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Questions tagged [injective-module]

A module $I$ over a ring $R$ is injective if $\hom_{R}({-},I)$ is exact. The notion of injective modules is dual to the notion of a projective module. In homological algebra injective modules are used for computing right derived functors.

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Is there a Baer's criterion for testing injectivity of sheaves of $\mathcal{O}_X$-modules?

In the important paper by Spaltenstein on resolving unbounded complexes, they turn their hand to sheaves. Let us fix a ringed space $(X;\mathcal{O}_X)$. In the proof of Lemma $4.3$ it is implicitly ...
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${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$.

For a field $k$, I am calculating ${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$, where $\epsilon^2 = 0$. However, there seems no complete explanation as far as I checked. ...
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$\operatorname{Hom}_k(R,k)$ is injective indecomposable $R$-module for $R$ local $k$-algebra of finite $k$-dimension

I am working on Exercise 3.1.22. from Bruns and Herzog's Cohen-Macaulay rings. The exercise in question is the following (the references I will use are of course from within the book): It appears ...
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Equivalent conditions of $\textrm{Ext}^2(A,B) = 0$ for all $A$ and $B$

I am studying homological algebra and I am having difficulty proving the equivalences of the following: (i) If $0 \to A \xrightarrow{f} B$ is exact and $B$ is projective, then $A$ is projective (ii) ...
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Prove that for $F$ an additive functor of abelian categories, $R^0F$ is exact iff $R^1F = 0$ iff $R^iF = 0$ for all $i > 0$

I am beginning to study homological algebra. Let $F: \mathcal{A} \to \mathcal{B}$ be an additive functor of abelian categories. Prove that the following are equivalent: The functor $R^0F$ is exact $R^...
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An Inquiry into the Properties of Injective Modules within Module Theory

Let $I$ be a left module over a ring with unity $R$. If for any exact sequence $$0 \to X \to Y \to Z \to 0$$ in the category of $R$-modules, we can get the following exact sequence in the category of ...
Liang Chen's user avatar
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Decomposition of $A$ torsion $R$-module over a PID $R$, where every element is of the order of some power of a prime $p\in R$

Let $A$ be a module over a PID $R$ such that $p^{n}A = 0$ and $p^{n-1}A \neq 0$ for some prime $p \in R$ and positive integer $n$. Let $a$ be an element of $A$ of order $p^{n}$. Then need to show ...
Dwaipayan Sharma's user avatar
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Is the quotient of an infinite product of fields by the direct sum injective?

Let $k$ be any field and consider $R=\prod_{\mathbb{N}}k$ the product of copies of $k$ indexed over the natural numbers. The direct sum $\sum_{\mathbb{N}}k=I$ is an ideal of $R$, hence we can form the ...
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Show that if $G$ is the right adjoint of restriction-of-scalars along any ring homomorphisms $A\to B$, then $G$ is always a monomorphism

Let $A$ and $B$ be rings. Show that the right adjoint of restriction-of-scalars functor along any ring homomorphism $f: A \to B$ preserves injectives and show that the unit of this adjunction is ...
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When $f\colon\mathbb{Z}^{+} \to\mathbb{R}$ satisfys $f(2x)=2f(x)$, then is it injective? [closed]

If $f\colon\mathbb{Z}^{+} \to\mathbb{R}$ satisfys $f(2x) = 2f(x)$. Is it injective? I’m new in analysis and can’t understand how to approach this problem. Should I find an analytic expression for $f(x)...
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Proposition 13, Section 4.3 of Hungerford’s Algebra

Proposition 3.12. Every unitary module $A$ over a ring $R$ with identity may be embedded in an injective $R$-module. Proposition 3.7. A direct product of $R$-modules $\prod_{i\in I} J_i$ is injective ...
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Lemma 8, Section 4.3 of Hungerford’s Algebra

Let $R$ be a ring with identity. A unitary $R$-module $J$ is injective if and only if for every left ideal $L$ of $R$, any $R$-module homomorphism $L\to J$ may be extended to an R-module homomorphism $...
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Left exact functors preserve injective objects

Let $G$ be a group and let $\operatorname{\textbf{Mod}}_G$ be the abelian category of $G$-modules, which has enough injectives. Let $A\in \operatorname{Mod}_G$ and let $(-)^G$ the functor \begin{...
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Does an exact sequence of holonomic D-modules imply an exact sequence in the solution space?

Here is a probably quite basic question on holonomic D-modules, but I am only a physicist so please bear with me. If I have the weyl algebra $D$ in $n$ variables, and an exact sequence of holonomic $D$...
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The injective hull of the ring of all eventually constant sequences of elements of $\mathbb{Z}_2$

Let $R$ be the ring of all eventually constant sequences $(x_n)_{n\in \mathbb{N}}$ of elements of $\mathbb{Z}_2$. It's known that the injective hull $E(R_R)$ of $R_R$ is $S:=\prod_{n\in \mathbb{N}} \...
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Is an injective module an injective object in every full subcategory that contains it?

Let $\mathcal{C}$ be a full subcategory of $\text{Mod}_A$, the category of modules over a commutative unital ring $A$ (I am mostly interested in the graded setting, but I doubt there is much ...
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Computing the injective hull, the quasi-injective hull, the quasi-continuous hull

Let $M_R$ be any module. Recall that $M_R$ is called quasi-injective if every $R$-homomorphism $N\to M$ from a submodule $N$ of $M$ extends to $M$, $M_R$ is called quasi-continuous if it's both CS (i....
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What is the injective hull of the ring $R=\mathbb{R}[x]/(x^2)$?

Let $R$ be a ring with unity and $M$ a right $R$-module. Recall that the injective hull $E(M_R)$ of $M_R$ is the maximal essential minimal injective extension of $M_R$. Let $R=\mathbb{R}[x]/(x^2)$. ...
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Injectivity of special orthogonal group

Let $GL_{n}(\mathbb{R})=\{A_{n\times n}\mid A \text{ invertible matrice}\}$, $SL_{n}(\mathbb{R})=\{A\in GL_{n}(\mathbb{R})\mid det(A)=1\}$ be a special linear group and $SO(n)=\{A\in O(n)\mid det(A)=1\...
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How Can We Prove Flatness from an Induced Exact Sequence?

I want to prove that for every short exact sequence $$ O \to A \xrightarrow{f} B \xrightarrow{g} C \to O $$ of $R$-module homomorphisms, if the induced sequence $$ O \to M\otimes_R A \xrightarrow{\...
Mr Prof's user avatar
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If a Short Exact Sequence is Split-Exact, Does that Mean it is Left-Split?

If a short exact sequence is split-exact, can we conclude it is left-split? Motivation: I am asking this question because I want to prove that if every $R$-module is projective, then every $R$-module ...
Mr Prof's user avatar
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How Should I Prove this Function is Well-defined?

I am trying to prove that the dual sum of P$_1$ and P$_2$ is projective iff P$_1$ and P$_2$ are projective. I am done with every aspect of the proof. I have this diagram which commutes: Now, let h: P$...
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Is this a valid proof of Well-definedness?

We are given $L, M$ and $N$ as unitary $R$-modules, and $f:M\rightarrow N$ as an isomorphism. We need to prove that $f^*:Hom_R(N,L)\rightarrow Hom_R(M,L)$ is an isomorphism. I started by defining the ...
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How Should I Prove this Function is an Isomorphism?

I have this question: Let $L$, $M$ and $N$ be unitary $R$-modules. Let $f: M \to N$ be an $R$-module isomorphism. Prove that the map $f_*: \text{Hom}_R(N,L) \to \text{Hom}_R(M,L)$ is an isomorphism. ...
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A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits. [duplicate]

A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits. I have seen these similar questions 1, 2, 3, but none contain a proof of this statement above. Here ...
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Question about proof that two modules are injectively equivalent.

Assume we have two exact sequences $$0 \to M \xrightarrow{\varphi} Q \xrightarrow{\psi} \mathcal{L} \to 0$$ and $$0 \to M \xrightarrow{\varphi'} Q' \xrightarrow{\psi'} \mathcal{L}' \to 0$$ where $M,Q,...
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Is $k[x, x^{-1}]$ a (graded) injective $k[x]$-module

Consider $k[x]$ with the usual grading, and the graded $k[x]$-module $k[x, x^{-1}]$. Is it injective? I suppose yes, because it is torsion free and graded divisible (i.e., divisible by homogeneous ...
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Confusion about injectivity of $\mathbb{Z}/m\mathbb{Z}$

It is well known that $\mathbb{Z}/n\mathbb{Z}$ is injective as a module over itself. And so because $\mathbb{Z}/n\mathbb{Z}$ is a PID (even if it isn't an integral domain, each ideal is generated by a ...
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Projective module on the ring of dual numbers

Let $\mathbb{K}$ be a field and consider the ring of dual numbers $R=\mathbb{K}[x]/<x^2>$. I have to prove that any projective $R$-module $P$ is injective. My idea is to use the Baer’s criterion....
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Constructing injective resolution on big sites.

Let's work over a category of varieties on some field. Let $\mathcal{F}$ be a presheaf on varieties that satisfies Zariski/etale sheaf conditions whenever restricted to the opens of a specific variety....
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Hom and Injective envelope

let $R$ be Noetherian ring and $m$ , $n$ are maximal ideals of $R$ s.t. $m\neq n$. can I say that ‎$$Hom_R(E_R(R/m),E_R(R/n))=0?$$ why?
pink floyd's user avatar
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chain map from a resolution to another injective resolution induced by linear map is unique up to chain homotopy equivalence

Let $0\rightarrow M\rightarrow E_0\rightarrow E_1\rightarrow...$ be a resolution of a $R-$module $M$. Let $0\rightarrow N\rightarrow I_0\rightarrow I_1\rightarrow...$ be a injective resolution of a $R-...
Ziqiang Cui's user avatar
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What is the importance of a projective cover and injective hull for a module?

My understanding of projective covers and injective hulls for modules over a (finite-dimensional) associative $\mathbb{C}$-algebra $A$ is as follows. $\bullet$ The projective cover of $M$ is an ...
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Finite injective resolution of a constant sheaf

Is it known if the injective resolution of the constant sheaf $\mathbb C_X$ on a smooth manifold $X$ is of finite length? I am asking this because the fine resolution of $\mathbb C_X$ in terms of the ...
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Difference between the injective object in the category of cochain complexes of modules and injective cochain complexes

There is a question completely ignored in the literature which is very strange considering how natural this question arises. I regard this as an omission and I hope to be able to clarify this matter ...
Flavius Aetius's user avatar
2 votes
1 answer
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Injectivity of the ring of formal power series over itself

It is well-known that the ring of formal power series $F[[t_1, ..., t_n]]$, where $F$ is a field of characteristic zero, is an injective abelian group. Is it true however that $F[[t_1, ..., t_n]]$ is ...
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2 votes
0 answers
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Injective module is stalkwise injective

Let $X$ be a Hausdorff locally compact topological space, $R$ be a sheaf of $\mathbb{C}_X$-algebras on $X$. If necessary, we can assume that $(X,R)$ is a complex manifold. If $M$ is an injective ...
Doug's user avatar
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Proof of Theorem II, 7.11 in "Residues and duality"

When reading the proof of Theorem II, 7.11 in "Residues and Duality" by R. Hartshorne, I have encountered some steps that are unclear to me. 1.Firstly, I am unsure about the notation $\...
Doug's user avatar
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Injective module iff is always a direct summand

I want to prove that an R-Module $J$ is injective if and only if whenever $J$ is a submodule of $M$, $J$ is a direct summand of $M$. One implication is easy: If $J$ is injective and is a submodule of $...
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Module homomorphisms that factor through an injective module

Let $R$ be a commutative unital ring, $M$ and $N$ be $R$-modules, and $u:M\to N$ be an $R$-linear map. Let $i_M:M\to I_M$ and $i_N:N\to I_N$ be injective hulls for $M$ and $N$ and set $\Sigma M=\text{...
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injective envelope and $Hom$

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. If $p$ is a prime ideal of $R$, is the following statement true: $$Hom_R(R/I,E_R(R/p))=‎ ‎\begin{cases} E_{R/I}(R/p)~~ if \hspace{....
pink floyd's user avatar
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example of injective module

Can you provide some examples of a commutative Noetherian local ring $(R,m,k)$ and an $R$-module $M$ that is not finitely generated, such that $id_R(M)\neq \sup \{i\in \mathbb{N}_0 \mid Ext_R^i(R/m,M)...
pink floyd's user avatar
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1 vote
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Finiteness of the Injective Hull

Please help me. I have read Finiteness of the Injective Hull by Alex Rosenberg and Daniel Zelinsky. An $R$-module $N$ has finite length if $N$ has a finite composition series. Alex Rosenberg and ...
YSA's user avatar
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Cardinal of the injective hull

Let $A$ be an abelian group that is countable. Is it true that the injective hull $E(A)$ of $A$ (as $\mathbb{Z}$-module) is also countable? For example, $E(\mathbb{Z})=\mathbb{Q}$ and for every prime ...
Doug's user avatar
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Indecomposable injective modules

Let $R$ be a ring with $1$. It's well known that if $M$ an indecomposable injective right $R$-module, then $M\cong E(R/\mathfrak{p})$ for some prime ideal $p\subset R$ where $E(R/\mathfrak{p})$ is the ...
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Injective modules in short exact sequences

Let me first state the question itself. Let $R$ be a ring with $1$. Let $0\to M\to Q\to L\to 0$ and $0\to M\to Q'\to L'\to 0$ be short exact sequences of $R$-modules where $Q$ and $Q'$ are both ...
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Injectivity of $\mathbb{Q}_{p}$ and $\mathbb{Z}(p^{\infty})$

Please help me. In the category of $R$-modules, we know that the Prüfer group $\mathbb{Z}(p^{\infty})$ is an injective envelope of $\mathbb{Z}/p\mathbb{Z}$ the ring integers modulo $p$ where $p$ is a ...
YSA's user avatar
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1 answer
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Two equivalent definitions of injective modules

According to my lecture notes, these two definitions of an injective module are equivalent: (Let $R$ be a ring, $Q$ an $R$-module.) For every injective $R$-module homomorphism $u:M’ \rightarrow M$ ...
dahemar's user avatar
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6 votes
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Is there an injective object in the category of all free abelian group?

The following notations is inherited from Homology. Suppose we have an exact sequence $$0\rightarrow B_{n}\rightarrow Z_{n}\rightarrow H_{n}(C)\rightarrow 0$$ where $B_{n}$ is the nth boundaries, and $...
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4 votes
0 answers
157 views

Injective Hull of $k$ as $k[x_1,..., x_n]$-module

Given a ring $A$ and an $A$-module $M$, there is a unique (up to isomorphism) injective $A$-module $E_A(M) $containing $M$, with the property that every injective $A$-module containing $M$ also ...
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