Commutative Algebra

David Speyer's notes on Jordan-Holder for finite groups: PDF

Idea is, look at a normal lattice $G$ containing large normal subgroups $N,N'$ containing $N\cap N'$ (intersection of normals is normal), applying induction to these subgroups. The point is the map $(G\cap N)/(N'\cap N) \to G/N'$ is always injective, so by simplicity of $G/N'$, it's either trivial or an isomorphism. Put another way, intersecting a simple filtration with $N$ is well-behaved: the successive quotients will either be preserved or eliminated.

In the abelian case or with modules (either over a commutative or non-commutative ring, former being useful in commutative algebra, algebraic number theory, and algebraic geometry and latter being useful in general representation theory), we can also go the other way (up instead of down): let $X = N+N'$ for small submodules $N,N'$, and apply induction to the quotients like $M/N$, $M/N'$, and $M/X$. Both intersection with $X$ and addition by $X$ are well-behaved: $$0 \to (M\cap X) / (N\cap X) \to M/N \to (M+X)/(N+X) \to 0$$ is short exact. The previous (group) approach with large $N,N'$ uses that $(M\cap X) / (N\cap X)$ is well-behaved; the alternative approach here uses that $(M+X)/(N+X)$ is well-behaved.

Flatness to-do

• Equivalent definitions of flatness, see e.g. Qing Liu... when is locally free a good way to think about it?
• Use both algebra (e.g. number theory) and geometry intuition
• MO: Why are flat morphisms "flat?"

My version/interpretation of Critch's answer: In topology, if a map $X\to Y$ is continuous, limiting behavior in $X$ corresponds to limiting behavior in $Y$. This can be useful if, say, $Y$ is simpler than $X$, and/or if we want to say something about $X$ based on properties of $Y$, say by pulling back a closed or open set of interest from $Y$ to $X$; or say (on the algebraic/analytic side) by pulling back functions of interest from $Y$ to $X$, like idempotents or curves or something.

Now, say $Y = \operatorname{Spec}(A)$ is flat over $X = \operatorname{Spec}(B)$; then e.g. the injection $A\to \operatorname{Frac}(A)$ (generic analysis/functions on base $Y$) induces an injection $B\to B\otimes \operatorname{Frac}(A)$ (allowing generic fiber analysis on $X$; see Qing Liu Chapter 4, Lemma 3.7).

Or say $A,B$ are local and we want to do differential calculus (tangent/cotangent spaces, Taylor series, ramification, etale, smoothness, etc.). Then the short exact sequence $$0\to \mathfrak{m}_y^r \to A \to A/\mathfrak{m}_y^r\to 0$$ induces a short exact sequence $$0\to \mathfrak{m}_y^r\otimes_A B \to B \to B/ \mathfrak{m}_y^rB\to 0.$$ In particular, as Critch mentions, the usually-just-a-surjective-map $$\mathfrak{m}_y^r\otimes_A B \to \mathfrak{m}_y^rB$$ is an isomorphism, so $\mathfrak{m}_y^rB$ should have "no surprise relations" (my currently attempted/preferred wording: it is an ideal of $B$ "cut out and/or described in terms of $A$, at least as much as one can hope for").

Remark: For all of the above, I'm not sure how much intuition (geometric or otherwise) is missing if one pretends $B$ is free over $A$.

Spectrum of a ring: basics

Essentially, go back to big picture: why the Zariski topology? Why is this useful? Why study this and not something else? Connect this with classical examples, e.g. Hilbert's nullstellensatz.

For those with an arithmetic or algebraic bent, it could be more inspiring to bring sheaves into the picture earlier, not sure.