theoryof algebraic monads, instead of referring to well-known properties of such

algebras, partly because we want to illustrate how the theory of modulesover algebraic monads works. Anyway, our proofs are quite short, comparedto those classical proofs given in terms of algebraic systems.

For example, we show that arbitrary projective and filtered inductivelimits of -modules exist and can be computed on the level of underly-ing sets. (Notice that the statement about filtered inductive limits usesalgebraicity of monad in an essential way.) Furthermore, arbitrary in-ductive limits of -modules exist. Monomorphisms of -modules are justinjective -homomorphisms, and strict epimorphisms are the surjective -homomorphisms. Since we have a notion of free -modules and free -

modules, we have reasonable notions of a system of generators and of a pre-sentation of -module M, and we can define finitely generated and finitelypresented -modules as the -modules isomorphic to a strict quotient ofsome free module (n), resp. to cokernel of a couple of homomorphisms(m) (n).


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These notions seem to have all the usual properties we expect from them,e.g. most of the properties of left modules over a classical associative ring,with some notable exceptions:

Not all epimorphisms are strict, i.e. surjective;

Direct sums (i.e. coproducts) MN are not isomorphic to direct prod-ucts M N;

Direct sum of two monomorphisms neednt be a monomorphism.0.4.26. (Scalar restriction and scalar extension.) Given any algebraic monadhomomorphism : , we obtain a scalar restriction functor :-Mod -Mod, such that = , where and are the for-getful functors into the category of sets. Conversely (cf. 3.3.21), any functorH : -Mod -Mod, such that H = , equals for some algebraicmonad homomorphism : . For example, the forgetful functor fromthe category Z-Mod of abelian groups into the category G-Mod of groupsis the scalar restriction functor with respect to canonical homomorphismG Z.

Furthermore, scalar restriction functor admits a left adjoint , calledthe scalar extension or base change functor. When no confusion can arise,we denote it by , or (), even if it is not immediatelyrelated to any tensor product. For example, if G is a group, then ZG G =Gab = G/[G, G]. Another example: RZ transforms a torsion-free Z-module A = (AZ, AR) into real vector space A(R) = AR. Finally, scalarextension functor with respect to F

is just the free -module functor

L : Sets -Mod.0.4.27. (Flatness and unarity.) Of course, these scalar restriction and exten-sion functors retain almost all their classical properties, e.g. scalar restrictioncommutes with arbitrary projective limits (hence is left exact), and scalarextension commutes with arbitrary inductive limits (hence is right exact).Notice, however, that scalar restriction : -Mod -Mod neednt pre-serve direct sums (i.e. coproducts), as illustrated by the case : F Z. Infact, commutes with finite direct sums iff is right exact iff is unary.This notion ofunarity is fundamental for the theory of generalized rings, butcompletely absent from the classical theory of rings, since all homomorphisms

of classical rings are unary.Similarly, we say that is flat, or that is flat over , if is (left) exact.

0.4.28. (Algebraic monads over topoi.) Chapter 4 treats the topos case aswell. There is a natural approach to algebraic monads over a topos E, based


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28 Overview

on algebraic inner endofunctors. However, the category of algebraic innerendofunctors turns out to be canonically equivalent to EN = Funct(N, E),i.e. an algebraic inner endofunctor over Ecan be described as a sequence{(n)}n0 of objects of Etogether with some transition morphisms () :(n)

(m), defined for each : n

m. Furthermore, the elementary

description of algebraic monads recalled in 0.4.6 transfers verbatim to toposcase, if we replace all sets by objects of topos E.

Applying this to the topos Eof sheaves of sets over a site Sor a topologicalspace X, we obtain a notion of a sheaf of algebraic monads over Sor X: it isa collection of sheaves of sets {(n)}n0, together with maps () : (n) (n), a unit section e (X, (1)) and evaluation maps (k)n : (k) (n)k (n), such that for any open subset U X (or any object U Ob S) (U, ) becomes an algebraic monad (over Sets). Sheaves of modulesover such are defined similarly. We can also define generalized ringedspaces (ringed spaces with a sheaf of generalized rings), their morphisms and

so on.

0.5. (Commutativity.) Chapter 5 is dedicated to another fundamental prop-erty we require from our generalized rings, namely, commutativity. In fact, wedefine a (commutative) generalized ring as a commutative algebraic monad.A classical ring is commutative as an algebraic monad iff it is commutativein the classical sense, so our terminology extends the classical one.

0.5.1. (Definition of commutativity.) Given an algebraic monad , twooperations t (n), t (m), a -module X, and a matrix (xij )1im,1jn,xij X, we can consider two elements x, x X, defined as follows. Letxi. := t(xi1, . . . , xin) be the element of X, obtained by applying t to i-th row

of (xij ), and x.j := t(x1j , . . . , xmj ) be the element, obtained by applying tto j-th column of (xij ). Then x

:= t(x1., . . . , xm.), and x := t(x.1, . . . , x.n).Now we say that t and t commute on X ifx = x for any choice ofxij X.We say that t and t commute (in ) if they commute on any -module X.

Notice that t and t commute iff the commutativity relation is fulfilledfor the universal matrix xij := {(i 1)n + j} in X = (nm). In this waycommutativity of t and t is equivalent to one equation x = x in (nm),which will be denoted by [t, t].

Given any subset S , we can consider its commutant S , i.e.the set of all operations of commuting with all operations from S. This

subset S turns out to be the underlying set of an algebraic submonadof , which will be also denoted by S. We have some classical formulas likeS = S, S S and so on.

Finally, the center of an algebraic monad is the commutant of itself, i.e. the set of all operations commuting with each operation of .


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0.5. Commutativity 29

We say that is commutative, or a generalized ring, if it coincides with itscenter, i.e. if any two operations of commute. If S is any system ofgenerators of , it suffices to require that any t, t S commute.0.5.2. (Why this definition of commutativity?) We would like to present

some argument in favour of the above definition of commutativity. Let be an algebraic monad, M and N be two -modules. Then the setHom(M, N) = HomSets(M, N) = N

M of all maps : M N admits anatural -module structure, namely, the product -structure on NM. Wecan describe this structure by saying that operations t (n) act on maps : M N pointwise:

t(1, . . . , n) : x t

1(x), . . . , n(x)

(0.5.2.1)

So far we have used only -module structure on N, but M might be justan arbitrary set. Now ifM is a -module, we get a subset Hom(M, N) Hom(M, N) = N

M

, and we might ask whether this subset is a -submodule.Looking at the case of modules over classical associative ring R, we seethat HomR(M, N) Hom(M, N) = NM is an R-submodule for all R-modules M and N if and only if R is commutative, and in this case theinduced R-module structure on HomR(M, N) is its usual R-module struc-ture, known from commutative algebra.

Therefore, it makes sense to say that an algebraic monad is commuta-tive iff Hom(M, N) Hom(M, N) = NM is a -submodule for all choicesof -modules M and N. When we express this condition in terms of opera-tions of , taking (0.5.2.1) into account, we obtain exactly the definition ofcommutativity given above in 0.5.1.

0.5.3. (Consequences: bilinear maps, tensor products and inner Homs.)Whenever M and N are two modules over a generalized ring (i.e. com-mutative algebraic monad) , we denote by Hom(M, N), or simply byHom(M, N), the set Hom(M, N), considered as a -module with respectto the structure induced from Hom(M, N) = NM.

Looking at the case of classical commutative rings, we see that we mightexpect formulas like

Hom(M N, P) = Bilin(M, N; P) = Hom(M, Hom(N, P))(0.5.3.1)

This expectation turns out to be correct, i.e. there is a tensor product functor on -Mod with the above property, as well as a notion of -bilinear maps : M N P, fitting into the above formula.

This notion of -bilinear map is very much like the classical one: a map : MN P is said to be bilinear if the map s(x) : N P, y (x, y),


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30 Overview

is -linear (i.e. a -homomorphism) for each x M, and d(y) : M P,x (x, y), is -linear for each y N. Multilinear maps are definedsimilarly.

In this way we obtain an ACU -structure on -Mod, which admitsinner Homs Hom. The unit object with respect to this tensor product is|| = L(1), the free -module of rank one.

When we apply these definitions to a classical commutative ring R, werecover classical notions ofR-bilinear maps, tensor product ofR-modules andso on. On the other hand, when we apply them to Z-Mod, we recover thedefinitions of Chapter 2, originally based on properties of maximal compactsubmonoids and symmetric compact convex sets. This shows that the tensorproduct we consider is indeed a very natural one.

0.5.4. (Algebras over a generalized ring.) Let be a generalized ring. Wecan define a (non-commutative) -algebra as a generalized ring togetherwith a central homomorphism f :

(i.e. all operations from the image

of f must lie in the center of ). Of course, if is commutative, all homo-morphisms are central, i.e. a (commutative) -algebra is a generalizedring with a homomorphism .0.5.5. (Variants of free algebras.) We have several notions of a free algebra,depending on whether we impose some commutativity relations or not:

Let be an algebraic monad, and consider the category of -monads,i.e. algebraic monad homomorphisms with source . We have a(graded) underlying set functor , which admits a left adjointS S. Indeed, if = T | E is any presentation of , then Scan be constructed as

S, T

|E

. Notice that this

S

is something like

a very non-commutative polynomial algebra over S: not only the in-determinates from S are not required to commute between themselves,they are even not required to commute with operations from !

Let be a generalized ring, i.e. a commutative algebraic monad, andconsider the category of all -algebras, commutative or not, i.e. centralhomomorphisms . The graded underlying set functor still admits a left adjoint, which will be denoted by S {S}. If = T | E, then {S} = T, S| E, [S, T], where [S, T] denotes theset of all commutativity relations [s, t], s S, t T. In other words,now the indeterminates from S still dont commute among themselves,

but at least commute with operations from .

Finally, we can consider the category of commutative algebras over a generalized ring , and repeat the above reasoning. We ob-tain free commutative algebras [S] = T, S| E, [S, T], [S, S], which


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are quite similar to classical polynomial algebras, especially when allindeterminates from S are unary.

0.5.6. (Presentations of -algebras.) In any of the three cases listed abovewe can impose some relations on free algebras, thus obtaining the notion of

a presentation of a -algebra. Consider for example the case of commutative-algebras, with = T | E any generalized ring. Then, given any N0-graded set S and any set of relations R [S] [S], we can constructthe strict quotient

[S| R] = [S]/R = T, S| E, [S, T], [S, S], R (0.5.6.1)This strict quotient is easily seen to be a commutative -algebra, and when-ever we are given a -algebra isomorphism = [S| R], we say that (S, R)is a presentation of (commutative) -algebra .

0.5.7. (Finitely generated/presented algebras.) If S can be chosen to befinite, we say that is a finitely generated -algebra, or a -algebra of finitetype, and if both S and R can be chosen to be finite, we speak about finitelypresented -algebras. These notions are quite similar to their classical coun-terparts.

0.5.8. (Pre-unary and unary algebras.) Apart from notions of finite gen-eration and presentation, which have well-known counterparts in classicalcommutative algebra, we obtain notions of pre-unary and unary -algebras.Namely, we say that a -algebra is pre-unary if it admits a system ofunary generators, and unary if it admits a presentation = [S| R], con-sisting only of unary generators (i.e. S

|

|= (1)) and unary relations

(i.e. R |[S] [S]|).These notions are very important for the theory of generalized rings, but

they dont have non-trivial counterparts in classical algebra, since any algebraover a classical ring is automatically unary.

0.5.9. (Tensor products, i.e. coproducts of -algebras.) Since any commu-tative -algebra i admits some presentation [Si | Ei] over , we can easilyconstruct coproducts in the category of commutative -algebras, called alsotensor products of -algebras, by

1 2 = [S1, S2 | E1, E2] (0.5.9.1)Using existence of filtered inductive limits and arbitrary projective limits ofgeneralized rings, which actually coincide with those computed in the cate-gory of algebraic monads, we conclude that arbitrary inductive and projectivelimits exist in the category of generalized rings.


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0.5.10. (Unary algebras as algebras in -category -Mod.) Notice that wemight try to generalize another classical definition of a -algebra and say thata -algebra A is just an algebra in ACU -category -Mod, with respect toour tensor product = , and define A-modules accordingly.

However, the category of such algebras in -Mod turns out to be equiv-alent to the category of unary -algebras, and we obtain same categories ofmodules under this correspondence.

This allows us to construct for example tensor, symmetric or exterioralgebras of a -module M first by applying the classical constructions in-side -category -Mod, and then taking corresponding unary -algebras.Monoid and group algebras can be also constructed in this way; in particu-lar, they are unary.

Unary algebras have some other equivalent descriptions, which correspondto some properties always fulfilled in the classical case. For example, isa (commutative) unary -algebra iff the scalar restriction functor with

respect to commutes with finite direct sums iff it is exact iff it admitsa right adjoint ! iff the projection formula (M N) = M Nholds.

0.5.11. (Affine base change theorem.) Another important result of Chap-ter 5, called by us the affine base change theorem (cf. 5.4.2), essentiallyasserts the following. Suppose we are given two commutative -algebras

and , and put := . Then we can start from a -module M,compute its scalar restriction to , and afterwards extend scalars to . Onthe other hand, we can first extend scalars to , and then restrict themto . The affine base change theorem claims that the two -modules thus

obtained are canonically isomorphic, provided either is unary or is flatover .

We also consider an example, which shows that the statement is not truewithout additional assumptions on or .

0.5.12. We have discussed at some length some consequences of commutativ-ity for categories of modules and algebras over a generalized ring . Now wewould like to discuss the commutativity relations themselves. At first glancecommutativity of two operations of arity 2 seems to be not too useful. Weare going to demonstrate that in fact such commutativity relations are verypowerful, and that they actually imply classical associativity, commutativity

and distributivity laws, thus providing a common generalization of all suchlaws.

0.5.13. (Commutativity for operations of low arity.) Lets start with somesimple cases. Here is some fixed algebraic monad.


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Two constants c, c (0) commute iff c = c. In particular, a gener-alized ring contains at most one constant.

Two unary operations u, u || = (1) commute iffuu{1} = uu{1},i.e. uu = uu in monoid

|

|. This is classical commutativity of multi-

plication.

An n-ary operation t commutes with a constant c iff t(c , c , . . . , c) = c.For example, a constant c and a unary operation u commute iff uc = cin (0). A binary operation and constant c commute iff c c = c.

An n-ary operation t commutes with unary u iff u t({1}, . . . , {n}) =t(u{1}, . . . , u{2}). For example, ift is a binary operation , this meansu({1} {2}) = u{1} u{2}, or u(x y) = ux uy, i.e. classical distribu-tivity relation.

Two binary operations + and commute iff (x+ y)(z + w) = (xz) +(y w). Notice that, while any operation of arity 1 automaticallycommutes with itself, this is not true even for a binary operation ,since (xy)(z w) = (xz)(y w) is already a non-trivial condition.

0.5.14. (Zero.) We say that a zero of an algebraic monad is just a centralconstant 0 (0). Since it commutes with any other constant c, we musthave c = 0, i.e. a zero is automatically the only constant of . Furthermore,we have u0 = 0 for any unary u, 00 = 0 for any binary , and t(0, . . . , 0) = 0for an n-ary t, i.e. all operations fix the zero.

Of course, a generalized ring either doesnt have any constants (then we

say that it is a generalized ring without zero), or it has a zero (and then itis a generalized ring with zero). We have a universal generalized ring withzero F1 = F[0[0]] = F0[0] = 0[0], called the field with one element.Furthermore, F1-algebras are exactly the algebraic monads with a centralconstant, i.e. with zero.

0.5.15. (Addition.) Let be an algebraic monad with zero 0. We say that abinary operation +[2] (2) is a pseudoaddition iff{1} + 0 = {1} = 0 + {1}.This can be also written as e + 0 = e = 0 + e or x + 0 = x = 0 + x. Anaddition is simply a central pseudoaddition. If admits an addition +, anyother pseudoaddition must coincide with +: indeed, we can put y = z = 0in commutativity relation (x + y) (z + w) = (x z) + (y z), and obtainx w = (x + 0) (0 + w) = (x 0) + (0 w) = x + w.

So let be an algebraic monad with zero 0 and addition +. Commu-tativity of + and unary operations u || means u(x + y) = ux + uy, i.e.distributivity. Now consider the commutativity condition for + and itself:


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34 Overview

(x + y) + (z + w) = (x + z) + (y + w). Putting here x = w = 0, we obtainy + z = z + y, i.e. classical commutativity of addition. Next, if we put z = 0,we get (x + y) + w = x + (y + w), i.e. classical associativity of addition.Therefore, classical commutativity, associativity and distributivity laws are

just consequences of our generalized commutativity law.

0.5.16. (Semirings.) Given any (classical) semiring R, we can consider cor-responding algebraic monad R, which will be an algebraic monad with ad-dition (hence also with zero). Conversely, whenever is an algebraic monadwith addition +, we get an addition + on monoid || = (1), commutingwith the action of all unary operations, i.e. R := || is a semiring. One cancheck that = R, i.e. classical semirings correspond exactly to algebraicmonads with addition. For example, semirings N0 and tropical numbers Tcan be treated as generalized rings. This reasoning also shows that any (gen-eralized) algebra over a semiring is also an algebraic monad with addition(since

is central, it maps the zero and addition of into zero and

addition of ), i.e. is also given by a classical semiring. Furthermore, anysuch -algebra is automatically unary.

Of course, we have a universal algebraic monad with addition, namely,

N0 =F1[+[2] | e+ 0 = e = 0 + e] (0.5.16.1)=F[0[0], +[2] | e+ 0 = e = 0 + e] (0.5.16.2)=0[0], +[2] | [+, +], e+ 0 = e = 0 + e (0.5.16.3)

0.5.17. (Symmetry.) A symmetry in an algebraic monad is simply a cen-tral unary operation , such that 2 = e in monoid ||, i.e. (e) =e, or (x) = x. Centrality of actually means t({1}, . . . , {n}) =t({1}, . . . , {n}) for any n-ary operation t. In particular, c = c for anyconstant c.

We have a universal algebraic monad with zero and symmetry, namely,

F1 = F1[[1] | (e) = e] = Z Z (0.5.17.1)Using the notion of symmetry, we obtain a finite presentation ofZ:

Z =F1[+[2] | x + 0 = x = 0 + x, x + (x) = 0] (0.5.17.2)=F[0[0],

[1], +[2]

|x + 0 = x = 0 + x, x + (

x) = 0] (0.5.17.3)

Reasoning as in the case of semirings, we see that classical rings can bedescribed as algebraic monads with addition + and symmetry , compatiblein the sense that x + (x) = 0. For example, if is a classical commutativering, and is a generalized -algebra, then algebraic monad admits a zero,


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a symmetry and an addition (equal to the images of those of ), compatiblein the above sense, hence is a classical ring, i.e. a classical -algebra. Inother words, we cannot obtain any new generalized algebras over a classicalcommutative ring.

0.5.18. (Examples of generalized rings.) Lets list some generalized rings.

Classical commutative rings, e.g. Z, Q, R, C, . . . Classical commutative semirings: N0, R0, T, . . . Algebraic submonads of generalized rings, e.g. Z R, Z() Q, or

AN := Z() Z[1/N] Q. Archimedian valuation rings V K, defined as follows. Let K be a

classical field, equipped with an archimedian valuation | |. Then

V(n) = {(1, . . . , n) Kn

: i

|i| 1} (0.5.18.1)

For example, Z() Q and Z R are special cases of this construc-tion.

Strict quotients of generalized rings. Consider for example the three-point Z-module Q, obtained by identifying all inner points of segment|Z| = [1, 1] (weve denoted this Q by F in 0.2.16), and denote byF the image of induced homomorphism Z END(Q). Then Fis commutative, i.e. a generalized ring, being a strict quotient of Z.Furthermore, it is generated over F

1by one binary operation

, the

image of (1/2){1} + (1/2){2} Z(2):F = F1[[2] | x(x) = 0, xx = x, xy = yx, (xy)z = x(yz)]

(0.5.18.2)

Cyclotomic extensions ofF1:F1n = F1[[1] | n = e] (0.5.18.3)

Modules over F1n are sets X with a marked element 0X and a permu-tation X : X

X, such that nX = idX and X (0X ) = 0X . This is

applicable to F1 = F12.

Affine generalized rings AffR R: take any classical commutative(semi)ring R, and put AffR(n) := {1{1} + + n{n} Rn |

i i =

1}. Then AffR-Mod is just the category of affine R-modules.


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36 Overview

Generalized ring := AffRR0 R. Set (n) is the standardsimplex in Rn, and -Mod consists of abstract convex sets.

0.5.19. (Some interesting tensor products.) We compute some interestingtensor products of generalized rings: Z

Z = Z, and similarly Z

Z

= R,

Z Z() = Q, if the tensor products are computed over F1, F1 or F.Furthermore, F F1 F = F and Z() F1 Z() = Z().0.5.20. (Tensor and symmetric powers.) Given any module M over any gen-eralized ring , we can consider its tensor and symmetric powers Tn(M) =Mn and Sn(M) = Tn(M)/Sn. They seem to retain all their usual proper-ties, e.g. they commute with scalar extension, and a symmetric power of afree -module M is again free, with appropriate products of base elementsof M as a basis.

0.5.21. (Exterior powers, alternating generalized rings and determinants.)On the other hand, exterior powers

n(M) are more tricky to define. First

of all, M has to be a module over a commutative F1-algebra , i.e. gen-eralized ring must admit a zero and a symmetry. Even if this conditionis fulfilled, these exterior powers n(M) dont retain some of their classicalproperties, e.g. exterior powers of a free module are not necessarily free, andin particular nL(n) = ||. In order to deal with these problems we in-troduce and study alternating generalized rings. Over them exterior powersof a free module are still free with the correct basis, e.g. if M is free withbasis e1, . . . , en, then n(M) is freely generated by e1 . . . en, so we writedet M := n(M), and use it to define determinants det u || of endomor-phisms u End(M). Then det(uv) = det(u) det(v), det(idM) = e, so anyautomorphism has invertible determinant. However, invertibility of det(u)doesnt imply bijectivity of u in general case. We need to introduce someadditional conditions (DET), which assure that invertibility det(u) indeedimplies invertibility of u.

Our general impression of this theory of exterior powers and determinantsis that one should avoid using exterior powers in the generalized ring context,and use symmetric powers instead whenever possible. Then we dont haveto impose

arakelov geometry

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