Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
9,806
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rational functions flat over $k[x]$
I was looking at the solution of an exercise, which proved the following statement:
The $k[x]$-module $k(x)$ of rational functions for $k$ a field is flat.
The solution goes as follows:
Let $\phi: N ...
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Flat Module from book of NS Gopala Krishnan
I was reading flat modules from book by NS Gopalakrishnan. In starting of faithfully flat algebra the below is written
Let $A$ be an $R$-algebra, $M$ and $N$ are $R$-modules. Then homomorphism
$\phi_M ...
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First syzygy is unique up to direct summand
Let $R:=K[x_0,...,x_n]$ be the polynomial ring over a field $K$ and $M$ be finitely generated $R$ module. Let $(m_i)_{i=1,...,k}$ be a generating set for $M$. Then we can define the free module $F_0 :=...
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Length and complete decomposability of $(\varphi,\Gamma)$-modules
I am following the following book on Galois representations and $(\varphi,\Gamma)$-modules.
Let $p$ be prime and $L$ a finite $\mathbb{Q}_p$ extension. Let $\Gamma= \mathrm{Gal}(L_\infty/L)$ be the ...
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Find the natural number n for which the set $A_n$ has exactly 323 integers
the problem
For n a natural number we define $A_n=\{x\in \Bbb R \,|\,\, |x+n+4| \leq 3n-4\}$. The natural number n for which the set $A_n$ has exactly 323 integers is....?
my idea
i absolutely have ...
- 748
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$\varphi$-modules that are not completely decomposable
Forgive me for I am not an expert in the area, and I fear that this may be a somewhat trivial question - so thank you for your patience!
I am reading Brinon-Conrad's notes on $p$-adic Hodge theoy, ...
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Connection between twisted graded modules and twisted sheaves
I came across the definitions of graded twisted modules while reading about the syzygy theorem. In the meantime I also attended an algebraic geometry course where twisted sheaves occured. For both ...
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Number of left ideals in the simple components of groups algebras
Let $G$ be a finite group and $K$ a field with characteristic zero. Suppose $G$ has $m$ irreducible $K$-representation $W_i$ with character $\chi_i$. $KG$ is semisimple algebra, and
$$KG=KGe_1 \times \...
- 2,307
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The residue class of a complete set of primitive orthogonal idempotents
I was studying the book Elements of the Representation Theory of Associative Algebras: Volume 1 and this question occurred to me:
In page 29, it says that
Because $\{e_1,…,e_n\}$ is a complete set of ...
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Let $R$ be a ring and $M$ a (left) $R$-module.
Let $R$ be a ring and $M$ a (left) $R$-module.
(a) Prove that for every $z \in Z(R)$ the set $zM = \{ zm \mid m \in M \}$ is a submodule of $M$.
(b) Give an example of a ring $R$ and an idempotent $e \...
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Let $\varphi: K_1 \rightarrow K_2$ be a ring homomorphism and $M$ a left $K_2$-module.
Let $\varphi: K_1 \rightarrow K_2$ be a ring homomorphism and $M$ a left $K_2$-module.
(a) Prove that $M$ becomes a left $K_1$-module if we introduce the operation of multiplication by elements from $...
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Free module and finitely generated submodule.
Let $A$ be a commutative ring, $M$ a free $A$-module and $N\subseteq M$ a finitely generated submodule.
Prove that there is a Sub-Noetherian Ring $A_0 \subseteq A$, a free $A_0$-submodule $M_0 \...
- 549
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Questions on the $\hom$ Functor and Free Groups
This question arises while learning about the $\hom$ functor. My algebra background is not that strong, so here is my question:
Let $G$ be a free group, and let $f\colon G \to G$ be a group
...
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Finiteness of a generalization of the class number of a number field
Let $f(x)$ be an irreducible monic integral polynomial with root $\alpha \in \mathbb{C}$. A classical result of Latimer and MacDuffee asserts that there is a bijection between similarity classes of ...
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Cohomology group for trivial group
Let $G$ be a group, $A, B$ be a $G$ - modulo. We can define the n-th Cohomology group of $G$ with coefficient in $A$.
$$H^n(G,A) =\text{Ext }_G^n(\mathbb{Z},A)$$
And the n-th Homology group of $G$ ...
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