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(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

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35 views

Units in the (stable) center of a Frobenius algebra [duplicate]

Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
2
votes
1answer
126 views

Why only consider decreasing filtrations on cochain complexes?

When reading various literature on spectral sequences one always comes across two setups: A chain complex with an increasing filtration A cochain complex with a decreasing filtration My question is ...
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vote
0answers
42 views

Direct limit of complexes from cocycle property up to homotopy

Suppose that we have a directed set $(I, \leq)$, and a set of maps \begin{equation} f_{i,j} : C^*(A_i) \to C^*(A_j), \quad i \leq j, \end{equation} of singular cochain complexes of topological spaces $...
8
votes
1answer
339 views

What is the interpretation of the Gerstenhaber bracket?

The homology of an $E_2$-algebra is a Gerstenhaber algebra. How precisely is the Gerstenhaber structure related to the $E_2$-structure? Obviously, the Gerstenhaber product is the commutative product ...
3
votes
1answer
155 views

Higher Extension Group Question

Suppose we have an associative unital ring $R$, and we have an $R$-module $M$ with a length 3 socle filtration, i.e. write $$soc(M) \text{ for the socle of } M,$$ $$soc^2(M) \text{ for the preimage ...
3
votes
1answer
90 views

Identity for $Ext^1$ for special algebras

Let $A$ be a finite dimensional algebra and assume all modules are also finite dimensional. A module $M$ is said to have dominant dimension at least $n$ in case the term $I_i$ for $i=0,1,...,n-1$ are ...
3
votes
0answers
251 views

Proof of 10.4.6 in Weibel [closed]

Let $I$ be a bounded below complex of injectives in some Abelian category, and $Z$ any bounded below complex. Suppose $u:I\to Z$ is a quasi isomorphism. We want to proof that $u$ is split injective up ...
3
votes
1answer
123 views

Invertible bimodules which are isomorphic in the stable module category

I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ ...
3
votes
1answer
346 views

What bigrading is used in this spectral sequence?

I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ...
12
votes
1answer
290 views

Obstruction of spin-c structure and the generalized Wu manifods

Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the $$ H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
2
votes
1answer
114 views

$Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$

Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the ...
4
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0answers
98 views

Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
7
votes
0answers
71 views

On constructible Hall algebra and instantons

I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
2
votes
1answer
160 views

Homology of bar complex vs homology of indecomposables

$\require{AMScd}$ Background: This question is about the bar and cobar constructions, and their relationship with the indecomposables of a dg-algebra. A brief summary of the bar and cobar ...
3
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1answer
166 views

Flat Model Structure on $\mathbf{Ch}(\mathbf{Mod}(\mathcal{O}_X))$ computes pullback / pushforward

Given a ringed space $(X,\mathcal{O})$ of can construct the flat model structure on chain complexes of $\mathcal{O}$-modules: Weak equivalences are quasi-isomorphisms The fibrations are epimorphisms ...
7
votes
2answers
271 views

Generalized Smith Theorem for the torsion of cokernels

Let $R$ be a (commutative) domain and let $Q$ be its fraction field. Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$. Let $I_k=(\det \...
7
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1answer
134 views

Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber?

The question is a special case of a previous question. Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection ...
5
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2answers
160 views

Is $C^{\infty}(M)$ a projective Frechet $C^{\infty}(N)$-module for a smooth map $M\to N$ between compact smooth manifolds?

Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\...
3
votes
0answers
78 views

Conjecture on tilting modules for an Auslander algebra

On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism classes of modules, occurring as the $i$-th summand of ...
11
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0answers
149 views

Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?

Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let $$AW_{A,B}\colon N_\ast(...
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0answers
70 views

Explicit description of periodic map $S : HC_{i} \to HC_{i-2}$ for dg and $A_\infty$ algebras

Let $A$ some associative unital $k$-algebra, let $HC_*(A)$ is cyclic homology of $A$ and $HH_*(A)$ is hochschild homology of $A$. Then we have Connes exact sequence: $$ ... \xrightarrow[]{} HH_n(A) \...
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1answer
378 views

“Rotated” version of the Atiyah-Hirzebruch spectral sequence

Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...
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138 views

What are the periodic Dyck paths?

Let $C$ be the Cartan matrix of a finite dimensional algebra $A$ with finite global dimension, then the Coxeter matrix is defined as $M=-C^{-1}C^T$. $A$ is called periodic in case $M^k=id$ for some $k ...
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0answers
66 views

On $Ext_A^2(S,A)$ for algebras $A$

For Nakayama algebras $A$ with a linear quiver and at most 7 points (which are 196 algebras) the following was true: $max \{ dim(Ext_A^2(S,A)) | S $ simple $\}= max \{ dim(Ext_A^2(X,A)) | X $ ...
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0answers
158 views

Does the first Tachikawa conjecture imply the Nakayama conjecture?

Let $A$ be a non-selfinjective algebra with dominant dimension at least one and $Af$ a minimal faithful projective-injective left $A$-module (but we use right modules in the following). The Nakayama ...
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0answers
52 views

Injective dimension of $A/AfA$

Let $A$ be an algebra of finite global dimension and $Af$ the direct sum of all indecomposable projective-injective left $A$-modules. Using right modules, left $g$ denote the injective dimension of $A/...
5
votes
2answers
175 views

Path algebras are formally smooth

In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2). I have two questions. First, how to show this claim and ...
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1answer
154 views

Categorification of monotone maps via tilting modules?

It is well known that for the algebra of $n \times n$-upper triangular matrices over a field the number of tilting modules is equal to the Catalan number $C_n$. This is just the (hereditary) Nakayama ...
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1answer
138 views

Eilenberg-Watts theorem for the derived category

Let $A$ and $B$ be $k$-algebras. And for convenience let's say $k$ is a field and both $A$ and $B$ are finite-dimensional. A well known theorem independently discovered by Eilenberg and Watts states ...
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1answer
231 views

Bisimplicial sets and homology

I'm not sure about the following result: Theorem (?): Let $f_{\bullet,\bullet}: X_{\bullet,\bullet}\rightarrow Y_{\bullet,\bullet}$ a map of bisimplicial sets such that for any natural number $n$, $...
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votes
2answers
181 views

Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data

Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step ...
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1answer
200 views

A precise definition of contractible Banach algebras

I asked this question at MSE but I did not received any answer. So I ask it here at MO I am sorry if this question is elementary: What is a precise definition of a contractible Banach ...
5
votes
1answer
209 views

Is the sheaf associated to a differential structure of a specific type?

On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...
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58 views

Isomorphisms between different completions of a module

Given a $R$-module $A$, suppose that $A$ has two different unbounded filtrations $\cdots \subseteq M_{i+1} \subseteq M_i\subseteq M_{i-1}\subseteq \cdots $ $\cdots \subseteq N_{i+1} \subseteq N_i\...
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1answer
128 views

Surjectivity of a map on inverse limits

(The following is crossposted from Math.SE, where the question did not receive any answers.) I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...
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0answers
70 views

Extending Beauville-Bogomolov orthogonal decomposition from variety to scheme

I'm seeking to understand the de Rham cohomology of a Hilbert scheme $K3^{[4]}$ of the K3 surface. By Beauville, this 8-dimensional compact manifold is Kaehler, irreducible, holomorphically symplectic ...
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1answer
74 views

In search of disconnected indecomposable self-injective finite-dimensional algebras

I wanted to know if it is possible to construct an indecomposable self-injective finite-dimensional algebra $\Lambda$ whose Auslander-Reiten quiver $\Gamma_\Lambda$ is not connected. I'd love to see ...
4
votes
1answer
135 views

Hochschild homology of a category of modules over an algebra

Suppose $A$ is an algebra over some field, say the complex numbers if that helps. Then we can consider the category $\mathbf{C}_A$ of finite-dimensional modules over $A$. This category can be seen as ...
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0answers
32 views

Periodic algebras from periodic simple modules

As continuation of the previous thread Example to periodic symmetric algebras , I have the following question: Is there a counterexample to the following: Let A be a symmetric algebra and W the ...
2
votes
1answer
74 views

Example to periodic symmetric algebras

In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric. Is there an easy counterexample for the following: In case $A$ is additionally a periodic ...
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0answers
47 views

When is a stable endomorphism ring selfinjective?

Let $A$ be a local symmetric finite dimensional algebra and $M$ an $A$-module with at least two non-isomorphic indecomposable non-projective summands. In case $\Omega^1(M) \cong M$ in the stable ...
3
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1answer
65 views

The kernel of the morphism from the Picard group to the stable Picard group of a self-injective algebra

Let $\Lambda$ be a finite-dimensional self-injective algebra (over an algebraically closed field, if necessary). Let $Pic(\Lambda)$ be the group of natural isomorphism classes of self-equivalences $...
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0answers
43 views

On the projective dimension of $D(A)$ as a bimodule

Let $A$ be a finite dimensional algebra over a field $K$ given by quiver with relations and assume for simplicity that it has finite global dimension. It is well known that the projective dimension of ...
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0answers
180 views

Definition of Non-commutative de-Rham-Cohomology

Let $A$ be a (not necessarily commutative, associative) $k$-algebra. The bimodule of non-commutative one-forms $\Omega^1_A$ is the free $A$-bimodule generated by symbols $da$, $a \in A$, subject to ...
2
votes
1answer
152 views

Injective dimension of $A/AeA$

Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $e$ the idempotent such that $eA$ is the direct sum of all indecomposable projective-injective $A$-modules. Do we have $g=...
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votes
0answers
184 views

On the Choice Content of Carathéodory's Conformal Mapping Theorem

The Schoenflies theorem, as a variant of the well-known Jordan curve theorem, states that the interior and the exterior planar regions determined by a simple closed curve (aka Jordan curve) in $\...
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0answers
72 views

Super global dimension

Let $R$ be a ring of finite global dimension. Define the small super global dimension as $sgl(R):= \sup \{ pd(X)+id(X) | X \in mod-R$ and indecomposable $\}$. Here $id(X)$ stands for the injective ...
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43 views

Questions on holonomic modules

An Auslander-Gorenstein ring is a noetherian ring R that has finite left and right selfinjective dimension and such that $fd(I_i) \leq i$ for all $i \geq 0$ for an injective coresolution of the ...
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1answer
188 views

question about infinite global dimension

I ask a question about $\prod k$ in Mathematics about several days.https://math.stackexchange.com/q/2766054/453628. And I have the following question: 1.What is the global dimension about $\prod k$? ...
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votes
3answers
346 views

Intuition behind the Canonical Projective Resolution of a Quiver Representation

Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective ...