# 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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### Concept of an exact ideal of a module category

**6**

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### 2-shifted Poisson bracket on Lie algebra cohomology

**3**

**1**answer

### RIng that is flat over a subring as a right module but not as a left module

**5**

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### Reference request: Étale base change of differential-graded algebras

**4**

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### Relation of the first Hochschild cohomology and the outer automorphism group

**4**

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### Quillen–Suslin theorem in a more general context

**4**

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### Is there a "Kunneth isomorphism" for internal hom of chain complexes?

**8**

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### Projective $BP_\ast$-dimension of the $BP$-homology of classifying spaces of finite groups

**4**

**1**answer

### Six functor formalism for quasi-coherent $D$-modules

**4**

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### An inequality for Ext

**4**

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### Which cocycles (function classes) compute $H^*(BG, A)$ for $G$ a compact Lie group?

**3**

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### On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $

**5**

**1**answer

### Why does $p_*p^! A$ deserve to be called homology with coefficients in $A$?

**0**

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### Cohomology spectral sequence of a CW complex filtered by its skeletons

**3**

**1**answer

### A non-projective rigid object in an abelian monoidal category

**1**

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### How to deduce Künneth from its relative version (in cohomology of sheaves)

**3**

**1**answer

### K-projectivity for rings of finite homological dimension

**4**

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### A projective module over a domain that is not faithfully flat?

**9**

**1**answer

### Which finite posets are Koszul self-dual?

**3**

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### Derived category supported in a Serre subcategory of a locally noetherian category

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### If a module is second syzygy and has no free summand , then it can be defined via a minimal resolution?

**1**

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### Classification of certain modules over the polynomial ring

**1**

**1**answer

### Differential of the Twisted complex for algebraic operads

**1**

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### Composition of faithfully flat ring extensions

**4**

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### Perfect dg-modules under faithfully flat extension

**10**

**1**answer

### Does the language of fibred categories gives the commutativity of the diagrams in Residues and Duality?

**3**

**1**answer

### Faithful flatness for rings

**6**

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### How are symmetric functions related to Koszul duality?

**2**

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### de Rham cohomology of a specific ring

**2**

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### Koszul differential of the complex $\bigwedge \mathfrak{g}^*$

**2**

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### Taking the homology of a chain complex, seen as a symmetric monoidal functor

**3**

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### Quasi-isomorphism preserves group hypercohomology

**5**

**1**answer

### Unsplitting sequence of vector bundles

**1**

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### Question on the classification of Cuntz algebras via their extension groups and via their K-theory

**5**

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### L-theory periodicity

**5**

**1**answer

### L-theory of additive category

**8**

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### What exactly goes wrong with $f_!$ outside of locally compact spaces?

**3**

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### Coxeter polynomials of graphs

**4**

**1**answer

### Cup product of hypercohomologies

**2**

**1**answer

### Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change

**4**

**1**answer

### Proof of derived tensor-hom adjunction

**5**

**1**answer

### What is a Serre-smooth algebra?

**8**

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### Chain complexes split in the derived category over rings of global dimension 1

**11**

**1**answer

### Unbounded resolutions for Grothendieck abelian categories

**6**

**1**answer

### Lifting of flat lci maps

**2**

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### Edge morphism of a particular spectral sequence

**4**

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### Why do we need the axiom MS3 for localizing categories?

**1**

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### Künneth formula for local cohomology with support

**8**

**2**answers

### Verdier duality under more general conditions

**4**

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