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Results tagged with homological-algebra
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user 24965
(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.
15
votes
4
answers
2k
views
What is the relation between the Lie bracket on $TX$ as commutator and that coming from the ...
Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence o …
- 9,019
15
votes
1
answer
3k
views
Is a locally free sheaf projective in the category of $\mathcal{O}_X$-modules when $X$ is an...
Let $X$ be an affine scheme and $\mathcal{E}$ a finitely generated locally free sheaf on $X$. It is obvious that $\mathcal{E}$ is a projective object in the category Qcoh$(X)$ since we can pass to rin …
- 9,019
14
votes
3
answers
758
views
Is $C^{\infty}(\mathbb{R}^{m+n})$ a flat module over $C^{\infty}(\mathbb{R}^{m})$?
For $m>0$ we consider the ring $C^{\infty}(\mathbb{R}^{m})$ of smooth functions on $\mathbb{R}^{m}$. For $n>0$ we consider the projection $\mathbb{R}^{m+n}\to \mathbb{R}^{m}$ hence $C^{\infty}(\mathbb …
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12
votes
1
answer
638
views
An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal
We know that for a curve $X$, any object $\mathcal{E}^{\bullet}$ in the derived category $D^b_{\text{coh}}(X)$ is formal, i.e. $\mathcal{E}^{\bullet}$ is quasi-isomporphic to the direct sum of its co …
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11
votes
1
answer
385
views
Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommu...
Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where
$$
C_n(A):=A^{\otimes n+1}
…
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10
votes
0
answers
204
views
Does the category of $G$-equivariant sheaves have enough injectives?
The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which …
- 9,019
9
votes
0
answers
196
views
Does a morphism which induces an isomorphism between Hochschild homology also induce an isom...
In a 1998 paper by B. Keller, the author consider the following problem in Section 1.4: Let $k$ be a commutative ring and $X$ a scheme over $k$. We can consider the cyclic homology as well as the Ho …
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8
votes
1
answer
318
views
Do chain homotopic maps between dg-algebras induce "same" maps on dg-modules
Let $A$ and $B$ be two dg-algebras over a field $k$. Let $f, g: A\to B$ be two maps between dg-algebras. We call $f$ and $g$ chain homotopic if there exists a degree $-1$ map $h: A\to B$ such that $f- …
- 9,019
8
votes
1
answer
2k
views
Is the derived category of perfect complexes idempotent complete?
Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any idempo …
- 9,019
7
votes
1
answer
219
views
Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bund...
The question is a special case of a previous question.
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection …
- 9,019
7
votes
0
answers
198
views
Is $\text{DGA}^{-}$ a monoidal model category?
Let $\text{DGA}^{-}$ denote the category of non-positively graded differential graded algebras with differentials of degree $+1$. It is well-known that $\text{DGA}^{-}$ has a model structure with
W …
- 9,019
7
votes
0
answers
96
views
Is the double orthogonal of a localizing Serre subcategory still a localizing Serre subcateg...
Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under i …
- 9,019
6
votes
1
answer
221
views
Is the existence of $A_{\infty}$-inverse a consequence of Homotopy Transfer Theorem?
Let $k$ be a field of characteristic $0$ and $(A,d_A)$, $(B,d_B)$ be two differential graded (dg) algebras over $k$. Let $f: A\to B$ be a closed degree $0$ map of dg-algebras and $g: B\to A$ be a map …
- 9,019
6
votes
0
answers
207
views
The meaning of a "subcomplex" of the Cartan-Eilenberg of a Lie algebra
Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex
$(\wedge^{\cdot} \mathfrak{g}^* ,\text{d}_{C …
- 9,019
6
votes
1
answer
247
views
Is the hom in derived category of a dg-algebra compatible with base field extension?
Let $k$ be a field and $A$ be an ordinary $k$-algebra. Let $M$ and $N$ be two left $A$-modules then we could consider the Ext group $\mathrm{Ext}^i_A(M,N)$ which is actually a $k$-vector space. Let $l …
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