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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 24th 2021

    Created:

    \tableofcontents

    Definition

    (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),

    and

    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.

    Properties

    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

    References

    v1, current

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