What is the significance of Hermitian forms on local rings?

Question

I am a first year math student at UNMSM, in Lima, Peru. My father is an 80-year-old man, a retired university professor, a Ph.D. in pure mathematics, and a passionate algebraist. He kept his mental faculties perfectly until a year ago when a stroke took away some parts of his memory and personality. His love for mathematics did not vanish instantly, but his interest in talking about it waned as he found it increasingly difficult to follow the ideas presented to him. He would usually refrain from making comments when I would share something that I find interesting about some math. He tells me "sure yes or sure no, but who cares" (pretty basic things, calculus or basic number theory, mostly).

In the past he spoke passionately about ring theory, homological algebra, among other topics. He finds impressive how structures and theories like these exist and the beautiful theorems that are proven in theorems he proved in them.

In particular, was on "Hermitian forms on local rings". Could someone give me the significance and interesting results of this field so I can try having conversations with my dad again?

Answer 1

A (commutative) local ring is a ring with exactly one maximal ideal; usually this is denoted as a pair $(R,\mathfrak{m})$.

An involution on a commutative ring $R$ is (non-trivial) ring homomorphism $J\colon R\to R$ satisfying $J\circ J=\operatorname{Id}_R$.

Example: The complex numbers $\mathbb{C}$ with complex conjugation.

More generally, a Galois extension $E/F$ with Galois group of order two always has such an involution. In particular any quadratic number field has.

Now a finitely generated module $M$ over a ring with involution which also admits an involution $J'\colon M\to M$ that satisfies $$ J'(rm)=J(r)J'(m) $$ for all $r\in R$ and $m\in M$ gives a Hermitian $(R,J)$-module.

Example: For $\mathbb{C}$ with complex conjugation, the vector space $\mathbb{C}^n$ has the Hermitian structure given by $J'(z_1,\ldots,z_n)=(\bar{z_1},\ldots,\bar{z_n})$.

Such Hermitian modules over Hermitian local rings occur in Number theory, for example when considering localisations of rings of integers in quadratic Galois extensions.