Questions tagged [filtrations]

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Dimension of the associated graded module at an ideal

Let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$. Define the associated graded ring $G_I(R):=\bigoplus_{n=0}^\infty I^n/I^{n+1}$. Then $G_I(R)$ is a Noetherian ring of the same ...
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1answer
62 views

Properties of filtrations preserved by a DG-algebra homomorphism

Suppose we have a homomorphism $f : A^{\bullet} \longrightarrow B^{\bullet}$ of differential graded algebras over a field $k$, and consider the filtration \begin{align*} A^{\bullet} \supseteq F^0A^{\...
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63 views

Filtration over tensor product

Let $$ M \supset M_1 \supset \ldots \supset M_n \supset \ldots \text{ and } N \supset N_1 \supset \ldots \supset N_n \supset \ldots$$ be exhaustive decreasing filtrations of modules over a commutative ...
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Nested subspaces of measurable functions through noise

Let $(X_t)_t$ be a Markovian semi-martingale generating the filtration for the stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$ on which a Brownian motion $(W_t)_t$ is defined. ...
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1answer
88 views

can we take skeletons of covering maps to give new covering maps?

Let $X$ be an $n$-dimensional cell complex. We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$. Take the universal cover (or a general covering space) $\tilde X'$ of $X'$. ...
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68 views

If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
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83 views

References about transfinite socle series

I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series. Let $R$ be an associative unital ring and $...
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1answer
147 views

Poset filtrations

Consider a chain complex $C$ and a poset $P$ so that there is a filtration by subcomplexes $C^p$ of $C$ where $p\in P$ in such a way that $p<q$ implies $C^p \leqslant C^q$. As a second option, ...
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49 views

Ordered sequence of elements of poset relevant to some filtration — highest weight category

Let $\mathcal{C}$ be a highest-weight category with $\Lambda$ as a interval-finite poset -- I'm using a definition of the highest-weight category given by Cline, Parshall and Scott and it is presented ...
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134 views

Standard reference/name for “initial ideals $\Leftrightarrow$ associated graded rings”

Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by ...
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75 views

If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?

Let $(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces $(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$ $(N_t)_{t\ge0}$ be a $\mathbb ...
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65 views

Condition for a map to carry over to Leray spectral sequences

I am trying to understand the conditions for two Leray spectral sequences to be related by a map. Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps of topological spaces (with ...
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297 views

Mysterious “raison d'être” of filtrations of congruence subgroups

I wonder for long why congruence subgroups seem to arise so naturally in certain filtrations. Everything below is on a local field $F_p$. Filtration for $GL_n$. Casselman and later Jacquet, Piatetski-...
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1answer
71 views

Whether the first moment two stochastic processes differ can be formulated in terms of filtrations?

I am, struggling to see whether the first moment when two processes are different (in terms of their finite dimensional distributions) can be defined in terms of their filtrations and would ...
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1answer
54 views

Writing $\sigma$-algebra in terms of predictble processes?

If $X_t$ is a semi-martingale, $\mathfrak{F}_t$ is the $\sigma$-field generated by $X_t$ and $L^2(Pred)$ is the set of all $\mathfrak{F}_t$-predictible processes. Then is it true that: $$ \mathfrak{G}...
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125 views

In which sense does the quadratic variation depend on the considered filtration?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $X$ be an almost surely ...
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167 views

Stopping time sigma-fields

Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence). Is it true that the associated sigma-fields satisfy $F_{S_n}\...
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1answer
92 views

Is the class of acyclic complexes deconstructible?

Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of $\mathcal{S}$...
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1answer
1k views

When is the hitting time of an open set a stopping time?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_t)_{t \in [0,T]}$ a filtration. Consider an adapted, right-continuous process $X$ taking values in $\mathcal{X}$ and let $B$ be ...
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99 views

What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
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100 views

When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ $$\mathcal{F}^{\...
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131 views

Is semistability of smooth Weil sheaf preserved under tensor product?

Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of ...
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204 views

Does martingale convergence hold for arbitrary time?

Let $\{\mathcal B_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\mathcal B_i\subset\...
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5answers
2k views

Properties preserved under passage to augmented filtration

Dear all, generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration. Recall that if $(\mathcal{F}_t)_{...
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1answer
4k views

Right-continuity of natural filtrations

My question: Is the natural filtration of a right-continuous process also right-continuous? I would say yes, but don't know where to start proving it. Thanks for your help/ideas!
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2answers
583 views

Does there exist a functorial splitting for the weight filtration (of singular cohomology)?

There are plenty of examples of varieties whose singular cohomology with rational coefficients considered as a mixed Hodge structure does not decompose as the direct sum of its pure (weight) factors. ...
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1answer
871 views

Do Deligne's decalage and two filtrations for mixed Hodge complexes have a conceptual explanation?

As far as I understood, there are two way to assign weights to a mixed Hodge complex (a way that does not depend on shifts, and another one that does). There is a clever way to relate these to ways ...
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2answers
403 views

Is the first filtration Hausdorff?

Maybe this is too technical and elementary, but I cannot make up my mind, nor find a reference. The situation is the following: let $X$ be a double cochain (right half-plane) complex of abelian ...
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1answer
660 views

Semistable filtered vector spaces, a Tannakian category.

Let $k$ be a field (char = 0, perhaps). Let $(V,F)$ be a pair, where $V$ is a finite-dimensional $k$-vector space, and $F$ is a filtration of $V$, indexed by rational numbers, satisfying: $F^i V \...
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2answers
720 views

If associated-graded of a filtered bialgebra is Hopf, does it follow that the original bialgebra was Hopf?

Warning: older texts use the word "Hopf algebra" for what's now commonly called "bialgebra", whereas now "Hopf" is an extra condition. So as to avoid any confusion, I'll ...
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4answers
3k views

What is the universal property of associated graded?

Given a filtered vector space (or module over a ring) $0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V$, you can construct the associated graded vector space $\mathrm{gr}\left(V\right)=\oplus_{i}V_{i+...