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Register a new userThe basic geometry of Witt vectors, I: The affine case
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Authors
James Borger
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We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for variants of these functors which are in a certain sense their analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications.
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This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, p-typical Witt vectors of p-adic schemes of finite type are already reasonably well understood. The main point here is to generalize this theory in two ways. We allow not just p-typical Witt vectors but those taken with respect to any set of primes in any ring of integers in any global field, for example. This includes the big Witt vectors. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of Buiums formal arithmetic jet functor, which is dual to the Witt functor. We also give concrete geometric descriptions of Witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors.
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 Edreis 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.
The subject matter of this paper is the geometry of the affine group over the integers, $mathsf{GL}(n,mathbb{Z})ltimes mathbb{Z}^n$. Turing-computable complete $mathsf{GL}(n,mathbb{Z})ltimes mathbb{Z}^n$-orbit invariants are constructed for angles, segments, triangles and ellipses. In rational affine $mathsf{GL}(n,mathbb Q)ltimes mathbb Q^n$-geometry, ellipses are classified by the Clifford--Hasse--Witt invariant, via the Hasse-Minkowski theorem. We classify ellipses in $mathsf{GL}(n,mathbb{Z})ltimes mathbb{Z}^n$-geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch-Jung continued fraction algorithm and the Morelli-Wl odarczyk solution of the weak Oda conjecture on the factorization of toric varieties. We then consider {it rational polyhedra}, i.e., finite unions of simplexes in $mathbb R^n$ with rational vertices. Markovs unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra $P$ and $P$ are continuously $mathsf{GL}(n,mathbb Q)ltimes mathbb Q^n$-equidissectable. The same problem for the continuous $mathsf{GL}(n,mathbb{Z})ltimes mathbb{Z}^n$-equi-dis-sect-ability of $P$ and $P$ is open. We prove the decidability of the problem whether two rational polyhedra $P,Q$ in $mathbb R^n$ have the same $mathsf{GL}(n,mathbb{Z})ltimes mathbb{Z}^n$-orbit.
In this paper, we introduce the definitions of signatures of braided fusion categories, which are proved to be invariants of their Witt equivalence classes. These signature assignments define group homomorphisms on the Witt group. The higher central charges of pseudounitary modular categories can be expressed in terms of these signatures, which are applied to prove that the Ising modular categories have infinitely many square roots in the Witt group. This result is further applied to prove a conjecture of Davydov-Nikshych-Ostrik on the super-Witt group: the torsion subgroup generated by the completely anisotropic s-simple braided fusion categories has infinite rank.
We give a $K$-theoretic account of the basic properties of Witt vectors. Along the way we re-prove basic properties of the little-known Witt vector norm, give a characterization of Witt vectors in terms of algebraic $K$-theory, and a presentation of the Witt vectors that is useful for computation.
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