Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories accessible adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nlab noncommutative noncommutative-geometry number number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 24th 2021




    (Definition 2.1 in Bhatt–Scholze.)

    Fix a prime pp. A δ-ring is a pair (R,Unknown characterUnknown character)(R,δ), where RR is a commutative ring and δ:RR\delta\colon R\to R is a map of underlying sets such that δ(0)=0\delta(0)=0, δ(1)=0\delta(1)=0,

    δ(xy)=x pδ(y)+y pδ(x)+pδ(x)δ(y),\delta(xy)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),


    δ(x+y)=δ(x)+δ(y)+(x p+y p(x+y) p)/p.\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.


    If (R,δ)(R,\delta) is a δ-ring, then the map ϕ:RR\phi\colon R\to R given by ϕ(x)=x p+pδ(x)\phi(x)=x^p + p\delta(x) is a ring homomorphism that lifts the Frobenius endomorphism on R/pR/p.

    For pp-torsionfree rings, the above correspondence between δ-structures and lifts of the Frobenius endomorphism on R/pR/p to RR is bijective. This motivates the identities in the definition of a δ-structure.

    Related entries


    v1, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)