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There are two ways a presheaf can fail to be a sheaf. It has local sections that should patch together to give a global section, but don't, It has non-zero sections which are locally zero. When di …

I'm a fan of The Geometry of Schemes by Eisenbud and Harris. Its great for a conceptual introduction that won't turn people off as fast as Hartshorne. However, it barely even mentions the concept of …

The Graded Nakayama's Lemma My intuition for Nakayama's lemma is rooted in the graded version. (Graded Nakayama's Lemma) Let $R$ be a $\mathbb{N}$-graded algebra, and let $R_+$ be the 'irrelevant' i …

The following should be pretty standard for any algebraic geometer. Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for eve …

I asked a similar question to Daniel Huybrechts some time ago, in the form of "If I have a derived equivalence between two varieties, what is this telling me about the relation between the two varieti …

From a practical perspective, putting a grading on an algebra usually organizes the algebra into a collection of finite-dimensional vector spaces, each indexed by a natural number. This opens the doo …

Background Let $S_{g,n}$ be an oriented surface of genus $g$, with $n$ punctures. We explicitly prohibit the non-hyperbolic cases: $g=0$, $n=0,1,2$. $g=1$, $n=0$. Let $\Gamma$ be the fundamental gr …

Let $R$ be a commutative algebra over a field $k$. Denote the $R$-module of Kahler differentials by $\Omega^1_kR$; this is the $R$-module generated by symbols of the form $da$, $a\in R$, and relation …

In a non-commutative ring, you need to be careful with what you even mean by a prime ideal, and usually there are very few two-sided ideals you might call prime. Oh, and even in the cases when there …

A morphism $f:X\rightarrow Y$ of schemes gives a faithful pullback functor $f^*$ exactly when the morphism $f$ is surjective on underlying sets. This can be seen in several steps. Note that the func …

Let $X$ be a quasi-affine scheme; that is, the natural map $$X\rightarrow \overline{X}:=Spec(\Gamma(X,\mathcal{O}_X))$$ is an inclusion. Each scheme has a quasi-coherent sheaf of Kahler differentia …

I have a passing familiarity with moduli theory, which gets me in trouble when I want to understand specific examples. The basic question I would like to understand is how to prove something is a ver …

There was an MSRI summer school on stacks and deformation theory a few years ago. The video of all the talks are online, at the workshop's webpage. There are several copies of notes around, I believ …

Let $V\rightarrow Y$ be a vector bundle of rank $n+1$ over $Y$, with $Y$ reasonably nice (I care about the case of smooth, irreducible affine). Let $X=\mathbb{P}(V)$ be the projectivization of $V$, so …

If you are looking for good first examples, Mumford's Red Book and Eisenbud and Harris's 'Geometry of Schemes' have some good pictures and examples. Its worth playing around with Spec(O_c), where O_c …