All Questions

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Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?
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Does the language suggest hard average cases?

\begin{equation*} \begin{aligned} \ \\ L & = \{ D \, | \, permuted \, C \, on \, its \, submatrices \, C_{i} \, \} \ \\ \ \\ C & = [\,C_{1}\, C_{2}\, ...\, C_{k-1} \, C_{k} \, C_{k+1} \, ... ...
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Davenport constant of class group

Denote $\mathsf{C(\Delta)}$ where $\mathsf{\Delta=b^2-4ac<0}$ with $\mathsf{gcd(a,b,c)=1}$ be class group of all equivalence classes of integral quadratic forms with discriminant $\mathsf{\Delta}$. ...
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Why should “small” P be preferred?

In contrast, of course, is the approach of finding an NP language of super-polynomial complexity. But why the overwhelming, obvious yet implicit favoritism? Has it anything to do with our ...
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An exercise in the Kaplansky's book

I saw the following exercise in the Kaplansky's book that is due to D. Lizard. Where can i find the main text for the proof of this exercise? Let $P$ be a prime ideal of $R$, $I$ the ideal generated ...
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Assume that following inequality holds $$\mathsf{w,x,y,z<AB,AC,AD,BC,BD,CD<ABC,ABD,ACD,BCD<wx,wy,wz,xy,xz,yz}$$ with $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$ ...
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The growth of a subset of a group

Let $S$ be a symmetric subset of a group $G$ containing the identity, and let $S^n$ be the set of all products of $n$ elements of $S$. If $S^3\subset gS$ for some translate $gS$ of $S$ then it ...
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Abelian varieties in DAG

This is (hopefully) a pretty simple question. There is the notion of a derived scheme, etc., in derived algebraic geometry. What is the analogue of an abelian variety in derived algebraic geometry? I ...
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$\mathsf{GCD}$ in arithmetic progression

Given $\mathsf{M\in\Bbb N}$, pick $\mathsf{r,s,A,B\in\Bbb N}$ randomly with $\mathsf{0<r<s<A<B<M}$ satisfying $\mathsf{gcd(A,B)=1}$. Given $\mathsf{c\geq1}$, what is the probability ...
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Derived global functions on (derived) stacks $BG$ and $G/G$

In Toen's Affine Stacks, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we ...
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Fell topology vs. convergence of matrix coefficients

My question is partially inspired by the following discussion: Topology on the Unitary Dual Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let ...
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Quantities associated to deformed sheaves

I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...
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Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer. Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...
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Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...
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Standard name / symbol for intersection in Brouwerian lattices

A Brouwerian lattice has a lower adjoint $\cdot - B$ to $B\lor\cdot$. It is called pseudodifference. The main reference is http://www.jstor.org/stable/1969038 Once you have pseudodifference, you can ...
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Is there an official name for the intersection of the join-irreducible representations of two lattice elements?

Given a lattice provided with a join-irreducible representation of its elements, there is a natural "intersection" operator $A \mathbin{\dot\cap} B$ that returns the join of the setwise intersection ...
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Making idempotent element by a relation [on hold]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?
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|(a,b)| = |R| ? [on hold]

I want to prove that any open interval (a,b) has the same cardinality of the real numbers (|(a,b)| = |R|). Do I have to find an function to prove it? or is there a theorem to prove it easier? or any ...
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Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$. Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...
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Cyclic faithfully flat modules

Iam looking for an example of a cyclic faithfully flat R-module but not projective. Could someone help me?
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Estimating the shift in the $\lambda_{max}$ of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, what ways do we have to estimate, $\lambda_{max}(A+D) - \lambda_{max}(A)$? (Feel free to make other assumptions about the matrices that they are all ...
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largest subgroup of $Out(\hat{F_2})$ which preserves the Nielsen invariant

Let $x,y$ be generators for the free group $F_2$. It's known that $Aut(F_2)$, and hence $Out(F_2)$ preserves the conjugacy class of the subgroup $\langle[x,y]\rangle$ generated by $[x,y]$ (This ...
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I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help. In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...
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Interchanging the tensor product with infinite product

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the ...
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Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected? Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
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Prime ideals decomposition [on hold]

How to prove the decomposition law for prime ideals in finite separable extensions of number fields? How we use the "conductor condition"?
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Doubling theorem for Alexandrov spaces

Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem? The only reference I am aware of is the original ...

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