3
votes
0answers
108 views

Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
2
votes
2answers
113 views

Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?
3
votes
1answer
71 views

Where can I find this result?

A Noetherian scheme $X$ over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent space $(\mathfrak{m}/\mathfrak{m}^2)^*$ ...
3
votes
2answers
160 views

Serre's Theorem for $\operatorname{Proj}$

Let $k$ be an algebraically closed field. If $S$ is a positively graded $k$-algebra which is finitely generated by $S_1$ over $S_0 = k$ then quasi-coherent sheaves on $\operatorname{Proj}S$ are ...
13
votes
2answers
397 views

Where do Chern classes live? $c_1(L)\in \textrm{?}$

If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like ...
4
votes
0answers
987 views

Proving that $\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong \operatorname{Pic}^0(X \times Y)$

Let $k$ be a field of arbitrary characteristic and let $X$ and $Y$ be projective varieties over $k$. I have come across the formula $$\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong ...
0
votes
0answers
57 views

Reference-Request: Symmetric Product Schemes

Is there a good reference for the theory of symmetric product schemes? (I only need a few basic things, the construction, etc.) Googling it turned up a lot of papers which use it as if it's common ...
1
vote
0answers
130 views

Why topological stratification is useful?

My main focus is on the applications of stratification in complex/abstract Algebraic geometry especially, from the scheme-theoretic viewpoint and (Added) Moduli spaces. I have a vague feeling that ...