I have precedently posted the same question on Math.Stackexchange (https://math.stackexchange.com/questions/4277856/quasi-compact-surjective-morphism-of-smooth-k-schemes-is-flat), but to no avail; I hope this is not too low-level for this site. In the article "The Greenberg functor revisited'' (h...

Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of all morphisms $X\rightarrow Y$ in $\mathcal{I}$. Then $\mathcal{I}(X,-)$ is a subfunctor of $\text{...

Given a filtration $\mathcal F_t$ on a probability space, we say a stochastic process $X$ is almost $\mathcal F_t$-adapted if there exists some $\mathcal F_t$-adapted process $Y$ such that $\underset{t \to \infty}{\lim} \mathbb \int_{t}^{\infty} \mathbb E[|X_s - Y_s|] ds = 0$. Let $W_t$ be a stan...

Let $\Omega$ be domain in $\mathbb{C}^n$. Suppose we have taken two distinct points from $\Omega$. Does there exist a domain $U$ in $\mathbb{C}$ such that there is a holomorphic function from $U$ to $\Omega$ whose range contains these two points? I tried to prove the identity theorem in several c...

While reading a paper, I encountered the following statement: Let $K$ be a convex compact subset of a locally convex topological vector space. If $\mu \in P(K)$ is a Radon probability measure on $K$, then there is a unique point $x_\mu \in K$ such that $$\int_K fd \mu = f(x_\mu)$$ for every cont...