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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Show that there is a $k \in \mathbb{N}$, for which applies: $f^{k}(V)=f^{k+l}(V)$ for all $l \in \mathbb{N}.$

I have a problem with the following task: Let $V$ be finite-dimensional $\mathbb{K}$ -Vector space and $f: V \rightarrow V$ an endomorphism. We define $f^{0}(V)=V$ and $f^{i+1}(V)=f\left(f^{i}(V)\...
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Lifting of meromorphic function along a finite morphism

I am currently reading the book "Geometry of algebraic curves II", by Arbarello, Cornalba and Griffiths, and I am having some difficulties understanding a passage p.105. The setting is the following:...
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Prove that $\mathbb{Q}[X]/I\cong Q\times Q$

Let $f(X)=(X^2-2)(X^4-X)$ and $g(X)=(X^2-1)X\in \mathbb{Q}[X]$. Let $I=(f,g)$ the ideal generated by $f$ and $g$. Prove that $\mathbb{Q}[X]/I\cong Q\times Q$ Using the reasoning of this answer I ...
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Morphisms and a polynom on which this morphism vanishes

I have a few problems with this exercise. Let $m > n$ be positive integers. Let $K$ be a field, and let $u : K^n \to K^m$ be a morphism defined with polynomials $f_1, \dots, f_m \in K[X_1, \...
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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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Equivalence between definitions of morphism of $k$-variety : Did I have understood well?

There is multiple definition of morphisms of projective $k$-varieties, and I want to make sure I'm okay with the equivalence between them. Actually, the definition I know for a morphism $\phi : V \...
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Conceitual difference between $\delta_{ij}$ and $\delta^i_j$

Question How is related $\delta_{ij}$ with $\delta^i_j$ ? Here $\delta_{ij}= \begin{cases} 1 \qquad\text{if} \qquad i=j \\0 \qquad \text{if} \qquad i\neq j\end{cases}$ Context I'm watching this ...
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What's a morphism between algebraic structures of different types?

Background I had originally posted this question as: Among morphisms there are homomorphisms that are structure-preserving maps between algebraic structures of the same type. What's a conventional ...
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Clarify the usage of symbols for “-morphism”.

I'm confused about the notation for "-morphism"! Even one author uses them differently in different books. For example, Some books use "$\simeq$" to represent both "path homotopic" and "homeomorphic" ...
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is it a stronger property of equalizers?

I understand that if $h$ is an eqalizer of $f,f:A\rightrightarrows B$ then $h$ is an isomorphism. Is it stronger to say that if an isomorphism $h$ is an equalizer of $f,f:A\rightrightarrows B$ then $...
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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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Composition of flat morphisms is flat

I do not understand in the snippet below why the composition of flat morphisms is flat. The full text (though in Czech) is given here. I do not understand these step given in bold, beginning from $q\...
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Determinant of endomorphism

I got the following exercise: Let $f$ be the endomorphism in $\mathbb{R}_3[T] := \{p \in \mathbb{R}[T]; \; deg(p) \leq 3\}$ given by $$\;f: \mathbb{R}_3[T] \rightarrow \mathbb{R}_3[T]$$ $$f(p) = T^4p'...
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When are two morphisms equal? [duplicate]

Trying to understand the very basics of category theory... While trying to solve a simple exercise on the subject I suddenly realized that I don't even know what morphism equality is! (The exercise ...
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The monoid of fractions associated with the submonoid of cancellable elements of a commutative monoid E

Let $E$ be a commutative monoid, $\Sigma$ the submonoid of cancellable elements of $E$, $E_{\Sigma}$ the monoid of fractions of $E$ associated with $\Sigma$ and $\varepsilon$ the canonical ...
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Do morphisms need to make full use of their domain (category theory)?

I'm new to category theory, and have a (I assume very basic) question that's probably best explained with an example. Say we have a category $C$ with pairs of sets as objects, e.g. $$\Sigma = (A,B) = ...
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A question about the exponential function endomorphism

I have the following problem : Is the function $f(x)=\mathrm{e}^{k \times x },$ with $k \in \mathbb{R}$ the only endomorphism/ function $f$ such that $$f : \begin{cases} (\mathbb{R}_+,+) \...
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Definition of locally of finite type morphism

Hartshorne defines the locally finite type morphism as follows: Definition: A morphism $f : X \to Y$ of schemes is locally of finite type if there exists a covering of $Y$ by affine open subsets $V_i ...
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Constant morphism from $\mathbb{P}^m_k$ to $\mathbb{P}^n_k$ as schemes [duplicate]

Let $k$ be a field and $m>n \in \mathbb{N}$. Then any morphism of schemes $\mathbb{P}^m_k \rightarrow \mathbb{P}^n_k$ is constant. What I know is that every morphism from a scheme $X$ to $\mathbb{...
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Endomorphism of a rings [duplicate]

I was trying to prove that a surjective Endomorphism $f:A \to A$ of a noetherian ring is also injective. I would like to know why this argument is not correct? $A/\rm{Ker}f \cong \rm{Im}f=A \...
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Spreading out a morphism of the generic fibers

Let $X$ and $Y$ be finite type schemes over $\mathrm{Spec} \mathbb{Z}$ and let $f_\xi : X_\xi \rightarrow Y_\xi$ be a morphism between the generic fibers. Then $f_\xi$ spreads out to a morphism $g_U : ...
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$X=Z(xy-1)\subset \mathbb{A}_k^2$ is not isomorphic to $\mathbb{A}^1_k$ [duplicate]

I need to prove that $X=Z(xy-1)\subset \mathbb{A}_k^2$ is not isomorphic to $\mathbb{A}^1_k$. I solved an exercise where I proved that, for instance, some $X\subset \mathbb{A}_k^3$ is isomorphic to $\...
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A simple question on endomorphisms of a noetherian module.

Let $M$ be a noetherian $R$-module and let $f$ be a non-zero endomorphism of $M$ such that $\frac{M}{\ker(f)}\cong M$. I want to show that $f$ is a monomorphism. In order to do it so, I want to use ...
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Monomorphisms and epimorphisms in full subcategories satisfying a certain property

Let $C$ be a category and $D$ be a full subcategory such that every object of $C$ has a monomorphism to some object of $D$ (or dually, an epimorphism from some object of $D$). Is it then true that ...
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Equivalent definitions of module homomorphism

Let $R$ be a commutative ring with 1. Let $f:R\times R\to R$ be a map. The claim: $f$ is an homomorphism of $R$-modules iff $\exists\,\alpha,\beta\in R: f((x,y))=\alpha x+\beta y\,\forall\,x,y\in R$. ...
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There are no non-constant morphisms from $\mathbb{A}^1$ to a cubic curve

Consider the cubic curve $X$ defined, over an arbitrary field $K$, by the equation $y^2 =x(x−1)(x−\lambda)$ in $\mathbb{A}^2$, where $\lambda \neq 0,1$. Show that there are no non-constant morphisms $...
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What do we call a “nearly” order-isomorphism?

Suppose $f:X\to Y$ preserves $x_1\geq x_2\implies y_1\geq y_2$ but does not necessarily preserve $x_1> x_2\implies y_1> y_2$. In other words, an $f$ satisfying this relation might, upon ...
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Blow up at point is finite?

Let $X$ be an affine algebraic curve with $0 \in X$ and $\tilde{X}$ the strict transform of $X$ w.r.t the blowup of $X$ at $0$. How to prove that $\pi \colon \tilde{X} \to X$ is finite? Is it even ...
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Example of a morphism that is not a function [duplicate]

Can someone please give me a simple example(I'm not a math professional) of a categorical morphism that is not a function?
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What are the pushouts and coequalizers in Hausdorff spaces?

Does the category of Hausdorff spaces have pushouts and coequalizers? I know that if we have pushouts and an initial object we have coequalizers, but do we have pushouts in this category?
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Linear subspaces of projective space.

I'm following a basic course in Algebraic Geometry where the lectures are based on the first chapter of Algebraic Geometry by Robin Hartshorne. Our lecturer gave an additional advanced exercise after ...
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Do subobjects in concrete categories correspond to subsets?

A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with ...
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Is there a notion of a transversal of subobjects?

If $a$ is an object in a category $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with codomain $a$. The subobjects of all the objects in $C$ partitions the class of ...
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${(F^*)}^{-1}(m)$ is maximal ideal where $m$ is maximal

Our main objective is to interpret $F:V \to W$ a morphism as a map $F:maxSpec \Bbb C[V] \to maxSpec \Bbb C[W]$, $V,W$ are algebraic varieties. Now from $F:V \to W$ using Hilbert Nullstelensatz we ...
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image of the neutral in group homomorphism

It it always true that the image of the neutral element in a group homomorphism $f$ is the neutral element of the codomain group regardless whether $f$ is injective/surjective? The answer is most ...
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Is there a general formalized method to define the homomorphism (structure preserving map) of a structure?

This is a follow-up to this question. There I asked for an intuition of what "structure preserving" means. My question here is, is there a universally applicable method (given the objects that define ...
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What is the colon ':' called in category theory morphism definitions?

Category theoretic morphism definition syntax, e.g.: $\varphi : G \to H$ uses the colon character rather than, for instance, the member-of sign e.g.: $\varphi ∈ G \to H$ What is the descriptive ...
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$\lambda$-pure morphisms in $\lambda$-accessible categories are monos, unclear proof

This is Proposition 2.29 from the book Locally Presentable and Accessible Categories by Jiří Adámek and Jiří Rosický. Above is a proof that $\lambda$-pure morphisms in $\lambda$-accessible categories ...
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In the proof of the Yoneda Lemma in “Categories & Sheaves” by Kashiwara & Schapira.

Our goal is to show that $\text{Hom}_{C^{\wedge}}(h_C(X), A) \simeq A(X)$. We first want to show a map from left to right. The book says: $$ \text{Hom}_{C^{\wedge}}(h_C(X), A) \to \text{Hom}_{\text{...
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Given a $G$-Set isomorphism from $G/H$ to $G/S$, can I make a morphism from $G/H$ to $G/K$ if $S<K$?

Let $G$ be a group and let $H,K$ be subgroups. I have an $S<K$ such that $S=g^{-1}Hg$. I'm asked to show that there is a morphism from $G/H$ to $G/K$. I have a theorem that says that since $H$ and $...
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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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Every monic is a kernel

This is part of Weibel's Exercise 1.2.2, where I have to show that in the category R-Mod, every monic is a kernel. A monic morphism is defined to be a map $i \colon A \to B$ such that if $g \colon A ...
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Kernel of a surjective module morphism $f : R^2 \rightarrow R$ is a free submodule

I have to prove this : Let $R$ be a commutative ring and : $f : R^2 \rightarrow R$ a morphism of $R$-module. I want to show : $\ker f$ is a free module. My first try is to say : $R^2/\ker(f) \...
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Understanding De/Suspension $\Sigma^{-1}(\Sigma{X})\neq X$

The question is about understanding suspension and desuspension, see also a previous question. Question: How do we define desuspension exactly? (Please see the comments below, people complain about ...
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Understand the suspension (topology) and some Lie group examples

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 ...