4
votes
1answer
128 views

$ mult(R/I) = d_1 \cdots d_r \quad \Rightarrow \quad f_1,\dots,f_r \quad \text{is a $R$-regular sequence?}$

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
2
votes
1answer
57 views

dg-flat complexes and their characters

Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X_n$ is flat for each n and ...
7
votes
1answer
148 views

Complexity of rational $\mathrm{GL}_{n(r)}$-modules

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
1
vote
0answers
95 views

Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects? What are some applications? For ...
1
vote
1answer
134 views

arrows in the injective representations of quivers

Let $Q$ be a quiver (a directed graph) and $E$ be a representation of $Q$ by R-modules, that is, for each vertex $v$ in $Q$ there exists an R-module $E(v)$ and for each arrow $a:v\to w$ there exists a ...
5
votes
1answer
373 views

Splitness of commutative diagrams

Consider the following commutative diagram in the category of $R$-modules where $R$ is an associative ring with identity and all modules are unital. $ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ ...
1
vote
0answers
128 views

Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...
1
vote
1answer
204 views

finitely presented representations

Let Q=(V,E) be a direct graph where V is the set of all its vertices and E denotes the set of all its arrows. $X$ is called a representation of Q by modules if it is a functor from Q to R-Mod. i.e. ...
1
vote
1answer
207 views

Pure monomorphism of functors-

Suppose that $F, G: Q\rightarrow R{\rm -Mod}$ be two covariant functors where $Q$ is an abelain category and $R$ is a commuatative ring. Also let $\eta: F\rightarrow G$ be a natural transformation ...
8
votes
1answer
205 views

Rings in which every module has an injective image

Consider the class of rings $R$ with identity such that any left $R$-module has a non-zero injective homomorphic image. Any such ring is clearly a left V-ring. Is it true that any such ring must be ...
0
votes
1answer
174 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 ...
3
votes
1answer
134 views

is every finitely n-presented (S^{-1})R-module a localization of a finitely n-presented R-module?

Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module. Then, more generally, is every finitely ...
3
votes
0answers
225 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; ...
2
votes
1answer
158 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 ...
1
vote
1answer
289 views

Drect limit of sequences

Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$. Let $$\{\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j\}$$ be a direct system of short ...
2
votes
1answer
290 views

An example of a tensor product consisting of only simple tensors?

Hy guys. I'm doing some independent analysis which makes use of the tensor product of modules (over commutative rings with unit 1, and ring homomorphisms map $1 \mapsto 1$). Let $\pi: A' \to A$ be a ...
1
vote
2answers
357 views

Let $M$ be a $R$-Bimodule that happens to be projective, is its associated left $R \otimes R^{op}$-module projective too?

Let $k$ be a commutative ring, a $R$-Bimodule $M$ over a $k$-algebra $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being $R$ itself ...
10
votes
1answer
568 views

Higher “Cartan-Eilenberg” Resolutions

I am a number theory graduate student learning a bit of homological algebra, and I am curious about higher complexes in abelian categories. I apologize if my post is slightly vague as I am not an ...
0
votes
1answer
212 views

Equivalent functors

Let $R$ be a commutative Noetherian ring, $M$ is a finitely generated $R$-module. If $F: Mod \to Mod$ is a left exact functor and $R^iF(E)=0$ where $E$ is injective module. Assume that $F(-) \cong ...
16
votes
7answers
2k views

“Sums-compact” objects = f.g. objects in categories of modules?

Hello, Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...
15
votes
4answers
4k views

Flat Module and Torsion-Free Module

All rings in this question are integral. It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see ...
26
votes
4answers
3k views

How to make Ext and Tor constructive?

EDIT: This post was substantially modified with the help of the comments and answers. Thank you! Judging by their definitions, the $\mathrm{Ext}$ and $\mathrm{Tor}$ functors are among the most ...
0
votes
1answer
136 views

Projectively splitting module

Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$ ?
3
votes
2answers
591 views

What is the characteristic of the module over Jacobson semisimple ring?

We know a ring R is semisimple ring iff every module over R is semisimple,a ring R is von-Neumann regular ring iff every module over R is flat,What about the Jacobson semisimple ring?
5
votes
1answer
258 views

Behavior of the projective dimension of modules in a continuous chain of extensions

Let $R$ be an arbitrary ring. Let $D$ be the class of $R$-modules of projective dimension less than or equal to a natural number $n$. If $L$ is the direct union of a continuous chain of submodules ...
2
votes
2answers
336 views

Homological dimensions of module

$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. ...