Almost purity for overconvergent Witt vectors

Christopher James Davis; Kiran Kedlaya

doi:10.1016/j.jalgebra.2014.08.055

Almost purity for overconvergent Witt vectors

Christopher James Davis, Kiran Kedlaya

Institut for Matematiske Fag

1257 Downloads (Pure)

Abstract

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p^-1] is "almost" finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p^-1] to a finite étale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustration, we construct (ϕ, Γ)-modules associated with Artin Motives over Q. The (ϕ, Γ)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work.

Originalsprog	Engelsk
Tidsskrift	Journal of Algebra
Vol/bind	422
Sider (fra-til)	373-412
ISSN	0021-8693
DOI	https://doi.org/10.1016/j.jalgebra.2014.08.055
Status	Udgivet - 5 jan. 2015

Adgang til dokumentet

10.1016/j.jalgebra.2014.08.055

AlmostPurityIndsendt manuskript, 423 KB

Citationsformater

@article{7bc4ee3868d845e7927dabf8b2d42191,

title = "Almost purity for overconvergent Witt vectors",

abstract = "In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite {\'e}tale extension of R[p-1] is {"}almost{"} finite {\'e}tale over R. Here, we use almost purity to lift the finite {\'e}tale extension of R[p-1] to a finite {\'e}tale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustration, we construct (ϕ, Γ)-modules associated with Artin Motives over Q. The (ϕ, Γ)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work.",

author = "Davis, {Christopher James} and Kiran Kedlaya",

year = "2015",

month = jan,

day = "5",

doi = "10.1016/j.jalgebra.2014.08.055",

language = "English",

volume = "422",

pages = "373--412",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press",

}

TY - JOUR

T1 - Almost purity for overconvergent Witt vectors

AU - Davis, Christopher James

AU - Kedlaya, Kiran

PY - 2015/1/5

Y1 - 2015/1/5

N2 - In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p-1] is "almost" finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p-1] to a finite étale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustration, we construct (ϕ, Γ)-modules associated with Artin Motives over Q. The (ϕ, Γ)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work.

AB - In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p-1] is "almost" finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p-1] to a finite étale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustration, we construct (ϕ, Γ)-modules associated with Artin Motives over Q. The (ϕ, Γ)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work.

U2 - 10.1016/j.jalgebra.2014.08.055

DO - 10.1016/j.jalgebra.2014.08.055

M3 - Journal article

SN - 0021-8693

VL - 422

SP - 373

EP - 412

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Almost purity for overconvergent Witt vectors

Abstract

Adgang til dokumentet

Fingeraftryk

Citationsformater