Questions tagged [homological-algebra]

(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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-5
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0answers
42 views

How to use this graph to determine intervals on which f is? [closed]

I have to have explanations of the calculations made to get the answer
2
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0answers
84 views

Functors with adjoints

I want to find a functor between abelian categories, which is faithful but not full. And this functor has left and right adjoint. I want to know a nontrivial example,which is not inducecd by a ring ...
-1
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37 views

Question about modules and the link with functor Hom [closed]

Let $R$ be a ring with unit and $\lbrace X_i\rbrace_{i\in I}$ be a family of $R$-modules Is $Hom(\prod_{i\in I}X_i,Q/Z)=\oplus_{i\in I}Hom(X_i,Q/Z)$?
2
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0answers
56 views

Tate module of elliptic curves; Commuting Hom functor and tensor product in the second coordinate

Let $\Lambda$ be the Iwasawa Algebra of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_{cyc}$ of $\mathbb{Q}$. Let $\widehat{\Lambda}$ be its Pontryagin dual (i.e the ...
4
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0answers
118 views

What does go wrong in Cellular homology if one considers projective limits of celullar complexes instead of CW-complexes?

Consider a nice topological space $X$ (e.g. the 3-sphere) and consider inside a decreasing sequence of compact subsets $(K_n)_{n\in\mathbb N}$ such that $K_\infty:=\bigcap_{n\in \mathbb N} K_n$ is 0-...
4
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0answers
74 views

Perfect modules for the Beilinson algebra

The Beilinson algebra $A=A_n$ is a finite dimensional quiver algebra that is derived equivalent to the category of coherent sheaves of $\mathcal{P}^n$. See for example https://link.springer.com/...
7
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3answers
357 views

Dimension of classifying space of a group

If $N$ is a normal subgroup of a group $G$ such that $G/N= \mathbb{Z}$. Suppose that the classifying space of $G$ is a finite CW-complex of dimension $n$. Does it follow that the classifying space of $...
6
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1answer
249 views

Which abelian groups are $\varprojlim^1$ groups?

Question 1: Let $\mathcal A$ be an abelian group. Does there exist an inverse system $(A^n)_{n \in \mathbb N} = (\cdots \to A^n \to A^{n-1} \to \cdots \to A^0)$ such that $\varprojlim^1 A^\bullet \...
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98 views

Tensor product and cohomology of dg categories

Let $\mathcal{C}$ and $\mathcal{D}$ be dg categories over a field $k$ of characteristic zero. Then one can form their tensor product $\mathcal{C} \otimes \mathcal{D}$: the objects of the tensor ...
6
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1answer
112 views

Endomorphism ring of trivial source modules for abelian p-groups

Bernhard Böhmler  (who is also on MO) and myself had the following idea: Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
2
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141 views

Understanding why $\pi_{3}(N) = \mathbb{Z}_{(2n, n^2)}$?

I am trying to understand the paper Arkowitz and Golasinski - Co-$H$ structures on Moore spaces of type $(G, 2)$: In section 4, titled "Homotopy elements of finite order", the authors say $\...
8
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0answers
62 views

Rlim versus tensor product

Let $R$ be a coherent ring, and let $(M_n)_{n\geq 1}$ and $(N_n)_{n\geq 1}$ be two inverse systems of finitely generated flat $R$-modules. If $R^1 \lim M_n=R^1 \lim N_n = 0$, is it true also that $R^1 ...
4
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52 views

Lie algebra “semi” coinvariants

In the process of my research, I've come across the need to understand the following construction: Let $\mathfrak{g}$ be a (finite-dimensional) complex Lie algebra, $\beta\in \mathfrak{g}^*$ a Lie ...
2
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1answer
264 views

Explicit automorphism map of ${\rm Spin}(8;\mathbb{R})$, ${\rm SO}(8;\mathbb{R})$, ${\rm PSO}(8;\mathbb{R})$

$\DeclareMathOperator{\SO}{\mathrm{SO}}\DeclareMathOperator{\Spin}{\mathrm{Spin}}\DeclareMathOperator{\Inn}{\mathrm{Inn}}\DeclareMathOperator{\Out}{\mathrm{Out}}\DeclareMathOperator{\Aut}{\mathrm{Aut}}...
3
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42 views

Questions on piecewise hereditary algebras

Let $A$ be a finite dimensional quiver algebra over a field $k$ that is quasi-tilted and representation-finite (this implies that $A$ is a tilted algebra). Assume that the Coxeter polynomial of $A$ is ...
2
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0answers
96 views

Are flasque sheaves exactly the retracts of “canonically” flasque sheaves?

Let $X$ be a topological space. Let $X^\delta$ denote the space whose elements are the points of $X$, and which is equipped with the discrete topology. There is a continuous map $i : X^\delta\to X$ ...
5
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0answers
131 views

Smoothness of a variety implies homological smoothness of DbCoh

I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions. It ...
11
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1answer
377 views

Embedding of a derived category into another derived category

I am considering the following two cases: Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...
2
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0answers
53 views

Bimodule Ext for Dynkin path algebras

Let $A=kQ$ be a path algebra of Dynkin type $Q$ and $B=A^{op} \otimes_k A$ the enveloping algebra of $A$. Note that $mod-B$ is just the category of $A$-bimodule and $A$ is a $B$-module. For a B-module ...
6
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1answer
203 views

An identity for Ext for rings

Let $A$ be a two-sided noetherian ring (which we should assume to be Gorenstein first so that everything is well defined, otherwise it is only well defined up to a conjecture, which states that every ...
3
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0answers
57 views

On Ext-duals of injective modules for commutative rings

Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$. Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$...
8
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1answer
288 views

Generalising the Union-closed sets conjecture from lattice to a larger class of posets

(edit: I decided to simplify the question and only pose it for bounded posets first) The Union-closed sets conjecture is equivalent for lattices P to: There exists a join-irreducible element $a$ with ...
3
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0answers
62 views

The union-closed sets conjecture for finite dimensional algebras

Say a finite dimensional algebra $A$ satisfies the right UC-condition if there exists an indecomposable projective module $P$ of $A$ such that $\operatorname{injdim}(\operatorname{top}(P))=1$ and $P$ ...
4
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0answers
264 views

A homological algebra approach to the Union-closed sets conjecture

I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
11
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322 views

Cohomology for distributive lattices

(edit: Here is a PDF with 5 examples of distributive lattices $L$ with the grades of every point of $L$: https://docdro.id/cUIOb2T . Another class of examples are the divisor lattice where the grade ...
5
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1answer
106 views

Covariant splittings of Hopf algebra projections

What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $...
5
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0answers
100 views

An intelligent ant living on a symmetric quiver algebra - Does it have a way to find out whether it lives on a trivial extension?

For a given algebra $B$ over a field $K$ the trivial extension $T(B)$ of $B$ is defined as follows: The underlying vectorspace is $T(B)=B \oplus D(B)$ where $D(B)=Hom_K(B,K)$ and the multiplication is ...
5
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1answer
356 views

What's the relationship between a $E_2$-Hochschild Cohomology module and a D-module?

Let's say for simplicity $A$ is a smooth algebra over a field $k$ ($A$ and $k$ are discrete commutative rings but from now on we are fully derived), and we will consider the $E_2$ algebra $HH^{\bullet}...
5
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0answers
63 views

Reference on two numbers associated to a module of finite homological dimension

Let $A$ be a finite dimensional algebra over a field $K$ with a module $M$ which has finite projective dimension and finite injective dimension. Let $n \geq 1$. Let $(P_i)$ be a minimal projective ...
7
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0answers
198 views

Why is Hochschild homology interesting if its cohomology groups are infinite-dimensional?

I am trying to understand Hochschild homology, in particular the Hochschild–Kostant–Rosenberg theorem. As far as I understand this result gives an isomorphism between the algebraic (Kähler) ...
11
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1answer
566 views

Any group is a quotient of an acyclic group?

As far as I know, for any group $G$ there exists an acyclic group $H$ such that $G$ is a subgroup of $H$. I am wondering about the dual situation. Is any group $A$ a quotient of an acyclic group $B$ ...
3
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0answers
58 views

Quiver algebras of Dynkin type

Let $kQ$ be one of the Dynin path algebras of type $A_n , D_n $ or $E_i$ for $i=6,7,8$. Question 1: How many (up to isomorphism) quiver algebras are there that are derived equivalent to $kQ$? ...
4
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0answers
43 views

Algebras derived equivalent to a hereditary algebra

Let $A=KQ/I$ be a quiver algebra with relations in $I$ having only coefficients 1 or -1. This implies that $A=FQ/I$ is defined over any other field $F$ (possibly of even another characteristic). ...
8
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239 views

Dyck paths of Dynkin type

(The conjecture is a homological algebra question, but question 2 is a pure combinatorics question given that the conjecture is true) A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...
13
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1answer
597 views

Who introduced the abstract definition of a DGA?

Differential graded algebras, or DGAs, are a basic object of study in many areas of modern mathematics. While they were present (implicitly at least) since the start of modern differential geometry, I ...
7
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0answers
98 views

When is an algebra derived indecomposable?

Call a finite dimensional (acyclic) quiver $K$-algebra A derived indecomposable in case $A$ is not derived equivalent to an algebra of the form $B \otimes_K C$. For example when the number of simples ...
4
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0answers
201 views

Finding local algebra and relations lottery

This can be seen as an attempt for a mini Polymath project on homological properties of (local) finite dimensional algebras. You only need to know what a finite dimensional algebra is and have GAP to ...
5
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1answer
125 views

Frobenius algebras from symmetric polynomials

Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
4
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0answers
138 views

Commutative algebras associated to simple Lie algebras

In Section 2 of the article https://www.sciencedirect.com/science/article/pii/S0021869307000385, the authors study the center $Z=Z_Q$ of certain preprojective like algebras associated to the simply ...
32
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1answer
360 views

Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$

Mac Lane homology is a homology theory for (not necessarily commutative) rings. Given a ring $A$, Eilenberg and Mac Lane define its cubical construction $QA$ to be a certain connective chain complex, ...
8
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2answers
952 views

Original proof of Hilbert's syzygy theorem

Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either: the theory of projective dimension and change of rings (plus a step ...
2
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0answers
65 views

Cone of a morphism of complexes that are concentrated in degree $0$ and $1$

Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...
3
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0answers
175 views

$E_{\infty}$-algebras à la Lurie

Let $D(\mathbb{F}_p)$ and $\mathcal{D}(\mathbb{F}_p)$ be the derived category and derived infinity-category of cochain complexes of $\mathbb{F}_p$-vector spaces. If $A$ is a sheaf of cdgas over $\...
3
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1answer
103 views

Smallness condition for augmented algebras

I'm not sure this question is research level question. Sorry in advance. Hypothesis $k$ is a commutative ring. $A$ is an augmented $k$-algebra. $A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...
1
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0answers
25 views

Coxeter period of representation-finite selfinjective algebras

Let $A$ be a representation-finite selfinjective (quiver) algebra, that we assume to be connected and non-semisimple. Define the Coxeter period $p_A$ of $A$ to be equal to the period of the Coxeter ...
0
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1answer
136 views

Commutativity of $\operatorname{Hom}$ and $\varprojlim$

Is it true the formula$\newcommand{\Hom}{\operatorname{Hom}}$ $\Hom(\varprojlim G_\alpha ;I) \simeq \varinjlim \Hom(G_\alpha ;I),$ if the group $I$ is injective? We can assume that $G_\alpha$ is ...
1
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0answers
115 views

To see that the fundamental class of a local complete intersection is independent of choice of regular sequence

In SGA 4½ ‘Cycle,’ Grothendieck defines (among other things) the fundamental class of a local complete intersection $Y\subset X$ ($X$ simply a noetherian scheme) of codimension $c$ locally as the cup-...
2
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0answers
32 views

Eilenberg–Zilber-type theorem for Map([n],A), where the degeneracy maps for [n] are forgotten

The following statement should be immediately implied by Eilenberg–Zilber theorem if the sequences $(i_0,\ldots,i_k)$ below are only monotone. But I need the strict monotone version which I believe to ...
2
votes
2answers
444 views

On the link between homology and homotopy

In the last semester I learned homological algebra and higher category theory/homotopy theory. But I am kind of confused when I try to really understand the link between the two subjects (this is ...
1
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0answers
62 views

Extension class of the pullback to a fiber of the pushforward of the universal extension of sheaves

Given $f: X \rightarrow Y$ a flat morphism between smooth projective varieties over $\mathbb{C}$ together with $F$ and $G$, two coherent $\mathcal{O}_X$-modules flat over $Y$. Assume the relative $E:=\...

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