Questions tagged [injective-modules]

For questions about injective modules over a ring and injective objects in related categories.

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Freeness of injective hull of finite module over Gorenstein Artinian ring

This is Lemma 4.1 in Brochard, Khare, Iyengar: Wiles defect and criteria for freeness. I have managed to understand the proof, except for the following part: Why is $F$, the injective hull of a finite ...
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Pontrjagin dual of modules [closed]

I am not sure whether this question is appropriate to appear here. If not, I apologize for that. Given an $R$-module $M$, we define its Pontrjagin dual as $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. ...
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Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring

Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
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  • 111
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273 views

Example of non vanishing Ext

Let $R$ be a commutative Noetherian ring and $I$ is a proper ideal of $R$. suppose that $M$ is a f.g. $R$-module. $\DeclareMathOperator\Ext{Ext}$I'm looking for an example that has this property: $$\...
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Finding an injective envelope containing another injective envelope

Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
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Do rationally contractible presheaves have rationally contractible injective resolution

Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\...
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On the finiteness of an Auslander-Reiten component

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7: And this is part of it's proof, in which the direction (2) $\Rightarrow $ ...
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-1 votes
1 answer
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infinite left degrees

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper: Definition: Let $f: X \rightarrow Y$ be an irreducible morphism ...
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About composition factors [closed]

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3 $A$ is assumed to be an Artin algebra and mod(A) the ...
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Question on simple modules and projective covers

I have the following question: Let $A$ be an Artin algebra. Let $S_1$ and $S_2$ be simple modules in $\text{mod}(A)$ and let $P(S_1)$ be the projective cover of $S_1$. Let $f: P(S_1) \rightarrow S_2$ ...
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2 answers
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Question on injective hulls

How can I show the following: Let $f: M \rightarrow N$ be a morphism in $\text{mod}(A)$, where $A$ is an Artin algebra. Suppose $f \neq 0$. Then there exists a simple module $S$ with its injective ...
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injective hull and projective cover of simple modules are indecomposable

Let $A$ be an Artinian algebra. Let $S$ be a simple module over $A$. Let $\pi: S \rightarrow I$ be the injective hull and $\tau: P \rightarrow S$ be the projective cover of $S$. Then $I$ and $P$ must ...
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Can we extract an injective envelope from a monomorphism?

Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a morphism of $A$-modules, where each $I_{j}$ is injective and indecomposable. If $f$ is a monomorphism, then can we ...
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Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
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2 answers
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Are there (enough) injectives in condensed abelian groups?

The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ? Does it, in fact, ...
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Bimodule resolutions

I have asked this question on Mathematics Stack exachange but didn't get any reply yet. So, I am asking it here. Let A be a finite-dimensional algebra. Let M be a left A-module and N be a right A-...
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injective hulls in mixed characteristic

Let $R=\underleftarrow\lim (R/\mathfrak m^i)$ be a complete local ring, with residue field $k=R/\mathfrak m$, and let's assume that $R$ is Noetherian. If $R$ is a $k$-algebra, then I believe that ...
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3 votes
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Is there a constructive proof of Baer's Criterion?

Baer's Criterion states than one can check injectivity of an $R$-module on inclusions of ideals. The proof, however, strikes me as very nonconstructive: it employs both Zorn's Lemma and LEM. Does ...
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On definitions and explicit examples of pure-injective modules

I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the ...
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2 votes
1 answer
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Injective Change of Rings

Sorry if this is too elementary, but when I was going to ask this question on math.stackexchange, I saw the same question with three up-votes and no answer. So I decided to post it here. I am doing ...
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Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective complexes themselves?

Let $R$ be an associative ring with unit and $I$ be a complex of $R$-modules. We call $I$ is h-injective if for any acyclic complex $T$ of $R$-modules, the mapping complex $\text{Hom}_R(T,I)$ is ...
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4 votes
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Explicit description of injective hulls

Let $k$ be a field, let $R:=k[x_1,\ldots,x_n]$, and consider the $R$-module $M:=R/{(x_1,\ldots,x_n)}\cong k$. Then the injective hull $I_M$ of $M$ admits the following explicit description: $$ I_M = k[...
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3 votes
1 answer
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Looking for example of quotient of group algebra by ideal of group ring which fails to be injective

I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...
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2 votes
1 answer
60 views

A generating set for injective envelope

Let $m$ be a maximal ideal of a commutative ring $R$ with $1$. Can we construct a generating set $\{x_i\}_{i\in I}$ for the injective envelope $E(R/m) $ of $R/m$ such that $R/m\not\subseteq\langle x_i\...
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3 votes
0 answers
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Decomposition of injective modules over Noetherian rings

Let $A=\mathbb{C}[x_1,\ldots,x_n]$ be a polynomial algebra over the complex numbers. I am interested in injective modules over $A$. Since $A$ is projective over itself, the $\mathbb{C}$-dual module $...
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1 answer
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Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\infty$?

Why does there exist a non-split sequence with the condition that $\mathrm{pd} M = \infty$? Remarks. I am reading Andrzej Skowroński, Sverre O.Smalø, Dan Zacharia: On the Finiteness of the Global ...
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3 votes
1 answer
106 views

Direct sum of K-injectives over a noetherian ring

Let $A$ be a noetherian ring, and let $\{I_i \mid i \in J\}$ be a collection of K-injective complexes over $A$. Is the direct sum $$ \bigoplus_{i \in J} I_i $$ also a K-injective complex over $A$? ...
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2 votes
0 answers
393 views

Could we prove the flat base change theorem for cohomology via injective resolution?

Let $X$ be a quasi-separated scheme over a base ring $A$ and $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $A\to B$ be a flat morphism and $X^{\prime}:=X\times_{\text{spec}(A)}\text{...
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5 votes
1 answer
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On some sense of representing an endofunctor of the category of modules over polynomial rings

If $R$ is commutative ring, $n\in\mathbb{N}$, $\mathsf{M}$ the category of $R[x_1,\dotsc,x_n]$-modules,and $F\colon\mathsf{M}\to\mathsf{M}$ an endofunctor of $\mathsf{M}$ which preserves all finite ...
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Fiberwise injective resolution of coherent sheaf

Let $k$ be an algebraically closed field (of characteristic zero) and $X, Y$ be projective $k$-varieties. Let $F$ be a coherent sheaf on $X \times_k Y$, flat over $Y$. Does there exists a coherent $\...
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1 answer
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Is $Hom_R(S_X^{-1}R, E)$ the minimal injective cogenerator of $S_X^{-1}R$?

Assume that $R$ is a commutative Noetherian ring with minimal injective cogenerator $E$. For a finite set of maximal ideals $X$ of $R$, define the multiplicative set $$S_X=R-\bigcup_{\mathfrak{m}\in X}...
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1 answer
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When is every injective module $\Sigma$-injective?

I have been looking for a couple of days for the answer to this question to no avail. Let me define what $\Sigma$-injective is. Let $R$ be a unital, not necessarily commutative ring. A left $R$-...
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1 answer
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injective modules and divisible modules

The following result is basic ( P.J.Hilton, U.Stammabach, a course in homological algebra ). Let $A$ be a principal ideal domain. Then a $A$ module is injective iff it is divisible. Now if the ...
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2 votes
1 answer
648 views

When is the pullback of an injective sheaf injective?

Let $X$ be a Gorenstein (not necessarily smooth) projective $\mathbb{C}$-scheme and $S$ another $k$-scheme. Let $I$ be an injective sheaf on $X$. Denote by $p:X \times_k S \to X$ the natural ...
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4 votes
1 answer
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locally noetherian categories and the category of quasi-coherent sheaves over a noetherian scheme

It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective if and only if every injective $R$-module is a direct sum of indecomposable injective $R$-...
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1 vote
1 answer
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Graded version of Baer's Criterion

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...
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6 votes
1 answer
551 views

Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective? This module is not dual, so standard arguments via flatness do not work. Injectivity is understood in the ...
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  • 1,567
6 votes
1 answer
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Injective flat module

Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such. ...
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Are injective modules flabby on basic open sets?

In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds: Statement: If $A$ is a commutative ring and $...
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2 votes
1 answer
167 views

Injective modules over noncommutative noetherian rings

Let $R$ be a left noetherian ring, then it is well known that Matlis proved that any injective $R$-module decomposes into a sum of indecomposable injectives, each of form $E(R/I)$ for some irreducible ...
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5 votes
2 answers
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How to characterize flasque sheaves in more functorial way?

The motivation to ask this question is some proposition of flasque sheaves. Let's recall the definition of flasque sheaf:A sheaf $F$ on a topological space $X$ is flasque if for every inclusion $V\...
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  • 1,516
1 vote
1 answer
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cofree modules and dual

1, Why do people pay special attention to Q/Z in the definition of cofree modules instead of ordinary abelian groups? 2, Over a PID, is every injective module cofree? Just like the relationship ...
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6 votes
0 answers
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Localisation of injectives

When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following properties:...
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3 votes
0 answers
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Rejects and injectives

Let $A$ be any ring and consider modules on the left. For $M$ $A$-module, the trace $Tr(M,A)$ is a two-sided ideal of $A$. If $A$ is a unitary ring then: $Tr(P,A)P=P$, for $P$ projective; $Tr(P,A)^2=...
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  • 425
2 votes
1 answer
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Is there an $A$ such that $B$ injective iff 1st Ext functor vanishes?

In the category of $\mathbb{Z}$-modules, there exists a module $A$---for instance $\bigoplus_{k=2}^\infty \mathbb{Z}/k\mathbb{Z}$---such that a $\mathbb{Z}$-module $B$ is injective iff $\operatorname{...
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8 votes
1 answer
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Injective objects in Mor(Ab)

Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}(\{0<1\},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are ...
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12 votes
1 answer
3k views

The direct sum of injective modules need not be injective

The Bass-Papp Theorem asserts that a commutative ring $R$ is Noetherian iff every direct sum of injective $R$-modules is injective. Thus every non-Noetherian ring carries a counterexample. If $$ I_1 ...
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1 vote
0 answers
411 views

Ring such that any submodule of an injective module is flat?

Does anyone know examples of rings $R$ with the property that any submodule of an injective (right) $R$-module is flat? If I'm not missing something, this class of rings includes the (Von Neumann) ...
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2 votes
1 answer
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Simple Question on Injective Hulls

Let $R$ be a noetherian local ring with maximal ideal $\mathcal m$ and denote by $E$ the injective hull of the residue field $k$. Then, as an $R-$module, what is the support of $E$?
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  • 609
5 votes
1 answer
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Self-injective basic algebras

Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes_k A$ is not self-injective? The algebra $A$ cannot be finite-dimensional, ...
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