For a proper flat map $f: X \to Y$ (of reasonable schemes) and a closed embedding $i: Y’ \to Y$, we know by EGA the base change quasi-isomorphism, where $f’$ and $i’$ are the pullbacks of $f$ and $i$: $$L i^* R f_* \mathcal O \sim R f’_* L i’^* \mathcal O$$ Here is my … Read more
I have a topological space X and two disjoint, closed subspaces Y and Z of X. I believe that in this situation, for any abelian sheaf F on X and any p∈N, there is a natural isomorphism HpY(X,F)⊕HpZ(X,F)→HpY∪Z(X,F) between local cohomology groups. I can obtain this by taking an injective resolution 0→F→I∙ of F, and … Read more
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, meaning $gcd(f_i,f_j) = 1$ for $i \neq j$. Recall that we can compute the sheaf cohomology of $\mathcal{O}_X$ using the pushforward $$ H^i(X,\mathcal{O}_X) = H^i(\mathbb{P}^n,i_*(\mathcal{O}_X)) $$ If $X$ … Read more
Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that ˇH2(X,F)≠H2(X,F). In the beginning he states that there exisits an exact sequence 0⟶ˇH2(X,F)⟶H2(X,F)⟶ˇH1(X,H1(F))⟶0 which results from the Cech-to-derived functor spectral sequence ˇHp(X,Hq(F))⇒Hp+q(X,F). My question ist: Does this short exact sequence exist for an arbitrary topological space X and a sheaf of abelian … Read more
Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss. So all proofs I can find factors through a particular statement, which goes to Kashiwara-Shapira, which leaves it to readers as exercise ((2.6.25) on page 114). The statement: Let f:X→Y be a map of “nice” spaces … Read more
This is basically the last step of problem 5.3.7 in Huybrechts’ Complex Geometry. Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. Denote by $ \hat{X} = \operatorname{Bl}_{\{x\}} X$ the blow-up of $X$ at $x$ and let $\sigma: \hat{X} \to X$ be … Read more
I have a flat, quasicompact, and separated map p:X→A1 and I know that Rip∗OX vanishes everywhere except possibly 0∈A1. Q1: Does “cohomology and base change” automatically hold here? I.e., is the natural map (Rip∗OX)s⊗OA1,sk(s)→Hi(Xs,OX,s) an isomorphism for all s∈A1? This is clear for all nonzero s and I believe it should follow for 0∈A1 automatically. … Read more
Suppose we have an inverse system of compact Hausdorff spaces {Xi,φij}i∈I and that each space has a presheaf Γi assigned to it in such a way that Γi(φij(U))=Γj(U) whenever i≤j. Then X:=lim←Xi has a presheaf Γ defined on it by Γ(U):=Γi(φi(U)) where φi:X→Xi is the map from X as an inverse limit; this is well-defined … Read more
Let (A,L) be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of L, so that d=χ(L)=dimH0(A,L) since L is ample. I’ve read in a lot of papers the sentence Let (A,L) be a polarized abelian variety of dimension g and of type (d1,…,dg), etc. I’ve seen the … Read more
I’m reading Mumfords’s Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3) and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON $\mathbb{P}^n$; p 47) dealing with yoga on coherent sheaves $F$ over pojective space $\mathbb{P}^n$ I found on page 52 a proof I not understand: Corollary 3: Given a coherent … Read more
