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Results tagged with homological-algebra
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user 22810
(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.
93
votes
3
answers
10k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid m …
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28
votes
4
answers
3k
views
Yoga of six functors for group representations?
I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can a …
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24
votes
1
answer
1k
views
About the abelian category of endofunctors of $\mathsf{Vect}$
Let $k$ be a field, $\mathsf{Vect}$ the category of finite dimensional vector spaces, and $\mathsf{C} = Fun(\mathsf{Vect},\mathsf{Vect})$ the abelian category of pointed endofunctors (sending $0$ to $ …
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23
votes
3
answers
2k
views
Where does one go to learn about DG-algebras?
The theory of differential graded algebras (in char 0) and their modules has numerous applications in rational homotopy theory as well as algebraic geometry.
I'm looking for a reasonably complete ref …
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15
votes
4
answers
3k
views
Why should the tensor product of $\mathcal{D}_X$-modules over $\mathcal{O}_X$ be a $\mathcal...
Let $R$ be a regular algebra over a field $k$ of char 0. Let $D$ be its corresponding algebra of differential operators.
As in the general setting of non-commutative algebra we can tensor right $D$- …
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11
votes
0
answers
551
views
The intrinsic meaning of abelian sheaf cohomology of a category
Basically my question is:
Suppose I meet an alien mathematician which understands everything through category theory and category theory alone. How would I convince said mathematician that certain …
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11
votes
1
answer
802
views
Understanding the purely formal part of the sheaf theoretic (cohomological) framework for re...
By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf t …
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9
votes
0
answers
502
views
Categorification of definitions in the context of the derived category of quasi-coherent she...
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned b …
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8
votes
1
answer
610
views
Functorial construction of ("pre"-)spectral sequences? (Or - what is the "higher structure" ...
Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the …
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7
votes
1
answer
1k
views
Can the homological dimension of a coherent sheaf explode along a formal deformation? (is th...
Let $X_0$ be a locally noetherian scheme and $\mathcal{F}_0$ a coherent $\mathcal{O}_{X_0}$-module. Let $C$ be an artin ring with residue field $k$ and let $X \to Spec C$ be a (flat) deformation of $X …
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5
votes
1
answer
548
views
Defining hom spaces in the derived category as limits of hom spaces in the homotopy category
Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof):
A. The following isomorphisms hold:
$$\li …
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4
votes
1
answer
875
views
When does derived pullback commute with infinite products?
Let $f:X \to Y$ be a morphism of reasonable schemes (qcqs). Let $f^*: D(Y) \to D(X)$ be the pullback defined on the derived unbounded categories of quasi-coherent sheaves.
Question: When does $f^ …
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3
votes
1
answer
347
views
Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?
Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators.
Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category …
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2
votes
2
answers
210
views
Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step filtr …
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2
votes
0
answers
178
views
Modern dictionary for "old" homological terms
I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors.
Let $X = SpecA$ be a spectrum of a local ring $(A, …
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