# Tagged Questions

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### On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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### Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1$

Let $\mathbb{P}^1$ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \$ we can define some quasi-coherent sheaves on $\mathbb{P}^1$. We can ...
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### Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact: If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...
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### Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
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### Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
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### Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
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### Existence of an isomorphism $\mathbb{P}^n\times\mathbb{P}^m \rightarrow \mathbb{P}^{n+m}$ [duplicate]

There exist an isomorphism of varieties? $$\mathbb{P}^n\times\mathbb{P}^m \rightarrow \mathbb{P}^{n+m}$$ I am considering $\mathbb{P}^n\times\mathbb{P}^m$ as the product in the category of ...
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### Leray spectral sequence for complexes

Let $f:X\rightarrow S$ be a morphism of schemes. Let $0\rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$ be an exact sequence of Abelian sheaves on $X$. Is there a general procedure to ...
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### How to define the natural map on the second page of a spectral sequence?

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
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### What are V(f) and D(f) in real practice of EGA

https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!355 would like to do and understand what is V(f) and D(f) where D(f) = SpecA - V(f) in the following diagram, it said V(f) is subset of p ...