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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.
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 …
- 9,019
1
vote
1
answer
325
views
Do we have the following isomorphism for $\mathcal{Ext}$?
Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times X$ be the diagonal embedding and $p_1: X\times X\rightarrow X, ~p_2: X\times X\rightarrow X$ be the projections to t …
- 9,019
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
2
votes
1
answer
565
views
Why $k[x,y]$ is not a formally smooth algebra?
We could talk about the formal smoothness of an algebra. See for example Ginzburg's lecture notes For an associative algebra $A$ over a field $k$ we define
$$
D(A)=T(A+\bar{A})/(\bar{ab}=a\bar{b}+\bar …
- 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
3
votes
1
answer
205
views
Is the image of a idempotent morphism in $\mathcal{K}(\mathcal{A})$ defined in the naive way?
Let $\mathcal{A}$ be an abelian category and $\mathcal{K}(\mathcal{A})$ be the homotopy category of chain complexes in $\mathcal{A}$. It is well-known that $\mathcal{K}(\mathcal{A})$ is idempotent com …
- 9,019
0
votes
0
answers
114
views
Is the pull back of a compact generator under field extension again a compact generator?
Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums. A compact ob …
- 9,019
2
votes
0
answers
858
views
How to prove that any perfect complex on an affine scheme is strictly perfect?
Let $(X,\mathcal{A})$ be a ringed space. A complex $\mathcal{S}^{\bullet}$ of $\mathcal{A}$-modules is $\textit{perfect}$ if for any point $x\in X$, there exists an open neighborhood $U$ of $x$ and a …
- 9,019
6
votes
0
answers
167
views
Could we characterize injective objects in the category of $G$-equivariant sheaves?
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 act on $X$ continuously from the left …
- 9,019
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 …
- 9,019
3
votes
0
answers
523
views
Could we prove the flat base change theorem for cohomology via injective resolution?
Let $X$ be a quasi-separated scheme over a base ring $A$ and $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $A\to B$ be a flat morphism and $X^{\prime}:=X\times_{\text{spec}(A)}\text{sp …
- 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
5
votes
2
answers
285
views
Is $C^{\infty}(M)$ a projective Frechet $C^{\infty}(N)$-module for a smooth map $M\to N$ bet...
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 of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\ …
- 9,019