All Questions
13 questions
52
votes
7
answers
5k
views
What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
- 1,056
12
votes
1
answer
1k
views
Fullness of pullback functor in algebraic geometry
Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can ...
- 6,131
6
votes
0
answers
607
views
On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes
Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
- 3,101
5
votes
1
answer
248
views
On the bounded derived category of sheaves with coherent cohomology
Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...
- 541
5
votes
0
answers
586
views
When is the cotangent complex perfect?
Let $X\rightarrow S$ be a proper flat morphism of schemes.
When is the cotangent complex $L_{X/S}$ perfect ?
It is well known, that for local complete intersections the cotangent complex is perfect, ...
- 51
4
votes
0
answers
113
views
Determining whether a morphism is the induced morphism?
Let $F\colon \mathcal A \to \mathcal B$ be a left exact functor between Grothendieck abelian categories. Given a morphism $f\colon A\to B$ in $\mathcal A$ and a morphism $g\colon RF(A)\to RF(B)$ in ...
- 3,079
3
votes
1
answer
378
views
Reference for comparison of heart cohomology with standard cohomology
I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations).
Let A,B be two hearts of ...
- 1,280
3
votes
0
answers
161
views
A Nakayama type of claim for countably generated modules on complex affine varieties
Let $U\subset \mathbb{A}^n_{\mathbb{C}}$ be any Zariski open affine subvarity. Let $M$ be an $\mathcal{O}(U)$-module. Suppose $M$ satisfies $M\overset{L}{\otimes}\mathbb{C}_{\mathfrak{M}}\cong 0$ for ...
- 163
2
votes
1
answer
361
views
If tensor product has finite length, do higher Tors also have finite length?
Let R be a local noetherian ring and let M, N be two finitely generated modules.
Is it true that if $M \otimes_R N$ has finite length, then $Tor_i^R(M,N)$ also has finite length for all i?
I know ...
- 1,889
1
vote
1
answer
169
views
perfect modules over polynomial algebra
This may be obvious. My question is short:
$R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...
- 117
1
vote
1
answer
124
views
Tensoring with complex of finite flat dimension in derived category
Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat ...
- 323
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
0
answers
359
views
Thickness of the category of perfect complexes with finite length homology
Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...
- 3,101