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Results tagged with homological-algebra
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user 7108
(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.
5
votes
1
answer
466
views
Uses for (Framed) E2 algebras twisted by braided monoidal structure
$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$
If $\C$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $ …
- 7,952
2
votes
0
answers
212
views
Do dg schemes have derived points?
Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced lo …
- 7,952
3
votes
Accepted
Perfect DG modules
Well by definition a DG module $X$ is perfect iff it is a direct summand of a module which is a finite iterated extension of free modules (it's then a nontrivial result that this property is the same …
- 7,952
2
votes
1
answer
166
views
Variant of co-Tor in a bimodule category
Say $\mathcal{C}$ is a strict monoidal abelian category and $A$ is a coalgebra object in $\mathcal{C}$, with left co-modules $M$ and right co-module $N$ (also in $\mathcal{C}$). Then we have a notion …
- 7,952
10
votes
2
answers
1k
views
periodic cyclic homology and tilting in the sense of Scholze
Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost s …
- 7,952
2
votes
1
answer
369
views
Nomenclature question: a morita-invariant way to say finite-dimensional?
Say $\mathcal{C}$ is the Abelian category of finitely-generated modules over some $k$-algebra $A$. Then an object $M\in \mathcal{C}$ is finite-dimensional over $k$ if and only if $\text{Hom}(P, M)$ is …
- 7,952
47
votes
1
answer
3k
views
A three-line proof of global class field theory?
There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern …
- 7,952
8
votes
2
answers
1k
views
Is there a finitely presented group with infinite homology over $\mathbb{Q}$?
Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H_i(G, \mathbb{Q})$, or equivalently $H_i(BG, \mathbb{Q})$ (topological homolog …
- 7,952
2
votes
How much of a variety can be reconstructed from codimension-zero data?
I realized the answer is almost certainly "no", so I asked a better version of this question at How much of a variety can be reconstructed from codimension-zero data?.
I think that if you take two hy …
- 7,952
14
votes
1
answer
796
views
How much of a variety can be reconstructed from codimension-zero data?
This is an improved version of this question. Maybe it should be an edit -- I'm not sure what the MO convention is.
I'm curious, more or less, how much information one can get out of the derived cate …
- 7,952
1
vote
1
answer
330
views
How much of a variety can be reconstructed from codimension-zero data?
Suppose $X$ is a nice (finite type over $\mathbb{C}$, smooth and proper if necessary) variety. Suppose we're given the $\mathbb{C}$-points of $X$ as a set and for any finite subset $S\subset X$ we kno …
- 7,952