English
There is a canonical IsScalarTower structure relating R, T, and the localized module LocalizedModule(S,M): for all r in R, x in T, and p in LocalizedModule S M, smul associativity holds: (r ⋅ x) ⋅ p = r ⋅ (x ⋅ p).
Русский
Существует каноническая структура IsScalarTower между R, T и локализованным модулем LocalizedModule(S,M): для всех r∈R, x∈T, p∈LocalizedModule S M выполняетсяAssoc запах: (r • x) • p = r • (x • p).
LaTeX
$$$\\forall r\\in R\\, \\forall x\\in T\\, \\forall p\\in \\mathrm{LocalizedModule}(S,M),\\ (r\\cdot x)\\cdot p = r\\cdot (x\\cdot p).$$$
Lean4
instance : IsScalarTower R T (LocalizedModule S M) where
smul_assoc r x
p :=
by
induction p with
| _ m s
rw [← IsLocalization.mk'_sec (M := S) T x, IsLocalization.smul_mk', mk'_smul_mk, mk'_smul_mk, smul'_mk, mul_smul]
