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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 ...
3
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1answer
343 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 ...
4
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1answer
126 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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0answers
98 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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0answers
51 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 ...
5
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0answers
139 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 ...
4
votes
1answer
131 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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3answers
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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0answers
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 ...
7
votes
1answer
249 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 ...
2
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0answers
97 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 ...
3
votes
2answers
221 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 $X$ ...
2
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0answers
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 ...
1
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2answers
199 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 ...
1
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0answers
148 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 ...
2
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1answer
130 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 ...
2
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1answer
182 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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votes
1answer
222 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 ...
1
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0answers
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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2answers
128 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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votes
1answer
77 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 ...
3
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0answers
122 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 ...
2
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0answers
192 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 ...
3
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0answers
80 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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0answers
129 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 ...
5
votes
0answers
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? ...
2
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1answer
73 views

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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1answer
247 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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0answers
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 ...
5
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1answer
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 ...
4
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1answer
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 ...
0
votes
1answer
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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0answers
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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2answers
331 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). ...
5
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1answer
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 ...
5
votes
1answer
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 ...
2
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0answers
79 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 ...
3
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1answer
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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0answers
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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0answers
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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0answers
127 views

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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0answers
221 views

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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0answers
74 views

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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0answers
98 views

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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0answers
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 ...
2
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1answer
155 views

Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...
2
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1answer
156 views

Exact triple yields a distinguished triangle in derived category

In Methods of Homological Algebra before Proposition III.3.5 there is a short comment: "The next proposition shows that any exact triple [of complexes] can be competed to a distinguished triangle". ...
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0answers
109 views

Reference needed: Homology of the blow-up

Given an algebraic variety $X$ over the complex numbers. Let $V$ be a subvariety of $X$ and $\pi_V \colon X' \rightarrow X$ be the blow-up of $X$ at $V$. It is posible in general to compute the ...
3
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1answer
110 views

Is the space of smooth functions with compact support a DF space?

Is there a good criterion, when a (nuclear) LF space is DF (DFN)? What are references to find out about that? If not, is there a known way, to construct a projective resolution of LF spaces? Prosmans ...