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        Stack Exchange Network

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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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        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 ...
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        votes
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        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 ...
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        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}(\...
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        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 ...
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        votes
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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 ...
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        votes
        1answer
        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, ...
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        1answer
        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 ...
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        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 ...
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        $Ext_{A^e}^i(D(A),A)$ 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>...
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        1answer
        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 ...
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        votes
        1answer
        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 ...
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        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 ...
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        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 ...
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        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 $H^3$, ...
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        votes
        2answers
        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 ...

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        山西福彩快乐十分钟
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