Search Results

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 …

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 …

A standard case where the quotient exists is when the group G is finite. In this case, if you start with the ring of functions O, and take the ring of G invariants O^G. Then O^G is finitely-generate …

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 …

Your idea of requiring that every irreducible closed subset have a generic point (or, equivalently, that you add a generic point to every irreducible closed subset) is called a 'sober space' (and the …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …