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Apr
19
comment “Parameterising” extensions $0\to E\to W\to F\to 0$ by $\mathbb P(H^1(E\otimes F^*))$?
I think you might be interested in the first section of the article by Mireille Martin-Deschamps, "Propriétés de descente des variétés à fibré cotangent ample, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 39–64.
Feb
28
comment Extension of ample vector bundles is ample
@Count Dracula. Sorry I commented too quickly. The fact that the extension of an ample bundle by another ample bundle is ample is not very difficult to prove (see Jason's reference) but it is a subtle fact that in char. 0, a non-trivial extension of an ample bundle by a trivial bundle is ample. This is also proven in Lazarsfeld's book (and it false in char. p>0).
Feb
26
comment Elementary proof of Chevalley's Theorem on constructible sets
I would say that the basic input in (one of the) proofs is Noether's normalisation lemma. I don't think that you can make it more elementary than that.
Feb
26
comment Extension of ample vector bundles is ample
@abx This is no true in positive characteristic. For the general theory, see the first part of Martin-Dechamps, Propriétés de descente des variétés à fibré cotangent ample." Annales de l'institut Fourier 34.3 (1984).
Feb
13
comment Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$
What about $\alpha_p:={\rm Spec}K[X]/X^p$?
Jan
10
comment Which sheaves satisfy cohomological purity?
About this, see middle of p. 124 of the book by Kiehl-Weissauer on Weil Conjectures, Perverse sheaves etc.
Sep
13
awarded  Necromancer
Aug
22
awarded  Yearling
Aug
3
comment Is this kind of scheme integral?
For smthg related to your question, see Prop 4.3.8 p. 137 in Liu book (this shows eg that if you assume that $X$ is flat over $R$ and $X_\eta$ is integral, then so is $X$).
Jun
25
accepted CM abelian varieties over the rationals
Jun
25
comment CM abelian varieties over the rationals
Nice answer... thank you for that. I wonder why this argument does not appear in any of the usual references for the theory of complex multiplication (unless I missed something ?).
Jun
24
asked CM abelian varieties over the rationals
Jun
17
comment Projective bundle formula
(you means $X={\bf P}(E)$ in your formula) If $X$ is smooth and proper, then by Hodge decomposition the formula is the same as for singular cohomology (see Griffiths-Harris, p. 606). If $X$ is smooth but not necessarily proper then the cohomology ring might be infinite-dimensional.
May
16
comment Reference request: Cohomology of Elliptic Curves
Why is the image of $\mu_\nu$ a normal subgroup of ${\rm Gal}(K^{\rm ab}|K)$ ? Sorry if I missed something obvious.
Apr
2
comment Connected vs Irreducible Subvarieties
What about normality ?
Feb
25
comment A proper smooth surface is projective
Several classical references are given on II, §4, p. 105 of Hartshorne's book.
Jan
25
comment References for general Hasse-Weil zeta function
I do not know any other coherent reference than "facteurs locaux"....
Jan
5
comment automorphisms of local rings vs local change of coordinates
I think you haven't actually stated which two objects you want to coincide.
Dec
14
comment Quotient of a (non-linear) algebraic group by a closed subgroup
Here is another reference: Raynaud, 'Passage au quotient par une relation d'équivalence plate', Proceedings of a conference on local fields (Driebergen, 1967), p. 82 Ex. a) (i).
Nov
29
comment Fermat's last theorem over larger fields
Sorry, your are right, Mazur's conjecture does not quite cover your question (but it gives a partial answer). The conjecture in Zarhin-Parshin corresponds precisely to what follows from Mordell-Lang + Mazur's conjecture.