Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with homological-algebra
Search options not deleted
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.
0
votes
0
answers
41
views
Is any deformation of an acyclic complex gauge equivalent to a trivial one?
This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator …
- 9,019
6
votes
0
answers
137
views
Does homotopy equivalence between deformations of cochain complexes imply gauge equivalence?
Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator $d+\epsilon: C^{\bullet}\otimes_k m\to C^{ …
- 9,019
2
votes
1
answer
231
views
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divis...
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By …
- 9,019
2
votes
0
answers
235
views
Do we have a left adjoint of $i^*$ for a closed immersion $i$?
Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.
My questions is: can we construct a left adjoint of $i^*$ in t …
- 9,019
2
votes
1
answer
422
views
Can we compute the Hochschild cohomology for $k[x]$ through the Hochschild complex?
For an algebra $A$ we can define its Hochschild cohomology $HH^{\bullet}(A,A)$ as in this wikipedia page.
Now let $A=k[x]$ be the polynomial ring where $k$ is a field. It is well-known that $HH^{0}(A, …
- 9,019
1
vote
1
answer
218
views
What is the cohomology of the $C^{\infty}$-Koszul complex on $\mathbb{C}^2$?
First we consider the holomorphic Koszul complex on $\mathbb{C}^2$:
$$
0\to \mathcal{O}(\mathbb{C}^2)\overset{\begin{pmatrix}-z_2\\z_1\end{pmatrix}}{\to} \mathcal{O}(\mathbb{C}^2)^{\oplus 2}\overset{( …
- 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 …
- 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 …
- 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
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
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
4
votes
Counterexamples to gluing complexes of sheaves
I'm not sure if you are still interested in this question. Actually for an open cover $\{U_i\}$ and complexes of sheaves on each $U_i$, we could give the "higher" descent data and "higher" cocycle con …
- 9,019
1
vote
0
answers
56
views
Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective co...
Let $R$ be an associative ring with unit and $I$ be a complex of $R$-modules. We call $I$ is h-injective if for any acyclic complex $T$ of $R$-modules, the mapping complex $\text{Hom}_R(T,I)$ is acycl …
- 9,019
4
votes
0
answers
171
views
Do we have criteria of strict localization of a Grothendieck category?
Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under formin …
- 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