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1answer
82 views

The magic of the morphisms

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, lets call it a spotted set. Given two spotted sets, then a morphism $\alpha ...
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22 views

If $\{v_1, v_2, …, v_n\}$ is a basis and $f$ is an injective morphism, show that $\{f(v_1), f(v_2), …, f(v_n)\}$ is linearly independent.

Let $V_1$, $V_2$ be two $K$-vector spaces with $dim_K V_1 = dim_K V_2 = n$, $f:V_1 \rightarrow V_2$ a morphism and $B = \{v_1, v_2, ..., v_n\}$ a basis for $V_1$. Now consider the set $T = \{f(v_1), ...
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0answers
25 views

Extension of projection from a point to a Blow Up

I feel like there's something obvious I'm missing here, and I'm not looking for a whole answer, but rather just a pointer in the right direction. Suppose you have the projection from a point ...
2
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1answer
36 views

If functions compose both ways to make automorphisms, are they isomorphisms?

Let's say that we have morphisms $f:A \to B$ and $g : B \to A$ such that $f \circ g$ and $g \circ f$ are both automorphisms (an automorphism is a morphism that is both iso and endo). Are $f$ and $g$ ...
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1answer
31 views

Morphism between affine scheme corresponding with ring homomorphism

Let $X = \text{Spec} R$ and $K = \text{Spec} S$ be two affine schemes with $f,g: K \rightarrow X$. I know that a morphism between affine schemes correspond with a ring homomrphism, so denote $\varphi: ...
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96 views

Are Monomorphisms injective?

In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective. Things are different for schemes. In fact I know that a scheme injective ...