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Results tagged with ag.algebraic-geometry
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user 1921
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
25
votes
Accepted
In what topology DM stacks are stacks
The rule of thumb is this: Your DM (or Artin) stack will be a sheaf in the fppf/fpqc topology if the condition imposed on its diagonal is fppf/fpqc local on the target ("satisfies descent").
In ot …
- 6,023
4
votes
Accepted
Homological smoothness implies projectivity?
When you write $HH_*(A)$ I will assume that you mean the $R$-linear Hochschild homology of $A$. (If you'd meant the absolute version, there would be no hope -- e.g., consider $A = R = \mathbb{Z}[x]/x …
CommunityBot
- 1
9
votes
Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper ...
Briefly re 2 and 4: the Ind-completion IndDCoh has shriek pullbacks and star pushforwards. Moreover, it has what one might call "derived h-" descent with respect to shriek pullback. This includes "d …
- 2,145
7
votes
Accepted
Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory?
Restricting to Aff is certainly enough, but Aff isn't small (there are e.g., polynomial algebras on arbitrary sets). If your DM stack is finitely presented over $k$ (which is probably good to include …
- 2,145
15
votes
Interdependence between A^1 homotopy theory and algebraic cobordism
The two topics are logically, if not morally, independent of one another. $\mathbb{A}^1$-homotopy encodes objects like motivic cohomology & it's relatives which are of interest regardless of the fram …
- 2,145
21
votes
Accepted
Where does the splitting principle come from and does it generalize
We can think of the splitting principle as a condition on a "cohomology theory" (of some sort) $E^*$, coming about when working with Chern classes for instance, and then ask: When does $E^*$ satisfy t …
- 2,145
8
votes
Flatness of relative canonical bundle
It sounds like you may want Exercise 9.7 in Hartshorne's "Residues and Duality". I paraphrase the statement:
Exercise 9.7 (RD):
Let $f: X \to B$ be a flat morphism of finite type of locally Noethe …
- 2,145
15
votes
Algebraic de Rham cohomology vs. analytic de Rham cohomology
If $X$ is smooth and proper, GAGA does in fact suffice (despite the observation that $d$ is not $\mathcal{O}_X$-linear: One obtains a comparison map of hypercohomology spectral sequences; it is an is …
- 2,145
8
votes
Gerbes for a cyclic group. (or maybe G_m too)
A bit of a response to your "Commentary":
As you point out, the failure of your construction to hit all $\mu_n$-gerbes is governed by the exact sequence
$H^1(X, \mathbb{G}_m) \to H^2(X, \mu_n) \to H …
- 2,145
25
votes
Accepted
Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective ...
The restriction-by-zero type arguments can actually be made to work, with some effort and an extra hypothesis. Suppose $X$ is locally Noetherian, $j: U \to X$ the inclusion of an open subscheme.
Let …
- 2,145