Plethystic algebra. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA

Transcription

1 Advances n Mathematcs 194 (2005) Plethystc algebra James Borger,1, Ben Weland 1 Department of Mathematcs, Unversty of Chcago, 5734 S. Unversty Avenue, Chcago, IL 60637, USA Receved 22 August 2003; accepted 16 June 2004 Communcated by Mchael Hopkns Avalable onlne 13 August 2004 Abstract The noton of a Z-algebra has a non-lnear analogue, whose purpose t s to control operatons on commutatve rngs rather than lnear operatons on abelan groups. These plethores can also be consdered non-lnear generalzatons of cocommutatve balgebras. We establsh a number of category-theoretc facts about plethores and ther actons, ncludng a Tannaka Kren-style reconstructon theorem. We show that the classcal rng of Wtt vectors, wth all ts concomtant structure, can be understood n a formula-free way n terms of a plethystc verson of an affne blow-up appled to the plethory generated by the Frobenus map. We also dscuss the lnear and nfntesmal structure of plethores and explan how ths gves Bloch s Frobenus operator on the de Rham Wtt complex Elsever Inc. All rghts reserved. MSC: 13K05; 13A99; 16W99; 14F30; 16W30; 19A99 Keywords: Rng scheme; Wtt vector; Wtt rng; Symmetrc functon; Balgebra; Lambda-rng; Delta-rng; Plethory; Brng Consder an example from arthmetc. Let p be a prme number. Recall that for (commutatve) rngs R, the rng W(R) of ( p-typcal) Wtt vectors s usually defned to be the unque rng structure on the set R N whch s functoral n R and such that the map (r 0,r 1,...) (r 0,r p 0 + pr 1,r p2 0 + pr p 1 + p2 r 2,...) Correspondng author. E-mal addresses: borger@math.uchcago.edu (J. Borger), weland@math.uchcago.edu (B. Weland). 1 Partally supported by the Natonal Scence Foundaton /$ - see front matter 2004 Elsever Inc. All rghts reserved. do: /j.am

2 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) s a rng homomorphsm, the target havng the usual, product rng structure. If R s a perfect feld of characterstc p, then W(R) s the unque complete dscrete valuaton rng whose maxmal deal s generated by p and whose resdue feld s R. However, n almost all other cases, W(R) s pathologcal by the usual standards of commutatve algebra. For example, W(F p [x]) s not noetheran. It s nevertheless an establshed fact that W(R) s an mportant object. For example, f R s the coordnate rng of a smooth affne varety over a perfect feld of characterstc p, there s a certan quotent of the de Rham complex of W(R), called the de Rham Wtt complex of R, whose cohomology s naturally the crystallne cohomology of R. But t s not at all clear from the defnton above what the proper way to thnk about W(R) s, much less why t s even reasonable to consder t n the frst place. The presence of certan natural structure, for example, a multplcatve map R W(R) and a rng map W(R) W(W(R)) adds to the mystery. And so we have a queston: s there a defnton gven purely n terms of algebrac structure rather than somewhat mysterous formulas, and s there a pont of vew from whch ths defnton wll be seen as routne and not the result of some ntangble nspraton? The purpose of ths paper s to dscuss an algebrac theory of whch a partcular nstance gves a formal answer to these questons and to wrte down some basc defntons and facts. For any (commutatve) rng k, we defne a k-plethory to be a commutatve k-algebra together wth a comonad structure on the covarant functor t represents, much as a k-algebra s the same as a k-module that represents a comonad. So, just as a k-algebra s exactly the structure that knows how to act on a k-module, a k-plethory s the structure that knows how to act on a commutatve k-algebra. It s not so surprsng that ths analogy extends further: Lnear/k Non-lnear/k k-modules M Commutatve k-algebras R k-k-bmodules N k-k-brngs S Hom k (N, M) Hom k-alg (S, R) N k M S k R k = -unt k[e] = -unt k-algebras A k-plethores P A-modules P-rngs A-A -bmodules P-P -brngs Ths s explaned n Secton 1. In fact, as Bergman has nformed us, ths pcture has been known n the unversal-algebra communty, under qute smlar termnology and notaton, snce Tall and Wrath s paper [19] n (See also [23,2].) For those famlar wth ther work, parts of the frst sectons wll be very famlar. The descrpton of the rng of Wtt vectors from ths pont of vew s that there s a Z-plethory Λ p, and W(R) s smply the Λ p -rng co-nduced from the rng R (whch observaton allows us to defne a Wtt rng for any plethory), and so the only thng left s to gve a natural constructon of Λ p. Ths s done by a process we call amplfcaton

3 248 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) and whch s formally smlar to performng an affne blow-up n commutatve algebra. We wll gve some dea of ths procedure below. In Secton 2, we gve some examples of plethores. The most basc s the symmetrc algebra S(A) of any cocommutatve balgebra A; n partcular, f A s a group algebra ZG, then S(A) s the free polynomal algebra on the set underlyng G. These plethores are less nterestng because ther actons on rngs can be descrbed entrely n terms of the orgnal balgebra A; for example, an acton of the plethory S(ZG) s the same as an acton of the group G. But even n ths case, there can be more maps between two such plethores than there are between the balgebras, and n some sense, ths s ultmately responsble for exstence of Λ p and hence the p-typcal Wtt rng. The rng Λ of symmetrc functons n nfntely many varables s a better example. The composton law of Λ s gven by the operaton known as plethysm n the theory of symmetrc functons and s what gves plethores ther name. An acton of Λ on a rng R s the same as a λ-rng structure on R, and n contrast to plethores of the form S(A), a Λ-acton cannot n general be descrbed n terms of a balgebra acton. We also gve an explct descrpton of Λ p, the plethory responsble for the p-typcal Wtt rng, n terms of symmetrc functons. Of course, ths descrpton s really qute close to a standard treatment of the Wtt rng and s stll a bt unsatsfyng. In Secton 3, we gve explct examples of P-Wtt rngs for varous plethores P. In Secton 4, we dscuss the restrcton, nducton, and co-nducton functors for a morphsm P Q of plethores, and we state the reconstructon theorem. As always, the content of such a theorem s entrely category theoretc (Beck s theorem). All the same, the result s worth statng: Theorem. Let C be a category that has all lmts and colmts, let U be a functor from C to the category of rngs. If U has both a left and a rght adjont and has the property that a map f n C s an somorphsm f U(f) s, then C s the category of P-rngs for a unque k-plethory P, and under ths dentfcaton, U s the forgetful functor from P-rngs to rngs. In Secton 7, we explan amplfcaton, the blow-up-lke process we mentoned above. Let O be a Dedeknd doman, for example the rng of ntegers n a local or global feld or the coordnate rng of a smooth curve. Let m be an deal n O, let P be an O-plethory, let Q be an O/m-plethory, and let P Q be a surjectve map of plethores. We say a P-rng R s a P-deformaton of a Q-rng f t s m-torson-free and the acton of P on R/mR factors through the map P Q. Theorem. There s an O-plethory P that s unversal among those that are equpped wth a map from P makng them P-deformatons of Q-rngs. Furthermore, P has the property that P-deformatons of Q-rngs are the same as P -rngs that are m-torsonfree. We say P s the amplfcaton of P along Q. In Sectons 8 11, we defne what could be called the lnearzaton of a plethory P. It nvolves two structures: A P, the set of elements of P that act addtvely on any P-rng, and C P, the cotangent space to the spectrum of P at 0.

4 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Theorem. Both A P and C P are (generally non-commutatve) algebras equpped wth maps from k, and under certan flatness or splttng hypotheses, the followng hold: A P s a cocommutatve twsted k-balgebra, there s a coacton of A P on the algebra C P, and the map A P C P s A P -coequvarant. We stop short of nvestgatng representatons of such lnear structures. If R R s a map of P-rngs wth kernel I, then all that remans on the conormal module I/I 2 of the acton of P s an acton of C P. In partcular, C P acts on the Kähler dfferentals of any P-rng. In the specal case when P = Λ p and R = W(S), for some rng S, ths addtonal structure s essentally a lft of Bloch s Frobenus operator on the de Rham Wtt complex. The fnal secton of the paper s the reason why the others exst, and we encourage the reader to look at t frst. Here, we consder Λ p and other classcal constructons from the pont of vew of the general theory. For example, we gve a satsfyng constructon of Λ p : Let F p e be the trval F p -plethory; ts balgebra of addtve elements has a canoncal deformaton to a Z-balgebra, and let P be the free Z-plethory on ths. Then Λ p s the amplfcaton of P along F p e. Essentally the same procedure, appled to rngs of ntegers n general number felds, gves at once ramfed and twsted generalzatons. An acton of ths amplfcaton on a p-torson-free rng R s, essentally by defnton, the same as a lft of the Frobenus endomorphsm of R/pR. The content of the statement that the Λ p -rng co-nduced by R agrees wth the classcal W(R) s ultmately just Carter s Deudonné Dwork lemma. Thus t would be accurate to vew amplfcatons as the framework where Joyal s approach to the classcal Wtt vectors [10] naturally lves. The last secton also has explct descrptons of the lnearzatons of Λ p, Λ, and smlar plethores. On a fnal note, ths paper does not even contan the bascs of the theory, and there are stll many smple mysteres. For example, the exstence of non-lnear plethores, those that do not come from (possbly twsted) balgebras, may be a purely arthmetc phenomenon: we know of no non-lnear plethory over a Q-algebra. For a broader example, the category of P-rngs s, on the one hand, a generalzaton of the category of rngs and, on the other, an analogue of the category of modules over an algebra. And so t s natural to ask whch notons n commutatve algebra and algebrac geometry can be generalzed to P-rngs for general P and, n the other drecton, whch notons n the theory of modules over algebras have analogues n the theory actons of plethores on rngs. It would be qute nterestng to see how far these analoges can be taken. 0. Conventons The word rng s short for commutatve rng, but we make no commutatvty restrcton on the word algebra. A k-rng s then a commutatve k-algebra. All these objects are assumed to be assocatve and untal, and all morphsms are untal. Rng k denotes the category of k-rngs.

5 250 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) We use the language of coalgebras extensvely; Dăscălescu, Năstăsescu, and Raanu s book [5] s more than enough. For categorcal termnology, we refer to Mac Lane s book [14]. In partcular, we fnd t convenent to wrte C(X, Y ) for the set of morphsms between objects X and Y of a category C. N denotes the set {0, 1, 2,...}. 1. Plethores and the composton product Let k, k,k be rngs. A k-k-brng s a k-rng that represents a functor Rng k Rng k. Composton of such functors yelds a monodal structure on the category of k-k-brngs. We then defne a k-plethory to be a monod n ths category, much as one could defne a k-algebra to be a monod n the category of k-k-bmodules. Fnally, the category of k-k-brngs acts on the category of k-rngs, and we defne a P-rng to be a rng together wth an acton of the k-plethory P. We spell ths out n some detal and gve a number of mmedate consequences of the defntons. We also gve many examples n ths secton, but they are all trval, and so the reader may want to look ahead at the more nterestng examples n Sectons 2 and A k-k -brng s a k-rng S, together wth a lft of the covarant functor t represents to a functor Rng k Rng k. Equvalently, t s the structure on S of a k -rng object n the opposte category of Rng k. Or n Grothendeck s termnology, ths s the structure on Spec S of a commutatve k -algebra scheme over Spec k. Explctly, S s a k-rng wth the followng addtonal maps (all of k-rngs except (3)): (1) coaddton: a cocommutatve coassocatve map Δ + : S S k S for whch there exsts a count ε + : S k and an antpode σ: S S, (2) comultplcaton: a cocommutatve coassocatve map Δ : S S k S whch codstrbutes over Δ + and for whch there exsts a count ε : S k, (3) co-k -lnear structure: a map β: k Rng k (S, k) of rngs, where the rng structure on Rng k (S, k) s gven by (1) and (2). Note that, as usual, ε +, σ, and ε are unque f they exst. Also note that omttng axom (3) leaves us wth the noton of k-z-brng. Fnally, n the case of k-plethores, we wll take k = k, but at ths pont t s best to keep the roles separate. A morphsm of k-k -brngs s a map of k-rngs whch preserves all the structure above. The category of k-k -brngs s denoted BR k,k.gvenamapk k,wecan vew a S as a k-k -brng, whch we stll denote S, somewhat abusvely. Let l and l be rngs, and let T be a l-l -brng. A morphsm S T of brngs s the followng data: a rng map k l, a rng map k l, and a map l k S T of l- k -brngs. The category of brngs s denoted BR. When necessary, we wll dstngush the structure maps of brngs by usng subscrpts: Δ + S,ε S, and so on. We wll also often use wthout comment the notaton Δ + p = p(1) p (2) and Δ p = p[1] p [2].

6 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Examples. (1) k tself s the ntal k-k -brng, representng the constant functor gvng the zero rng. (2) Let k e denote the k-k-brng that represents the dentty functor on Rng k. Thus k e s canoncally the rng k[e] wth Δ + (e) = e 1+1 e, Δ (e) = e e, β(c)(e) = c (and ε + (e) = 0,ε (e) = 1, σ(e) = e). (3) If k s fnte, then the collecton of set maps k k s naturally a k-k -brng. The k-rng structure s gven by pontwse addton and multplcaton, and the corng structure s gven by the rng structure on k. For example, Δ + s the composte k k k k k = k k k k, where the frst map s gven by addton on k.ifk s not fnte, there are topologcal ssues, whch could surely be avoded by consderng pro-representable functors from Rng k to Rng k. Recall that the acton of a k-algebra A on a k-module M can be gven n three ways: as a map A k M M, asamapm Mod k (A, M), orasamapa Mod k (M, M). In fact, we have the same choces when defnng the multplcaton map on A tself. The Wtt vector approach to operatons on rngs follows the second, comonadc model, but we wll follow the frst, monadc one. The thrd approach encounters the topologcal problems mentoned n the example above. We now defne the analogue of the tensor product Functor k : BR k,k Rng k Rng k. Take S BR k,k and R Rng k. Then S k R s defned to be the k-rng generated by symbols s r, for all s S,r R, subject to the relatons (for all s, s S,r,r R,c k ) and ss r = (s r)(s r), (s + s ) r = (s r) + (s r), c r = c (1.3.1) s (r + r ) = Δ + S (s)(r, r ) := (s (1) r)(s (2) r ), s (rr ) = Δ S (s)(r, r ) := (s [1] r)(s [2] r ), s c = β(c)(s). (1.3.2) Ths operaton s called the composton product and s clearly functoral n both R and S. As n lnear algebra, where a tensor a b remnds us of the formal composton of operators a and b or the formal evaluaton of an operator a at b, the symbol s r s ntended to remnd us of the composton s r of possbly non-lnear functons or the formal evaluaton of a functon s at r. Thus the meanng of (1.3.1) s that rng operatons on functons are defned pontwse, and the meanng of (1.3.2) s that there

7 252 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) s extra structure on our rng of functons that controls how they respect sums, products, and constant functons. For example, f S s the brng of 1.2(3), the evaluaton map S k k k gven by s r s(r) s a well-defned rng map Proposton. Let S be a k-k -brng. The functor S k s the left adjont of Rng k (S, ). In other words, for R 1 Rng k, R 2 Rng k we have Rng k (S k R 2,R 1 ) = Rng k (R 2, Rng k (S, R 1 )). The proof s completely straghtforward. We leave t, as well as the task of specfyng the unt and count of the adjuncton, to the reader Examples. (1) There are natural dentfcatons S k k e =S, k e k R = R, S k k = k, and k k R = k. (2) If k l s a rng map, then l e k R = l k R. (3) k-l -brng structures on S compatble wth the gven k-k -brng structure are the same, under adjuncton, as maps S k l k of k-rngs. (4) If k l s a rng map, we have (l k S) k R = l k (S k R). (5) The composton product dstrbutes over arbtrary tensor products: ( S ) k R = (S k R), ( ) S k R = (S k R ) If R s not only a k -rng but a k -k -brng, then the functor Rng k (S k R, ) = Rng k (R, Rng k (S, )) naturally takes values n k -rngs, and so S k R s naturally a k-k -brng. One can also see ths drectly n terms of the structure maps Δ + and so on by usng the fact that the composton product dstrbutes over tensor products. If k = k = k, the composton product gves a monodal structure on the category of k-k-brngs wth unt k e =k[e] of 1.2. As s generally true wth composton or the tensor product of bmodules, ths monodal structure not symmetrc Remark. Note that, n contrast to the analogous statement for bmodules, t s generally not true that a k-k -brng structure on R nduces k -k -brng structure on the k-rng Rng k (S, R).

8 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) A k-plethory s a monod n the category of k-k-brngs, that s, t s a brng P equpped wth an assocatve map of brngs : P k P P and unt k e P.For example, k e = k[e] wth taken as n 1.5(1) (that s, composton of polynomals) s a k-plethory. The mage of e under the unt map k e P s denoted e (or e P ); together wth, t gves the set underlyng P a monod structure. The rng k s called the rng of scalars of P. If P s a k -plethory, a morphsm P P of plethores s a morphsm k k plus a morphsm φ: P P of brngs whch s also a morphsm of monods. Ths s equvalent to requrng that k e k P k P φ 1 P k P P k k e k P 1 1 φ P k P k e k P k k P φ P be a commutatve dagram of k -k-brngs. If k = k, the dagram smplfes to the obvous one. If we are already gven a map k k, then we wll always assume the map of scalars s the same as the gven map. It s easy to see that k e s the ntal k-plethory and Z e s the ntal plethory A(left) acton of P on a k-rng R s defned as usual n the theory of monodal categores; n ths case t means a map : P R R such that (α β) r = α (β r) and e r = r for all α, β P,r R. We also denote α r by α(r). AP-rng s a k-rng equpped wth an acton of P. (There s no danger of a conflct n termnology wth a rng equpped wth a rng map from P because we never use such structures n ths paper.) A morphsm of P-rngs s a map of rngs that makes the obvous dagram commute; equvalently, t s a map of rngs that s P-equvarant as a map of sets acted on by the monod (P, ). The category of P-rngs s denoted Rng P. If S s a k-k -brng, we say P acts on S as a k-k -brng f : P S S s a map of k-k -brngs. Such an acton s the same as a functoral collecton of k -rng structures on the sets Rng P (S, R) such that the maps Rng P (S, R) Rng k (S, R) are maps of k -rngs. A rght acton of a k -plethory P on a k-k -brng s a map : R k P R of k-k -brngs compatble wth and e n the obvous way. A map of rght P -rngs s P -equvarant map of k-k -brngs. A P-P -brng s a k-k -brng equpped wth a left acton of P as a k-k brng and a commutng rght acton of P. The category of P-P -brngs s denoted BR P,P, morphsms beng maps of brngs that are both

9 254 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) P-equvarant and P -equvarant. A P-P -brng s the same as a represented functor Rng P Rng P A k-plethory structure on a k-k-brng P s the same as a monad structure on the functor P k and, by adjuncton, also the same as a comonad structure on the functor Rng k (P, ). An acton of P on R s the same as the structure on R of an algebra over the monad or a coalgebra over the comonad. Thus Rng P has all lmts and colmts, the forgetful functor U: Rng P Rng k preserves them, and the functors P k and Rng k (P, ) lft to gve left and, respectvely, rght adjonts to U. (These functors could well be called restrcton, nducton, and co-nducton for the map k e P. We postpone the treatment of these functors for general maps of plethores untl secton four.) In partcular, the underlyng k-rng of a (co)lmt of P-rngs s the (co)lmt n that category and there exsts a unque compatble P-rng structure on t. We gve a converse to all ths n Secton 4. We often denote the functor Rng k (P, ) by W P ( ) and call the P-rng W P (R) the P-Wtt rng of R. The reason for ths termnology wll be made clear n Secton Examples. (1) If k s fnte, the brng of set maps k k s a k-plethory, wth gven by composton of functons. In partcular, 0 s a plethory over the rng 0. It s the termnal plethory, and of course the only 0-rng s 0. (2) A plethory P clearly acts on tself on the left (and also the rght). It s n fact the free P-rng on one element: morphsms n Rng P from P to another object are the same as elements of the underlyng rng, a map φ: P R correspondng to the element φ(e) n R, and an element r R correspondng to the map α α(r). The morphsms P k correspondng to r = 0 and r = 1 are ε + and ε. More generally, the morphsm P k correspondng to c k s β(c). (3) The dentfcaton P k k = k s an acton of P on k, and f R s any P-rng, the structure map k R s a map of P-rngs smply by the thrd relaton of (1.3.2). Therefore, k s the ntal P-rng. Smlarly, the dentfcaton k k P = k gves k the structure of a P-P-brng, and t s the ntal P-P-brng. (4) If k s a P-rng, the natural k -map (k k P) k k = k k (P k k ) k gves (by 1.5) k k P the structure of a k -k -brng. We wll see below that k k P even has a natural k -plethory structure Proposton. Let P be a k-plethory. Then the k-rng morphsms Δ + P, Δ P, ε+ P, and ε P are n fact P-rng morphsms. For any A Rng P, the unt η A : k A and multplcaton m A : A k A A are P-rng morphsms. Proof. The unt and counts were dscussed n 1.11(3) and (2). Multplcaton s the coproduct of the dentty wth tself.

10 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) By 1.11(2), the P-rng P represents the forgetful functor U from Rng P to the category of sets and P k P represents the functor U U. But these factor through the category of rngs, and so there are natural transformatons U U U, one for addton and one for multplcaton. Thus there are maps P P k P n Rng P. The one for addton s the map that sends e to 1 e + e 1, and thus sends α to Δ + (α)(1 e, e 1) = Δ + (α). Smlarly, the one for multplcaton s Δ Base change of plethores. If k s a P-rng, then the k -k-brng k k P has a k -k -brng structure (1.11). Even further, the k -rng map (usng 1.5(4)) (k k P) k (k k P) = k k (P k (k k P)) 1 k k (k k P) k k P descends to a map (k k P) k (k k P) k k P, whch gves k k P the structure of a k -plethory. Conversely, f k P s a k -plethory, then P acts on k by way of k P. Note that not only does the plethory structure on k P depend on the acton of P on k, there may not exst even one such acton. For example, there s no acton of the Z-plethory Λ p (of 2.13) on F p. We leave t as an exercse to show that a k P -acton on a k -rng R s the same as a P-acton on the underlyng k-rng compatble wth the gven acton on k. 2. Examples of plethores Before contnung wth the theory, let us gve some basc examples of plethores Free plethory on a brng. Let k be a rng, and let S be a k-k-brng. There s a plethystc analogue of the tensor algebra: a k-plethory Q, wth a k-k-brng map S Q, whch s ntal n the category of such plethores. Put Q = n0 S n. The system of maps S S j S (+j) (s 1 s ) (t 1 t j ) s 1 s t 1 t j

11 256 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) nduces a map Q Q =,j S S j n S n = Q, whch s clearly assocatve. Ths gves Q the structure of a k-plethory wth a map k e =S 0 Q of k-plethores. A Q-acton on a rng R s then the same as a map S R R of rngs Free plethory on a cocommutatve balgebra. Frst, let A be a cocommutatve coalgebra over k; denote ts comultplcaton map by Δ and ts count by ε. The symmetrc algebra S(A) of A, vewed as a k-module, s of course a k-rng, but the followng gves t the structure of a k-k-brng: Coaddtve structure: The coaddton map Δ + s the one nduced by the lnear map A S(A) S(A), a a a. The addtve count ε + : S(A) k s the map nduced by the zero map A k. Comultplcatve structure: Δ s the map nduced by the lnear map A Δ A A S(A) S(A), where the rght map s the tensor square of the canoncal ncluson. The multplcatve count ε : S(A) k s the composte map Co-k-lnear structure: The map gves S(A) a k-k-brng structure by 1.5. S(A) S(ε) S(k) = k e ε k e k. S(A) Z k k e Z k k e k k = k 2.3. Isomorphsm S(A) S(B) S(A B) of k-k-brngs. Let B be another cocommutatve k-coalgebra, and let R be a k-rng. Then we have Rng k (S(A) S(B),R) = Rng k (S(B), Rng k (S(A), R)) = Mod k (B, Mod k (A, R)) = Mod k (A B,R) = Rng k (S(A B),R)

12 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) and hence a natural somorphsm S(A) S(B)S(A B) of k-rngs. Explctly, [a] [b] corresponds to [a b], where [a] denotes the mage of a under the natural ncluson A S(A) and lkewse for [b]. We leave the task of showng ths s a map of k-k-brngs to the reader It follows that the comultplcaton and the count nduce maps S(A) S(A) S(A), S(A) k e that gve S(A) the structure of a commutatve comonod n BR k,k Now suppose A s a balgebra, that s, A s equpped wth maps A A A, k A of k-coalgebras makng A a monod n the category of k-coalgebras. By the dscusson above, ths makes S(A) a monod n the category of cocommutatve comonods n BR k,k. It s n partcular a k-plethory. (It could reasonably be called a cocommutatve bmonod n BR k,k ts addtonal structure s the analogue of the structure added to an algebra to make t a cocommutatve balgebra but because s not a symmetrc operaton on all of BR k,k, ths termnology could be confusng.) 2.6. Remark. Gven a k-rng R, an acton of the plethory S(A) on R s the same as an acton of the balgebra A on R. We leave the precse formulaton and proof of ths to the reader. It may be worth notng that any k-rng admts an S(A)-acton n a trval way. Ths s true by the prevous remark or by usng the natural map S(A) k e of k-plethores. It s false for general plethores Examples. (1) If A s the group algebra kg of a group (or monod) G, then S(A) s the free polynomal algebra on the set underlyng G. For any g G, the correspondng element n S(A) s rng-lke : Δ + (g) = g g and Δ (g) = g g. An acton of the plethory S(A) on a rng R s the same as an acton of G on R. (2) Let g be a Le algebra over k, and let A be ts unversal envelopng algebra. Then for all x g, the correspondng element x S(A) s dervaton-lke : Δ + (x) = x x and Δ (x) = x e + e x. Ifg s the one-dmensonal Le algebra spanned by an element d, then S(A) = k[d N ]:=k[e, d, d d,...], and S(A)-rngs are the same as k-rngs equpped wth a dervaton Remark. Because of the dentfcaton S(A) k S(B) S(A B), there s a natural somorphsm S(A) k S(B) S(B) k S(A) of k-k-brngs gven by the canoncal

13 258 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) nterchange map on the tensor product. Explctly, t exchanges [a] [b] and [b] [a], where a A, b B. There s no functoral map S T T S for k-k-brngs S and T that agrees wth the prevous map when S and T come from balgebras. For example, take S = Z[d N ] and T = Λ p below Hopf algebras. An antpode s: A A gves a map S(A) S(A) of k-k-brngs, makng S(A) what could be called a cocommutatve Hopf monod n BR k,k Symmetrc functons and λ-rngs. Let Λ be the rng of symmetrc functons n countably many varables,.e., wrtng Λ n for the sub-graded-rng of Z[x 1,...,x n ] (deg x = 1) of elements nvarant under the obvous acton of the n-th symmetrc group, we let Λ be the nverse lmt of Λ n Λ n 1. n the category of graded rngs. The map above sends x n to 0 and sends any other x to x. Of course, Λ s the free polynomal algebra on the elementary symmetrc functons [15, I.2], but there are many other free generatng sets, and makng ths or any other partcular choce would leave us wth the usual formulac mess n the theory of λ-rngs and Wtt vectors. The rng Λ naturally has the structure of a plethory over Z. Because all the structure maps are already descrbed at varous ponts n the second edton of MacDonald [15], we gve only the brefest descrptons here: Coaddtve structure [15, I.5 ex. 25]: For f Λ, consder the functon Δ + (f ) = f(x 1 1, 1 x 1,x 2 1, 1 x 2,...) n the varables x x j,(,j1). It s symmetrc n both factors, and so Δ + s a rng map Λ Λ Z Λ. The count ε + : Λ k sends f to f(0, 0,...). Comultplcatve structure [15, I.7 ex. 20]: Smlarly, consder the functon Δ (f ) = f(...,x x j,...) n the varables x x j. As before, t s symmetrc n both factors, and so Δ s a map Λ Λ Z Λ. The count ε : Λ k sends f to f(1, 0, 0,...). Monod structure [15, I.8]: For f, g Λ, the operaton known as plethysm defnes f g: Suppose g has only non-negatve coeffcents, and wrte g as a sum of monomals wth coeffcent 1 n the varables x. Then f g s the symmetrc functon obtaned by substtutng these monomals nto the arguments x 1,x 2, of f. Ths gves a monod structure wth dentty x 1 +x 2 + on the set of elements wth non-negatve coeffcents, and ths extends to a unque Z-plethory structure on all of Λ.

14 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Remark. By the theorem of elementary symmetrc functons [15, I2.4], we have Λ = Z[λ 1, λ 2,...], where λ 1 = x 1 + x 2 +, λ 2 = x 1 x 2 + x 1 x 3 + x 2 x 3 +,... are the elementary symmetrc functons. Any Λ-rng R therefore has unary operatons λ 1, λ 2,...Itsan exercse n defntons to show that n ths way, a Λ-rng structure on a rng R s the same as a λ-rng structure (whch, n Grothendeck s orgnal termnology [1], s called a specal λ-rng structure). Ths was n fact one of the prncpal examples n Tall and Wrath s paper [19]. Let ψ n denote the nth Adams operaton: ψ n = x n 1 + xn 2 +. The elements w 1,w 2,... of Λ determned by the relatons ψ n = d n dw n/d d for all n N (2.11.1) also form a free generatng set. Ths s easy to check usng the followng dentty: n0 ( 1) n λ n t n = 1 (1 x t) = exp n1 1 n ψ n tn = n1(1 w n t n ). The w are responsble for the Wtt components, as we wll see n the next secton Remark. There s also a descrpton of Λ n terms of the representatons of the symmetrc groups [15, I.7]. Let R n denote the representaton rng of S n, the symmetrc group on n letters. The maps S n S m S n+m, S n S n S n, and S n S m = S n S n m S mn nduce maps between the R n by restrcton and nducton, and these make up a plethory structure on n0 R n agreeng wth that on Λ. Ths s one natural way to vew Λ when studyng ts acton on Grothendeck groups (see, e.g. [6]). We do not yet know f smlar constructons n other areas of representaton theory also yeld plethores p-typcal symmetrc functons. Let p be a prme number, and set F = ψ p. Then Z F :=Z[e, F, F F,...] s a subrng of Λ, and because F s rng-lke, t s actually a sub-z-plethory. It s also the free plethory on the balgebra assocated to the monod N. We wll denote t Ψ p, and we wll see later that t accounts for the ghost components of the p-typcal Wtt vectors.

15 260 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Now let Λ p be the subrng of Λ consstng of elements f for whch there exsts an N such that p f Ψ p. Then Λ p s a sub-z-plethory of Λ, and s what we call the plethory of p-typcal symmetrc functons. For all n N, let θ n = w p n. Then (2.11.1) becomes F n = θ pn 0 + +p n θ n (2.13.1) and therefore θ 0, θ 1,... le n Λ p. Conversely, because we have we see Λ p = Z[θ 0, θ 1,...]. Λ = Z[θ 0, θ 1,...][w n n s not a power of p], Bnomal plethory. Because Λ s a Z-plethory, the rng Z of ntegers s a Λ-rng. The deal n Λ of elements that act as the constant functon 0 s generated by the set {ψ n e n1}. The quotent rng s stll a plethory, and an acton of t on a rng R s the same as gvng R the structure of a Λ-rng whose Adams operatons are the dentty. Ths has been shown by Jesse Ellott (unpublshed) to be the same as a bnomal λ-rng structure [11, p. 9] on R. Ths plethory can also be nterpreted as the set of functons Z Z that can be expressed as polynomals wth ratonal coeffcents [2]. 3. Examples of Wtt rngs Let k be a rng. Recall that f P s a k-plethory, then W P (R) denotes the P-rng Rng k (P, R). Because W P s the rght adjont of the forgetful functor from P-rngs to rngs, there s a natural map W P W P (W P (R)), whch n the case of the classcal plethores s sometmes called the Artn Hasse map Balgebras. Let P be the free k-plethory (2.2) on a cocommutatve k-balgebra A. Then we have W P (B) = Mod k (A, B). If A s fntely generated as a k-module, W P (B) s just B k A, where A denotes the dual balgebra Mod k (A, k). We leave t to the reader to verfy that, n ths case, the map W P (B) W P (W P (B)) s nothng but the comultplcaton map on ths balgebra. For example, f A s the group algebra of a fnte group G, then we have W P (B) = B G and the map above s the map B G B G G = B G B B G nduced by the multplcaton on G Symmetrc functons. Because Λ = Z[λ 1,...], the set W Λ (B) s just n>0 B, and t s easy to check that, as a group, we have W Λ (B) = 1 + xb[[x]], where the group operaton on the rght s multplcaton of power seres. It s also true that f 1 + xb[[x]] s gven a Λ-rng structure as n [1, 1.1], then the dentfcaton above s an somorphsm of Λ-rngs,.e., W Λ (B) s the Λ-rng of bg Wtt vectors. The proof of ths s very straghtforward but nvolves, of course, the somewhat unpleasant

16 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) defnton of the Λ-rng structure on 1 + xb[[x]]. Because the whole pont of ths paper s to move away from such thngs, we wll leave the argument to the reader. The generatng set {w 1,w 2,...} of 2.11 allows us to vew an element of W Λ (B) as a ( bg ) Wtt vector n the tradtonal sense [8, ]. Under ths dentfcaton, the map W Λ (B) W Λ (W Λ (B)) agrees wth the usual Artn Hasse map [8, 17.6]. If Ψ denotes the sub-plethory Z[ψ n n1] of Λ, then W Ψ (B) s just n>0 B as a rng, and under ths dentfcaton, the map W Λ (B) W Ψ (B) s the ghost-component map. Some early references to the bg Wtt vectors are Carter [4] and Wtt ([12] or [22, pp ]) p-typcal symmetrc functons. Because Λ p = Z[θ 0,...], the set W Λp (B) = Rng Z (Λ p,b) s naturally bjectve wth B N.IfwevewB N as the set underlyng the rng of p-typcal Wtt vectors [21], [8, ], then ths bjecton s an somorphsm of rngs. One can wrte down the correspondng Λ p -acton on B N, and we recover the p-typcal Artn Hasse map as we dd above. Also as above, f Ψ p denotes the plethory Z[ψ N p ], then the natural map W Λ p (B) W Ψp (B) s the p-typcal ghost-component map. The Techmüller lft can be constructed by consderng the monod algebra ZB on the multplcatve monod underlyng B. The rng ZB has no addtve p-torson, and the map F : [b] [b p ]=[b] p ([ ] denotng the multplcatve map B ZB) reduces to the Frobenus map modulo p. The rng Z[B] therefore (3.4) admts a unque Λ p -rng structure where F s the above map. The canoncal rng map ZB B then nduces by adjontness a map ZB W Λp (B). In the standard descrpton, t s [b] (b, 0, 0,...), whch s of course the Techmüller lft of b. The followng lemma mples that a Λ p -rng s the same as what Joyal calls a δ-rng. (A comonadc verson of ths statement s stated qute clearly n Joyal [10]; we nclude t only because we wll use t later.) 3.4. Lemma. The R be a p-torson-free rng. Gven an acton of Λ p on R, the element F gves an endomorphsm of R such that F(x) x p mod pr. Ths s a bjecton from the set of actons of Λ p on R to the set of lfts of the Frobenus endomorphsm of R/pR. Proof. Because R s p-torson-free, (2.13.1) mples that any acton of Λ p s determned by the endomorphsm F, and so we need only show every Frobenus lft comes from some acton of Λ p. Gven a Frobenus lft f : R R, Carter s Deudonné Dwork lemma [13, VII Secton 4] states there s a rng map R W Λp (R) such that the composte R W Λp (R) W Ψp (R) sends r to (r,f(r),f(f(r)),...). Ths gves a map Λ p R R; to show t s an acton we need only check t s assocatve. Because R s p-torsonfree t suffces to check the nduced map of Ψ p R R s an acton. But the Deudonné Dwork lemma mples ths map sends F r to f (r), whch s clearly assocatve.

17 262 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Reconstructon and recognton In preparaton for the reconstructon theorem, we generalze the notons of brng and plethory from Rng k to Rng P for non-trval plethores P. Ths gves us P-P -brngs and P-plethores, whch reduce to k-k -brngs and k-plethores when P = k e and P = k e. Let P be a k-plethory and P a k -plethory, where k and k are arbtrary rngs Functor P : BR P,P Rng P Rng P. Take S BR P,P and R Rng P. Then S P R s defned to be the coequalzer of the maps of P-rngs S k P k R S k R s α r (s α) r s α r s (α r) Lemma. Let S be a P-P -brng. Then the functor S P : Rng P Rng P s the left adjont of the functor Rng P (S, ). We leave the proof to the reader Proposton. Let P Q be a map of plethores. Then the restrcton functor Rng Q Rng P preserves lmts and coequalzers and has a left adjont ( nducton ) Q P. If the map P Q s an somorphsm on scalars, t has a rght adjont ( co-nducton ) Rng P (Q, ) and preserves all colmts. Proof. Because Q s a Q-P-brng, Q P s left adjont (by 4.2) to Rng Q (Q, ), whch s the forgetful functor Rng Q Rng P.IfP Q s a map of k-plethores, Q s a P-Q-brng, so Rng P (Q, ) s rght adjont to Q Q, the forgetful functor. It follows that the forgetful functor preserves lmts and, when the rngs of scalars agree, colmts. It remans to show t always preserves coequalzers. Consder the commutatve dagram of forgetful functors Rng Q Rng kq Rng P Rng kp. The upper functor preserves colmts, and the rght-hand functor preserves coequalzers. The lower functor reflects somorphsms and preserves colmts. It then follows that the left-hand functor preserves coequalzers.

18 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Remark. If k P k Q s not an somorphsm, ε + wll fal to descend. Thus, Q wll not be a k P -k Q -brng, let alone a P-Q-brng A P-plethory s defned to be a plethory Q equpped wth a map P Q of plethores whch s an somorphsm on scalars. A morphsm Q Q of P-plethores s a morphsm of plethores commutng wth the maps from P Proposton. P makes BR P,P nto a monodal category wth unt object P. Monods n ths category are the same as P-plethores. An acton of such a monod Q on a P-rng s the same as an acton of Q on the underlyng k-rng such that the acton of Q restrcted to P s the gven one. Proof. The frst statement requres no proof. Gven a monod Q, the structure maps gve map Q k Q Q P Q Q and P Q makng t a k-plethory. Conversely, a map P Q of k-plethores makes Q a P-P-brng and the assocatvty condton Q k Q k Q Q k Q Q mples that Q k P k Q Q k Q Q commutes, so composton descends to Q P Q Q. Smlarly, an acton of Q on the underlyng k-rng of a P-rng A s a map Q k A A, and t descends to a P-acton Q P A A because Q k P k A Q k A A commutes Now let C be a category that has all lmts and colmts, and let U: C Rng P be a functor that has a left adjont F. We also assume U reflects somorphsms, that s, a morphsm f s an somorphsm f and only f U(f) s an somorphsm. Set Q = UF (P ). Let U be the composte of U wth the forgetful functor from Rng P to the category of sets k-plethory structure on Q when U has a rght adjont. Suppose U has a rght adjont W. The functor UW s represented by Q: UW(A) = Rng P (P, UW(A)) = Rng P (UF (P ), A), and ths gves Q the structure of a P-P-brng (1.9). The composte UW of adjonts s a comonad, and so ts adjont Q P s a monad. By 4.6, Q s a k-plethory wth a map P Q. Gven an object A of C, the adjuncton gves an acton of UF( ) = Q on U(A), and hence we have a functor C Rng Q between categores over Rng P Theorem. If U has a rght adjont W, then the functor C Rng Q s an equvalence of categores over Rng P. Proof. Beck s theorem [14] Let k be the P-rng UF (k), and let P be the k -plethory k k P. Because F(k) s the ntal object, U factors as a functor U : C Rng P followed by the forgetful functor V : Rng P Rng P. The functor U has a left adjont F gven by descent: f A s a P -rng, then F V (A) has two maps from F(k ) = FUF(k), one from applyng

19 264 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) FV to the ntal map k A and the other gven by the composte FUF(k) F(k) F V (A), where the frst map s the adjuncton and the second s the ntal map. Let F (A) denote the coequalzer of F(k ) F V (A) Theorem. If P Q s a map of plethores and U s the forgetful functor Rng Q Rng P, then U of 4.10 has a rght adjont. Conversely, suppose U : C Rng P has a rght adjont, and let Q be the k -plethory U F (P ) of 4.8. Then the functor C Rng Q s an equvalence of categores over Rng P. Proof. Apply 4.9 to U Remark. In practce, t s qute easy to check the exstence of F and W usng Freyd s theorem from category theory. 5. P -deals Let P be a k-plethory, and let P + denote the kernel of ε + : P k An deal I n a P-rng R s called a (left) P-deal f there exsts an acton of P on R/I such that the map R R/I of rngs s a map of P-rngs. If such an acton exsts, t s unque, and so beng a P-deal s a property of, rather than a structure on, a subset of R Proposton. Let I be an deal n a P-rng R. Then the followng are equvalent: (1) I s a P-deal; (2) I s the kernel of a morphsm of P-rngs; (3) P + I I; (4) I s generated by a set X such that P + X I. The proof s n 5.6. Gven any subset X of P, t s therefore reasonable to call the deal generated by P + X the P-deal generated by X Elements of P P gve bnary operatons on any P-rng R by (α β)(r, s) = α(r)β(s) and extendng lnearly Lemma. Let R be a P-rng, I an deal n R and X a subset of R. Assume that for all x X and f P +, we have f(x) I. Then for all t P P + and all (r, ) R I, we have t(r,) I.

20 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Proof. Snce t P P +, t may be expressed as t = t j t j t j preserves I. Then for (r, ) R I, t(r,) = t j (r)t j Typcal applcatons wll use X = I, a P-deal. wth t j P +, so that () I Lemma. LetSbeak-Z-brng. Then Δ + (S + ) s contaned n S + S + S S +, and Δ (S + ) s contaned n S + S +. Proof. S s a rng object n the opposte of Rng k ; the rng dentty = 0 translates nto the dentty (ε + ε + ) Δ + = ε +, whch s clearly equvalent to the frst statement. The second statement s smlarly just a coalgebrac translaton of a rng dentty. Let W denote the rng object correspondng to S n the opposte category. Then the commutatvty of the followng two dagrams s equvalent: W W W S Δ S S 0 W W. d 0 0 d k ε + S S d ε + ε + d But the commutatvty of the frst s just a restatement of the rng dentty 0 x = x 0 = 0. We therefore have ( ) Δ (S + ) ker S S S S = S + S Proof of 5.2. (1) (2) and (3) (4) are clear. (2) (3): P + preserves the set {0} n k and, thus, n any P-rng; t therefore must preserve ts premage under a morphsm of P-rngs. (3) (1): IfI s preserved by P +, we must put a P-rng structure on R/I so that R R/I s a morphsm of P-rngs. The acton must be p(r + I) = p(r) + I; ts necessary only to check that ths s well defned. The kernel of d P ε + : P P P s P P +, and so by the count condton, we have Δ + p p 1 P P + for all p P. For any I, wehavep(r + ) p(r) = (Δ + p p 1)(r, ). By 5.4, the rght-hand sde of ths equalty s n I, and so the acton s well defned. (4) (3): Consder the set J of elements of I that are sent nto I by all elements of P +.Iff P +, then Δ + f P + P + P P +. Thus for j,k J, Lemma 5.4 mples f(j + k) I and hence j + k J. Smlarly, Δ f P + P + P P +, and so for r R and j J,wehavef(rj) I and hence rj J. Therefore J s an deal, and f a generatng set for I s sent by P + nto I, wehavei = J. So all of I s preserved by P Proposton. Let I and J be P-deals n a P-rng A. Then IJ s a P-deal.

21 266 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) Proof. It s suffcent to check f(xy) IJ for all f P +, x I, and y J because such xy form a generatng set. We can wrte Δ f = f [1] f [2] wth f [1],f [1] P +, and so we have f(xy)= f [1] (x)f [2] (y) IJ. 6. Two-sded deals Let P be a k-plethory, and let P be a k -plethory An deal J n a k-k -brng S s called a k-k -deal f the quotent k-rng S/J admts the structure of a k-k -brng. Ths s clearly equvalent to S/J beng, n the opposte of Rng k, a sub-k -rng object of S, and so f S/J admts such a structure, t s unque. Ths s also equvalent to the exstence of a generatng set X of J such that, n the notaton of 1.1, we have 1. Δ + S (X) S J + J S, 2. Δ S (X) S J + J S, and 3. β S (c)(x) = 0 for all c k A k-k -deal J n a P-P -brng S s called a P-P -deal f there exsts a P-P -brng structure on the quotent k-k -brng S/J such that S S/J s a map of P-P -brngs. If such an acton exsts, t s unque, and so as was the case for P-deals, beng a P-P -deal s a property, rather than a structure Proposton. Let J be a k-k -deal n a k-k -brng S. Then the followng are equvalent: 1. JsaP-P -deal; 2. J s the kernel of a map of P-P -brngs; 3. P + J P J ; 4. J s generated by a set X such that P + X P J. The asymmetry n (3) s due to the tradtonal defnton of deal. If we took a more categorcal approach and consdered, nstead of kernels of maps R S of k-rngs, the fber products R S k, the P + n (3) would become a P. Proof. As n 5.2, the only mplcaton that requres proof s (4) (1). So, assume (4). By 5.2, J s a P-deal; and by assumpton, J s a k-k -deal. Therefore S/J s a P-k -brng. For all s S,j J,f P,wehave (s + j) f = s f + j f s f mod J, and so the rght P -acton descends to S/J.

22 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) If J s a P-P-deal n P tself, then ths proposton mples P/J s a P-plethory n the sense that the P-P-brng structure on P/J extends to a unque P-plethory structure on P/J Proposton. The category BR P,P of P-P -brngs has all colmts, and the forgetful functor BR P,P Rng P preserves them. Proof. Gven a dagram C of P-P -brngs, ts colmt S n the category of P-rngs has the property that for any P-rng R, the set Rng P (S, R) s the lmt of the sets Rng P (T c,r), where c ranges over C. Because each Rng P (T c,r) s a P -rng and the maps are P -equvarant, Rng P (S, R) s a P -rng. Thus, by a remark n 1.9, S has a unque P-P -brng structure makng the maps T c S maps of P-P -brngs, whch wastobeproved Free plethory on a ponted brng. The free P-plethory Q on a P-P-brng S can be constructed as n 2.1. It comes equpped wth a map P Q of k-plethores. Now let f : P S be a map of P-P-brngs. (Ths s equvalent to specfyng an element s 0 S such that p s 0 = s 0 p for all p P.) Then the free plethory on the ponted brng S s coequalzer (6.5) of the two Q-Q-brng maps Q P Q Q nduced by sendng e α e, on the one hand, to α P = S 0 and, on the other, to f(α) S 1. By 6.4, Q s a k-plethory. It s the ntal object among P-plethores P equpped wth a map S P such that the composte P S P agrees wth the structure map P P. An acton of ths plethory on a k-rng R s the same as an acton of P on R together wth a map S R R such that f(p) r p(r) for all p P,r R. At ths pont, t s qute easy to gve an explct constructon of Λ p that does not rely on symmetrc functons. Let S = Z[e, θ 1 ] be the Z e -ponted Z-Z-brng determned by p 1 Δ + : θ 1 θ θ 1 =1 ( ) 1 p e e p, (6.6.1) p Δ : θ 1 e p θ 1 + θ 1 e p + pθ 1 θ 1. (6.6.2) Then Carter s Deudonné Dwork lemma mples Λ p s the free Z-plethory on S. Of course, ths s just a plethystc descrpton of Joyal s approach [10] to the p-typcal Wtt vectors The followng asymmetrc varant of ths constructon wll be used n Secton 7. Let P 0 be a k-plethory, let P be a P 0 -plethory, let S be a P 0 -P-brng, and let g: P S be a map of P 0 -P-brngs. Let Q denote the free P 0 -plethory on S vewed as a ponted P 0 -P 0 -brng. Then we have two maps of P 0 -P 0 -brngs S P0 P Q

23 268 J. Borger, B. Weland / Advances n Mathematcs 194 (2005) gven by s α s g(α) S 2 and s α s α S 1. These then nduce two maps of Q-Q-brngs Q P0 S P0 P P0 Q Q. The coequalzer T of these maps s a P 0 -plethory (6.4), but the two maps P Q become equal n T, and so T s n fact a P-plethory. An acton of T on a rng R s the same as an acton of P on R together wth a map S P R R such that g(α) r α r. 7. Amplfcatons over curves Let O be a Dedeknd doman, and let m be an deal; let k denote the resdue rng O/m, and let K denote the subrng of the feld of fractons of O consstng of elements that are ntegral at all maxmal deals not dvdng m. The m-torson submodule of an O-module M s the set of m M for whch there exsts an n N such that m n m = 0. We say an O-module s m-torson-free f ts m-torson submodule s trval, or equvalently, f t s flat locally at each maxmal deal dvdng m. Now let P be an O-plethory that s m-torson-free, let Q be a k-plethory, and let f : P Q be a surjectve map of plethores agreeng wth the canoncal map on scalars. A P-deformaton of a Q-rng s an m-torson-free P-rng R such that the acton of P on k R factors through an acton of Q on k R. (Note that because P Q s surjectve, t can factor n at most one way.) A morphsm of P-deformatons of Q-rngs s by defnton a morphsm of the underlyng P-rngs. The purpose of ths secton s then to construct an O-plethory P, the amplfcaton of P along Q, such that m-torson-free P -rngs are the same as P-deformatons of Q-rngs. It s constructed smply by adjonng m 1 I to P, where I s the kernel of the map P Q, and so t s analogous to an affne blow-up of rngs. Note however that there are some mnor subtletes nvolved n adjonng these elements because a plethory nvolves co-operatons, not just operatons, and because we need to know how to compose elements of P wth elements of m 1 I, butp may not even act on K, let alone preserve m Theorem. The P-plethory P of 7.6 s m-torson-free, and the forgetful functor from the full category of m-torson-free P -rngs to Rng P dentfes t wth the category of P-rng deformatons of Q-rngs. Furthermore, P has the followng unversal property: Let P be a P-plethory whose underlyng P-rng s a P-deformaton of a Q-rng. Then there s a unque map P P of P-rngs commutng wth the maps from P, and ths map s a map of P-plethores Corollary. Let P be a P-plethory wth the property that the forgetful functor from the full category of m-torson-free P -rngs to Rng P dentfes t wth the category of P-rng deformatons of Q-rngs. Then there s a unque map P P of P-rngs; ths map s a map of P-plethores, and t dentfes P wth the largest m-torson-free P -rng quotent of P. We prove these at the end of ths secton. Note that ether the theorem or the constructon of 7.6 mples amplfcaton s functoral n P and Q.