MathOverflow

Some hopefully not too nonsensical questions on Daniel Kriz' work (from just revisiting it after a very long delay, my excuses in advance):
Could one view Kriz’s period sheaves as prismatic crystals and connect his work to Fargues-Scholze geometrization via local Shimura varieties?...

Let $\{S_i \}_{i=2}^\infty$ be a collection of rings. For each $i\geq 2$ let $e_i \in \mathbb{M}_i(S_i)$ be the idempotent whose (1,1)- and (2,2)-entries are $1$ and all other entries are $0$. Then $e:=\{e_i \}_{i=2}^\infty$ is an idempotent of $R:=...

Let $A, B, A', B' \in B(H)$ be self-adjoint operators on a Hilbert space. Suppose $\|aA + bB\| = \|aA' + bB'\|$ for all $a,b \in \mathbb{R}$. Does this imply that $\|A + iB\| = \|A' + iB'\|$?
I think a positive solution would answer this question about operator systems....

I'm sure this is a very silly question, but I'm not an expert so clarification would be greatly appreciated. Consider the path fibration $\Omega X \to PX \to X$. In his paper Twisted Tensor Products, I, Brown constructs a twisting cochain$$\tau : C_...