Intereting Posts

fundamental period
Consider the “infinite broom”
How to do $\frac{ \partial { \mathrm{tr}(XX^TXX^T)}}{\partial X}$
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
How to prove that algebraic numbers form a field?
Question about Holomorphic functions
If $f^N$ is contraction function, show that $f$ has precisely one fixed point.
Showing $$ is nilpotent.
Correspondence between eigenvalues and eigenvectors in ellipsoids
$C^{1}$ function such that $f(0) = 0$, $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$ and $\int_{0}^{1}f(x)\, dx = 1$
Dominated convergence theorem for absolutely continuous function
Prove by induction: $2^n = C(n,0) + C(n,1) + \cdots + C(n,n)$
Determining the generators of $I(X)$
Extending the ordered sequence of 'three-number means' beyond AM, GM and HM
Why do we limit the definition of a function?

I am trying to understand the intuition of thinking about number theoretic ideas in terms of geometric ones. For example, ramification is something that happens when a “covering” space of a Riemann surfaces collapses down to a point. I can sort of see the analogy with ramification of primes like this:

Let $L/K$ be an extension of number fields, and $P$ a prime of $\mathcal{O}_K$. If $P$ splits in $\mathcal{O}_L$, then I imagine each of the $n = [L:K]$ primes as lying in $n$ distinct “layers” of some “covering” of the primes of $K$. Then when we have ramification, say $P = Q^e \prod Q_i$, the $Q_i$ remain as expected in the covering, but we have a degeneration at $Q$ giving us $e$ layers collapsing into just one above $P$, like in the case of Riemann surfaces.

So my question is: using this type of geometric analogy, where is the inertia hiding? I don’t yet have enough background to understand etale morphisms yet, but I’ve heard that these give a nice analogue of covering maps in this kind of setting. Is there some concrete way I can try to build intuition about first? Maybe is there a way to interpret the residue field extension $k_Q/k_P$ using a corresponding unramified extension of the local fields $L_Q/K_P$ and give a uniform way of looking at this “covering” interpretation so that only ramification really feels like the “bad” points where we have less than expected cardinality of the fibers?

- Principal ideals having embedded components
- Basic question regarding degrees of algebraic sets
- Closed points in projective space correspond to which homogenous prime ideals in $k$
- Why degree of a reducible projective variety is the sum of the degree of its irreducible components
- Proving algebraic sets
- Smoothness of $A \subseteq C$ implies smoothness of $B \subseteq C$? where $A\subseteq B \subseteq C$

- Hypersurfaces meet everything of dimension at least 1 in projective space
- Direct way to show: $\operatorname{Spec}(A)$ is $T_1$ $\Rightarrow$ $\operatorname{Spec}(A)$ is Hausdorff
- Question on calculating hypercohomology
- tensor product of sheaves commutes with inverse image
- Defining/constructing an ellipse
- Proving whether ideals are prime in $\mathbb{Z}$
- Extension of regular function
- Is $\mathbb Z$ Euclidean under some other norm?
- Finite etale maps to the line minus the origin
- A chain ring with Krull dimension greater than one

For a geometric picture, I would say that inertia means there’s fewer points $Q_i$ above a given point $P$ in the covering because some of the points got “fat”, in the sense that

$$|\mathcal{O}_L/Q_i|>|\mathcal{O}_K/P|$$

(Indeed, the inertial degree is the degree of the extension $[\mathcal{O}_L/Q_i:\mathcal{O}_K/P]$.)

I like to think of an analogy with conservation of momentum. The extension $L/K$ has “energy” $n$, where $n=[L:K]$. At any given point (prime) $P$ of $K$, the energy can go into

- pushing $n$ different primes of the same size at the same speed (totally split)
- pushing a single prime of size $n$ at the same speed (totally inert)
- pushing a single prime of size $1$ at $n$ times the speed (totally ramified)

or some mixture of these effects. The fundamental relation $n=\sum_{i=1}^r e_if_i$ just expresses conservation of momentum.

Here’s an image from Eisenbud and Harris’ *The Geometry of Schemes* of $\mathrm{Spec}(\mathbb{Z}[\sqrt{3}])$. Note that $(5)$ and $(7)$ are inert in $\mathbb{Z}[\sqrt{3}]$.

Here’s a long excerpt explaining what’s going on in the image.

- Geometric meaning of the determinant of a matrix
- Understanding Reed-Solomon as it applies to Shamir secret sharing
- Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge?
- Why is the set of commutators not a subgroup?
- Intuition or figure for Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$ (Abbott p 11 q1.2.5)
- Chinese Remainder Theorem Interpretation
- Proofs that every natural number is a sum of four squares.
- Alternating and special orthogonal groups which are simple
- Why is a straight line the shortest distance between two points?
- Prove that there exist linear functionals $L_1, L_2$ on $X$
- Intuition behind Cantor-Bernstein-Schröder
- If $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$, then find $f(2)$
- How do I calculate $\lim_{x\rightarrow 0} x\ln x$
- A normal matrix with real eigenvalues is Hermitian
- Existence of d-regular graphs