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Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the Zaris …

I remember the following two quotes about flatness (I forgot who said/wrote this): For every geometric description of flatness there is a counterexample. Flatness is one of the few notions in algebr …

First of all, I am aware of the questions about the Zariski topology asked here and I am also aware of the discussion at the Secret Blogging Seminar. But I could not find an answer to a question that …

If $k$ is algebraically closed and $X$ is a $k$-scheme locally of finite type, then the $k$-rational points are precisely the closed points. (See EGA 1971, Ch. I, Corollaire 6.5.3). More generally: i …

In the 1971 edition of EGA (this is a revised version of the original 1960 EGA) you can find the following remark in the foreword (avant-propos): Signalons enfin, par rapport à la première édition, un …

For an overview of tropical geometry perhaps: A. Gathmann, Tropical algebraic geometry, Jahresbericht der DMV 108 It's available here.

In my opinion, all answers go a little too far. In this (non-topological!) setting I think about this as follows: The aim is to analyze the lack of right-exactness of a left-exact functor $\Gamma:C \r …

Let $G$ be an algebraic group over an algebraically closed field $k$. Then G/H is a quasi-projective homogeneous G-variety for any closed subgroup $H$. Now, several times I have seen something like "L …

Let $k$ be an algebraically closed field and let $X$ be a Cohen-Macaulay variety over $k$, i.e. all local rings are Cohen-Macaulay (perhaps this can later be generalized). What is the dualizing sheaf …

In Qing Liu's book Algebraic geometry and arithmetic curves I came across several confusing definitions. Several times he defines a notion only for a subclass of schemes/morphisms but later he is neve …

This question is an addition to my question on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group o …

Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the can …

Perhaps this is too special but ramification of primes in number fields is a nice motivation, here you can also draw funny pictures of curves over Spec(Z). The step from here to Dedekind schemes is im …

This is just about primary decomposition! There are several CAS which can do that, for example Singular.