0
votes
1answer
29 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
58 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
0
votes
1answer
112 views

$\operatorname{inj.dim}_R N= \operatorname{inj.dim}_R \widehat{N}$?

$(R,m)$ is a local ring. For an $R$-module $N$, we know that $\operatorname{inj.dim}_R N= \operatorname{inj.dim}_\widehat{R} \widehat{N}$. Is it true that $\operatorname{inj.dim}_R N= ...
1
vote
0answers
17 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
51 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
39 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
231 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
102 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
48 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
63 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
40 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
34 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
39 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
48 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
66 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
51 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
62 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, ...
1
vote
1answer
44 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
28 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$A$: $$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$ ...
2
votes
2answers
62 views

Represent localization as a direct limit

Let $A$ be a commutative ring with identity, $S\subset A$ a multiplicatively closed subset and $1\in S$. Does the equation $$S^{-1}A=\varinjlim_{s\in S}A_s$$ make sense? Here $A_s$ is the ...
0
votes
1answer
76 views

Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
1
vote
1answer
32 views

can the projective dimension be read from any projective resolution?

Let $P_{\bullet}, P'_{\bullet}$ be two projective resolutions of an $R$-module $M$. Denote their differentials by $d,d'$ respectively. Define $M_i = \operatorname{ker} d_{i-1}, M'_i = ...
2
votes
1answer
45 views

homotopy equivalence of projective resolutions

Let $P_{\bullet}$ and $P'_{\bullet}$ be projective resolutions of a module $M$ over a commutative ring $R$. Then $P_{\bullet}$ and $P'_{\bullet}$ are homotopy equivalent (see e.g. Matsumura, CRT, ...
5
votes
1answer
94 views

On Gorenstein ring of dimension zero

Let $R$ be an Artinian local ring. Then $R$ is a Gorenstein ring (i.e., $R$ is an injective $R$-module) iff for any ideal $I$ of $R$, Ann$($Ann$(I))=I$. Why? (We call $R$ Gorenstein if injective ...
3
votes
0answers
45 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
2
votes
1answer
59 views

Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
1
vote
1answer
45 views

Behaviour of Betti tables with exact sequences

Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of finitely generated graded $S$-modules, where $S=k[x_1, \ldots, x_n]$ is a polynomial ring in $n$ variables. Let $\beta_{i,j}(M)$ denote the ...
2
votes
1answer
48 views

Is the derived category of a commutative ring monoidal?

Let $A$ be a commutative ring, and consider the derived category $D(A)$. Is this a symmetric monoidal category? We have an obvious product, that is $-\otimes^L_A - $, and it is clear that we have an ...
3
votes
1answer
71 views

vanishing of Tor and regular sequences

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Let $x=x_1,\dots,x_n$ be an $R$-sequence such that it is also an $M$-sequence and let $I=(x_1,\dots,x_n)$. Question: Is it true that ...
0
votes
0answers
13 views

The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
3
votes
1answer
37 views

Question from Cartan-Eilenberg, Chapter 6, exercise 5

The exercise problem is this; consider a unital ring $A$. For each right $A$-module $M$ and left ideal $I$ of $A$, TFAE. (a) For each relation $\:\sum _{i} a_iu_i=0 \:(a_i\in M, u_i\in I)$ there ...
0
votes
1answer
37 views

degree zero term of minimal free resolution

Let $R=k[x_{1},\ldots,x_{n}]$ where $k$ is a field, and let $I$ be a homogenous ideal. Suppose that $\cdots\to R_{1}\to R_{0}\to R/I\to 0$ is a (the) graded minimal free resolution of $R/I$. Is it ...
2
votes
2answers
79 views

Finite injective dimension

Let $A$ be a commutative noetherian ring. Is it true that if $A$ is regular then any module over it has a finite injective dimension? What if $A$ is Gorenstein? Any reference who discuss this?
1
vote
0answers
65 views

Spectral sequences to involve together two ideals of a ring

I'm looking for spectral sequences to involve together two ideals of a ring. For instance, let $I,J$ be two ideals of Noetherian ring $R$ and $M$ be a finite $R$-module then we have the following ...
5
votes
1answer
128 views

Ext of an $\mathfrak{m}$-primary ideal

Let $(A,\mathfrak m,k)$ be a Noetherian local ring, $M$ a finitely generated $A$-module, and $I$ an $\mathfrak{m}$-primary ideal. If $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m},M)=0$ then ...
3
votes
0answers
65 views

Depth for intersection of prime ideals

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over field $K$. How can one compute $\operatorname{depth}(R/\bigcap_{i=1}^r p_j)$, where each $p_j$ is generated by some variables $x_i$ and have a ...
1
vote
1answer
71 views

Dedekind ring characterization via projective modules

I am looking for a book or course notes proving the following result: Let $R$ be an integral domain. Then $R$ is a Dedekind ring if and only if every submodule of a projective $R$-module is ...
5
votes
2answers
161 views

Why is $ \hbox{Ext}_R^* (M,M) = H^*(\hbox{Hom}_R^*(P^*,P^*))$?

Let me first fix some notation and conventions. Let $ R$ be a ring and $ M$ a left $R$-module. Given chain complexes $P^*$ and $Q^*$ in $R$-mod, define $ \hbox{Hom}^*_R(P^*,Q^*)$ to be the graded ...
2
votes
2answers
89 views

Explicit computation $\operatorname{Tor}(M,N)$

Let $R=\mathbb{C}[t]/t^2$ the ring of dual numbers. Using the homomorphism $\phi:R \to \mathbb{C}=R/(t)$ we have that $\mathbb{C}$ is a $R$-module, infact we have $$\psi: \mathbb{C} \times ...
2
votes
2answers
91 views

Eisenbud's proof of right-exactness of the exterior algebra

I'm trying to understand the proof in Eisenbud's Commutative Algebra that, given a right exact sequence $$K \to N \to M \to 0$$ of $R$-modules, we have an exact sequence $$K \otimes \wedge N \to ...
0
votes
1answer
29 views

on the extension module of a pair (some module, a finite free module)

Let $A$ be a commutative ring and $M$ an $A$-module that admits a projective resolution. According to my understanding it is true that $\operatorname{Ext}^i(M,A^n) \cong ...
0
votes
1answer
33 views

Topological interpretation of a zero map.

I have a pair of complexes of modules $A_{\bullet}$ and $B_{\bullet}$, and I want to create a new complex pretty trivially by shifting one complex by 1, and then taking the direct sum of the ...
1
vote
1answer
103 views

existence of finite free resolutions of finite modules over polynomial rings

Theorem 2.8 in Chapter XXI of Lang's Algebra says Theorem 2.8. Let $R$ be a commutative Noetherian ring. Let $x$ be a variable. If every finite $R$-module has a finite free resolution, then every ...
3
votes
0answers
49 views

on the proof of a basic theorem on double complexes

This question concerns my attempt to prove the following theorem and a point where i get stuck. My working category is that of modules over a commutative ring. Theorem [B.2, Matsumura, CMR, ...
2
votes
0answers
35 views

Does $M$ finitely presented and $N$ finitely generated imply Hom$_R(M,N)$ f.g. when $R$ is not Noetherian? [duplicate]

If $R$ is a non-Noetherian ring, $M$ is a finitely presented $R$-module, and $N$ is a finitely generated $R$-module, does it hold that Hom$_R(M,N)$ is a finitely generated $R$-module? We tried ...
1
vote
1answer
97 views

Global dimension.

What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$? What is the global dimension of ...
1
vote
1answer
119 views

augmented algebras and their morphisms

Let $R$ be a commutative unital ring and $A$ an associative (unital) $R$-algebra. What is an augmented $R$-algebra? A (unital) $R$-algebra $A$, together with a (unital) ring morphism $\varepsilon: ...
7
votes
1answer
132 views

Derived category of certain ring

I'm interested in the structure of $D^b(R)$, where $R=k[x]/(x^n)$. How one can describe this category? What is the list of indecomposable objects in this category?
2
votes
1answer
48 views

Are automorphisms of extensions trivial?

Here is a statement for abelian categories which seems so basic I'm feeling embarrassed to have to ask whether it's true in general. Suppose $0 \to A \to B \to C \to 0$ is an exact sequence with ...
5
votes
2answers
174 views

characterization of projective/injective/flat modules via $\operatorname{Hom}$ and $\otimes$

Let $R$ be a commutative unital ring and $M$ an $R$-module. Then $M$ is projective iff $\operatorname{Hom}(M,-)$ is exact, injective iff $\operatorname{Hom}(-,M)$ is exact, and flat iff $M\otimes-$ is ...