Search Results

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 …

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 …

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 …

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 …

I believe you need to be careful if $A$ is not over a perfect field. When $A$ is over a perfect field, the Jacobian ideal is the $r$th fitting ideal of the module of differentials, and so it is canon …

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 …

2) I'm not sure what you are asking with the first half of the question. Are you talking about the category of subsets of a fixed Euclidean space with unique inclusions, or are you talking about a mo …

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 …

Theres graded local duality which works just like local duality; however, it requires that $A_0$ is a field. I've had some luck making things work when A_0 is not a field, but then the local duality …

Given any $R$-module $M$, there is a scheme which corresponds to the 'total space' of $M$, given by $$ Tot(M):=Spec( Sym_R(M*))$$ where $M*$ is the dual module $Hom_R(M,R)$ and $Sym_RM*$ is the symme …

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 …

Your intuition is confusing the 'fiber over a point' with `restriction to a closed subscheme'. In general these can be very different, even if they come from the same place conceptually. Rational fu …

My guess is this is the kind of algebra you don't care about (since they aren't subrings of real-valued functions), but algebras of the form $\mathbb{R}[x]/x^n$ will have your property. When looking …

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 …

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 …