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ring_theory.witt_vector.identities

Identities between operations on the ring of Witt vectors #

In this file we derive common identities between the Frobenius and Verschiebung operators.

Main declarations #

References #

The composition of Frobenius and Verschiebung is multiplication by p.

Verschiebung is the same as multiplication by p on the ring of Witt vectors of zmod p.

theorem witt_vector.coeff_p_pow {p : } {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] (i : ) :
(p ^ i).coeff i = 1
theorem witt_vector.coeff_p_pow_eq_zero {p : } {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] {i j : } (hj : j i) :
(p ^ i).coeff j = 0

The “projection formula” for Frobenius and Verschiebung.

theorem witt_vector.mul_char_p_coeff_zero {p : } {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] (x : witt_vector p R) :
(x * p).coeff 0 = 0
theorem witt_vector.mul_char_p_coeff_succ {p : } {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] (x : witt_vector p R) (i : ) :
(x * p).coeff (i + 1) = x.coeff i ^ p

Iteration lemmas #

theorem witt_vector.iterate_verschiebung_coeff {p : } {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] (x : witt_vector p R) (n k : ) :
theorem witt_vector.iterate_frobenius_coeff {p : } {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] (x : witt_vector p R) (i k : ) :
theorem witt_vector.iterate_verschiebung_mul_coeff {p : } {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] (x y : witt_vector p R) (i j : ) :

This is a slightly specialized form of Hazewinkel, Witt Vectors 6.2 equation 5.