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5h

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Failure of universal flatification
There is an appropriate Quot space that is proper over $S$ whenever the support of $M$ is proper over $S$ and $M$ is locally finitely presented. So counterexamples should have the support of $M$ not proper (or, in the nonNoetherian setting, $M$ not locally finitely presented). 
8h

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Some examples of non trivial principal bundles
@AliTaghavi: "where is my mistake?" There are two group homomorphisms. There is $q:\mathbb{S}^1\to \mathbb{S}^1\times \mathbb{S}^1$, $x\mapsto (x,0)$. For any $\mathbb{S}^1$bundle $E$ on a manifold $M$, there is an associated pushout $\mathbb{S}^1\times \mathbb{S}^1$bundle, $q_*E$. Next there is a group homomorphism $r:\mathbb{S}^1\times \mathbb{S}^1\to \mathbb{S}^1$, $(x,y)\mapsto x$. So for any $\mathbb{S}^1\times \mathbb{S}^1$bundle $F$, there is an associated $\mathbb{S}^1$bundle $r_*F$. Of course $r_*(q_*E)$ equals $E$. Thus, if $q_*E$ is trivial, then also $E$ is trivial. 
1d

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Pull back of a semistable vector bundle to a product is semistable?
Yes, the proof is very similar to the proof of Bertini's theorem. I believe that Gabber  Liu  Lorenzini prove a version of this in a very general setting. However, the case you need is an easy parameter count: with respect to a projective embedding $X\times X \hookrightarrow \mathbb{P}^n$, just consider complete intersections in the linear system $\mathcal{O}(d)$ such that the Hilbert function of the fibers of the projection evaluated at $d$ are strictly larger than the dimension of $X$. 
1d

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Reference request: category of sheaves of Omodules with coherent cohomology
In ThomasonTrobaugh, this is Proposition 2.3.1(e), which is attributed there to SGA 6, Expose II. 
1d

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Reference request: category of sheaves of Omodules with coherent cohomology
Isn't this in ThomasonTrobaugh? 
1d

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does Gorenstein imply reduced?
No, Gorenstein does not imply reduced. The scheme $X=\text{Spec}\ k[x]/\langle x^2 \rangle$ is Gorenstein (and projective), yet not reduced. If your scheme is generically reduced and CohenMacaulay, then it is reduced. 
2d

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Some examples of non trivial principal bundles
@AliTaghavi: "How this group homomorphism $x\mapsto (x,1)$ define an action by $S^1\times S^1$"?" I am afraid that I do not understand your question. Are you asking what is the definition of the group homomorphism $q:\mathbb{S}^1\to \mathbb{S}^1\times \mathbb{S}^1$? Are you asking about the definition of the associated pushout principal $G$bundle associated to a principal $H$bundle and a group homomorphism $H\to G$? For either of those questions, I recommend that you look this up in a reference. 
2d

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Some examples of non trivial principal bundles
@AliTaghavi: Please look again at the link you included: that math.stackexchange question is only about the onetorus, not the twotorus. 
2d

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Some examples of non trivial principal bundles
. . . The pullback to $\mathbb{S}^1\times \mathbb{S}^1$ of the nontrivial $\mathbb{S}^1$bundle over $\mathbb{S}^2$ has first Chern class that is nonzero. Now, turn this into a principal $\mathbb{S}^1 \times \mathbb{S}^1$bundle via the standard group homomorphism $\mathbb{S}^1 \to \mathbb{S}^1 \times \mathbb{S}^1$, $x\mapsto (x,0)$. This $\mathbb{S}^1\times \mathbb{S}^1$bundle over $\mathbb{S}^1\times \mathbb{S}^1$ is nontrivial, yet the fibers give a foliation as you request. 
2d

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Some examples of non trivial principal bundles
Here is an example of Thomas Rot's suggestion. Begin with the Hopf fibration, $\mathbb{S}^1 \hookrightarrow \mathbb{S}^3 \twoheadrightarrow \mathbb{S}^2$. The basic invariant of this $\mathbb{S}^1$principal bundle is its first Chern class, which is a generator of $H^2(\mathbb{S}^2,\mathbb{Z})\cong \mathbb{Z}$. Let $f:\mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^2$ be a continuous map of positive topological degree, e.g., the standard $2$to$1$ map obtained by the involution $(x,y) \sim (x,y)$ with respect to the standard group structure on $\mathbb{S}^1\times \mathbb{S}^1$ . . . 
Nov
28 
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Smooth algebraic curves through smooth points
(The original formulation of Bertini's theorem did include base points, and it is worth learning.) This result is proved by induction on the dimension of your smooth, quasiprojective variety $X$. If the dimension equals $1$, you are done. Thus, by way of induction, assume that $\text{dim}(X)$ is $>1$ and the result is known for smaller dimensions. Consider the linear system of hyperplane sections $D$ that contain a specified point $p$. A general member $D$ is smooth away from $p$, by Bertini. Also $D$ is smooth at $p$ so long as it does not contain the tangent space of $X$ at $p$. 
Nov
28 
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Smooth algebraic curves through smooth points
Yes, that follows from Bertini's theorem. 
Nov
28 
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Is every commutative ring a quotient ring of an integral domain?
Yes: just take $A$ to be the free commutative $\mathbb{Z}$algebra generated by the underlying set of $R$ with its canonical $\mathbb{Z}$algebra homomorphism to $R$. 
Nov
28 
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Homeomorphism of fibers of holomorphic maps
That is not true for holomorphic morphisms of complex analytic varieties. For $X=\mathbb{C}\times D^*\setminus\{(0,1/n) : n\in \mathbb{Z}_{>0} \}$ with projection to $D^*$, that fails. 
Nov
27 
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Is stable map space $\overline{M_{0,n}}(\mathbb{P}^n,d)$ is irreducible for all $n,d$?
I would like to say: Yi Zhu used and extended the results of KimPandharipande in a serious way to prove rational simple connectedness (not only connectedness of $\overline{M}_{0,n}(X,\beta)$, but also simple connectedness) for all projective homogeneous spaces $X$. This had been proved for projective homogeneous spaces of Picard number $1$ by Xuhua He, but a full typefree proof of Serre's Conjecture II (the current proofs are very typedependent) will require the case of arbitrary $X$ (and probably also wonderful compactifications as well). 
Nov
27 
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Is stable map space $\overline{M_{0,n}}(\mathbb{P}^n,d)$ is irreducible for all $n,d$?
This is implicit in Kontsevich's original paper on "Enumeration of Rational Curves via Torus Actions" (in my opinion). It is explicit in an article by Bumsig Kim and Rahul Pandharipande, "The Connectedness of the Moduli Space of Maps to Homogeneous Spaces". It also follows from Kuznetsov's paper, "Laumon's Resolution ..." 
Nov
26 
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Show that $G$ a group?
That sounds like a homework exercise. 
Nov
26 
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Fpoints of product of closed subgroups vs. product of Fpoints, F a local field, reference?
That is a good point. 
Nov
26 
answered  Fpoints of product of closed subgroups vs. product of Fpoints, F a local field, reference? 
Nov
26 
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Fpoints of product of closed subgroups vs. product of Fpoints, F a local field, reference?
Let $K/F$ be a finite Galois extension, and let $m\in M(K)$, $h\in H(K)$ be elements such that $m\cdot h^{1}$ equals $g$, for $g\in G(F)$. Consider the (nonAbelian) $1$cocycle for $\Gamma = \text{Aut}(K/F)$ by $(m^{1}\cdot \gamma(m))_{\gamma\in \Gamma}$. This equals the $1$cocycle $(h^{1}\cdot \gamma(h))_{\gamma\in \Gamma}$. By your hypothesis, this $1$cocycle in $M\cap H$ is a coboundary, i.e., there exists $r\in (M\cap H)(K)$ such that $mr^{1}$ and $rh^{1}$ are Galois invariant. These are the $F$points of $M$, resp. $H$, that multiply to $g$. 