All Questions
39 questions
14
votes
0
answers
1k
views
Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
- 1,514
9
votes
1
answer
692
views
Equivalence of definitions of Cohen-Macaulay type
I know that the Cohen-Macaulay type has these two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim_k ...
- 113
8
votes
2
answers
2k
views
Original proof of Hilbert's syzygy theorem
Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either:
the theory of projective dimension and change of rings (plus a step ...
- 14.7k
8
votes
1
answer
356
views
Homological conjectures for finite dimensional commutative algebras
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
- 26.5k
7
votes
2
answers
917
views
Is this exact sequence known?
$\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\...
- 14.1k
7
votes
1
answer
911
views
The Mittag-Leffler condition as necessary and sufficient
Let $A_1\leftarrow A_2\leftarrow A_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p_{ij}\colon A_j\to A_i$, $j\ge i$. The derived functor of projective limit $\...
- 15.6k
7
votes
0
answers
228
views
Terminology for vanishing of Hochschild homology with symmetric coefficients?
In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying $H^n(A,M)=0$...
- 25.8k
5
votes
1
answer
225
views
Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
- 26.5k
5
votes
1
answer
674
views
Resolution of a module as an $A_\infty$ module over resolution of an algebra
The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...
- 1,545
5
votes
0
answers
87
views
Reference request: Étale base change of differential-graded algebras
I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here.
I'm looking for a reference for the following fact, which I've recently stumbled upon:
...
- 171
5
votes
0
answers
132
views
On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber
This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...
- 51
5
votes
0
answers
331
views
Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
4
votes
1
answer
227
views
Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
4
votes
1
answer
472
views
Original sources for two theorems by Bass, Matlis and Papp
It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...
- 6,700
4
votes
1
answer
158
views
Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences
Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.
Then, given an exact sequence $\eta: 0\to X \to Z_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $...
- 642
4
votes
0
answers
218
views
map of Koszul cohomology
I am reading paper "Standard systems of parameters and their blowing-up rings", J. Reine Angew. Math. 344 (1983), 201--220 of Peter Schenzel. In proof of Theorem 3.9, page 209-the second diagram, he ...
- 2,773
3
votes
3
answers
714
views
Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
3
votes
1
answer
591
views
Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...
- 31
3
votes
1
answer
173
views
$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
3
votes
1
answer
496
views
Regular ring is smooth when the field is perfect
Take $A$ a (not necessarily local) commutative algebra over a field $k$ which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ...
3
votes
1
answer
153
views
Artinian Tor modules (Reference request)
I am looking for a reference for the following basic fact:
Let $R$ be a noetherian ring, let $M$ be an artinian $R$-module, let $N$ be a finitely generated $R$-module, and let $i\in\mathbb{N}$. ...
- 6,700
3
votes
1
answer
422
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 ...
- 165
3
votes
0
answers
248
views
Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
3
votes
1
answer
177
views
Quiver and relations for ADE singularities in dimension one
Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...
- 26.5k
3
votes
0
answers
69
views
On Ext-duals of injective modules for commutative rings
Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$.
Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$...
- 26.5k
2
votes
2
answers
2k
views
Torsion-free modules over a general ring
I want to know how to prove that a torsion-free module over a general ring is flat. In Lectures on Rings and Modules, T.Y. Lam proves this in the case where your ring is an integral domain. Can you ...
- 17
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
2
votes
1
answer
194
views
Reference request: example of pointwise free module which is not projective
I am looking for a reference for examples showing the following phenomena:
Let $A$ be a commutative noetherian ring, and let $F$ be an $A$-module such that for all $p \in Spec(A)$ it holds that $F_p$ ...
2
votes
0
answers
124
views
Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 21
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
2
votes
0
answers
93
views
Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ...
- 480
2
votes
0
answers
128
views
On the generalization of a Cech-to-sheaf type spectral sequence
Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
- 541
1
vote
1
answer
379
views
Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring
In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for ...
- 642
1
vote
1
answer
359
views
Cohomology after completion
I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if it'...
- 1,049
1
vote
0
answers
106
views
Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
1
vote
0
answers
108
views
On Serre's condition and singular locus of determinantal rings
Let $R$ be a Commutative Noetherian ring. Let $\mathbf X:=[X_{ij}]_{1\le i \le r, 1 \le s \le t}$ be a matrix of indeterminates. Let $t>1$ be an integer, and $I_t(\mathbf X)$ denote the ideal in $...
- 413
1
vote
0
answers
132
views
On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)
I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
- 642
1
vote
0
answers
161
views
Projective dimension of a principal ideal
Let $R$ be a polynomial ring, $I$ a homogeneous ideal, and let $\operatorname{pd}(R/I)$ denote its projective dimension. Is there a characterization of homogeneous elements $a\in R\setminus I$ for ...
- 878
1
vote
0
answers
113
views
Reference request. The adjunction $\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$
We have the adjunction
$$\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$$
where $CDGA$ is the category of commutative diffferential graded algebras and $CA$ is the category of ...
- 1,615