# Questions tagged [morphism]

In category theory, a morphism is a structure-preserving map, such as continuous mappings on topological spaces, measurable functions, and linear maps.

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### Equality of morphisms $f,g:K\rightarrow X$ of schemes, where $K$ is a reduced scheme.

Suppose that $f,g:K\rightarrow X$ are morphisms of schemes, where $K$ is a reduced scheme. I want to show that $f=g$ if and only if for all $x\in K$, $f(x)\equiv g(x)$. Here $f(x)\equiv g(x)$ means ...
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### Separating Points and Tangent Vectors (real curves)

In [Hartshorne, Proposition 7.3.] as well as in [Görtz & Wedhorn, Rem. 13.55] and [Vakil Notes, around 19.2] the following is said: If $X$ is a curve over (let's say) $\mathbb{C}$ (algebraically ...
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### Prove $f(X)=(f(X))$. Is necessary for $f$ to be an epimorphism?

Let $R,S$ two rings, and let $f:R\rightarrow S$ be a homomorphism. Prove thar if $f$ is an epimorphism, then for any subset $X\subseteq R$ we have that $f(X)$ is the ideal generated by itself, ...
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### Lifting a Frobenius endomorphism under an étale morphism

Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
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### Can two morphisms with equal mappings be distinct in category theory?

I'm trying to get an understanding of equality in category theory, and found some answers to this question on SE (here and here), but I didn't really feel like I fully grasped the notion of equality ...
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### Something went wrong with that morphism and I don't know what

I have a quick question that is bugging me. Let $D_{2n}$ be the dihedral group of the $n-$gon (write $\rho$ the "elementary" rotation, and $\sigma$ a reflection), and let $\mathbb{Z}_2$ be the ...
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### Coproduct of Group Homomorphisms

I've been trying to work through the following problem: Let $H$, $G$, and $G'$ be groups, and let $f:H\to G$ and $g:H\to G'$ be two homomorphisms. Define the notion of coproduct of these two ...
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### Bijective comorphism of a polynomial map

Let $f:G\rightarrow H$ be a polynomial map, between algebraic groups $G,H$ (ie defined by polynomial equations), such that the comorpism $f^*:K(H)\rightarrow K(G),\ g\mapsto g\circ f$ is bijective. ...
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### Understanding De/Suspension $\Sigma^{-1}(\Sigma{X})\neq X$
Let $\Sigma$ denotes a suspension $$\Sigma X =S^1 \wedge X \equiv (S^1\times X)/(X\vee S^1)$$ where $\wedge$ and $\vee$ are the smash product and the wedge sum (one point union) of pointed ...