I'm trying to understand whether there's a fully faithful functor $LRS \supset FormalSch \to IndSch$ and in what sense. Here's my progress so far: Let $\mathsf{A}$ be the category of adic rings. The objects are topological rings whose topology is generated by a descending filtration of ideals wh...

Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$. Assume that for every $i$ and for every $j$ the polynomial $f$ belongs to the ideal $\langle P_i, Q_j \rangle$. Is it true that...

(This is inspired by Algebraic geometry examples.) I want to collect here (counter)examples in arithmetic geometry. Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. It is a nontrivial element of the Tate–Shafarevich group of the elliptic curve $3\cdot4\cdot5\cdo...

A Banach space $X$ is said to have w-FPP (weak fixed point property) if for every non-empty, weakly compact and convex subset $K\subseteq X$; every non-expansing mapping $T:K\longmapsto K$ i.e. $$\|Tx-Ty\|\leq \|x-y\|\quad\quad \forall\, x,y \in K$$ has a fixed point. Recall that if $X$ has an...

Suppose that $b$ is a braid. Then $b$ can be uniquely written as $D_{RL}(b)^{-1}N_{RL}(b)$ where $D_{RL}(b),N_{RL}(b)$ are the unique positive braids such that $b=D_{RL}(b)^{-1}N_{RL}(b)$ and where $D_{RL}(b)^{-1}\wedge_{L}N_{RL}(b)=e$ where $r\wedge_{L}s$ denotes the left gcd of the positive bra...

Suppose from distance $d$, while driving at speed $v_0$, I notice that there's a red traffic light in front of me. Suppose that there are no other vehicles, my vehicle has perfect brakes, my maximum acceleration is $a$ and the red light will turn green according to some $\mu$ distribution. My goa...

Given a finite dimensional algebra $A$ with Jacobson radical $J$ (defined as the intersection of all maximal right ideals of the algebra) and let $J^i$ be the powers of this ideal. Let $injdim(M)$ denote the injective dimension of a module $M$. Questions: Do we have $injdim(J) \geq in...

I am looking for a reference on continuity of (proximal) subdifferentials. For a continuous function $F: \mathbb R^n \rightarrow \mathbb R$, a vector $v$ is called proximal subgradient at $x$ if there exist $r>0, \delta >0$ s. t. $$\forall y \; \text{s. t.} \; \|y-x\| \le r \quad F(y) \le F(x) +...