0
votes
0answers
24 views

The injectivity of $f\mapsto f\circ v$ on $\hom(M'',N)$ implies that $v$ is surjective [duplicate]

I'm an undergrad getting familiar with some notions of commutative algebra by reading Atiyah-McDonald. On the exact sequences part, a part of the proof of (2.9) is proving that if ...
0
votes
0answers
46 views

Injective dimension and Krull dimension of a module

Let $R$ be a regular local ring and $M$ an $R$-module (not necessarily finite), then the injective dimension $\operatorname{id}_R(M)$ of $M$ is finite. When $M$ is finitely generated, we have ...
2
votes
2answers
67 views

Minimal injective resolution of a module

Let $R$ be a commutative Noetherian ring and $M$ an $R$-module. Let $0\rightarrow M \rightarrow E^{\bullet}$ be a minimal injective resolution of $M$ and $0\rightarrow M\rightarrow I^{\bullet}$ be an ...
0
votes
1answer
35 views

Example of Tor-Rigid Module

Let $R$ be a commutative ring (with 1) and $M$ a finitely generated $R$-module. We say that $M$ is rigid if for every finitely generated $R$-module $N$ whenever Tor$_i^R(M,N)=0$ then Tor$_j^R(M,N)=0$ ...
0
votes
1answer
36 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
2
votes
1answer
56 views

Possible examples where the Zero Divisor Conjecture does not hold

Given a ring $R$ with a nonzero zero divisor $x$, it is easy to show that if $M$ is a nonzero $R$-module, then there exists $y\in R-\{0\}$ such that $ym=0$ for some $m\in M-\{0\}$. I was ...
0
votes
1answer
53 views

Characterization of the kernel and cokernel of the natural homomorphism between a module and its double dual. [closed]

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Suppose $$ G \overset{\varphi}{\rightarrow} F \to M \to 0$$ is exact where $F,G$ are finite free modules. Suppose ...
0
votes
0answers
39 views

Grade of an ideal greater than the projective dimension of quotient of another one

We know that the grade of an ideal $I$ in a Noetherian ring $R$ is the infimum of the set of all $i$ with $Ext^i(R/I,R)$ nonzero. Also, the projective dimension of an $R$-module $M$ is at most $s$ if ...
4
votes
1answer
62 views

From a vector bundle to a Koszul complex

Let $k = \mathbb C$. Given a commutative $k$-algebra $A$, an $A$-module $M$ and a homomorphism of $A$-modules $s:M \to A$, we can construct the Koszul dg algebra. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ ...
5
votes
1answer
98 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
0
votes
1answer
39 views

Relation between faithfully flatness and map of $Spec$

I'm stuck on this exercise ( from Bosch ) : Let $\phi :R \to R' $ a flat ring morphism. Show that $\phi$ is faithfully flat if and only if the associated map $Spec(R') \to Spec(R)$ , ...
1
vote
0answers
28 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
1
vote
2answers
108 views

Examples of Noetherian local rings which are not Gorenstein

Can anyone give me an example of a Noetherian local ring which is not a Gorenstein ring?
3
votes
3answers
66 views

Injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module

What is the injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module? Can one use the well-known fact that $id(M)$ is less than or equal to $i$ iff $Ext^{i+1}(N,M)=0$ for all $N$? Thanks in ...
0
votes
0answers
39 views

Direct limits commute with $\mathrm{Tor}$ functor

How one could prove that direct limits commute with the functor $\mathrm{Tor}$? Of course, I know that $\mathrm{Tor}$ with its first $0$ index is the same as tensor product which does commute ...
2
votes
1answer
83 views

Finite projective dimension may lead to projectiveness!

Assume a ring $R$ is injective as an $R$-module. If the projective dimension of an $R$-module $P$ is finite could one conclude that $P$ is a projective $R$-module? Probably one should start with ...
0
votes
2answers
32 views

induced sequence exact

If $D$ is a multiplicatively closed subset of $R$. I'm trying to come up with an example where $$0\to L \to M \to N \to 0$$ is not exact, but the induced sequence $$0 \to D^{-1}L \to D^{-1}M \to ...
0
votes
1answer
27 views

Comparing injective dimensions in a short exact sequence

If $0→A→B→C→0$ is an exact sequence in the category of $R$-modules ($R$ commutative having unity) with injective dimensions of $A$ and $C$ both $≤n$, is that of $B$ also $≤n$? It seems to me that ...
1
vote
1answer
66 views

if $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about m zcn's comment on my question Projective dimension of all principal ideals is finite. Is R an integral domain?. It's a good point. so i ask it for use of everybody: if ...
6
votes
0answers
76 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
0
votes
1answer
90 views

Localization of Gorenstein ring

Let $R$ be a Gorenstein local ring and $S=R \setminus Z(R)$. I want to prove $S^{-1}R =⊕_{ht\ p=0} R_p$ and $S^{-1}R$ is injective $R$-module. I can see the above $p$'s are minimal, $id_{R_p} R_p=0$ ...
2
votes
2answers
82 views

The Relationship Between Cohomological Dimension and Support

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) ...
1
vote
1answer
41 views

Inequality amongst projective dimensions!?

Assume $φ : R\to S$ is a ring homomorphism between commutative rings sending unity to unity. Taking any $S$-module $M$ as an $R$-module, is it true that always $$\operatorname{pd}_R (M)\le ...
5
votes
2answers
310 views

Exercise 2.27 Atiyah-Macdonald, absolute flatness

A commutative ring $R$ is absolutely flat if every $R$-module is flat. Prove that the following are equivalent: 1) $R$ is absolutely flat 2) Every principal ideal of $R$ is idempotent 3) Every ...
1
vote
1answer
54 views

Tor for graded modules over a graded ring

I am confused about how this Tor is defined. Suppose $R$ is a graded ring, $M,N$ graded modules over $R$. What is $\operatorname{Tor}_{st}^R(M,N)$? I am confused about the subscripts. I realize ...
2
votes
1answer
78 views

Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
1
vote
1answer
48 views

Exercise from Atiyah about flatness

This is an exercise from Atiyah. Let $N$ be a flat $B$-module, and $B$ a flat $A$-algebra where $A$ is a commutative ring with unit. Then $N$ is flat as $A$-module Any hint ?
1
vote
2answers
37 views

Some projective property of projective resolution

In the proof of lemma 3.2 in this PDF, it said: Let $0 → A → A′ → A′′ → 0$ be a short exact sequence. $P′′_{*}→ A′′$ be a projective resolution. (i.e. $ ··· P_{1}′′→ P_{0}′′ → A′′ → 0$ is exact ...
2
votes
1answer
91 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is $1$. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that ...
3
votes
1answer
79 views

Why we can consider both modules as modules over $R_{(p)}$? (Bruns and Herzog, Theorem 1.5.9)

I'm reading Bruns-Herzog's book Cohen Macaulay rings and have a probably elementary question. Why we may consider both modules as modules over $R_{(p)}$ in this theorem? ... i know that ...
0
votes
1answer
51 views

Injective dimension and depth

Here is Bruns and Herzog's book Cohen-Macaulay Rings, Theorem 3.1.17: Let $R$ be a Noetherian local ring, and $M$ a finite $R$-module of finite injective dimension. Then $\operatorname{inj\ ...
0
votes
1answer
74 views

History of five lemma

I am interested in the history of five lemma. Who was first to prove it and What was the purpose of proving it ? http://en.wikipedia.org/wiki/Five_lemma
3
votes
2answers
71 views

How can I compute $\operatorname{Tor}(\mathbb Z_{p},\mathbb Z_{q})$?

I am self-studying Vick's Homology Theory, and now it is on the topic of free resolutions. Since I am not familiar with it, I have little ideas about how to compute $$\operatorname{Tor}(\mathbb ...
0
votes
1answer
141 views

Is it true that $\operatorname{inj.dim}_R R= \operatorname{inj.dim}_R \widehat{R}?$ Is there a one-sided inequality?

$(R,m)$ is a local ring. Is it true that $\operatorname{inj.dim}_R R= \operatorname{inj.dim}_R \widehat{R}?$ Is there a one-sided inequality? (Here $\operatorname{inj.dim}$ denotes the injective ...
1
vote
0answers
18 views

Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...
1
vote
1answer
61 views

Local Cohomology - Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings

This question arises in the context of Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings. Let $(R,m)$ be a local complete Cohen-Macaulay ring of dimension $d$. Denote by $H_m^d(-),\omega_R$ ...
2
votes
0answers
43 views

Bounds dimension, scheme and projective dimension

Is the dimension of a (commutative unital associative) algebra always bounded above by its protective (injective) dimension? If not is it always bounded above by its global dimension?
4
votes
1answer
249 views

Mayer-Vietoris sequence for local cohomology

Update 7:35pm UTC 3/23/14: I've reposted this quesion on MathOverflow here. As an assignment in my commutative algebra class, I need to prove the Mayer-Vietoris sequence for local cohomology: Let ...
2
votes
0answers
128 views

Does $\operatorname{id} M =\dim R$ hold for finite modules of finite injective dimension?

When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for ...
0
votes
0answers
59 views

An easy infinite free resolution

I'm doing exercise 1.23 on Eisenbud's Commutative algebra, and I have the following situation: let $k$ be a field and $R = k[x]/(x^n)$. They ask for a free resolution of $R/(x^m)$, for some $m \leq ...
1
vote
2answers
71 views

Existence of module of finite injective dimension

At p. 107 of the book Cohen-Macaulay Rings by Bruns and Herzog, the authors write "any module of finite projective dimension (over a Gorenstein ring $R$) has finite injective dimension as well, ...
1
vote
1answer
44 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map ...
2
votes
0answers
36 views

Under what conditions are the resolutions of two modules subcomplexes of the resolution of the tensor product?

I have that $S=k[x_1, \dots, x_n]$, $I$ is a lattice ideal, and $J$ is a monomial ideal. I am interested in the resolution of $S/(I+J)\cong S/I\otimes S/J$. In particular, I am interested in knowing ...
2
votes
0answers
45 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
3
votes
1answer
61 views

tensor, symmetric, exterior power of a module over a PID

Let $R$ be a PID and $M\cong R^r\!\oplus\bigoplus_{i=1}^s\!R/Ra_i$. Denote the tensor, symmetric, exterior power of $M$ by $T^nM=\bigotimes_{k=1}^nM$ and $S^nM= T^nM/\langle ...
4
votes
0answers
83 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
3
votes
1answer
75 views

Determinant of long exact sequence

Let the following be a long exact sequence of free $A$-modules of finite rank: $$0\to F_1\to F_2\to F_3\to...\to F_n\to0$$ I want to show that $\otimes_{i=1}^n (\det F_i)^{-1^{i}} \cong A$, where ...
3
votes
0answers
66 views

Mapping cones and resolutions

Let me preface my question by acknowledging the vagueness of it. I am hoping to find some information in the form of references as opposed to a hard and fast solution. Suppose that $S=k[x_1, \dots, ...
2
votes
1answer
46 views

$K(A)\cong \mathbb Z$ for a PID $A$

In Atiyah and Macdonald, chapter 7, exercise 26, iii), it's required to show the Grothendieck group $K(A)\cong \mathbb Z$ for a PID $A$. By ii) of this problem, it's easy to show that $K(A)$ is ...
3
votes
1answer
40 views

Exact sequence induces exact sequences for free parts and torsion parts?

Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$: $$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$ ...