0002132人目の素数さん2021/04/04(日) 16:06:54.42ID:F2zKij9S Def: R: commutative ring A: R-algebra M: A-module
R-derivation of A into M is a linear map
d: A → M s.t. d(ab) = bd(a) + ad(b)(∀a, b∈A).
We denote the set of R-derivation of A into M by Der_R(A, M). 0003132人目の素数さん2021/04/04(日) 16:13:01.89ID:F2zKij9S Remark: For all r∈R, we have dr = 0.
∵ dr = d(1r) = rd(1) + dr (∵Leibniz rule) ∴ rd(1) = 0
dr = rd(1) (∵d is a R-linear map)
∴ dr = 0. 0004132人目の素数さん2021/04/04(日) 16:14:26.42ID:F2zKij9S>>2 > R-derivation of A into M is a linear map → R-derivation of A into M is a R-linear map 0005132人目の素数さん2021/04/04(日) 16:24:24.93ID:F2zKij9S Def: R: commutative ring A: R-algebra
The module of relative differential forms of A over R is a A-module Ω_A/R endowed with an R-derivation
d: A → Ω_A/R
satisfying the following universal properties:
∀M: A-module, ∀d': R-derivation of A into M, ∃φ: Ω_A/B → M: A-homomorphism s.t.
the module of relative differential forms (Ω_A/R, d) exists and is unique up to isomorphisms. 0009132人目の素数さん2021/04/04(日) 22:24:56.03ID:ZbEAKX7J>>8 Proof: The uniqueness follows from the universal property.
Let us show the existence. Let F be a free module generated by elements of the form da (a∈A). Let E be a submodule of F generated by elements of the form:
* dr * d(a + b) - da - db * d(ab) - bda - adb
(r∈R, a, b∈A). Set Ω_A/R = F/E and d: A → Ω_A/R sends a∈A to the image of da∈Ω_A/R. (Ω_A/R, d) satisfies the universal property.