All Questions
9 questions
20
votes
2
answers
9k
views
Does module Hom commute with tensor product in the second variable?
Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...
- 1,906
10
votes
2
answers
2k
views
Lemma 1 from Beilinson's "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", intuition?
Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: \mathcal{...
user61522
5
votes
1
answer
313
views
Is the dual of a compact generator also a compact generator of the derived category of a variety?
Let $X$ be a variety (or more generally a quasi-compact, separated scheme) and $D(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with quasi-coherent cohomologies. Let $\mathcal{E}$...
- 9,019
4
votes
1
answer
694
views
Proper mapping theorem
Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules.
In particular, the coherent ...
user120812
4
votes
0
answers
258
views
Generators of unbounded derived categories of (quasi-)coherent sheaves
An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
- 263
3
votes
2
answers
1k
views
Some questions on vanishing of Ext sheaves
Let $X$ be a complex manifold of dimension $3$ and $\mathcal{E}$ be a coherent sheaf such that $\dim supp(\mathcal{E})=1$. In this situation, I would like to know why we have the Ext sheaf $\mathcal{...
- 31
3
votes
0
answers
169
views
equivalence of two categories
I am new to algebraic geometry and category theory. I am wondering about the following functor is equivalence of categories or not.
Let $X$ be irreducible scheme and $x$ be its unique generic point. ...
- 613
2
votes
0
answers
314
views
Derived category of coherent sheaves with a codimension $\geq$ 1 support
Let $X$ be some smooth algebraic variety. I would like to understand the relation between the following two categories:
$D^b_{cd,1}\text{Coh}(X) \subset D^b\text{Coh}(X)$: the full subcategory of the ...
- 988
1
vote
0
answers
125
views
Realization of a formal duality isomorphism via integration
This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of $X$...
- 1,331