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1d
revised Weil restriction
added 20 characters in body
1d
comment Weil restriction
It is still false, for instance, if $k$ equals $\mathbb{Q}$, if $X$ equals $\text{Spec}(\mathbb{Q})$, if $Y$ equals $\text{Spec}(\mathbb{Q}[i])$, and if $G$ equals $\mathbb{G}_m$. Since you can get the previous counterexample from this one by basechange, there cannot be an isomorphism.
1d
answered Weil restriction
Aug
24
comment Characteristic polynomials of reductive subgroup over C
For one direction, presumably it would suffice to know that every maximal torus of $H$ extends to a maximal torus of all of $\textrm{SL}(n,\mathbb{C})$, since you already know the morphism is finite on maximal tori of $\textrm{SL}(n,\mathbb{C})$.
Aug
23
comment $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$
I suggest you look up "factorial ring" / "unique factorization domain".
Aug
22
comment A lifting problem
Denoting the rank of $E'$ over $B'$ by $n'$ and the rank of $E$ over $B$ by $n$, then there is a (real) Grassmannian bundle $\text{Grass}(n',E)$ over $B$ with fibers isomorphic to the real Grassmannian $\text{Grass}(n',\mathbb{R}^n)$. Given a homotopy $\{g_t\}$ of $g_0$ to $g_1$, you are asking whether this lifts to a homotopy of the induced morphisms $\widetilde{f}_i:B'\to \text{Grass}(n',E)$.
Aug
22
answered Non-reducedness of schemes and projective morphisms(revisited)
Aug
22
comment How does the Atiyah-Singer index theorem in a relative setting related to “ringed spaces and pseudocoherent complexes of finite tor-dimension”?
This is not really what is being asked, but I also find the article of Thomason-Trobaugh a good introduction to pseudocoherence.
Aug
22
comment Pseudo-automorphisms on Fano varieties
When you write "birational map of degree > 1", what precisely do you mean? Do you mean "rational self-map" rather than a birational map?
Aug
22
comment Pseudo-automorphisms on Fano varieties
Can't you just apply Hartog's theorem / S2 extension to sections of (positive) tensor powers of the anticanonical bundle?
Aug
21
comment Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)
Just to clarify, when you say that the Ceresa cycle is homologically trivial, are you saying that the image of the cycle in etale cohomology of $J(X)\otimes_K \overline{K}$ is trivial? If so, are you asking about the cycle in etale cohomology of $J(X)$?
Aug
21
comment Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)
Could you say more about the cycle class map you are using? Are you looking at the image of the algebraic cycle in Deligne cohomology, or something like that?
Aug
21
comment Is there a higher Grothendieck ring?
A true loss ...
Aug
20
comment What is known about the Brauer group of an arithmetic surface?
You should read Grothendieck's exposes in "Dix Exposes ..."
Aug
20
comment Degree of a smooth curve in an abelian variety
Your argument is also valid for curves of higher genus. There are plenty of self-maps of $A$, and (with rare exceptions) the composition of such self-maps with a closed immersion of a curve will be another closed immersion with higher $\mathcal{L}$-degree.
Aug
20
answered Hilbert scheme of an infinitesimal neighborhood of a subvariety
Aug
19
comment Open subset of the moduli space of stable sheaves on a noetherian scheme
If your schemes are proper, then local freeness should be open, and I am sure it is written down somewhere in EGA. First of all, using "openness of flatness", the locus in the total space of your family where the universal coherent sheaf is flat is open. Thus its complement is closed. Since the family is proper over the base, the image in the base of this closed subset is also a closed subset. The open complement is the open subset that you want.
Aug
18
comment Open subset of the moduli space of stable sheaves on a noetherian scheme
How do you define the Hilbert polynomial for a sheaf with non-proper support? If you scheme is non-proper, how can any (nonzero) locally free sheaf have proper support?
Aug
18
comment Open subset of the moduli space of stable sheaves on a noetherian scheme
Is your scheme proper? How are you defining stable? For coherent sheaves on a non-proper scheme, "locally free" is not always an open condition.
Aug
17
comment FIltrations on a vector bundle on a curve
You can think of this as an example of "bend-and-break". If you denote by $(\pi:P\to X,\pi^*E\to \mathcal{O}_P(1))$ the universal invertible quotient of the pullback of $E$, then the problem is to give an upper bound on the $\mathcal{O}_P(1)$-degree of a section. Since $\omega_\pi$ is $\pi^*\text{det}(E)\otimes\mathcal{O}_P(-\text{rank}E)$, and since bend-and-break breaks sections until their anticanonical degree is at most $2$, this precisely gives that the minimal $\mathcal{O}_P(1)$-degree of a section is no greater than $(\text{deg}(E)+2g)/\text{rank}(E)$.