# Questions tagged [injective-modules]

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

48
questions

**6**

votes

**1**answer

101 views

### 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 $ ...

**-1**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**1**

vote

**1**answer

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 $f: P(S_1) \rightarrow S_2$ ...

**3**

votes

**2**answers

149 views

### 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 ...

**0**

votes

**1**answer

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 ...

**-1**

votes

**1**answer

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 ...

**4**

votes

**1**answer

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 ...

**15**

votes

**1**answer

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,...

**2**

votes

**0**answers

52 views

### 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-...

**4**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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 ...

**4**

votes

**1**answer

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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**4**

votes

**0**answers

163 views

### 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[...

**3**

votes

**1**answer

527 views

### 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 ...

**2**

votes

**1**answer

58 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\...

**3**

votes

**0**answers

125 views

### 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 $...

**4**

votes

**1**answer

210 views

### 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 ...

**3**

votes

**1**answer

98 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 $A$?
...

**2**

votes

**0**answers

309 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{...

**5**

votes

**1**answer

140 views

### 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 ...

**0**

votes

**0**answers

153 views

### 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 $\...

**4**

votes

**1**answer

248 views

### 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}...

**5**

votes

**1**answer

200 views

### 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 $R$-...

**0**

votes

**1**answer

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 ...

**2**

votes

**1**answer

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 ...

**4**

votes

**1**answer

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 $R$-...

**1**

vote

**1**answer

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 ...

**6**

votes

**1**answer

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 ...

**5**

votes

**1**answer

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.
...

**4**

votes

**0**answers

135 views

### 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 $...

**2**

votes

**1**answer

163 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 ...

**5**

votes

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821 views

### 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\...

**1**

vote

**1**answer

692 views

### 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 ...

**5**

votes

**0**answers

696 views

### 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:...

**3**

votes

**0**answers

287 views

### 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=...

**2**

votes

**1**answer

310 views

### 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{...

**8**

votes

**1**answer

514 views

### 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 ...

**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 ...

**1**

vote

**0**answers

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) ...

**2**

votes

**1**answer

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 $E$?

**5**

votes

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448 views

### 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, ...

**4**

votes

**2**answers

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 ?

**8**

votes

**1**answer

996 views

### 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 ...

**3**

votes

**0**answers

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 ...

**4**

votes

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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 ...