Fontaine's θ map #
In this file, we define Fontaine's θ map, which is a ring
homomorphism from the Witt vector 𝕎 R♭ of the tilt of a perfectoid ring R
to R itself. Our definition of θ does not require that R is perfectoid in the first place.
We only need R to be p-adically complete.
Main definitions #
fontaineTheta: Fontaine's θ map, which is a ring homomorphism from𝕎 R♭toR.
TODO #
Establish that our definition (explicit construction of θ mod p ^ n) agrees with the
deformation-theoretic approach via the cotangent complex, as in
[Bhatt, Lecture notes for a class on perfectoid spaces. Remark 6.1.7]
(https://www.math.ias.edu/~bhatt/teaching/mat679w17/lectures.pdf).
Tags #
Fontaine's theta map, perfectoid theory, p-adic Hodge theory
Reference #
θ as a ring homomorphism #
Let 𝔭 denote the ideal of R generated by the prime number p. In this section, we first
define the ring homomorphism fontaineThetaModPPow : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1).
Then we show they are compatible with each other and lift to a
ring homomorphism fontaineTheta : 𝕎 R♭ →+* R.
To prove this, we define fontaineThetaModPPow as a composition of the following ring
homomorphisms.
𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/p) --gh_n-> R/p^(n+1)
Here, the ring map gh_n fits in the following diagram.
𝕎(A) --ghost_n-> A
| |
v v
𝕎(A/p) --gh_n-> A/p^(n+1)
The lift ring map gh_n : 𝕎(A/p) →+* A/p^(n+1) of the n-th ghost component
𝕎(A) →+* A along the surjective ring map 𝕎(A) →+* 𝕎(A/p).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Fontaine's theta map modulo p^(n+1).
It is the composition of the following ring homomorphisms.
𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/p) --gh_n-> R/p^(n+1)
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Fontaine's θ map from 𝕎 R♭ to R.
It is the limit of the ring maps fontaineThetaModPPow n from 𝕎 R♭ to R/p^(n+1).