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This post is related to this one: http://mathimatikoi.org/forum/viewtopic.php?f=4&t=88&hilit=biholomorphic In the above post we saw that the following sets are not conformally equivalent: The unit disc $ \mathbb{D} $ and the complex plane $ \mathbb{C} $. The punctured unit disc $ \mathbb{D}^...

1) If X is compact then is not subspace 2)The functions in Hahn-Banach theorem are linear 1) As $X$ is a subset of $ \mathbb{R}^{n} $, it is naturally a subspace; We additionally require it to be compact. 2) Hahn Banach Theorem/ Formulation What can you say about the solution of the exercise? How i...

Let $X$ be a complete variety and let $ \mathcal{L} $ be a line bundle on $X$. The following are equivalent:

1. $ \mathcal{L} $ is trivial.
2. $ \mathrm{H}^{0}(X,\mathcal{L}) \neq 0 \neq \mathrm{H}^{0}(X,\mathcal{L}^{*}) $.

Definition : Let $Y$ be a topological space. A function $ \varphi \ \colon Y \longrightarrow \mathbb{Z} $ is called upper semicontinuous if for every $y \in Y$ there exists an open neighborhood $U$ of $y$ such that $ \varphi(y) \geq \varphi(y^{\prime}) $ for all $ y^{\prime} \in U $. Show that a fu...

Can the tensor product of a free and a non-free module be a free module?

Show that a local ring cannot be a direct sum of two other rings.

Consider the morphisms of schemes $ f \ \colon X \longrightarrow Y $ and $ g \ \colon Y \longrightarrow Z $. If $ g \circ f $ is étale and $g$ is unramified, then show that $f$ is étale.

Let $M$ be a smooth manifold. For a point $p \in M$, set \[ \mathfrak{m}_{p} = \left\{ \ f \in C^{\infty}(M) \ \big| \ f(p) = 0 \ \right\} \]Show that $\mathfrak{m}_{p}$ is a maximal ideal of $C^{\infty}(M)$. Any derivation on $ C^{\infty}(M) $ is determined by its values of $ \mathfrak{m}_{p} $. Th...

Show that the determinant line bundle $ \det \left( T^{*} \mathbb{S}^{n} \right) $ is trivial.

Show that a closed, injective, continuous map is a (topological) embedding. Give an example to show that an injective immersion can fail to be an embedding. Show that an injective immersion $ F \ \colon M \longrightarrow N $ (between smooth manifolds) is a (smooth) embedding if either $M$ is compac...