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I will just sum up the situation as I see it (too big for the comment box). One important goal is to set up a good intersection theory for cycles without quotienting by rational equivalence, and usin …

If I'm not mistaken, the kernel of $\alpha$ is necessarily flat, and indeed either equal to $\mu_p$ or $0$ if $S$ is connected. The key is that $\mu_p$ is of multiplicative type, and we have the follo …

Luc Illusie has written two great expository texts on étale vanishing cycles and other topics in SGA7, one in English (with an associated set of slides) and one in French (unfortunately). Illusie has …

A small suggestion : the deformation to the normal cone is a nice construction that I would have liked to see in a first course. It illustrate the use of blow-ups, the degeneration of a family with co …

The first thing to consider is the case of affine curves : let $k$ be an algebraically closed field, $C/k$ a smooth affine curve, $\bar{C}/k$ its smooth projective compactification, $\bar{C}=C\cup{p_0 …

A small but illuminating exemple : isolated singularities consisting of affine $k$ lines meeting at the origin in $\mathbb{A}^n$. One can show easily that the analytic type of the singularity depends …

Here are some comments about the use of topologies in motivic homotopy theory. This is based on the discussion in Morel-Voevodsky's "A^1-homotopy theory of schemes" p.94-95 (MV below), I only add some …

The short answer is that they are very different, but become quite similar if you 1) stabilize, i.e invert smash product by $\mathbb{P}^1_k$ on the homotopy side and invert tensor product by the Tate …

Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ? Details: Let $S$ be a scheme and $f:S'\rightarrow S$ a finit …

I don't know much about this topic, but I was recently recommended the paper An extension of the Grothendieck Galois theory of Grothendieck by Joyal and Tierney as an enlightening abstract generalisat …

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the …

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. obtain …