(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.

**5**

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52 views

### Derived invariant acyclic algebras

Call a connected quiver algebra $A=KQ/I$ (finite quiver Q and admissible ideal I) derived-invariant in case every quiver algebra derived equivalent to $A$ is even isomorphic to $A$.
For example local ...

**3**

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**0**answers

109 views

### How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...

**3**

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**0**answers

97 views

### Induced Homomorphism on Cohomology of Symmetric Group 3

For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...

**3**

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**0**answers

56 views

### $\Omega^2(S) \cong \tau(S)$ for simple modules

Let $A$ be an Artin algebra with $\Omega^2(S) \cong \tau(S)$ for each simple module $S$. Is $A$ a quasi-Frobenius (=selfinjective) algebra?
Here $\tau$ denotes the Auslander-Reiten translate, which is ...

**7**

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80 views

### DG functors along which contractions can be lifted

For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...

**6**

votes

**1**answer

366 views

### “Exactness” of operadic cohomology

There are two somewhat widely known theorems which say
if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_{Harr}(A, A) \to H^*_{Hoch}(A, ...

**6**

votes

**1**answer

230 views

### Understanding the functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point.
Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...

**5**

votes

**1**answer

172 views

### Manifold generators of O-bordism invariants

If I understand correctly, I can obtain the $O$-cobordism group of
$$
\Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4,
$$
The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...

**2**

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**0**answers

42 views

### $Ext_{A^e}^i(D(A),A)$ for finite dimensional algebras

Let $A$ be a finite dimensional non-semisimple algebra over a field $K$ with enveloping algebra $A^e=A^{op} \otimes_K A$. Let $D(A)=Hom_K(A,K)$.
Question:
Is there always a positive integer $i>...

**4**

votes

**1**answer

101 views

### Reference for isomorphism $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$

Let $A$ be a finite dimensional algebra and $X,Y$ indecomposable modules and $n \geq 2$.
We know that $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$.
Now such an isomorphism should be given by ...

**9**

votes

**1**answer

337 views

### Hochschild homology with coefficients in a certain bimodule

Let $A$ be a finite-dimensional $k$-algebra and $U$ and $V$ two finite-dimensional projective $A$-modules (maybe neither the finiteness nor projectivity has to play a role, but these requirements are ...

**8**

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**0**answers

87 views

### Is there a homological interpretation for the cokernel of the kernel of a map between complexes induced by tensor product?

Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$.
Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...

**5**

votes

**0**answers

81 views

### Questions on group and Nakayama algebras from a book

Recall that a Nakayama algebra (also called serial algebras in the literature) is an algebra such that every indecomposable module has a unique composition series.
In the book "Classical artinian ...

**4**

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**0**answers

203 views

### Generalized Postnikov square

Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...

**8**

votes

**2**answers

197 views

### Künneth formulas/theorem for bordism groups and cobordisms?

We are familiar with Künneth theorem:
The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...