# Questions tagged [injective-modules]

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

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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) $\Rightarrow$ ...
1answer
118 views

### 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 ...
1answer
111 views

### 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 ...
1answer
157 views

### 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 $f: P(S_1) \rightarrow S_2$ ...
2answers
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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 ...
1answer
88 views

### 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 ...
1answer
62 views

### 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 ...
1answer
165 views

### 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 ...
1answer
2k views

### 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,...
0answers
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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-...
1answer
134 views

### 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 ...
0answers
245 views

### 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 ...
1answer
219 views

### 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 ...
1answer
197 views

### 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 ...
0answers
48 views

### 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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1answer
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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 $R$-...
1answer
861 views

### 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 ...
1answer
537 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 ...
1answer
434 views

### 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 $R$-...
1answer
253 views

### 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 ...
1answer
523 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 ...
1answer
503 views

### 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. ...
0answers
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1answer
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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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### 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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1answer
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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 ...
1answer
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 ...
0answers
405 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) ...
1answer
307 views

### 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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### 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, ...
2answers
806 views

### About injective hull

Let $M$ be an $A$-module. Is its injective hull affected by whether I regard $M$ as an $A$-module or $A/\mbox{Ann}(M)$-module ?
1answer
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### Direct sum of injective modules over non-Noetherian rings

By the Bass-Papp theorem, if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists an injective module over $R$ non-Noetherian, that ...
0answers
506 views

### Finite generatation of Ext

If $A$ is a Noetherian ring and $M$, $N$ are finitely generated modules over $A$, it is easy to see that $\mbox{Ext}_{A}(M,N)$ is finitely generated by taking a finitely generated projective ...
2answers
1k views

### Injective modules and Pontrjagin duals

Forgive me for this naive question. We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784. Every module is a submodule of an ...