Non-additive geometry

Transcription

1 Composto Math. 143 ( do: /s x M. J. Sha Haran Abstract We develop a language that makes the analogy between geometry and arthmetc more transparent. In ths language there exsts a base feld F, the feld wth one element ; there s a fully fathful functor from commutatve rngs to F-rngs; there s the noton of the F-rng of ntegers of a real or complex prme of a number feld K analogous to the p-adc ntegers, and there s a compactfcaton of Spec O K ; there s a noton of tensor product of F-rngs gvng the product of F-schemes; n partcular there s the arthmetcal surface Spec O K Spec O K, the product taken over F. Contents Introducton F, the feld wth one element The category F Varants F ± The algebrac closure F of F F-rngs, varants, examples Defnton of F-rngs F t, F ±, F λ, F, F λ -rngs Examples of F-rngs Modules Defntons and examples A-submodules and equvalence A-modules Operatons on submodules Operatons on modules Ideals and prmes H -deals and prmes S-prme deals and H -E-deals Localzaton and structural sheaf O A Localzaton Structural sheaf O A Schemes Locally F-rng spaces Zarsk F-schemes F-schemes and the compactfed Spec Z Fbred products Fbred sums of F-rngs Fbred product of F-schemes, the case of Spec Z Spec Z Receved 28 June 2006, accepted n fnal form 13 September Mathematcs Subject Classfcaton 11G99. Keywords: arthmetcal surface, compactfed Spec Z, Remann hypothess, ABC conjecture. Ths journal s c Foundaton Composto Mathematca 2007.

2 8 Monods F-monods Modules over an F-monod Functoral operatons on modules References 688 Introducton The ancent dea of makng arthmetc nto geometry engaged the mnds of great mathematcans such as Kummer, Kronecker, Dedeknd, Hensel, Hasse, Mnkowsk, and especally Artn and Wel. It s a beautful quest nspred by the smlarty between the rng of ntegers Z, and the rng of polynomals Z = k[x] over a feld k; for closer smlarty the functon feld case s relevant where k = F q s a fnte feld. There s nduced smlarty of the fracton felds, the feld of ratonal numbers Q and the feld of ratonal functons Q = k(x. For a prme p of Z, we have the p-adc ntegers Z p = lm Z/p n, and ts feld of fractons [ ] 1 Q p = Z p, p wth dense embeddngs Z Z p and Q Q p. The geometrc analogues are the power seres rng Z f = lm Z/f n = k f [[f]], and the feld of Laurent seres [ ] 1 Q f = Z f = k f ((f, f for f a prme of Z, where k f = k[x]/(f, and the embeddngs Z Z f (respectvely Q = k(x Q f correspond to expandng a polynomal (respectvely a ratonal functon nto a power seres (respectvely a Laurent seres n f. Fnte extensons of Q = k(x correspond one-to-one wth the smooth projectve curves Y defned over fnte extensons of k, and fnte extensons of Q are the number felds. There are two man dffcultes wth ths analogy that we are gong to descrbe, the problem of the real prme of Q, and the problem of the arthmetcal surface, that s defnng for Spec(Z the analogue of the geometrc surface Y k Y. From geometry we know that, n order to have theorems, we must pass from affne to projectve geometry, n partcular we need to add the pont at nfnty to the affne lne, P 1 k = A1 k { }. Ths corresponds to the rng [ ]/( 1 1 n [[ ]] 1 Z = lm k = k, x x x and ts fracton feld (( 1 Q = Z [x] =k ; x the embeddng Q Q s the expanson of a ratonal functon as a Laurent seres n 1/x. The analogue of for Q s the real prme, whch we denote by η. The assocated feld s Q η = R, the real numbers. But there s no analogue Z η of Z. For fnte prmes p, Z p = {x Q p, x p 1}. We have to carry remander when we add elements of Z p unlke the smple addton of power seres n Z f or Z. We carry the remander from the larger scale p j to the smaller scale p j+1, hence x + y p max{ x p, y p }, 619

3 M. J. Sha Haran and Z p s closed under addton. In contrast, when we add real numbers, we carry the remander from the smaller to the larger scale, we have only the weaker trangle nequalty x + y η x η + y η, and {x Q η, x η 1} =[ 1, 1] s not closed under addton. The second problem s that n geometry we have products, n partcular the affne plane A 2 = A 1 A 1, wth the rng of polynomal functons k[x] k k[x] =k[x 1,x 2 ], the tensor product ( sum n the category of k-algebras of Z wth tself. When we try to fnd the analogous arthmetcal surface, we fnd Z Z = Z. The ntegers Z are the ntal object n the category of rngs, so ts tensor product ( sum n the category of rngs wth tself s just Z. For any geometry that s based on rngs, Spec Z wll be the fnal object, and Spec Z Spec Z = Spec Z, whch means the arthmetcal surface reduces to the dagonal! Motvated by the Wel conjectures, Grothendeck developed the modern language of algebrac geometry, the language of schemes [EGA], based on commutatve rngs. Grothendeck came from a background of functonal analyss, where the paradgm of geometry = commutatve rngs was frst set. It s the famous Gelfand Namark theorem on the equvalence of the category of (compact, Hausdorff topologcal spaces and the category of commutatve (untal C -algebras. Ths equvalence s gven by assocatng wth the topologcal space X the algebra C(X ={f : X C,f contnuous}, usng addton and multplcaton (and conjugaton, and norm of C to defne the smlar structure on C(X, gvng rse to the structure of rng (and C -algebra structure on C(X. The axoms of a commutatve C -algebra are generalzatons of the axoms of C: when X = { } reduces to a pont, C( = C. It s clear that there s no connecton between addton and multplcaton of C and the geometry of X. The language of rngs (and commutatve C -algebras s just one convenent way n whch to encode geometry. Wth the goal of fndng the arthmetcal surface, the dea of abandonng addton has recently appeared n the lterature. Soulé [Sou04] talks of the feld wth one element F, and tres to defne F-varetes as a subcollecton of Z-varetes. Kurokawa, Ocha and Wakayama [KOW03] were the frst to suggest abandonng addton, and workng nstead wth the multplcatve monods. Ths dea was further descrbed n Detmar [De05], but note that the spectra of monods always looks lke the spectra of a local rng: the non-nvertble elements are the unque maxmal deal. For Kurokawa there s also a zeta world of analytc functons that encode geometry, where the feld F s encoded by the dentty functon of C; see Mann [Man95]. Here we take our clues from the problem of the real prme to understand F, and then develop the language of geometry based on the concept of F-rng. Denote by x η the eucldan norm of x =(x 1,..., x n R n,.e. x η = x 2 η. We have the fundamental Cauchy Schwartz nequalty x y η = x 1 y x n y n η x η y η. Hence [ 1, 1] wll contan x 1 y x n y n = x y, whenever x η, y η 1, although t s not closed under addton. Moreover, unlke addton, matrx multplcaton behaves well n the real prme: a b η a η b η for real or complex matrces a, b where η s the operator norm. Wthn matrx multplcaton there s encoded addton, but we have to take matrx multplcaton as the more fundamental operaton. We add also the operatons of drect sum and of tensor product of matrces. Our analogue of Z p (respectvely of the localzaton Z (p for the real prme η s the category O R,η 620

4 (respectvely O Q,η wth objects the fnte sets and morphsms from X to Y gven by the Y X matrces wth real (respectvely ratonal coeffcents and wth operator norm 1; these matrces are closed under the operatons of drect sums and tensor products (but are not closed under addton. Rememberng that the quantum area n physcs started wth Hesenberg s dscovery of matrx multplcaton as the fundamental operaton descrbng the energy levels of mcroscopc systems, perhaps n the future also physcs wll beneft from the language of non-addtve geometry. The contrbuton to arthmetc s evdent: the real ntegers Z η become a real object, and the arthmetcal surface exsts and does not reduce to the dagonal. Some well-known conjectures of arthmetc (Remann hypothess, ABC,... are easy theorems n the geometrc analogue of a curve C over a fnte feld. Ths s because we can form the surface C C. The knowledge of the frst nfntesmal neghborhood of the dagonal C wthn C C,.e. of dfferentals, s often suffcent to prove theorems n geometry whose arthmetc analogues are deep conjectures. Therefore, the further study of the arthmetcal surface F(Z F F(Z, ts compactfcaton usng F(Z F Z η, and the arthmetc frst nfntesmal neghborhood of the dagonal are mportant challenges. Here we gve only the foundatons of the language of non-addtve geometry. In 1 we decpher what s the feld wth one element F. The dea s that, whle F degenerates nto one (or two elements, there s a whole category of F-valued matrces. There are varous degrees of structures one can mpose on F. In 2 we gve the basc noton of an F-rng. As mportant examples of F-rngs we have: F(A, the F-rng attached to a commutatve rng A; O η, the F-rng of ntegers at a real or complex prme η of a number feld; and ts resdue feld F η, the F-rng of partal sometres. In 3 we gve the elementary theory of modules over F-rngs, and dscuss (fbred sums and products, kernels and cokernels, free modules, tensor products, and base change. A novelty of the non-addtve settng s the connecton between submodules and equvalence modules of a gven module. In 4 we gve the elementary theory of deals and prmes. We assocate wth any F-rng A ts spectrum Spec A, a compact sober space wth respect to the Zarsk topology. (A topologcal space s sober f every closed rreducble subset has a unque generc pont. In 5 we gve the theory of localzaton. It gves rse to a sheaf of F-rngs over Spec A. By glung such spectra we get Zarsk F-schemes. In 6 we gve the theory of F-schemes whch are the pro-objects of Zarsk F-schemes. As mportant examples we gve the compactfcaton of Spec Z and of Spec O K, K a number feld. Ths s our soluton to the problem of the real prme. In 7 we gve the tensor product, the (fbred sum n the category of F-rngs, and we obtan the (fbred product n the categores of Zarsk F-schemes and of F-schemes. As an mportant example we defne and descrbe the fbred product F(Z F F(Z, ts compactfcaton, and ts generalzaton for number felds. Ths s our soluton to the problem of the arthmetcal surface. In 8 we work over a fxed F-rng F, and repeat the above constructons n the category of monod objects n F-modules. Everythng goes through, the tensor product of F-monods s just ther tensor product as modules, so we avod the complcated product of 7, but the functor from commutatve rngs to F-monods s not fully fathful. 1. F, the feld wth one element We defne a category F wth objects the fnte sets endowed wth two symmetrc monodal structures and. The unt element [0] for s the ntal and fnal object of the category, and s dstrbutve over. 621

5 1.1 The category F M. J. Sha Haran We consder F-vector spaces as fnte sets X wth a dstngushed zero element 0 X X, and set X + = X \{0 X }. For a commutatve rng A, we let A X = x X + A x denote the free A-module wth bass X +, and thnk about A X as A F X obtaned by base extenson from F to A. We let F[A] Y,X = Hom A (A X, A Y, the Y + X + matrces wth values n A. The base extenson of X from F to Z η and to Q η = R gves Z η X and Q η X: Q η X s the real vector space wth bass X +, and Z η X s the subset of Q η X of vectors wth norm 1 n the nner product gven by decreeng X + to be an orthonormal bass. We have the Y + X + real-valued matrces, and F[Q η ] Y,X = Hom Qη (Q η X, Q η Y, (Z η Y,X = {f F[Q η ] Y,X,f(Z η X Z η Y } = {f, f η 1}, where f η denotes the operator norm on F[Q η ] Y,X. A map of fnte sets ϕ : X Y, preservng the zero elements ϕ(0 X =0 Y, nduces an A-lnear map ϕ A : A X A Y, ϕ A F[A] Y,X. For ϕ Qη : Q η X Q η Y to map Z η X nto Z η Y t s necessary and suffcent that ϕ s an njecton of X \ ϕ 1 (0 Y nto Y. Thus we set F Y,X = {ϕ : X Y, ϕ(0 X = 0 Y,ϕ X\ϕ 1 (0 Y njectve}, (1.1 and we vew F as the category wth objects fnte sets wth a dstngushed zero element, and wth arrows F Y,X = Hom F (X, Y. In practce, we shall gnore the dstngushed elements, and vew F as the category wth objects fnte sets (wthout a dstngushed zero element, and wth arrows the partal bjectons It s clear that and s an somorphsm of categores F Y,X = {ϕ : V W bjecton,v X, W Y }. (1.1 X X + := X \{0 X } ϕ {ϕ : X \ ϕ 1 (0 Y ϕ(x \ ϕ 1 (0 Y } F Y,X F Y +,X +. We shall dentfy F wth F. Thus from now on the objects of F are fnte sets wthout a dstngushed zero element. Alternatvely, F Y,X are the Y X matrces wth entres 0, 1 and wth at most one 1 n every row and column. We have a functor gven by the dsjont unon of sets. More formally, for sets X, Y we let : F F F (1.2 X Y = {(z, {0, 1}; =0 z X, =1 z Y } (

6 and for f 0 F X,X,f 1 F Y,Y, we have f 0 f 1 F X Y,X Y gven by f 0 f 1 (z, = (f (z,. (1.4 (Note that n the verson of F where the objects have a dstngushed zero element, X Y s obtaned from the dsjont unon X Y by dentfyng 0 X wth 0 Y. We have for f 0 F X,X,f 1 F Y,Y, and (f 0 f 1 (f 0 f 1 =(f 0 f 0 (f 1 f 1 (1.5 d X d Y = d X Y. (1.6 The operaton makes F nto a symmetrc-monodal category. The dentty element s the empty set [0] (or the set wth only the dstngushed zero element, whch s the ntal and fnal object of the category F. There are canoncal somorphsms n F: X [0] l X X r X [0] X. (1.7 The commutatvty somorphsm c X,Y F Y X,X Y s gven by c X,Y (z, = (z,1. (1.8 The assocatvty somorphsm a X,Y,Z F X (Y Z,(X Y Z s gven by a((w, 0, 0 = (w, 0, a((w, 1, 0 = ((w, 0, 1, a(w, 1 = ((w, 1, 1. We shall usually abuse notaton and vew l X,r X,c X,Y,a X,Y,Z as dentfcatons; thus e.g. for f F X,X we wrte f 0 f 1 = f 1 f 0 nstead of We have a functor (1.9 c X 0,X 1 (f 0 f 1 = (f 1 f 0 c X0,X 1. (1.10 : F F F (1.11 gven by the product of sets X Y = {(x, y x X, y Y }, and for f 0 F X,X, f 1 F Y,Y, we have f 0 f 1 F X Y,X Y gven by f 0 f 1 (x, y =(f 0 (x,f 1 (y. (1.12 (Note that workng wth the verson of F where the objects have a dstngushed zero element, X Y s obtaned from the product X Y by dentfyng (x, 0 Y and (0 X,y wth (0 X, 0 Y for all x X, y Y. We have for f 0 F X,X,f 1 F Y,Y, and (f 0 f 1 (f 0 f 1 =(f 0 f 0 (f 1 f 1 (1.13 d X d Y = d X Y. (1.14 The operaton also makes F nto a symmetrc monodal category. The dentty element s the set wth one element [1] (or the set wth a dstngushed zero element 0, and another element [1] = {0, 1}. We have agan somorphsms n F: X [1] l X X r X X [1], (

7 M. J. Sha Haran and c X,Y F Y X,X Y, c X,Y (x, y =(y, x, (1.16 a X,Y,Z F X (Y Z,(X Y Z, a X,Y,Z((x, y,z = (x, (y, z. (1.17 We have as well the dstrbutvty somorphsm d X0,X 1 ;Y F (X0 Y (X 1 Y,(X 0 X 1 Y d X0,X 1 ;Y ((x,,y = ((x, y,, {0, 1}. (1.18 We abuse notaton and vew lx,r X,c X,Y,a X,Y,Z,d X 0,X 1 ;Y as dentfcatons; thus e.g. for f F X,X,g F Y,Y we wrte whch should be read as (f 0 f 1 g =(f 0 g (f 1 g (1.19 d X 0,X 1 ;Y [(f 0 f 1 g] = [(f 0 g (f 1 g] d X0,X 1 ;Y. (1.19 We note that there s a natural nvoluton F Y,X FX,Y,f f t. (1.20 When vewng F Y,X as the partal bjectons f : V W, V X and W Y, f t s the nverse bjecton, f t = f 1 : W V. When we vew F Y,X as 0, 1 matrces, f t s the transpose matrx. We have (g f t = f t g t, ( (d X t = d X, ( (f t t = f, ( (f 0 f 1 t = f t 0 f t 1, ( (f 0 f 1 t = f t 0 f t 1. ( Remark. Whenever we use the notaton for composton f g t wll always be mplctly assumed that the doman of f s the range of g; thus e.g. f we have (f 0 f 1 g and f F X,X, t s mplctly assumed that g has range X 0 X Varants F ± The model F for the feld wth one element s the one we shall use here, but there s a varant F ± whch s mportant, and leads to a tghter theory. The objects of the category F ± are fnte sets X together wth an acton of the group {±1}, wthout fxed ponts (or wth a unque fxed pont the zero element. A subset X + X wll be called a bass f X s the dsjont unon of X + and X + = { x x X + }. The maps f F ± Y,X are partal bjectons f : V W, V X, W Y, V = V, W = W, that commute wth the acton f( x = f(x. Fxng bass X + X, Y + Y, we can dentfy the elements of F ± Y,X wth the Y + X + matrces of entres 0, 1, 1, wth at most one non-zero term n each row and column. The map f s dentfed wth the matrx M(f, where for x X +, y Y +, M(f y,x = 0 (respectvely, 1, 1 f f(x ±y (respectvely, y, y. We have functors, : F ± F ± F ±, (1.22 X Y = dsjont unon of X and Y, wth ts natural {±1} acton, ( X Y = X Y/ (x,y ( x, y, wth {±1} acton : (x, y =( x, y =(x, y. (

8 Wrte x y for the mage of (x, y n X Y ; we have for f 0 F ± X,X,f 1 F ± Y,Y, f 0 f 1 (x y =f 0 (x f 1 (y. (1.23 The unt for s [0], the ntal and fnal object of F ±. The unt for s [±1]. The analogue of formulas (1.5 to (1.19 reman true for F ±. We have an nvoluton F ± F ±,f f t, where f t s the nverse bjecton (or transpose of an Y + X + matrx, and formulas ( reman true for F ±. Defnton. Let X be an object of F ± and let X + X be a bass. The number of elements of X + : d =#X + wll be called the dmenson of X, and denoted d = dm X. For n =1,..., d let where s the equvalence relaton P n (X ={x 1 x n X X x ±x j for j} (1.24 n (X =P n (X/ x σ(1 x σ(n sgn(σ x 1 x n, for σ S n. (1.25 Wrte x 1 x n for the mage of x 1 x n P n (X n n (X. A map f F ± Y,X nduces a map P n (f F ± P n (Y,P n (X, whch nduces n turn a map n (f F ± n (Y, n (X, n (f(x 1 x n =f(x 1 f(x n. (1.26 For n>dwe have n (X = [0], and by defnton we let 0 (X =[±1]. Thus we have a sequence of functors n : F ± F ±,n=0, 1,..., (1.27 n (f g = n (f n (g, ( n (d X = d n (X, ( n (f t =( n (f t. ( There are natural somorphsms n F ± whch we vew as dentfcatons n (X Y = j (X n j (Y. (1.28 0jn Remark When we consder the objects F of the category F (respectvely F ±, we assume that t contans [n],n 0 (respectvely [±n], and that t contans X Y, X Y (respectvely and n (X whenever t contans X, Y. Hence we may assume F and F ± are countable sets. On the other hand, we shall not use the actual realzaton of F n most of what follows. All we need s a category F wth two symmetrc monodal structures and, the unt element [0] for s the ntal and fnal object of F, s dstrbutve over and t respects [0]: X [0] = [0]. Ths opens up the possblty of ntroducng quantum deformatons. 1.3 The algebrac closure F of F We can smlarly work over the algebrac closure F of F, whch n arthmetc means adjonng all roots of unty µ = Q/Z. The objects of F are sets X wth µ-acton, satsfyng the followng two propertes: ( set X decomposes nto a fnte unon of µ-orbts X = X 1 X d, X = µ x ; (

9 M. J. Sha Haran ( for x X there s a natural number N and a fnte set of prmes {p 1,..., p l }, p N, such that the stablzer of x n µ s gven by Here µ N = {ζ µ ζ N =1} and µ p = n µ p n. {ζ µ ζ x = x} = µ N µ p 1 µ p l. ( Let x j X, 1 j d, be such that X j = µ x j for each j. Then the subset X + = {x 1,..., x d } X representng the µ-orbts wll be called a bass for X, and d =#X + = dm X the dmenson of X. The maps n the category F from an object X to an object Y are gven by µ-covarant partal bjectons F Y,X = {f : V W V X, W Y, V = µ V, W = µ W, f(ζx =ζf(x, x V, ζ µ}. ( We have functors, : F F F, (1.31 X Y = dsjont unon of X and Y, wth ts natural µ-acton, ( X Y = X Y/ (x,y (ζx,ζ 1 y, wth µ-acton ζ (x, y =(ζx, y =(x, ζy. ( We wrte x y for the mage of (x, y n X Y. For f F Y,X we have and we have f 0 f 1 F Y0 Y 1,X 0 X 1,f 0 f 1 (z, = (f (z,, =0, 1, f 0 f 1 F Y0 Y 1,X 0 X 1,f 0 f 1 (x 0 x 1 =f 0 (x 0 f 1 (x 1. Both and make F nto symmetrc monodal category; the unt for s the empty set [0] whch s the ntal and fnal object of F; the unt for s [1] = µ. The analogue of formulas (1.5 (1.19 reman true for F. We have an nvoluton on F satsfyng (1.21. We have λ-operatons: for an object X of F of dmenson d = dm X and for n =1,..., d we let P n (X ={x 1 x n X X x ζx j for j, ζ µ}, n (X = P n (X/ xσ(1 x σ(n sgn(σ x 1 x n,σ S n. (1.32 We wrte x 1 x n for the mage of x 1 x n n n (X,x X. A map f F Y,X nduces a map n (f F n (Y, n (X by n (f(x 1 x n =f(x 1 f(x n, Doman( n (f = {(x 1 x n x Doman(f}. (1.33 We let 0 (X = [1] = µ, 1 (X =X, and n (X = [0] for n>dm X. Thus we have a sequence of functors n : F F,n =0, 1, 2,..., and (1.28 remans vald. A novelty of F s that we have a sequence of functors gven by Adam s operators ψ n : F F, n = ±1, ±2,..., ψ n (X = the set X wth the new µ-acton ζ (n x = ζ n x (1.34 (we can take n n {n =(n p Ẑ = p Z p,n p Z p for all but fntely many p}. These functors satsfy ψ n (X Y =ψ n (X ψ n (Y, ψ n (X Y =ψ n (X ψ n (Y, (1.35 and are the analogue n our settng of the Frobenus endomorphsms n the theory of varetes over F q. (Indeed, the acton on K-theory of the Frobenus endomorphsm for such varetes s gven by ψ q. 626

10 2. F-rngs, varants, examples We gve the defnton of F-rngs and of Rng category. We gve varous varants of F-rngs wth nvoluton or wth λ-rng structure. We then gve our man examples. 2.1 Defnton of F-rngs Defnton 2.1. An F-rng s a category A wth objects the fnte sets F, and arrows A Y,X = Hom A (X, Y contanng F Y,X,.e. we have a fathful functor F A whch s the dentty on objects. We assume [0] s the ntal and fnal object of A. We have two functors, : A A A, whch agree wth the gven functors on F, and whch make A nto a symmetrc monodal category wth the gven dentty (l X, r X ; l X, r X, commutatvty (c X,Y ; c X,Y, assocatvty (a X,Y,Z; a X,Y,Z somorphsms of F. We assume that s dstrbutve over usng the somorphsm d X0,X 1 ;Y of F. Thus n explct terms, an F-rng s a set wth operatons satsfyng A = Y,X F A Y,X, (2.2 : A Z,Y A Y,X A Z,X, (2.2.1 : A Y0,X 0 A Y1,X 1 A Y0 Y 1,X 0 X 1, (2.2.2 : A Y0,X 0 A Y1,X 1 A Y0 Y 1,X 0 X 1, (2.2.3 f (g h =(f g h; (2.3.1 d Y f = f = f d X, f A Y,X ; (2.3.2 (f 0 f 1 (g 0 g 1 = (f 0 g 0 (f 1 g 1, g A Y,X,f A Z,Y ; (2.4.1 d X d Y = d X Y ; (2.4.2 f 0 f 1 = f 1 f 0 ; (2.4.3 f 0 (f 1 f 2 = (f 0 f 1 f 2 ; (2.4.4 f d [0] = f; (2.4.5 (f 0 f 1 (g 0 g 1 = (f 0 g 0 (f 1 g 1, g A Y,X,f A Z,Y ; (2.5.1 d X d Y = d X Y ; (2.5.2 f 0 f 1 = f 1 f 0 ; (2.5.3 f 0 (f 1 f 2 = (f 0 f 1 f 2 ; (2.5.4 f d [1] = f; (2.5.5 f (g 0 g 1 = (f g 0 (f g 1. (2.6 Remark. We remnd the reader that we omt the wrtng of the canoncal somorphsms of F. Thus e.g (2.4.3 should be wrtten (f 0 f 1 c X1,X 0 = c Y1,Y 0 (f 1 f 0, f A Y,X. (2.4.3 We assume that F Y,X A Y,X, and that the above operatons,, agree wth the gven operatons on F. In partcular, we have the zero map 0 Y,X A Y,X, whch s the unque map that factors through [0]. 627

11 M. J. Sha Haran We note that on A [1],[1], the operatons of composton and of tensor product nduce the same operaton, makng A [1],[1] nto a commutatve monod: f d def = f g =(f d [1] (d [1] g =f g = g f, f, g A [1],[1]. (2.7 The set A [1],[1] has a unt 1 comng from the map d [1],[1] : [1] [1], and a zero element 0 comng from the map 0 [1],[1] : [1] [1], z 0 [1]. The set A [1],[1] acts on the sets A Y,X, Ths acton satsfes f g def = f g, f A [1],[1], g A Y,X. (2.8 (f 0 f 1 g = f 0 (f 1 g,f A [1],[1], g A Y,X ; ( g = g; ( g =0 Y,X ; (2.8.3 f (g h =(f g h = g (f h; (2.8.4 f (g 0 g 1 =(f g 0 (f g 1 ; (2.8.5 f (g 0 g 1 =(f g 0 g 1 = g 0 (f g 1. (2.8.6 Defnton 2.9. Let A, B be F-rngs. A functor ϕ : A B s a homomorphsm of F-rngs f ϕ(a Y,X B Y,X, (2.9.1 ϕ(f =f for f F Y,X, (2.9.2 ϕ(f g =ϕ(f ϕ(g, (2.9.3 ϕ(f 0 f 1 =ϕ(f 0 ϕ(f 1, (2.9.4 ϕ(f 0 f 1 =ϕ(f 0 ϕ(f 1. (2.9.5 Thus ϕ s a functor over F that respects and. It s clear that f ϕ : A B, ψ : B C are homomorphsms of F-rngs, then ψ ϕ s a homomorphsm of F-rngs, hence we have a category F-Rngs, wth F as an ntal object. Remark. A (commutatve rng category A s a category wth a symmetrc monodal structure : A A A, wth assocatvty (respectvely commutatvty, unt somorphsms a (respectvely, c, u, wth the unt object for, denoted by [0], beng the ntal and fnal object of A, and another symmetrc monodal structure : A A A, wth assocatvty (respectvely commutatvty, unt somorphsms a (respectvely, c,u, the unt object for s denoted by [1], and dstrbutve somorphsms d Y ;X0,X 1 : Y (X 0 X 1 (Y X 0 (Y X 1 functoral n Y, X 0,X 1 A, and compatble wth a, c, u, a,c,u. That s, we have commutatve dagrams. For X,Y objects of A, we have the followng. 628

12 (a Y ((X 0 X 1 X 2 d Y (X 0 X 1 Y X 2 d a Y (X 0 (X 1 X 2 Y X 0 Y (X 1 X 2 d d d d d (Y X 0 Y X 1 Y X 2 a Y X 0 (Y X 1 Y X 2 (a (Y 1 Y 0 (X 0 X 1 a Y 1 (Y 0 (X 0 X 1 d d Y 1 ((Y 0 X 0 (Y 0 X 1 d (Y 1 Y 0 X 0 (Y 1 Y 0 X 1 a a Y 1 (Y 0 X 0 Y 1 (Y 0 X 1 d (c Y (X 0 X 1 d c Y (X 1 X 0 d (Y X 0 (Y X 1 (u Wth [1] denotng the unt object for, c (Y X 1 (Y X 0 d [1] (X 0 X 1 u d u u ([1] X 0 ([1] X 1 X 0 X 1 (u The canoncal map gves somorphsm Y [0] [0], and we have Y (X [0] d u Y X d d [0] (Y X (Y [0] (Y X [0] u A homomorphsm of commutatve rng categores ϕ : A A s a functor respectng, a, c, u,, a,c,u,d. Thus an F-rng s a homomorphsm of commutatve rng categores ϕ : F A whch s a bjecton on objects. Most of what we do n the followng works more generally for commutatve rng categores, but workng wth F-rngs s easer and allows the suppresson of the somorphsms a, c, u, a,c,u,d. On the other hand t wll be nterestng to work more generally wth braded rng categores, replacng the symmetrc monodal structure by a braded monodal structure; ths mght lead to the quantum geometry behnd [Har01] and [Har06]. 2.2 F t, F ±, F λ, F, F λ -rngs Remark We can defne F t -rngs to be F-rngs wth nvoluton A Y,X A X,Y, f f t, (

13 M. J. Sha Haran agreeng wth the gven nvoluton on F, and satsfyng (f g t = g t f t, ( f tt = f, ( (f 0 f 1 t = f t 0 f t 1, ( (f 0 f 1 t = f t 0 f t 1. ( A homomorphsm of F t -rngs s a homomorphsm of F-rngs ϕ satsfyng Thus we have a category of F t -Rngs. ϕ(f t = ϕ(f t. Remark One defnes F ± -rngs A as a category wth objects F ±, wth [0] as an ntal and fnal object, and wth symmetrc monodal structures, : A A A, ( wth [0], [±1] as denttes, wth dstrbutve over, and wth a functor F ± A whch s the dentty on objects and respects the symmetrc monodal structures and. A homomorphsm ϕ : A B of F ± -rngs s a functor over F ± whch respects the symmetrc monodal structures and. Thus we have the category F ± -Rngs. Replacng F ± by F, and [±1] by µ, one obtans the defnton of the category F-Rngs. We can smlarly defne F ±,t -rngs to be F ± -rngs wth nvoluton, agreeng wth the gven nvoluton on F ±, and respectng and. Maps of F ±,t -rngs are maps of F ± -rngs respectng the nvoluton, hence we have a category F ±,t -Rngs. Smlarly we have the category F t -Rngs. Defnton An F λ -rng A s an F ± -rng, together wth functors such that k : A A, k =0, 1,... ( k : A Y,X A k (Y, k (X ( k (d X = d k (X ( and moreover k agree wth the gven operaton on F ± cf. (1.26, and 0 (f = 1, ( (f =f, ( k (f g = j (f k j (g. ( jk One smlarly defnes an F λ,t -rng to be an F ±,t -rng and an F λ -rng such that k (f t = k (f t. ( Smlarly replacng F ± by F one defnes an F λ -rng to be an F-rng A together wth functors ( satsfyng ( ( Smlarly, F λ,t -rngs are F λ -rngs wth nvoluton satsfyng ( Remark. It s possble to add further axoms (e.g. the ones correspondng to specal λ-rngs. Here we shall only note the followng. For X a fnte set wth {±} acton, an orentaton on X s a choce of an somorphsm ε :[±1] d (X, d = dm X, (

14 .e. t s a choce of one of the two (non-zero elements ε(1 d (X. For A an F λ -rng, and for a A X,X we have det X (a A [±1],[±1], defned by det X (a =ε 1 d (a ε. ( It s ndependent of the choce of ε F d (X,[±1] A d (X,[±1], and t satsfes det X (a a = det X (a det X (a, ( det X (d X = 1(= d [±1], ( det X1 X 2 (a 1 a 2 = det X1 (a 1 det X2 (a 2. ( The choce of the orentaton ε on X gves also the dualty somorphsm unquely determned by For a A X,X, we have a adj A X,X, defned by ε : X d 1 (X (and ε : j (X d j (X, ( x ε(x =ε(1, x X. ( a adj = ε 1 d 1 (a t ε, ( where ε F d 1 (X,X A d 1 (X,X. It s ndependent of the choce of ε, and t satsfes (a b adj = b adj a adj, ( (d X adj = d X. ( It s useful to have the expanson of the determnant by rows/columns, a a adj = a adj a = det(a d X. ( As a corollary of ( we have that a A X,X s nvertble (.e. there exsts a 1 A X,X wth a a 1 = a 1 a = d X f and only f det X (a A [±1],[±1] s nvertble. Indeed, f a s nvertble det(a s always nvertble wth nverse det(a 1, and conversely, f det(a s nvertble then by ( a tself s nvertble wth nverse det(a 1 a adj. Remark. For an F-rng A (or an F ± or F-rng, we let GL X (A denote the group of nvertble elements n A X,X, We have homomorphsms, GL X (A ={a A X,X a 1 A X,X,a a 1 = a 1 a = d X }. ( GL X1 (A GL X2 (A GL X1 X 2 (A, (a 1,a 2 a 1 a 2, ( GL X1 (A GL X2 (A GL X1 X 2 (A, (a 1,a 2 a 1 a 2. ( In partcular, we have the homomorphsms hence the drect lmt GL [n] (A GL [n+1] (A, a a d [1], GL (A = lm GL [n] (A. ( We can then defne the hgher K-groups of A followng Qullen [Qu73]: K n (A =π n+1 (BGL (A +. ( Note that for an F-rng A assocated wth a commutatve rng B, A = F(B (see example 1 below, we have GL (A = lm GL n (B and K n (A =K n (B. 631

15 M. J. Sha Haran 2.3 Examples of F-rngs Example 0. F s an F-rng. Example 1. wth Let A be a commutatve rng (always wth dentty. We denote by F(A the F-rng F(A Y,X = Hom A (A X, A Y =Y X matrces wth values n A, (2.13 where s the usual composton of A-lnear homomorphsms (or multplcaton of A-valued matrces, and where s the usual drect sum, and the tensor product. Note that a homomorphsm of commutatve rngs ϕ : A B nduces a map of F-rngs F(ϕ :F(A F(B, hence we have a functor F : Rngs F-Rngs. ( Moreover let ϕ : F(A F(B be a map of F-rngs. For a F(A Y,X wrte a y,x = j t y a j x A = F(A [1],[1] ( for ts matrx coeffcents, where j x,j t y are the morphsms of F gven by and where j t y j x : [1] X, j x (1 = x X, : Y [1] s the partal bjecton {y} { 1}. ( Snce ϕ s a functor over F, and jy,j t x F, we have ϕ(a y,x = ϕ(a y,x and ϕ s determned by ϕ : A = F(A [1],[1] B = F(B [1],[1]. Ths map s multplcatve, ϕ(a 1 a 2 =ϕ(a 1 ϕ(a 2,ϕ(1 = 1, and moreover t s addtve, [ ϕ(a 1 + a 2 =ϕ (a 1,a 2 Thus the functor F s fully fathful. ( ] 1 =(ϕ(a 1 1,ϕ(a 2 ( 1 = ϕ(a 1 1 +ϕ(a 2. ( Example 2. Let M be a commutatve monod wth a unt 1 and a zero element 0. Thus we have an assocatve and commutatve operaton and 1 M s the (unque element such that and 0 M s the (unque element such that M M M, (a, b a b, a (b c =(a b c, a b = b a, (2.14 a 1=a, a M, ( a 0 = 0, a M. ( Let F M denote the F-rng wth F M Y,X the Y X matrces wth values n M wth at most one non-zero entry n every row and column. Note that ths s ndeed an F-rng wth the usual multplcaton of matrces (there s no addton nvolved only multplcaton n M, drect sum, and tensor product. Denotng by Mon 0,1 the category of commutatve monods wth unt and zero elements, and wth maps respectng the operaton and the elements 0, 1, the above constructon yelds a functor Ths s the functor left-adjont to the functor Mon 0,1 F-Rngs, M F M. ( F-Rngs Mon 0,1, A A [1],[1], 632

16 namely Hom F-Rngs (F M,A = Hom Mon0,1 (M, A [1],[1]. ( As a partcular example, take M = M q to be the free monod (wth zero generated by one element q, Then M q = q N {0}. Hom F-Rngs (F M q,a=a [1],[1]. Example 3. Let S denote the F-rng of sets. The objects of S are the fnte sets of F, and we let S Y,X be the partally defned maps of sets from X to Y, S Y,X = {f : V Y V X}. ( Notce that f A s an F-rng, the opposte category A op s agan an F-rng, snce F op = F and snce the axoms of an F-rng are self-dual. In partcular, we have the F-rng S op wth S op Y,X = {f : V X V Y }. ( We have the F-rng of relatons R that contans both S and S op, wth The composton of F R Y,X and G R Z,Y s gven by and G F R Z,X. R Y,X = {F Y X a subset}. ( G F = {(z, x Z X y Y wth (z, y G, (y, x F }, ( The sum F 0 F 1 R Y0 Y 1,X 0 X 1 of F R Y,X s gven by the dsjont unon of F 0 and F 1, and the product F 0 F 1 R Y0 Y 1,X 0 X 1 F 0 F 1 = {((x,, (y, (x, y F }, ( s gven by F 0 F 1 = {((x 0,x 1, (y 0,y 1 (x 0,y 0 F 0, (x 1,y 1 F 1 }. ( Equvalently, R Y,X are the Y X matrces wth values n {0, 1}, and, are the drect sum and the tensor product of matrces (but does not correspond to matrx multplcaton. We have the F-subrng of F(Z consstng of matrces wth values n N; we denote t by F(N. Ths F-rng also contans S and S op, but composton n F(N s matrx multplcaton. We can summarze these basc F-rngs n the followng dagram, where A s a commutatve rng. F(A F(Z R S F F(N S op (

17 M. J. Sha Haran Example 4. Let k be a rng and η : k C an embeddng (e.g. η a real or complex prme of a number feld. For X F, let k X denote the free k-module wth nner product havng X as an orthonormal bass. Thus for a =(a x k X we have ts norm ( a η = η(a x, 2 ( x X and for a k-lnear map f Hom k (k X, k Y we have ts operator norm We have Let O k,η denote the F-rng wth and wth the usual operatons,,. For X 0,X 1 F, denote by the natural ncluson, and by ts transpose. For an F-rng A we get maps and f η = sup f(a η. ( a η1 f g η f η g η, ( f g η = max{ f η, g η }, ( f g η = f η g η. ( (O k,η Y,X = {f Hom k (k X, k Y, f η 1} (2.16 j : X X 0 X 1, j (x =(x,, ( j t : X 0 X 1 X, j t (x, = { x, = 0, ( A Y,X0 X 1 A Y,X0 A Y,X1, f (f j 0,f j 1, ( A X0 X 1,Y A X0,Y A X1,Y, f (j0 t f, jt 1 f. ( We say that A s a matrx rng f these maps are always njectons. Equvalently, A s a matrx rng f every element s determned by ts coeffcents, that s we have an njecton A Y,X (A [1],[1] Y X, f {j t y f j x} y Y,x X, (2.17 wth j x,j t y as n ( and ( The above examples 0, 1, 2, 3 (except for R and 4, all consttute matrx rngs. The followng gves examples of F-rngs whch are not matrx rngs (they are the resdue F-feld of the F-rngs of Example 4 (2.16. Example 5. Let k be a rng and let η : k C be an embeddng, and for X F, let k X denote the free k-module wth bass X and wth the nner product havng X as an orthonormal bass. Let F k,η denote the F-rng of partal sometres, wth (F k,η Y,X = {f : V W, wth V k X, W k Y k-submodules and f s a k-lnear sometry}. (2.18 For f =(f : V W (F k,η Y,X,g =(g : W U (Fk,η Z,Y, we have g f =(g f : f 1 (W W g(w W (F k,η Z,X ; (

18 and for f =(f : V W (F k,η Y,X, we have f 0 f 1 =(f 0 f 1 : V 0 V 1 W0 W 1, ( f 0 f 1 =(f 0 f 1 : V 0 V 1 W0 W 1. ( We wll see n Example 4.21 below that F k,η s ndeed the resdue feld of O k,η. Remark. All of the above examples (except S and S op have a natural nvoluton makng them nto F t -rngs. Moreover, all the examples have obvous analogous F ± -rngs. For example, for a commutatve rng A, we have the F ± -rng F ± (A wth F ± (A Y,X = Hom A (A X, A Y, where A X denotes the free A-module wth bass {(x x X}, dvded by the A-submodule generated by {(x+( x x X}: A X = x X A(x/ ( x (x ; alternatvely, A X s the free A-module wth bass X +, where X + X s a bass of the ±1-set X. Then A F ± (A s a fully fathful functor from Rngs to F ± -Rngs. All the above examples of F ± - rngs are F ±,t -rngs wth respect to transposton. Moreover, exteror powers gve them the structure of F λ -rngs. For a commutatve rng A that contans all the roots of unty, together wth a fxed map µ µ(a from our abstract group µ onto the group of roots of unty µ(a A (ths map could have kernel µ p f A has characterstc p, we have the F-rng F(A wth F(A Y,X = Hom A (A X, A Y, where A X denotes the free A-module wth bass X dvded by the A-submodule generated by {ζ (x (ζ x x X, ζ µ}. Then A F(A s a fully fathful functor from µ-rngs to F-Rngs, where µ-rngs s the category of such commutatve rngs A together wth the map µ µ(a, and rng homomorphsms preservng these maps. The F(A has an nvoluton makng t an F t -rng. Moreover, exteror powers gve F(A the structure of F λ -rngs. Remark. The categores F-Rngs (respectvely F ± -Rngs, F t -Rngs, F λ -Rngs, F-Rngs, F t -Rngs, F λ -Rngs have fbred products. Gven homomorphsms of F-rngs ϕ : A B, =0, 1, we have the F-rng A 0 B A1, wth ( A 0 A 1 = {(a 0,a 1 A 0 Y,X A 1 Y,X ϕ 0 (a 0 =ϕ 1 (a 1 }. B Y,X Smlarly we can construct arbtrary products A, and arbtrary nverse lmts lm A, where A s a functor from a small category to F-Rngs (respectvely F t -Rngs, etc.. Defnton Let A be an F-rng. An equvalence deal E s a collecton of subsets E = E Y,X, wth E Y,X A Y,X A Y,X, such that Y,X F E Y,X s an equvalence relaton on A Y,X. (

19 M. J. Sha Haran For (a, a E Y,X, and for b 1 A Y,Y,b 2 A X,X, For (a,a E Y,X, =0, 1, For (a, a E Y,X, and for b A W,Z, Gven an equvalence deal E of A, let b 1 (a, a def b 2 =(b 1 a b 2,b 1 a b 2 E Y,X. ( (a 0,a 0 (a 1,a 1 def =(a 0 a 1,a 0 a 1 E Y 0 Y 1,X 0 X 1. ( b (a, a def =(b a, b a E W Y,Z X. ( A/E = Y,X F A Y,X /E Y,X, and let π : A A/E denote the canoncal map whch assocates wth a A Y,X ts equvalence class π(a A Y,X /E Y,X. It follows from ( (respectvely (2.19.3, ( that we have well-defned operatons on A/E, π(f π(g =π(f g (respectvely π(f π(g =π(f g,π(f π(g =π(f g, makng A/E nto an F-rng such that π : A A/E s a homomorphsm of F-rngs. ( Gven a homomorphsm of F-rngs ϕ : A B denote by KER(ϕ = KER Y,X (ϕ, Y,X F KER Y,X (ϕ ={(a, a A Y,X A Y,X ϕ(a =ϕ(a }. (2.20 It s clear that KER(ϕ s an equvalence deal of A, and that ϕ nduces an njecton of F-rngs ϕ : A/KER(ϕ B, such that ϕ = ϕ π,.e. ϕ A B π ϕ A/ KER(ϕ (2.21 s a commutatve dagram. Thus every map ϕ of F-rngs factors as an epmorphsm (π followed by an njecton (ϕ. Example Let A = O Z[1/N ],η be the F-rng of Example 4, (2.16, wth k = Z[1/N ]. For a prme p not dvdng N there s a surjectve homomorphsm ϕ p : A F(F p, We have the equvalence deal E p = KER(ϕ p. ϕ p (a =a(mod p. Smlarly, there s a surjectve homomorphsm ϕ η : A F Z[1/N ],η, cf. Example 4.21 below, wth F Z[1/N ],η the F-rng of Example 5, (2.18, and we have the equvalence deal E η = KER(ϕ η. 3. Modules We defne the noton of an A-module for an F-rng A. Snce we gave up addton we cannot defne drectly the quotent M/N where N s a sub-a-module of M. We can dvde A-modules only by 636

20 an equvalence A-module, and we study the relatonshp between sub-a-modules and equvalence A-modules. We descrbe the standard operatons on A-modules and gve many examples. 3.1 Defntons and examples Defnton 3.1. Let A be an F-rng. An A-module M s a collecton of sets M = {M Y,X } Y,X F, together wth maps A Y,Y M Y,X A X,X M Y,X, (a, m, a a m a, (3.1.1 A Y0,X 0 M Y1,X 1 M Y0 Y 1,X 0 X 1, (a, m a m, (3.1.2 M Y0,X 0 M Y1,X 1 M Y0 Y 1,X 0 X 1, (m 0,m 1 m 0 m 1. (3.1.3 We assume M [0],X = {0 X }, M Y,[0] = {0 t Y }, and we have a dstngushed zero element 0 Y,X M Y,X, such that 0 m =0, m 0 = 0, a 0 a =0, a 0 = 0, 0 0 = 0. (3.1.4 The maps,, satsfy: for a, a, a, a,a,a A, m, m M, a (a m a a =(a a m (a a, (3.1.5 d Y m d X = m, (3.1.6 (a 0 a 1 (m 0 m 1 (a 0 a 1=(a 0 m 0 a 0 (a 1 m 1 a 1, (3.1.7 m 0 m 1 = m 1 m 0, (3.1.8 m 0 (m 1 m 2 = (m 0 m 1 m 2, (3.1.9 m 0 [0] = m, ( (a a (a 0 m (a a =(a a 0 a (a m a, ( a 0 (a 1 m = (a 0 a 1 m, ( d [1] m = m, ( (a 0 a 1 m =(a 0 m (a 1 m, ( a (m 0 m 1 = (a m 0 (a m 1. ( In partcular, (3.1.2 nduces an acton of the monod A [1],[1] on M Y,X va (a, m a m. Example Let A be a commutatve rng, F(A the assocated F-rng. For an A-module M let F(M Y,X denote the Y X matrces wth values n M. Then F(M has natural operatons (3.1.1, (3.1.2, (3.1.3 makng t nto an F(A-module. Note that for M = A we obtan the F-rng F(A vewed as an F(A-module. We have, for A-modules M 1,M 2, F(Hom A (M 1,M 2 Y,X = Hom A (M 1 A A X, M 2 A A Y. (3.2.1 Example For a fnte set V let F(V Y,X denote the Y X matrces wth values n V {0} such that every row and every column contans at most one non-zero term. Then F(V has natural operatons (3.1.1, (3.1.2, (3.1.3 makng t nto a module over the F-rng F. For V = [1] we obtan F([1] whch s just F vewed as an F-module. We have, for fnte sets V 1,V 2, F(Hom F (V 1,V 2 Y,X = Hom F (V 1 X, V 2 Y. (3.2.2 For an F-module W, such that W Y,X s a fnte set for all X, Y F, we say t has dmenson dm F W over F f the followng lmt exsts (where n, m go to nfnty ndependently of each other: dm F W = lm n,m 1 nm log W [n],[m]. Thus f V s a fnte dmensonal vector space over the fnte feld F q, and F(V the assocated 637

21 F(F q -module vewed as F-module, we have wth dm F F(F q = log q. M. J. Sha Haran dm F F(V = dm F F(F q dm Fq V For a fnte set V, the assocated F-module F(V s zero dmensonal n the above sense, dm F F(V = 0. We can use a dfferent dmenson functon, Dm F W for W an F-module (wth W [n],[m] fnte for all n, m, gven by Dm F W = lm x,y 1 xy log n,m0 For the F-module W = F(V, V a fnte set, t gves 1 Dm F F(V = lm x,y xy log n,mk ( W [n],[m] xn n! ( n k ( m k y m m!. k!( V k xn n! 1 = lm log exp(x + y + xy( V = V. x,y xy Indeed, to gve an arbtrary element of F(V [n],[m] we have to choose k rows (respectvely, k columns, and there are ( ( n k (respectvely, m k choces, then we have to choose a bjecton between these rows and columns (there are k! possbltes for such a bjecton, and fnally we have to fll n the k chosen entres wth elements of V (and there are ( V k such choces, hence F(V [n],[m] = ( ( n m k!( V k. k k kn,m Example Let k be a feld, η : k C an embeddng, and let V be a k-vector space wth an nner product (, V and assocated norm V. Let F(V Y,X denote the Y X matrces wth values n V,v = (v y,x, such that for a = (a x k X, b = (b y k Y, we have (cf. ( V b y v y,x a x a η b η. x,y The set F(V has natural operatons (3.1.1, (3.1.2, (3.1.3 makng t nto an O k,η -module. For V = k we obtan F(k whch s O k,η vewed as an O k,η -module. Defnton 3.3. Let A be an F-rng, and M, M be A-modules. A collecton of maps ϕ = {ϕ Y,X : M Y,X M Y,X Y, X F } s a homomorphsm of A-modules f t respects the operatons y m m! ϕ(a m a =a ϕ(m a, (3.3.1 ϕ(a m =a ϕ(m, (3.3.2 ϕ(m 0 m 1 =ϕ(m 0 ϕ(m 1. (3.3.3 The collecton of A-modules and homomorphsms form a category A-Mod. It has an ntal and fnal object 0 = {{0 Y,X }} Y,X F. For a commutatve rng A, the constructon of Example gves a functor F : A-Mod F(A-Mod, M F(M. (3.4.1 As n ( we see that ths functor s fully fathful. Smlarly, the constructon of Example gves us a functor F : F F-Mod, V F(V. (

22 For a feld embeddng η : k C, let (k,η-vec denote the category whose objects are k-vector spaces wth an nner product and morphsms are k-lnear maps wth operator norm at most 1; the constructon of Example gves a functor 3.2 A-submodules and equvalence A-modules F :(k,η-vec O k,η -Mod, V F(V. (3.4.3 Defnton 3.5. Let A be an F-rng, M an A-module. An A-submodule M of M s a collecton of subsets M = {M Y,X M Y,X} whch s closed under the operatons,, : A M A M, A M M, M M M. (3.5.1 We denote by sub A (M the collecton of A-submodules of M. The ntersecton of A-submodules s agan an A-submodule. An A-submodule of A s called an deal. Let be a homomorphsm of A-modules. We have an A-submodule of M: It s the kernel of ϕ n the category A-Mod. We have also an A-submodule of N: The homomorphsm ϕ nduces maps ϕ : M N (3.6 ϕ 1 (0 = {m M ϕ(m = 0}. (3.6.1 ϕ(m ={ϕ Y,X (M Y,X } Y,X F. (3.6.2 ϕ : sub A (M sub A (N, ϕ : sub A (N sub A (M, M ϕ M def = ϕ(m, (3.6.3 N ϕ N def = ϕ 1 (N. (3.6.4 The category A-Mod has fbred products. Gven A-Mod homomorphsms we have the A-module wth and the operatons ( M 0 M 1 M Y,X ϕ 0 : M 0 M M 1 : ϕ 1 (3.7 M 0 M M 1 = {(m 0,m 1, m (M Y,X, ϕ 0 (m 0 =ϕ 1 (m 1 }, (3.7.1 a (m 0,m 1 a =(a m 0 a,a m 1 a, a (m 0,m 1 =(a m 0,a m 1, (3.7.2 (m 0,m 1 (m 0,m 1 = (m 0 m 0,m 1 m 1. In partcular we have products M 0 M1. We can smlarly form arbtrary products λ M λ, and arbtrary nverse lmts { lm M λ = (m λ } M λ ϕ λ,λ(m λ =m λ, (3.7.3 where λ M λ s a functor from a small category to A-Mod. 639

23 M. J. Sha Haran Let ϕ : M N be a homomorphsm of A-modules. Let { ( KER(ϕ Y,X = (m, m M } M ϕ(m =ϕ(m = M Y,X N M. (3.8 Then KER(ϕ s an A-submodule of M M such that, for all Y, X F, KER(ϕ Y,X equvalence relaton on M Y,X. s an Defnton 3.9. Let M be an A-module. An equvalence A-module of M s an A-submodule E of M M, such that E Y,X s an equvalence relaton on M Y,X. We denote by equv A (M the collecton of equvalence A-modules of M. For E equv A (M we can form the equvalence classes (M/E Y,X = M Y,X /E Y,X. There s an nduced A-module structure on M/E such that the canoncal map π : M M/E s a homomorphsm. We have Hom A-Mod (M/E,N={ϕ Hom A-Mod (M, N KER(ϕ E}. (3.9.1 We have one-to-one order-preservng correspondence and a natural somorphsm equv A (M/E = {E equv A (M E E}, E /E E, (3.9.2 (M/E/(E /E = M/E. (3.9.3 Defnton For an equvalence A-module E of M, a submodule M 0 M s called E-stable f for all (m, m E, m M 0 m M 0. We have a one-to-one order-preservng correspondence sub A (M/E = {M 0 sub A (M M 0 s E-stable}, M 0 /E M 0. (3.9.5 Every homomorphsm of A-modules ϕ : M N factors as (njecton (surjecton, as n the dagram. M ϕ N M/ KER(ϕ ϕ(m Defnton For an equvalence A-module E M M let Z(E =π 1 (0 = {m M (m, 0 E} = E It s an A-submodule of M. For an A-submodule M 0 M let (3.9.6 ( M {0}. ( E(M 0 M M ( be the equvalence A-module of M generated by {(m, 0 m M 0 },.e. E(M 0 s the ntersecton E of all equvalence A-modules E of M such that M0 {0} E. We wrte M/M 0 for M/E(M 0. For a homomorphsm of A-modules ϕ : M N we have ts cokernel, Coker(ϕ =N/ϕ(M =N/E(ϕ(M. (

24 Lemma We have M 0 M 0 A-submodules of M E(M 0 E(M 0, ( E E M M equvalence A-modules of M Z(E Z(E, ( Proof. The proof s straghtforward. Corollary We have Hence we have we denote ths set by E-sub A (M. Smlarly, we have M 0 Z(E(M 0, ( E(Z(E E. ( E(M 0 =E(Z(E(M 0, ( Z(E =Z(E(Z(E. ( {Z(E E equv A (M} = {M 0 M M 0 = Z(E(M 0 }; ( {E(M 0 M 0 MA-submodule} = we denote ths set by Z-equv A (M. Moreover, there s an nduced bjecton { E M M E-sub A (M Z-equv A (M, M 0 E(M 0, Z(E E. }; E = E(Z(E ( Lemma Let M 0 M be an A-submodule, and let E Y,X (M M Y,X denote the collecton of pars (m, m such that there exsts a path m = m 0,m 1,..., m l = m, where for j =0,..., l 1, {m j,m j+1 } has the form {a (n n 0 a,a (n 0 a } for some a, a A, n M, n 0 M 0. Then E(M 0 =E. Proof. Note that for a, a A, n M, n 0 M 0, we have (m 0, 0 E(M 0, (m, m E(M 0 and snce E(M 0 M M s a submodule we get (a (n n 0 a,a (n 0 a E(M 0. Thus f there s a path m = m 0,..., m l = m as above, then (m, m E(M 0 ; so E E(M 0. For the reverse ncluson note that E Y,X s an equvalence relaton on M Y,X. Moreover, E s an A-submodule of M M. For (m, m E there exsts a path m = m 0,..., m l = m as above, hence for a, a A, s a path from a m a to a m a, hence Smlarly, for a A Y,X, a m 0 a,..., a m l a (a m a,a m a E. a m 0,..., a m l 641

25 s a path, whch shows that To show ths use M. J. Sha Haran (a m, a m E. a (a (n n 0 a =(a a ((d X n (d X n 0 (d X a, d X n 0 M 0, d X 0 = 0. If also (m, m E, we can assume the path m = m 0,..., m l = m has the same length l (by addng denttes n 0 =0,a= d,a = d, and then s a path, whch shows that To show ths use m 0 m 0,..., m l m l (m m, m m E. (a (n n 0 a (a (n n 0 a =(a a ((n n (n 0 n 0 (a a. Thus E s an equvalence A-submodule of M, and snce {(m 0, 0 m 0 E(M 0 E. Corollary Let M 0 M be an A-submodule. We have f and only f, for all m 0 M 0, m M, a, a A, Z(E(M 0 = M 0 M 0 } E, we have a (m m 0 a M 0 a (m 0 a M 0, ( e. M 0 E-sub A (M f and only f M 0 s E(M 0 -stable. Proof. Assume ( holds. By Lemma 3.13 f (m, m E(M 0 there exsts a path m = m 0,..., m l = m, and we have hence m j M 0 m j+1 M 0, m M 0 m M 0. Takng m =0 M 0, we get (m, 0 E(M 0 mples m M 0. Thus Z(E(M 0 M 0, and snce the reverse ncluson always holds we get Z(E(M 0 = M 0. Conversely, assume Z(E(M 0 = M 0, then a (m m 0 a M 0 (a (m m 0 a, 0 E(M 0, ( a (m 0 a M 0 (a (m 0 a, 0 E(M 0. ( Usng the fact that E(M 0 Y,X s an equvalence relaton, and that for m 0 M 0 (a (m m 0 a,a (m 0 a E(M 0, we see that the statements n ( and ( are equvalent, hence ( holds. For submodules M 0 sub A (M, M E-sub A (M, we have M M 0 f and only f M s E(M 0 -stable. We get a one-to-one order-preservng correspondence and a natural somorphsm E-sub A (M/M 0 ={M E-sub A (M M M 0 }, M /M 0 M ( (M/M 0 /(M /M 0 = M/M. (

26 An A-submodule of A s called an deal, and an equvalence A-module of A s called an equvalence deal. Thus we have the maps E, Z between deals and equvalence deals satsfyng Lemma 3.11 and Corollary Elements of E-sub A (A wll be called E-deals. Example For A = O Z[1/N ],η, wth the notaton of Example 2.22, we have for p N: EZ(E p E p E η. 3.3 Operatons on submodules For a famly {M } of A-submodules of M, we have the ntersecton M sub A (M. Note that f M E-sub A (M then M E-sub A (M. We have also M the A-submodule generated by the M,.e. t s the ntersecton N taken over all submodules N that contan all the M. It can be descrbed explctly as ( { ( } M = a m a a A Y, Y, a A X,X, m (M Y,X. (3.16 Y,X Indeed the rght-hand sde wll be contaned n any submodule N whch contans all the M, t tself contans the M, and s closed under the module operatons ( ( ( b a m a b =(b a m (a b, ( ( ( ( ( a m a b b ( a ( m m a = (d Y a ( b =(a b (m m (a b, ( ( (b m (d X a,b A Y,X. ( More generally, gven any subset {m I} M, wth m M Z,W, the A-submodule t generates A m can be descrbed explctly as ( A m Y,X = { a ( d X m a a A Y, (X Z,a A (X W,X }. ( Gven an A-module M and an deal a A we have ther product a M whch s an A-submodule of M, (a M Y,X { ( } = b (a m b b A Y, (Y Z,b A (X W,X,a a Y,X,m M Z,W. Gven A-submodules M 0,M 1 of M we can form ther quotent It s easly checked that (M 0 : M 1 s an deal of A. 3.4 Operatons on modules ( (M 0 : M 1 ={a A a m M 0 m M 1 }. ( Sums. Gven A-modules M 0,M 1, we frst construct the sum (coproduct M 0 M1 n the category A-Mod. We form (M 0 M1 = {(a, m 0,m 1,a a A Y,Y0 Y 1,a A X0 X 1,X,m (M Y,X }/ ( Y,X 643