All Questions
Tagged with local-rings homological-algebra
25 questions
2
votes
0
answers
66
views
Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
- 412
4
votes
2
answers
286
views
Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 412
1
vote
0
answers
136
views
Local cohomology and image of $1$ under the canonical map from Ext to local cohomology
Let $R$ be a commutative Noetherian local ring, and $S$ be an $R$-algebra. Let $x_1,\dots,x_t$ be elements, in the maximal ideal of $R$, which is a regular sequence on both $R$ and $S$, and let $I$ be ...
- 412
2
votes
1
answer
112
views
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
1
vote
0
answers
85
views
Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
- 480
5
votes
1
answer
191
views
Are module finite algebras over semiperfect rings again semiperfect?
Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
- 412
2
votes
1
answer
171
views
Is $\mathrm{Hom}_R(M,R)\neq \mathfrak m \mathrm{Hom}_R(M,R)$ if $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and $M$ be an $R$-module such that $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$. Then, is it true that $\mathrm{Hom}_R(M,R)\neq \...
- 413
3
votes
1
answer
239
views
Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated
Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
- 413
4
votes
1
answer
160
views
DG algebra structure on minimal free resolution of modules over regular local ring
Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...
- 412
1
vote
0
answers
132
views
A question concerning cancellation of ideals
I am working on a number theory project, and at one stage, I encounter a commutative algebra problem. Vaguely speaking, my hope is to show that two ideals are equal. Now I shall explain the data I am ...
- 11
2
votes
0
answers
111
views
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}...
- 361
4
votes
1
answer
224
views
For a local complete intersection ring $(R,\mathfrak m)$ with $\mathfrak m^3=0\ne \mathfrak m^2$, $\mathfrak m$ can be generated by two elements
Let $(R,\mathfrak m,k)$ be a local complete intersection ring with $\mathfrak m^3=0\ne \mathfrak m^2$. As $0\ne \mathfrak m^2 \subseteq \text{soc}(R)$ and $R$ is Gorenstein, so we get $\mathfrak m^2 =\...
- 73
1
vote
1
answer
106
views
On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings
Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n) $ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...
- 73
1
vote
1
answer
228
views
On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal
$\DeclareMathOperator\Ext{Ext}$Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak ...
- 412
3
votes
1
answer
422
views
Vanishing of $\operatorname{Ext}_R(\operatorname{Tr} M,N)$ and freeness criteria
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$I am investigating the interplay between freeness ...
- 183
3
votes
3
answers
486
views
On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\Hom_R(\_,R)$. There is a natural map
\begin{align} \Phi_M: M \...
- 361
4
votes
2
answers
458
views
Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$
Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ...
- 279
4
votes
1
answer
158
views
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 ...
- 43.2k
3
votes
1
answer
239
views
commutative ring satisfying descending chain condition on radical ideals
Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
- 1,209
2
votes
0
answers
178
views
Modern dictionary for "old" homological terms
I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors.
Let $X = SpecA$ be a spectrum of a local ring $(A,...
- 7,799
2
votes
0
answers
145
views
Question about Ext$^1$ in local commutative algebras
Given a local commutative (commutative only if needed...) selfinjective (non-semisimple) finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{...
- 26.5k
5
votes
0
answers
209
views
Ext^1 for a local finite dimensional selfinjective algebra
Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here:
http://arxiv.org/pdf/1609.00588.pdf.
There it is ...
- 26.5k
5
votes
1
answer
415
views
Inverse limit of Gorenstein local rings is again Gorenstein?
If we have the system of surjective ring homomorphisms
$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$
for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put
$R^{\...
- 51
1
vote
2
answers
552
views
A Question About Free Resolutions
I would warmly appreciate it if someone could tell me whether the following question has an affirmative answer. I am new to the field of commutative algebra, so I am simply trying to fill in some (...
- 816
2
votes
1
answer
286
views
Modules with small support have big depth - reference wanted
Hello,
I would appreciate an exact reference / proof of the following fact, which I am almost able to prove, but not really:
Let $A$ be a regular Noetherian comm. ring, of finite Krull dimension. ...
- 5,562