Definition 10.131.1 (00RN)—The Stacks project

Processing math: 100%

Definition 10.131.1. Let $\varphi : R \to S$ be a ring map and let $M$ be an $S$ -module. A derivation, or more precisely an $R$ -derivation into $M$ is a map $D : S \to M$ which is additive, annihilates elements of $\varphi (R)$ , and satisfies the Leibniz rule: $D(ab) = aD(b) + bD(a)$ .

Comments (2)

Comment #1849 by Peter Johnson on February 27, 2016 at 15:11

Maybe too fussy, but since you implicitly use left modules, should say something somewhere about bimodules/right action to explain D(a)b.

A matter of taste, but I would prefer to see something like "R-algebra" rather than $\varphi: R \to S$ , with "R-linear" put in the def. rather than "additive, annihilates elements of $\varphi(R)$ ", giving hint to calculate D(1) = D(1.1). [Even without assuming the fussier detail of unitary action 1m=m, still get 1D(1)=0, then D(1)=0, then D(r.1) = 0.]

Comment #1886 by Johan on April 01, 2016 at 19:26

Thanks, fixed here.

There are also:

14 comment(s) on Section 10.131: Differentials

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (2)

Post a comment