Bass’ conjecture implies the Parshin’s conjecture

In the appendix of this paper. It is proved that Bass’ conjecture for $ K_n$ implies the rational Beilinson-Soulé conjecture for $ K_n$ . Then at the end the author claims that the same method can be applied to prove that the Bass’ conjecture implies the Parshin’s conjecture but I can’t figure it out. The general idea behind the Beilinson-Soulé conjecture is that he proves the Theorem A.1 for fields then uses the Quillen spectral sequence to prove it for the general regular scheme $ X$ which I can’t see how this idea can be used for the Parshin conjecture. I wonder whether this proof is written anywhere with more details. I’d appreciate if anyone of the experts in the field can explain the sketch of the proof for the Parshin’s conjecture or answer this question:

Is the finite generation of $ K_0$ required to imply the Parshin’s conjecture or the finite generation of $ K_n$ for $ n\geq 1$ is enough?

Proper and flat morphism implies finitely presented?

I´ve been reading the Deligne-Mumford construction of the moduli of curves with a given genus and I have some questions about the article

1) When the authors talk about a scheme what are they referring to? In the EGA, a scheme is what we call separated scheme, so I don´t know if they are working on the category of separated schemes or just schemes (in our terminology).

2) My second question is about Definition 1.1. They say that a stable curve is a proper flat morphism of schemes $ f:X\rightarrow S$ whose geometric fibers are reduced, connected, 1-dimensional schemes such that:

  • $ X_{s}$ has only ordinary double points,

  • If $ E$ is a non-singular rational component of $ X_{s}$ then $ E$ meets the other components of $ X_{s}$ in more than 2 points;

  • $ \rm{dim}\rm{H}^{1}(\mathcal{O}_{X_{s}})=g$

In general, a relative curve is defined as a flat finitely presented morphism of schemes $ X\rightarrow S$ of relative dimension 1. My question is if proper+flat in this particular case implies finitely presented. It is the same true if $ f:X\rightarrow S$ is a proper and flat morphism whose geometric fibers are complete integral algebraic curves of arithmetic genus $ g$ ?

Thank you for your time.

Connected boundary implies $\pi_1(M,\partial M)=0$.

I have two questions: Let $ M$ be a compact connected manifold with boundary.

1, If the boundary $ \partial\tilde{M} $ of universal covering $ \tilde{M}$ is connected, is $ \partial M$ connected? How about converse direction, if not, any counterexamples?

2, Does connectedness of boundary $ \partial M$ imply $ \pi_1(M,\partial M)=0$ , if not, any counterexamples?

If $ M$ is not necessarily compact, will it be different?

Thanks for your help.

Proving non-zero derivative implies growing/decreasing function using Bolzano-Weierstrass Theorem

Using the Bolzano-Weierstrass theorem (for every bounded sequence, there exists a convergent subsequence), how would one go about proving that $ f'(x)>0$ (or $ f'(x)<0$ ) on a closed interval $ [a,b]$ implies that $ f(x)$ is increasing (decreasing) on $ [a,b]$ . I’ve been instructed to proceed with a proof by contradiction, i.e. $ f'(x)>0 \implies f(x_1)\geq f(x_2)$ for every $ x_1 \leq x_2$ , yet I’m stuck on where to begin.

A constraint that implies convexity

Let $ f:\mathbb{R} \to \mathbb{R} $ be a function such that $ \forall x<y, \exists z\in(x, y) $ with $ (y-x) f(z) \le (y-z) f(x) +(z-x) f(y) $ .
a) Give an example of a non-convex function $ f$ which has this property.
b) Prove that a continuous function $ f$ which has this property is convex.
For a) it is obvious that we must search for a discontinuous $ f$ with this property, but I can’t find one. For b), I tried to assume that $ f$ is not convex, which means that $ \exists u<v$ and $ a \in [0,1]$ such that $ f(au+(1-a)v)>af(u)+(1-a)f(v)$ ,but here I am stuck.