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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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(Definition 2.1 in Bhatt–Scholze.)
Fix a prime p. A δ-ring is a pair (R,Unknown characterUnknown character), where R is a commutative ring and δ:R→R is a map of underlying sets such that δ(0)=0, δ(1)=0,
δ(xy)=xpδ(y)+ypδ(x)+pδ(x)δ(y),and
δ(x+y)=δ(x)+δ(y)+(xp+yp−(x+y)p)/p.If (R,δ) is a δ-ring, then the map ϕ:R→R given by ϕ(x)=xp+pδ(x) is a ring homomorphism that lifts the Frobenius endomorphism on R/p.
For p-torsionfree rings, the above correspondence between δ-structures and lifts of the Frobenius endomorphism on R/p to R is bijective. This motivates the identities in the definition of a δ-structure.
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