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### What is the geometric fixed points of an (equivariant) Eilenberg Maclane Spectrum?

The following was posted to math.stackexchange to no avail: http://math.stackexchange.com/questions/908756/an-exercise-in-homology-computation-what-is-the-geometric-fixed-points-of-an-e
The question ...

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60 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 $I$ ...

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73 views

### Relation between long exact sequences and Derived functors [on hold]

I know that if i have a short exact sequence of chain complexes
$$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$
then i can extend it to long exact sequence of homology groups as
...

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votes

**1**answer

100 views

### Is a inverse limit of indecomposable again indecomposable?

In truth, I do not need in the general case.
Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$.
If $\mathbb{N}$ is the ...

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130 views

### What is the abutment filtration of the second spectral sequence of hypercohomology?

I have been recently learning about spectral sequences, following mainly Illusie's notes and EGA, and I am about to write some expository notes, but there are still some points that I was not able to ...

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**1**answer

346 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

131 views

### $ mult(R/I) = d_1 \cdots d_r \quad \Rightarrow \quad f_1,\dots,f_r \quad \text{is a $R$-regular sequence?}$

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...

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99 views

### Definition of 'Koszul Ring' (in BGS)

In the paper 'Koszul Duality Patterns in Representation Theory' by Beilinson et. al, they give the definition of a Koszul Ring:
A Koszul ring is a positively graded ring $A = \bigoplus_{j \geq 0} ...

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52 views

### Change of relative base

If $k$ is a commutative ring, $A$ a $k$-algebra and $\phi: k \rightarrow k'$ is a morphism of rings then how (/under what conditions) can the relative homology functors $Ext_{A/k}(-,-)$ and ...

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142 views

### Can we “complete” model categories to compute derived functors in the usual way?

Suppose we have a functor between two model categories $F\colon \mathcal{C}\to \mathcal{D}$. Suppose we don't know that it is a part of a Quillen pair. Nevertheless, it can still happen that the ...

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**1**answer

137 views

### AdicCompletion$\dashv$Torsion adjunction on spectra?

It seems to me that in slight paraphrase the central result of the article
Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386)
(theorems 6.11 and ...

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**3**answers

235 views

### The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$

Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible ...

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35 views

### Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones

This question is strictly related to this other one.
Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let
be a diagram in $Z^0(\mathcal A)$, where the rows are ...

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**1**answer

250 views

### Is there a finitely presented group with infinite homology over $\mathbb{Q}$?

Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H_i(G, \mathbb{Q})$, or equivalently $H_i(BG, \mathbb{Q})$ (topological ...

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98 views

### Unbounded derived category that is not left-complete

Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...

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**2**answers

229 views

### Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact:
If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...

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106 views

### Are open immersions in analytic geometry transverse?

lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...

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200 views

### If an abelian category $\mathcal{A}$ has enough injectives then so does $\mathrm{Ch}^{\geq 0}(\mathcal{A})$

Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if
...

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153 views

### On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...

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**1**answer

133 views

### Interpretation of Hochschild Homology groups

In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc....
In ...

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**1**answer

183 views

### Castelnuovo Mumford Regularity

Let
$$
0 \rightarrow E_{n-1} \rightarrow ... \rightarrow E_1 \rightarrow E_0 \rightarrow I \rightarrow 0 ,
$$
be a minimal free resolution of ideal
$I$,
where
$$
E_p = ...

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**1**answer

225 views

### Motivation behind the definition of hochschild cohomology

For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...

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77 views

### generalisation of the universal coefficient spectral sequence

Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective ...

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129 views

### Smooth Affine algebras are Calabi-Yau

Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/

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79 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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124 views

### Pursuing an abelian categorical proof of the Zassenhaus Lemma

Fix an abelian category and suppose $M',M,N',N$ are sub-objects of a given object such that $M'\subset M$ and $N'\subset N$. Then there exists a canonical isomorphism
$\frac{M'+(M\bigcap ...

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193 views

### Tensor product of pullbacks of abelian categories

Does the tensor product of abelian categories commute with pullbacks? In more detail:
Let $k$ be a field. We consider $k$-linear small abelian categories ...

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82 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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131 views

### Coherent sheaves and Mitchell's embedding theorem

Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...

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108 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

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### Does semi-free behave well under totalization

Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this ...

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**1**answer

249 views

### Known norm varieties and the Bloch-Kato conjecture

The Bloch-Kato conjecture states that
$K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$.
A important part in the proof of the Bloch-Kato conjecture is to ...

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95 views

### How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture:
If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...

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**1**answer

160 views

### Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question.
Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k).
The Arason-Pfister-Hauptsatz states:
"If $\varphi$ is any anisotropic class in ...

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**1**answer

128 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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**1**answer

70 views

### Bounded algebras of finite global dimension

Let $k$ be a field, $Q$ an acyclic finite quiver and $I$ an admissible ideal of $kQ$.
I am looking for a reference for the fact that the bounded algebra $kQ/I$ has finite global dimension.

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76 views

### A factorization system on ${\rm Ch}(R)$

This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Ch}(R)$ be the category of chain complexes of $R$-modules (eventually bounded).
...

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207 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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333 views

### Homotopy factorization of morphisms of chain complexes

This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...

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**1**answer

288 views

### DG enhancements of $\ell$-adic derived categories

This question is similar in flavor to Existence of dg realization for 6 functors
Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...

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**1**answer

138 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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80 views

### How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?

First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of ...

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**1**answer

129 views

### Can André–Quillen homology detect the property of being Gorenstein?

Let $(A,m,k)$ be commutative noetherian local ring.
Can one detect if $A$ is a Gorenstein ring from the André–Quillen homologies $H_n(A,k,-)$?

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127 views

### Trivial extensions by torsion-free groups

Let $A$ be an abelian group. Recall that $A$ is
($\bullet$) a Whitehead group if $\text{Ext}(A,\mathbb Z)=0$,
($\bullet$) a free abelian group if $\text{Ext}(A,D)=0$ for every abelian group $D$.
...

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84 views

### Dimension of category of sheaves [duplicate]

Let $k$ be a field. Consider the category $Shv(\mathbb{R}^n)$ of sheaves of $k$-vector spaces on $\mathbb{R}^n$. What is the cohomological dimension $d$ of $Shv(\mathbb{R}^n)$? I know that $d \in ...

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### What is a morphism of $B_\infty$ algebra

Let $k$ be a commutative ring and $C$ a $ \mathbb{Z}$ graded $k$ module. A $B_\infty$ structure on $C$ is the datum of a differential and a multiplication on $ BC:= \bigoplus^\infty_{i =0} ...

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### Cohomological dimension of the category of sheaves

Let $X$ be an $n$-dimensional manifold. Then for any sheaf $\mathcal{F}$ on $X$, the cohomology $H^i(X; \mathcal{F})$ vanishes for $i > n$.
Let $k$ be a field, and let $\mathrm{Shv}_k(X)$ be the ...

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### Definition of modules over $C_\infty$-algebras (“commutative $A_\infty$-algebras”)

Let $\Lambda$ be a finite-dimensional associative algebra. We can think of this as an $A_\infty$-algebra with vanishing $m_i$ for $i \ne 2$ and consider $A_\infty$-modules $M$ over it. A list of ...

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### Hochschild cohomology and bar resolutions

I asked the following question on mathstack but didn't receive any comments, so I thought I'd try my luck here.
Let $A$ be an associative algebra over a field $k$. One can define $HH^n (A,A)$ as $ ...

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### 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 ...