Noncommutative algebra in Algebraic Geometry.

Question

I'm a undergraduate student with an interest on Algebraic Geometry, and I already took 3 courses on Algebra, from Basic Group/Ring Theory to Galois Theory(never studied modules though, only saw one thing or another about it), but I feel the need for a revision(especially Galois Theory, I forgot many things about it), so I'm planning to study algebra again.

But I'd like to double focus on what would be useful for AG, be it Varieties or Schemes(Planning to reach them someday). I already know that Commutative Algebra is vital, but what of noncommutative algebra would be good to know?

Thanks in advance.

Answer 1

You don't need any non-commutative algebra at all for basic algebraic geometry.
For example the two basic references for Algebraic Geometry in the last half-century, Hartshorne's Algebraic Geometry and the thousands of pages of Grothendieck-Dieudonné's monumental EGA, contain (as far as I'm aware) no noncommutative algebra at all.
Even at an expert level, noncommutative algebra is only needed in certain very specialized topics like Brauer groups and $K$-theory of schemes.

The bad (?) news is that you will need a frightening lot of commutative algebra.
Even proving intuitively obvious statements like the fact that the locus cut out by a single equation on a variety reduces its dimension by one requires tough commutative algebra.

Apart from commutative algebra the most useful preparation would be to review the advanced calculus of several variables, especially the notion of differential of a map and the implicit function theorem, and its natural extension: the elementary theory of differential manifolds.