# Tagged Questions

**0**

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35 views

### a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...

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212 views

### Learning roadmap in Algebra [closed]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas:
a) Commutative Algebra
b) Field Theory and Galois Theory
c) Homological Algebra
My question is ...

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**1**answer

138 views

### Bounds for Betti numbers

Why the graded Betti numbers of ideal $I \subset k[x_1 , \cdots , x_n]$, are bounded by the graded Betti numbers of $\mathrm{gin}(I)$?
(Where $\mathrm{gin}(I)$, is the generic initial ideal of $I$ ...

**4**

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**1**answer

355 views

### How to prove a Proposition of Rouquier?

Proposition 7.15 in Rouquier's paper (see Publication paper) "Dimension of triangulated categories J. K-theory, 1 (2008) 193-256" as follows, for details please see Rouquier's paper (or arXiv):
``Let ...

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**1**answer

218 views

### Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be
$$
\text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!,
$$
where $ P_{M} $ denotes the Hilbert ...

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**1**answer

84 views

### Formally smooth map from a regular ring

Let $A,B$ be two commutative noetherian rings. Let $f:A\to B$ be a formally smooth homomorphism. If $A$ is a regular ring (in the sense that all its localizations are regular local rings), does this ...

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86 views

### When does a commutative DGA have a finitely generated quasi-free resolution?

Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question ...

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110 views

### Vanishing of Andre-Quillen homology and injective dimension

Let $(A,m,k)$ be a commutative local ring.
Assume that for all $n\ge 3$, the Andre-Quillen homology modules $H_n(A,k,k)$ vanish.
Does this imply that $A$ has finite injective dimension over itself?
...

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**1**answer

133 views

### Interpretations of differentials in hypercohomology spectral sequences as Yoneda products

I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups.
More ...

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208 views

### Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit ...

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**1**answer

147 views

### Is a Gorenstein ring a quotient of a local complete intersection

The title says it all - Suppose you are given a noetherian Gorenstein local ring $(A,m,k)$ of finite Krull dimension.
Does there exist a local complete intersection ring $B$ such that $A$ is a ...

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63 views

### Injective dimension over enveloping algebra

Let $k$ be a field, and let $A$ be a commutative noetherian $k$-algebra.
If a finitely generated $A$-module $M$ has finite injective dimension over $A$, does this imply that $M\otimes_k M$ has finite ...

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259 views

### Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...

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114 views

### interpretation of homology of “non-commutative Koszul complex”

Let $A = Sym^*(V)$ be a polynomial ring. The Koszul complex
$\cdots \to \wedge^2 V \otimes A \to V \otimes A \to A$
gives a resolution of the residue field $k$, so for any $A$-module $M$, the ...

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267 views

### Are dualizable modules finitely generated?

Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module.
...

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**1**answer

96 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 ...

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**1**answer

240 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 ...

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**2**answers

245 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 ...

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**1**answer

184 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 ...

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267 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 ...

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**1**answer

195 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 ...

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**1**answer

84 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 ...

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96 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 ...

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247 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 ...

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165 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 ...

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**1**answer

103 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 ...

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**1**answer

214 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
...

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184 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 ...

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**1**answer

116 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?

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**1**answer

480 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 ...

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**3**answers

389 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 ...

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**1**answer

435 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 ...

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167 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**

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311 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, ...

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**1**answer

182 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 ...

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**1**answer

136 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 ...

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**1**answer

357 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 ...

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136 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 $A$ ...

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**1**answer

209 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, ...

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369 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, ...

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**1**answer

217 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

203 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
...

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votes

**1**answer

163 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 ...

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**1**answer

307 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 ...

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154 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 ...

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**1**answer

144 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. ...

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151 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 ...

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**1**answer

605 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.
...

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**3**answers

373 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 ...

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votes

**2**answers

1k 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 ...