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This tag is used if a reference is needed in a paper or textbook on a specific result.

14 votes
0 answers
1k views

Is there a slick proof of the fundamental theorem of dimension theory?

The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of in …
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6 votes
1 answer
248 views

Universal model category as a $\text{sSet}$-enriched co-completion

It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it. I'm sure this can be done not only for $\text{Se …
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6 votes
0 answers
479 views

“Cohomological equation” in dynamical systems

Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$ Then, being able to find a suitable $h$ i …
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4 votes
1 answer
280 views

When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?

Let $\mathcal{M}$ be a locally finitely presentable model category, cofibrantly generated by two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial cofibrations with presentable domain …
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4 votes
0 answers
173 views

Is the Reedy model structure cofibrantly generated?

I am reading about the Reedy model structure from Hovey's book and I was wondering if the Reedy model structure on $\mathcal{M}^{\Delta}$ is cofibrantly generated by a small set of arrows and permits …
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3 votes
1 answer
370 views

Universal property of the category of $\mathcal{S}$-sheaves and the definition of Topos

Let $\mathcal{C}$ be a small category; let $\mathcal{S}$ be any family of maps in $\text{Psh}\left(\mathcal{C}\right)$. Call $X\in \text{Psh}\left(\mathcal{C}\right) $ an $\mathcal{S}$-sheaf when $\ …
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