# Tagged Questions

**10**

votes

**1**answer

169 views

### Homological criteria for finite generation and finite presentation of modules?

(I'm new here; if I'm doing something wrong please help me out.)
In short, my question is: There are some results on the behavior of finite generation and finite presentation in exact sequences (of ...

**2**

votes

**2**answers

194 views

### local cohomology mayer-vietoris sequence

(I originally asked this question on Math.SE here. As suggested on meta.MathOverflow (posting an unanswered Math.SE question on MathOverflow), I've waited about a week before reposting it here. Note ...

**2**

votes

**1**answer

95 views

### Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...

**5**

votes

**2**answers

219 views

### Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$?
Is there any ...

**3**

votes

**1**answer

176 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 ...

**2**

votes

**1**answer

59 views

### Kernel of the induced map of the wedge product

Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of ...

**2**

votes

**0**answers

91 views

### syzygy of a generalized cohen-macaulay module

Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized ...

**1**

vote

**1**answer

239 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 ...

**3**

votes

**0**answers

154 views

### Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let ...

**0**

votes

**1**answer

90 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 ...

**4**

votes

**1**answer

205 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
...

**4**

votes

**1**answer

173 views

### Algebra structure $Tor(A,A)$

This is a question i asked on math.stackexchange but i didn't get any answer.
Let $A$ be algebra over commutative ring $k$ and $P_{\bullet}=(P_i,d_i)\rightarrow A$, $k$ projective resolution. Then we ...

**4**

votes

**1**answer

113 views

### Localizations of hereditary rings

It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?

**5**

votes

**1**answer

357 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 ...

**6**

votes

**3**answers

349 views

### zero homology of augmented Koszul complex implies the sequence is regular?

Let $A$ be a Noetherian ring, $M$ a finite $A$-module and $I=(y_1,\cdots,y_n)$ an ideal of $A$ such that $M \neq IM$. Denote by $H_i(y_1,\cdots,y_n;M)$ the homology at dimension $i$ of the augmented ...

**13**

votes

**1**answer

369 views

### History of Koszul complex

This is a question about history of commutative algebra. I'm curios why Koszul complex from commutative algebra is called Koszul complex? All Koszul's early papers are about Lie algebras and Lie ...

**4**

votes

**0**answers

151 views

### What is known about this short exact sequence in Lie algebra cohomology?

In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact
$ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...

**4**

votes

**1**answer

282 views

### Computing cotangent complex

I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?
In my precise situation, ...

**0**

votes

**1**answer

169 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

**1**answer

130 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 ...

**4**

votes

**1**answer

329 views

### example of Local cohomology

Let $S=k[x_1,...,x_n]$ be a polynomial ring over field $k$ with maximal ideal $m=(x_1,...,x_n)$. I wanna make a $3$-dimensional $S$-module $M$ such that $H^0_m(M)=H^1_m(M)=0$ and $H^2_m(M)\neq 0$ be ...

**3**

votes

**0**answers

127 views

### Is Hochschild cohomology finitely generated?

I am sorry if this is a naive question.
Let $k$ be a field, and let $A$ be a finitely generated commutative $k$-algebra.
Let $M$ be a finite $A$-module.
Consider the Hochschild cohomologies of $A$ ...

**6**

votes

**1**answer

199 views

### When are MCM ideals principal?

Suppose $(R,\mathfrak m, k)$ is a $d$-dimensional Cohen-Macaulay local ring with canonical module $\omega_R$ and $d>1$. Suppose $I\subset R$ is an ideal which is MCM (=maximal Cohen-Macaulay, ...

**7**

votes

**0**answers

349 views

### Grothendieck trace on local cohomology?

Let R be an augmented regular local ring over a field $k$ with maximal ideal m. There is the Grothendieck residue symbol:
$$Res: H^n_m(\Omega^n) \to k$$
If $k=\mathbb{C}$ and $R$ is affine space, ...

**3**

votes

**1**answer

207 views

### finiteness of local cohomology

Well-known Theorem:
Let $a$ be an ideal of the noetherian ring $R$ and let $M$ be a finitely generated
$R$-module. Let $i \in \Bbb N_0$ be such that $H^j_a
(M)$ is finitely generated for all $j < ...

**1**

vote

**2**answers

201 views

### Decomposition of a quotient module

Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let
$f:R^2 \rightarrow R^2$ denote the map given by the matrix
$$M=\begin{pmatrix}
v & y \\
x & z
...

**2**

votes

**1**answer

142 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 ...

**2**

votes

**1**answer

269 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 ...

**2**

votes

**0**answers

130 views

### Coordinate free Koszul-Tate resolution

Tate's original construction of the Koszul-Tate resolution involved choosing cocycles representing the cohomology to be killed. Where is it written in a coordinate free treatment, perhaps via a ...

**2**

votes

**1**answer

129 views

### Compatibility of connecting homomorphisms for Tor/Ext

This is a simple question about Tor and Ext functors.
Let $R$ be a commutative ring, and let $0 \to M' \to M \to M'' \to 0$ and $0 \to N' \to N \to N'' \to 0$ be short exact sequences of $R$-modules. ...

**2**

votes

**0**answers

141 views

### About free resolutions of graded commutative algebras

Hi, I'm having troubles in adapting certain algebraic constructions to graded cases.
We know that if $A$ is a commutative ring and $a_1,...,a_k$ are elements on $A$, there is a construction of the ...

**1**

vote

**1**answer

598 views

### question on an exercise on homological algebra?

Suppose R has finite global dimension n, N is a f.g. module, F is a free module and Ext^n(N,F) is not equal to zero, then Ext^n(N, R) is not trival either.
Note, HERE R is not Noetherian necessarily.
...

**2**

votes

**3**answers

362 views

### Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.
Let us assume we are given a group $G$ and a ...

**5**

votes

**2**answers

981 views

### Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...

**3**

votes

**1**answer

367 views

### Vanishing of Tor

Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module. Let $x_1,...,x_t$ be an $M$-regular sequence and $I = (x_1,...,x_t)$. Is it true that
...

**6**

votes

**3**answers

409 views

### A lost lemma about periodicity in a grid of long exact sequences?

This is a question about finding references and hopefully a larger
context for a lemma in homological algebra I proved recently.
The motivation is to understand properties of characteristic
classes of ...

**4**

votes

**1**answer

549 views

### (geometric/intuitive) interpretation of ext

Hi folkz,
In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, but I don't have a feeling about the meaning of ext.
Is there a ...

**2**

votes

**1**answer

152 views

### Proving indecomposability of special modules

Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
On page 6 there are the definitions (in a diagrammatical way) of some $A_n$ modules, whereupon ...

**0**

votes

**1**answer

175 views

### Representation dimension of a special algebra

Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
I've come across a piece of information, which I don't understand, and wanted to ask, if I ...

**4**

votes

**1**answer

139 views

### Turning a resolution of an algebra into a resolution of its field of fractions

Is there a way to turn a free resolution of a $k$-algebra $A$ into a resolution of the field of fractions $Q(A)$?
Specifically, I'm interested in the ring of polynomials in two variables: $A = ...

**2**

votes

**2**answers

454 views

### vanishing of local cohomology $H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$

Edited:
I guess
$$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$
We know that if $\operatorname{Supp} H^i_I(M)\subseteq V(I)\cap \operatorname{Supp}(M)$, then
$$\operatorname{Supp} ...

**0**

votes

**2**answers

169 views

### Projective dimension of cohomology over regular rings

Suppose $R$ is a regular ring and $F^{\bullet}: 0\to F^0 \to F^1 \to \dots \to F^d \to 0$ is a complex of finite rank free $R$-modules. Is is true that $\mathrm{projdim}H^i(F^{\bullet}) \leq d$ for ...

**3**

votes

**0**answers

131 views

### Minimal Koszul-Tate resolutions

In what generality of commutative associative algebras does there exist a minimal Koszul-Tate resolution? Or what is the most general condition known?

**11**

votes

**3**answers

666 views

### Serre's theorem about regularity and homological dimension

One of the nicest results I know of is (Auslander-Buchsbaum-)Serre's theorem asserting that a (commutative!) local ring is regular iff it has finite global dimensional.
I'd like to ask a somewhat ...

**3**

votes

**1**answer

158 views

### Do the classes of cotorsion and strongly cotorsion modules coincide?

Cotorsion modules
A module M is called cotorsion if for all flat modules X, $Ext_R^1(X,M)=0$ .
Strongly cotorsion modules
M is called strongly cotorsion if for all modules X of finite flat ...

**12**

votes

**2**answers

1k views

### Why are injective modules more complicated than projective modules?

For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. For example, for finitely generated modules over a noetherian ...

**10**

votes

**1**answer

617 views

### When tensor reflects exact sequences?

Is there a characterization of modules $N$ for which the functor $N\otimes-$ reflects exact sequences?

**4**

votes

**2**answers

498 views

### Why depth, dimension, etc?

Recently I came across a paper by Stephen McAdam "GRADE SCHEMES AND GRADE FUNCTIONS" (Transactions of the American Mathematical Society, Vol. 288, No. 2 (Apr., 1985), pp.
563-590). So far I have just ...

**14**

votes

**0**answers

347 views

### Refinement of concept of support of a module

My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...

**12**

votes

**3**answers

641 views

### Injective dimension of graded-injective modules.

In "Existence theorems..." Van den Bergh proposes the following "pleasant excercise in homological algebra":
Let $A$ be a connected graded noetherian $k$-algebra (that is, $\mathbb N$-graded with ...