Questions tagged [injective-modules]
For questions about injective modules over a ring and injective objects in related categories.
69 questions
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Is $(\mathbb{R}, +)$ still injective as long as $(\mathbb{Q},+)$ is?
It is known that the existence of nontrivial injective abelian groups is independent of choice in ZF (or, rather, ZFA). In particular, $\mathbb{Q}$ is not provably injective, much less $\mathbb{R}$, ...
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1
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Lemma of Harada and Sai on sums of modules with a "chain" of monomorphisms between them
I am trying to get a contradiction from the following set of hypotheses:
Let $R$ be a ring. Let $M$ be a direct sum of non-zero $R$-modules $M_1$, $M_2$, $\dotsc$. For each $i\ge1$, let $f_i:M_i\to M_{...
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Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?
I know that for such ...
- 480
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Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ...
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Does every SES of injective bounded cochain complexes split?
Question: Does every short exact sequence of injective bounded cochain complexes, $0\rightarrow I^\bullet\rightarrow J^\bullet\rightarrow K^\bullet\rightarrow 0$, split?
I am interested in a discrete ...
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2
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Enough injectives in the category of quasi-coherent sheaves on a stack
For a scheme $X$, I have a reference - https://stacks.math.columbia.edu/tag/077P - that says there are enough injectives in the category $\text{QCoh}(X)$. I am looking for a reference that says the ...
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A nonzero injective module with no nonzero uniform submodule
Recall that a module $M_R$ is called uniform if the intersection of any two nonzero submodules of $M$ is nonzero. I need an example of a nonzero injective module with no nonzero uniform submodules. ...
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2
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1
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Does anyone have a good example of an injective resolution?
I'm learning about injective resolutions and derived functor sheaf cohomology, and it seems that every source on injective resolutions gives no examples. I feel like just one good example would make ...
- 137
4
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0
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111
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Flatness of $C_0(S)$-module $L_\infty(S,\mu)$
Let $S$ be a locally compact Hausdorff space. By $C_0(S)$ we denote the space of continuous functions vanishing at infinity. Let $\mu$ be a finite Borel regular measure om $S$, then consider $L_\infty(...
- 1,697
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Nuclearity of $C(\partial_F \mathbb{G})$
Let $\mathbb{G}$ be a discrete quantum group, and consider the non-commutative Furstenberg boundary $\partial_F\mathbb{G}$ with function algebra $C(\partial_F \mathbb{G}) = I_{\mathbb{G}}(\mathbb{C})$....
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The eventual number of generators of modules of which $M$ is a subquotient
Let $R$ be a (commutative) ring and let $M$ be an $R$-module. Say that $M$ is subfinitely generated if $M$ is a submodule of a finitely-generated module. Write
$$\mathcal F(M) = \{ M \rightarrowtail N ...
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When do faithfully semiinjective complexes exist?
Question:
For which (perhaps noncommutative but always unital and associative) rings $R$ do faithfully semiinjective complexes of right or left $R$-modules exist?
Hopefully the answer is: "for ...
- 1,020
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For a pure-injective module $M$ does the property "$\operatorname{Hom}(-,M)$ is surjective" commute with certain limits?
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a pure-injective module. Then $\Hom(\varphi,M)$ is surjective for a pure-mono $\varphi$. It is well-known that $\varphi$ is a direct limit of split monos $\...
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Analogy between quasi-injective modules & extensible Banach spaces
Let $X$ be a module. $X$ is said to be quasi-injective if every homomorphism $h:A\to X$ from any submodule $A\subseteq X$ has an extension to an endomorphism $\tilde{h}:X\to X$.
A module $X$ is quasi-...
- 2,605
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Indecomposable injectives over Weyl algebras
Let $A=A_n(\mathbb{C})$ be the $n$-th Weyl algebra over the complex field. Then $A$ is a left Noetherian noncommutative ring. Is there a complete classification of indecomposable injective $A$-modules?...
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When does a local Cohen–Macaulay ring admit a non-zero finitely generated maximal Cohen–Macaulay module of finite injective dimension?
Let $(R,\mathfrak m)$ be a local Cohen–Macaulay ring. Then, it is well- nown that there exists a non-zero finitely generated $R$-module of finite injective dimension; for instance $\operatorname{Hom}...
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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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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
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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1
vote
1
answer
201
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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
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203
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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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1
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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 ...
user144185
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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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1
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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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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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470
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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 ...
- 16.9k
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1
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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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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:
$$
I_M = k[...
- 43.2k
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Looking for example of quotient of group algebra by ideal of group ring which fails to be injective
I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...
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A generating set for injective envelope
Let $m$ be a maximal ideal of a commutative ring $R$ with $1$. Can we construct a generating set $\{x_i\}_{i\in I}$ for the injective envelope $E(R/m) $ of $R/m$ such that $R/m\not\subseteq\langle x_i\...
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Decomposition of injective modules over Noetherian rings
Let $A=\mathbb{C}[x_1,\ldots,x_n]$ be a polynomial algebra over the complex numbers.
I am interested in injective modules over $A$.
Since $A$ is projective over itself, the $\mathbb{C}$-dual module $...
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Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\infty$?
Why does there exist a non-split sequence with the condition that $\mathrm{pd} M = \infty$?
Remarks.
I am reading
Andrzej Skowroński, Sverre O.Smalø, Dan Zacharia: On the Finiteness of the Global ...
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1
answer
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Direct sum of K-injectives over a noetherian ring
Let $A$ be a noetherian ring, and let $\{I_i \mid i \in J\}$ be a collection of K-injective complexes over $A$.
Is the direct sum
$$
\bigoplus_{i \in J} I_i
$$
also a K-injective complex over $A$?
...
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Could we prove the flat base change theorem for cohomology via injective resolution?
Let $X$ be a quasi-separated scheme over a base ring $A$ and $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $A\to B$ be a flat morphism and $X^{\prime}:=X\times_{\text{spec}(A)}\text{...
- 9,019
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1
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On some sense of representing an endofunctor of the category of modules over polynomial rings
If $R$ is commutative ring, $n\in\mathbb{N}$, $\mathsf{M}$ the category of $R[x_1,\dotsc,x_n]$-modules,and $F\colon\mathsf{M}\to\mathsf{M}$ an endofunctor of $\mathsf{M}$ which preserves all finite ...
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Fiberwise injective resolution of coherent sheaf
Let $k$ be an algebraically closed field (of characteristic zero) and $X, Y$ be projective $k$-varieties. Let $F$ be a coherent sheaf on $X \times_k Y$, flat over $Y$. Does there exists a coherent $\...
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Is $Hom_R(S_X^{-1}R, E)$ the minimal injective cogenerator of $S_X^{-1}R$?
Assume that $R$ is a commutative Noetherian ring with minimal injective cogenerator $E$. For a finite set of maximal ideals $X$ of $R$, define the multiplicative set $$S_X=R-\bigcup_{\mathfrak{m}\in X}...
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290
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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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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 ...
3
votes
1
answer
970
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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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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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