All Questions
13 questions
4
votes
1
answer
227
views
Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
4
votes
2
answers
285
views
Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 412
11
votes
1
answer
513
views
When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 412
3
votes
0
answers
120
views
Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
4
votes
1
answer
267
views
A particular morphism being zero in the singularity category
Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
- 361
3
votes
0
answers
106
views
Multiplication map by a ring element on an object vs. all its suspensions in singularity category
Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
- 412
1
vote
0
answers
111
views
Kunneth formula for hypercohomology
Let $A_{\bullet}$ and $B_{\bullet}$ be two bounded complexes of sheaves over a variety $X$. Is there a Kunneth-like formula for the hypercohomology of the tensor product $A_{\bullet}\otimes B_{\bullet}...
- 494
1
vote
1
answer
149
views
Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?
Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
- 73
8
votes
1
answer
413
views
Chain complexes split in the derived category over rings of global dimension 1
Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post When do chain complexes decompose as a direct sum?) that any chain complex over $R$ is equivalent ...
- 937
4
votes
1
answer
558
views
derived tensor product and finite projective dimension
Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules.
Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ ...
- 361
6
votes
1
answer
175
views
Relative Ext of Avramov-Martsinkovsky as a derived Hom
Avramov-Martsinkovsky (http://mathserver.neu.edu/~martsinkovsky/Relative.pdf) have defined an exotic version of Ext between two modules over (for simplicity) Gorenstein rings. The basic idea of their ...
- 3,758
10
votes
0
answers
312
views
Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
user20948
16
votes
1
answer
753
views
When is every "solid" perfect complex faithful?
Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy.
...
- 466