# Questions tagged [decomposition-theorem]

In mathematics, especially algebraic geometry the decomposition theorem is a set of results concerning the cohomology of algebraic varieties.

48
questions

-3
votes

0
answers

35
views

### Primary decomposition thm proof [migrated]

I'm studying the proof of primary decomposition thm in linear algebra, hoffman and kunze. Page 221, in the process of proving that minimal polynimial for T_i is (p_i)^r_i, I don't understand why g(T_i)...

0
votes

0
answers

63
views

### A question to the Wedderburn-Mal’cev decomposition

Excuse me, I saw the result on the Wedderburn-Mal’cev decomposition of unital compact rings which M.I. Ursul and A. Tripe introduced in the attached file. However, I cannot contact them because ...

1
vote

1
answer

61
views

### Adding valid cuts for integer feasibility problem under Benders decomposition framework?

Traditional infeasibility cut is derived under the assumption that the feasibility problem is LP instead of ILP and relies on the dual form of the LP.
Is there a systematic way of adding valid cuts ...

3
votes

1
answer

413
views

### Decomposition theorem over more general base schemes

The BBDG decomposition theorem says that if $f\colon X \to Y$ is a projective morphism of finite type $\mathbf{C}$-schemes and $X$ is smooth of (pure) dimension $d$ then $\mathbf{R}f_*\mathbf{Q}_\ell[...

2
votes

1
answer

85
views

### Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them

I'm looking for an elegant way to show the following claim.
Claim: Let $m_1, m_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are ...

4
votes

1
answer

271
views

### Decomposition of direct image of a smooth morphism, Deligne's theorem, motives

Let $f : X \to Y$ be projective and smooth morphism of complex algebraic varieties. Here we care about the algebraic topology of $X$ and $Y$, so use classical topology for simplicity.
I can take the ...

2
votes

0
answers

92
views

### Are a.e. derivatives of continuous $VBG_*$ functions Denjoy–Perron integrable?

I would like to ask a question pertaining to the Denjoy–Perron (Henstock–Kurzweil) theory of integration. It is simple enough that I have entertained the idea that perhaps an answer is known, but I ...

2
votes

0
answers

371
views

### Etale cohomology of a nodal (cuspidal) curve

Let $k$ be a separably closed field, and $X/k$ be a curve (not necessarily complete) with a single singularity, a simple node $x$. Suppose $\ell$ is a prime number invertible in $k$, how do we compute ...

3
votes

0
answers

144
views

### Intersection homology of toric resolutions

I'm interested in the intersection homology of toric varieties associated to a polytope $P$ with proper faces F, and a subdivision $P'$ of P. Let $X_P$ be the toric variety associated to the polytope $...

12
votes

2
answers

525
views

### What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?

Let $f\colon X\to Y$ be a surjective morphism of smooth projective varieties. If the decomposition theorem for $f$ is given by $$Rf_*\mathbb{C} \simeq \bigoplus_i R^if_*\mathbb{C}[-i],$$ what are the ...

1
vote

1
answer

261
views

### Is it possible to prove the Jordan decomposition starting from Schur's decomposition?

Schur's decomposition says any matrix $A$ is similar to a upper triangular matrix $U$ i.e., there exists unitary $Q$ such that $A = Q^{-1}UQ$. If we split $U$ as $D+N$ where $D$ is the diagonal part ...

1
vote

1
answer

380
views

### Generate a two-variable polynomial from its "roots [closed]

I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros.
But I want know if is ...

5
votes

1
answer

367
views

### What is the elementary proof of Weil's polynomial theorem of decomposition?

André Weil sometimes glosses his Theorem of Decomposition in a simplified polynomial form:
If $P(x,y)$ and $Q(x,y)$ are homogeneous polynomials algebraically
prime to each other, with integer ...

1
vote

0
answers

62
views

### Being additive map using spectral decomposition theorem

Let $(\mathcal{A},\tau_\mathcal{A})$ and $(\mathcal{B},\tau_\mathcal{B})$ be semifinite von Neumann algebras with normal semifinite faithful traces $\tau_\mathcal{A}$ and $\tau_\mathcal{B}$. I defined ...

1
vote

0
answers

44
views

### Efficient scissors congruence between efficiently describable convex polytopes and simplex?

Is there a convex polytope in $\mathbb R^n$ describable by only $O(poly(\log n))$ half-plane inequalities with positive volume (so at least $n+1$ vertices) such that the standard simplex has a ...

3
votes

1
answer

138
views

### Connections between eigenvectors after matrix multiplication

Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...

2
votes

1
answer

103
views

### Equality or inequality for determinant of $A_{n \times m} D_{m \times m} A^T_{m \times n}$

Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n}$. $D$ is a positive diagonal matrix and $m > n$.
Is there any equality or inequality over $|B|$, $|AA^...

2
votes

1
answer

838
views

### Find a minimum set of paths that cover all pairs of dependent vertices

Let $G=(V,A)$ be a simple directed acyclic graph. A set consisting of two vertices is called dependent if there is a directed path from one of the vertices to the other. The question is to find ...

6
votes

0
answers

97
views

### What is this matrix decomposition called and does it exist always? - II

Given a rank $2r$ matrix $M\in\Bbb Q_{\geq0}^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank at most $r$ such that $M=M_+-...

4
votes

1
answer

281
views

### What is this matrix decomposition called and does it exist always?

Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank $r$ such that $M=M_+-M_-$ holds?
...

2
votes

1
answer

176
views

### Singular Value decomposition of huge dimensional matrix

i would like to consider singular value decomposition of such type of matrix
creation of matrix from small sample is not big issue, i have ready code for this
...

9
votes

1
answer

221
views

### Decomposition of Henstock-Kurzweil-integrable functions

Let $f:[a,b]\to\mathbb R$ be a Henstock-Kurzweil-integrable function (short: HK-integrable).
Can $f$ always be written as a sum of a Lebesgue-integrable function and a function which has a ...

5
votes

1
answer

256
views

### The closure of cyclic modules under direct sums and direct summands

Cohen and Kaplansky have proven that a commutative ring $R$ has the property
C: Every $R$-module is a direct sum of cyclic $R$-modules.
if and only if $R$ is an Artinian principal ideal ring. Can ...

0
votes

1
answer

215
views

### How to find $K,W,S$ in the Mostow decomposition theorem?

The Mostow decomposition theorem states:
Let $Z$ a nonsingular complex matrix, then $Z$ can be factored as:
$$Z=We^{iK}e^S$$ where: $W$ is unitary, $K$ is real and skew symmetric and $S$ is real and ...

4
votes

0
answers

181
views

### Optimal instructions for the modular construction of rectlinear Lego structures

Let $X$ be a compact (or periodic) union of integer translates of unit cubes such that the interior of $X$ is connected. (If it makes any difference, suppose that the dimension $n$ of $X$ is 3.) I am ...

1
vote

0
answers

526
views

### Structure theorem for infinitely generated modules over a PID

This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain.
The question is all in the title: is there ...

15
votes

1
answer

2k
views

### Gabber's original proof of his purity theorem

Gabber's purity theorem is the statement that if $\mathscr{F}$ is a pure perverse sheaf on an open subvariety $j : U \hookrightarrow X$ then so is $j_{!*} \mathscr{F}$.
It is remarkable because it ...

2
votes

1
answer

531
views

### Simple decomposition of $K_{2n}-I$ into hamiltonian cycles

http://mathworld.wolfram.com/HamiltonDecomposition.html
In the 1890s, Walecki showed that complete graphs K_n admit a Hamilton decomposition for odd n, and decompositions into Hamiltonian cycles ...

4
votes

1
answer

600
views

### Is there an elementary proof of the polar factorization theorem for vector-valued function?

I have recently learned the polar factorization theorem for vector-valued functions due to Brenier. Namely, given a probability space $(X,\mu)$ and a bounded domain $\Omega\subset \mathbb{R}^n$ with ...

6
votes

0
answers

514
views

### What's the relationship between the different versions of the BBD decomposition theorem?

I have a few questions relating to the BBD decomposition theorem.
I have come across the following two versions of the decomposition theorem.
Version 1. Let $f : X \to Y$ be a proper map of ...

0
votes

1
answer

149
views

### Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?

Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space.
Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action ...

14
votes

1
answer

2k
views

### When does a perverse sheaf occur in the decomposition theorem?

Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...

5
votes

2
answers

386
views

### A Krull-Schmidt Theorem for Lie groups?

I wondered whether there is an analogue of the Krull-Schmidt theorem for real Lie-groups. More precisely, what conditions do you have to impose on a connected finite dimensional Lie group $G$ so that ...

1
vote

1
answer

219
views

### Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix ...

3
votes

1
answer

330
views

### Trace of multiplied positive definite matrices

I have to compute $Tr(K^{-1}\Sigma)$ where both $K$ and $\Sigma$ are symmetric positive definite matrices.
Question is considering that I have computed the Cholesky, $L_1$ of $K$ previously, is there ...

7
votes

0
answers

183
views

### Divisibility of all entries in an intersection form

What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer?
More precisely: $F \subset S$ is a proper connected (usually reducible) half-...

1
vote

1
answer

989
views

### Eigenfunctions and eigenvalues of the product of two exponential kernels

Consider the following exponential kernel:
$k(x_1, x_2) = \exp\left(\frac{|x_1 - x_2|}{L}\right)$,
which is symmetric and non-negative definite. By virtue of Mercer's theorem, we have
$k(x_1, x_2) =...

1
vote

1
answer

369
views

### Do signed measures on sigma-rings always have a Hahn decomposition?

Let $X$ be a set.
Let $\mathcal{R}$ be a set of subsets of $X$ such that
$\{\} \in \mathcal{R}$
and
For all members $A$ and $B$ of $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} \;\;$...

2
votes

3
answers

1k
views

### Decomposing a discrete signal into a sum of rectangle functions

Hello mathoverflow community !
I have a simple question that seems to have a non trivial answer.
Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar (rectangular)...

20
votes

4
answers

44k
views

### complexity of eigenvalue decomposition

what is the computational complexity of eigenvalue decomposition for a unitary matrix?
is O(n^3) a correct answer?

1
vote

0
answers

758
views

### Which statement do people usually call the Decomposition Theorem, and what is the precise reference for it?

Which statement is usually called the Decomposition Theorem (for perverse sheaves)? Is this (roughly): a proper pushforward of an intersection complex could be decomposed into a direct sum of (shifted)...

10
votes

2
answers

4k
views

### Iwasawa Decomposition & Polar Decomposition related how ?

In an earlier post (Use Lie Sub-Groups of GL(3, R) for elastic deformation ? here), I mentioned polar decompositions as in F = RU where R in SO(3) & U in symmetric positive-semidefinite matrices. ...

5
votes

1
answer

731
views

### Easy special cases of the decomposition theorem?

The decomposition theorem states roughly, that the pushforward of an IC complex,
along a proper map decomposes into a direct sum of shifted IC complexes.
Are there special cases for the decomposition ...

4
votes

1
answer

593
views

### Morphisms between pure complexes of sheaves

I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...

24
votes

2
answers

3k
views

### Why is the decomposition theorem awesome?

I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...

22
votes

4
answers

4k
views

### Examples for Decomposition Theorem

There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...

13
votes

4
answers

2k
views

### How to do Computations Using the Decomposition Theorem for Perverse Sheaves

This is a follow-up to this post on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.
My question is how does one use the ...

6
votes

2
answers

1k
views

### Explicit Direct Summands in the Decomposition Theorem

Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...