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Witt Vectors, Semirings, and Total Positivity 275 Witt vectors, semirings, and total positivity James Borger∗ Abstract. We extend the big and p-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the p-typical case, it uses positivity with respect to an apparently new basis of the p-typical symmetric functions. We also give explicit descriptions of the big Witt vectors of the natural numbers and of the nonnegative reals, the second of which is a restatement of Edrei’s theorem on totally positive power series. Finally we give some negative results on the relationship between truncated Witt vectors and k-Schur positivity, and we give ten open questions. 2010 Mathematics Subject Classification. Primary 13F35, 13K05; Secondary 16Y60, 05E05, 14P10. Keywords. Witt vector, semiring, symmetric function, total positivity, Schur positivity. Contents 1 Commutative algebra over N,thegeneraltheory 281 1.1 The category of N-modules ...................... 281 1.2 Submodulesandmonomorphisms . 281 1.3 Products,coproducts. 281 1.4 Internal equivalence relations, quotients, and epimorphisms . 282 1.5 Generatorsandrelations. 282 1.6 Hom and ⊗ ............................... 282 1.7 N-algebras................................ 283 1.8 Commutativityassumption . 283 1.9 A-modules and A-algebras....................... 283 1.10 HomA and ⊗A ............................. 284 1.11 Limits and colimits of A-modules and A-algebras . 284 1.12 Warning: kernelsandcokernels . 284 arXiv:1310.3013v2 [math.CO] 9 Sep 2015 1.13 Basechange,inducedandco-inducedmodules . 284 1.14 Limits and colimits of A-algebras................... 284 1.15 Basechangeforalgebras. 285 1.16 Flatmodulesandalgebras. 285 1.17 Examplesofflatmodules . 285 ∗Supported the Australian Research Council under a Discovery Project (DP120103541) and a Future Fellowship (FT110100728). 274 J. Borger 2 The flat topology over N 285 2.1 Flatcovers ............................... 286 2.2 Thefpqctopology ........................... 286 2.3 Faithfullyflatdescent ......................... 286 2.6 Algebraic geometry over N ....................... 287 2.7 Extending fibered categoriesto nonaffine schemes . 288 2.8 Additively idempotent elements . 288 2.9 Cancellativemodules.......................... 288 2.10 Strongandsubtractivemorphisms . 289 2.11 Additively invertible elements . 289 3 Plethystic algebra for N-algebras 290 3.1 Models of co-C objects in AlgK .................... 290 3.2 Co-L-algebra objects in AlgK ..................... 290 3.3 Plethysticalgebra............................ 291 3.4 Example: composition algebras and endomorphisms . 292 3.5 Models of co-C objects in AlgK .................... 292 3.6 Flat models of co-C-objects in AlgK ................. 292 3.7 Modelsofcompositionalgebras . 293 4 The composition structure on symmetric functions over N 293 4.1 Conventionsonpartitions . 293 4.2 ΨK .................................... 294 4.3 Symmetricfunctions .......................... 294 4.4 Remark: ΛN is not free as an N-algebra ............... 294 4.5 ElementaryandWittsymmetricfunctions . 295 4.8 Explicit description of a ΛN-action .................. 297 4.9 Example: the toric ΛN-structureon monoidalgebras . 298 4.10 Example:theChebyshevline . 298 4.11 Flatness for ΛN-semirings ....................... 298 4.12 Example: convergent exponential monoid algebras . 298 4.13 Example: convergentmonoidalgebras . 299 4.14 Remark: non-models for ΛZ over N .................. 299 5 The Schur model for ΛZ over N 300 5.1 Schur functions and ΛSch ....................... 300 5.3 Remark ................................. 301 5.4 Remark: ΛSch is not free as an N-algebra .............. 301 5.5 Remark: explicit description of a ΛSch-action ............ 302 5.6 Example................................. 302 5.7 Example................................. 302 5.8 Frobenius lifts and p-derivations ................... 302 5.9 Remark: thenecessityofnonlinearoperators . 303 5.10 Remark: composition algebras over number fields . 303 5.11 Remark: representation theory and K-theory . 304 Witt vectors, semirings, and total positivity 275 6 Witt vectors of N-algebras 305 6.1 W and W Sch .............................. 305 6.2 Theghostmapandsimilarones. 306 6.4 Remark ................................. 307 6.5 CoordinatesforWittvectorsofrings . 307 6.6 Witt vectors for semirings contained in rings . 308 6.7 Example: some explicit effectivity conditions . 308 6.8 Coordinates for Witt vectors of semirings . 309 6.9 Topologyandpro-structure . 309 6.10 Teichm¨uller and anti-Teichm¨uller elements . 310 6.11 The involution and the forgotten symmetric functions . 310 6.12 Example: the map N → W Sch(A) is injective unless A =0..... 311 7 Total positivity 311 7.2 Totalpositivity ............................. 312 Sch 7.6 Remark: W (R+) and W (R+) as convergent monoid algebras . 313 7.7 Remark: W (R+)andentirefunctions . 313 7.9 Remark ................................. 314 7.10 Counterexample: W doesnotpreservesurjectivity . 314 7.11Remark ................................. 314 8 A model for the p-typical symmetric functions over N 314 8.1 p-typical Witt vectors and general truncation sets. 314 8.2 Positive p-typicalsymmetricfunctions . 315 8.5 p-typical Witt vectors and Λ-structures for semirings . 317 8.6 Remark: a partition-like interpretation of the bases . 318 8.7 Relation to the multiple-prime theory . 318 8.8 Some explicit descriptions of W(p),k(A)................ 319 8.9 W(p),k(A) when A is contained in a ring . 319 8.10 Counterexample: The canonical map W(p),k+1(A) → W(p),k(A) is notgenerallysurjective. 320 8.11 Semirings and the infinite prime . 320 9 On the possibility of other models 321 10 k-Schur functions and truncated Witt vectors 321 Sch 10.1 k-Schur functions and Λk ...................... 322 10.3Remark ................................. 323 10.4 Truncated Schur–Witt vectorsfor semirings . 323 Sch 10.5 Counterexample: Λk is not a co-N-algebraobject . 323 11Remarksonabsolutealgebraicgeometry 324 276 J. Borger Introduction Witt vector functors are certain functors from the category of rings (always com- mutative) to itself. They come in different flavors, but each of the ones we will consider sends a ring A to a product A × A × · · · with certain addition and multi- plication laws of an exotic arithmetic nature. For example, for each prime p, there is the p-typical Witt vector functor W(p),n of length n ∈ N ∪ {∞}. As sets, we n+1 have W(p),n(A)= A . When n = 1, the ring operations are defined as follows: p−1 1 p (a ,a ) + (b ,b ) = (a + b , a + b − ai bp−i) 0 1 0 1 0 0 1 1 p i 0 0 i=1 p p X (a0,a1)(b0,b1) = (a0b0, a0b1 + a1b0 + pa1b1) 0=(0, 0) 1=(1, 0). For n ≥ 2, the formulas in terms of coordinates are too complicated to be worth writing down. Instead we will give some simple examples: ∼ n+1 W(p),n(Z/pZ) = Z/p Z, ∼ n+1 i+1 W(p),n(Z) = hx0,...,xni ∈ Z | xi ≡ xi+1 mod p . In the second example, the ring operations are performed componentwise; in par- ticular, the coordinates are not the same as the coordinates above. For another example, if A is a Z[1/p]-algebra, then W(p),n(A) is isomorphic after a change of coordinates to the usual product ring An+1. This phenomenon holds for other kinds of Witt vectors—when all relevant primes are invertible in A, then the Witt vector ring of A splits up as a product ring. The p-typical Witt vector functors were discovered by Witt in 1937 [46] and have since become a central construction in p-adic number theory, especially in p-adic Hodge theory, such as in the construction of Fontaine’s period rings [16] and, via the de Rham–Witt complex of Bloch–Deligne–Illusie, in crystalline coho- mology [20]. Also notable, and related, is their role in the algebraic K-theory of p-adic rings, following Hesselholt–Madsen [19]. There is also the big Witt vector functor. For most of this chapter, we will think about it from the point of view of symmetric functions, but for the moment, what is important is that it is formed by composing all the p-typical functors in an inverse limit: W (A) = lim W(p ),∞(W(p ),∞(··· (W(p ),∞(A)) ··· )), (0.0.1) n 1 2 n where p1,...,pn are the first n primes, and the transition maps are given by projections W(p),∞(A) → A onto the first component. (A non-obvious fact is that the p-typical functors commute with each other, up to canonical isomorphism; so in fact the ordering of the primes here is unimportant.) This has an adelic flavor, and indeed it is possible to unify the crystalline cohomologies for all primes Witt vectors, semirings, and total positivity 277 using a (“relative”) de Rham–Witt complex for the big Witt vectors. However the cohomology of this complex is determined by the crystalline cohomologies and the comparison maps to algebraic de Rham cohomology; so the big de Rham–Witt cohomology does not, on its own, provide any new information. But because of this adelic flavor, it is natural to ask whether the infinite prime plays a role. The answer is yes—big Witt vectors naturally accommodate positiv- ity, which we will regard as p-adic integrality at the infinite prime. On the other hand, we will have little to say about other aspects of the infinite prime, such as archimedean norms or a Frobenius operator at p = ∞. To explain this in more detail, we need to recall some basics of the theory of symmetric functions. The big Witt vector functor is represented, as a set-valued functor, by the free polynomial ring ΛZ = Z[h1,h2,... ]. So as sets, we have W (A) = HomAlgZ (ΛZ, A)= A × A × · · · . If we think of ΛZ as the usual ring of symmetric functions in infinitely many formal variables x1, x2,... by writing hn = xi1 ··· xin , i1≤···≤X in then the ring operations on W (A) are determined by two well-known coproducts in the theory of symmetric functions. The addition law is determined by the + n coproduct ∆ on ΛZ for which the power-sum symmetric functions ψn = i xi are primitive, + P ∆ (ψn)= ψn ⊗ 1+1 ⊗ ψn, and the multiplication law is determined by the coproduct for which the power sums are group-like, × ∆ (ψn) = (ψn ⊗ 1)(1 ⊗ ψn)= ψn ⊗ ψn.
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