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Results tagged with homological-algebra
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user 2191
(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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Any examples known of $K^b(B)$ localized by a set of morphisms (i.e. of complexes of length 1)?
I would like to understand the following setting: for an additive $B$ localize $K^b(B)$ by a set of $B$-morphisms (i.e. by a thick triangulated subcategory generated by some set of two-term complexes) …
- 16.9k
10
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2
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566
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Does a triangulated category that possesses a subcategory $B$ of generators with no extensio...
Suppose that a triangulated category $C$ contains a full additive subcategory $B$ of (strong) generators (i.e. there does not exist a proper strict triangulated subcategory $C'\subset C$ that contains …
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9
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Chain homotopy: Why du+ud and not du+vd?
Amusingly, I have considered the category of complexes where maps of the form du+vd are killed in: Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for mo …
- 16.9k
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A conservative, non faithful functor between triangulated categories
You can start with the derived category of graded polarizable Hodge structures or with the category of Hodge modules (over a complex variety $X$). These categories possess natural weight structures (a …
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11
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1
answer
263
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What are the projective dimensions of big fraction fields?
Let $A$ be an integral domain, $B$ is its fraction field. Can the projective dimension of the $A$-module $B$ be greater than $1$? This surely cannot happen if the spectrum of $A$ is countable (since t …
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1
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Recollement of multiple $t$-structures
I didn't check that thoroughfully, but it seems that in the ("abstract") setting you are interested in there exist (exact) "projections" $j^{i*}:D\to D^i$ and $j^{i!}:D\to D^i$ ($D$ is the "big" trian …
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4
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3
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531
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When an exact embedding of abelian categories induces a full embedding of their derived cate...
Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also?
I would be interested in any necessary or sufficient c …
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3
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distinguished triangles and cohomology
You might be interested in the paper: Vaknin A., Virtual Triangles// K-Theory, 22 (2001), no. 2, 161--197.
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3
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3
answers
2k
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Homology or cohomology?
How do people call an additive functor from a triangulated category $C$ to an abelian one that converts distinguished triangles into long exact sequences. Does one usually call a covariant functor o …
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6
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1
answer
535
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Exceptional collections of objects in topological triangulated categories?
People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in …
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What is the purpose of section 3 of BBD?
I am not quite sure that this question is appropriate for Mathoverflow, yet I would be deeply grateful for any hint: what happens in section 3 of Beilinson A., Bernstein J., Deligne P., Faisceaux perv …
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Accepted
Is the dual of a compact generator also a compact generator of the derived category of a var...
Let me try to sketch an argument (though I am not quite sure in details).
A theorem of Neeman implies that a compact object $M$ in a compactly generated triangulated category $T$ is a generator if an …
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13
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Is there an additive functor between abelian categories which isn't exact in the middle?
As far as I remember, there is an important example of a functor that transforms mono- and epimorphisms into mono- and epimorphisms, respectively, but is not half-exact; this is the functor of interm …
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6
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1
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240
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For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My questi …
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DG enhancements of $\ell$-adic derived categories
Q1. So, I suggest you the following plan of the proof.
Note that any Verdier localization of a triangulated category possessing a differential graded enhancement possesses a differential graded enha …
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