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

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### Why only consider decreasing filtrations on cochain complexes?

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### Direct limit of complexes from cocycle property up to homotopy

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### What is the interpretation of the Gerstenhaber bracket?

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### Higher Extension Group Question

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### Identity for $Ext^1$ for special algebras

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### Proof of 10.4.6 in Weibel [closed]

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### Invertible bimodules which are isomorphic in the stable module category

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### What bigrading is used in this spectral sequence?

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### Obstruction of spin-c structure and the generalized Wu manifods

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

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

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### On constructible Hall algebra and instantons

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### Homology of bar complex vs homology of indecomposables

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

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### Generalized Smith Theorem for the torsion of cokernels

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

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

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### Conjecture on tilting modules for an Auslander algebra

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

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

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### “Rotated” version of the Atiyah-Hirzebruch spectral sequence

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### What are the periodic Dyck paths?

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

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### Does the first Tachikawa conjecture imply the Nakayama conjecture?

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### Injective dimension of $A/AfA$

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### Path algebras are formally smooth

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### Categorification of monotone maps via tilting modules?

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### Eilenberg-Watts theorem for the derived category

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### Bisimplicial sets and homology

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

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### A precise definition of contractible Banach algebras

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

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### Isomorphisms between different completions of a module

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### Surjectivity of a map on inverse limits

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### Extending Beauville-Bogomolov orthogonal decomposition from variety to scheme

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

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### Hochschild homology of a category of modules over an algebra

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### Periodic algebras from periodic simple modules

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### Example to periodic symmetric algebras

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### When is a stable endomorphism ring selfinjective?

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

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

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### Definition of Non-commutative de-Rham-Cohomology

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### Injective dimension of $A/AeA$

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### On the Choice Content of Carathéodory's Conformal Mapping Theorem

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### Super global dimension

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### Questions on holonomic modules

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### question about infinite global dimension

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