Search Results

The highbrow way of reformulating your question is as follows. Consider the category $Sch$ of all schemes endowed with the Zariski topology. There is a fully faithful embedding of the category of aff …

Actually, there are two different restriction maps: The first one (the one you correctly say is neither surjective nor injective in general) is that on sections: for $\mathcal{F}$ a sheaf on a schem …

You say that a ring homomorphism $\phi: A \to B$ is étale (resp. smooth, unramified), or that $B$ is étale (resp. smooth, unramified) over $A$ is the following two conditions are satisfied: $A \to B …

The first example is the twisted cubic in $\mathbb{P}^3$.

UPDATE: My answer essentially just gives the definition of Kahler differentials and differential forms and misses the point of the question. Georges' answer addresses the relationship between the two. …

The Kähler cone of a del Pezzo surface of degree 6 is not simplicial: see section 6 of these notes.

As Kevin said, the higher cohomology groups of constant sheaves on irreducible varieties are zero when working with the Zariski topology. Also, "fibre bundles aren't locally trivial" and "the inverse …

Look at the affine pieces: over the open subset $u \neq 0$, you have a local coordinate $z = v/u$ and your equations can be written as $y = zx$ and $xy = x^6 + y^6$. Substituting $y$ in the second equ …

Here's a basic (and often used!) example: the zero locus of a homogeneous polynomial of degree $d$ in $\mathbb{P}^n$. For concreteness, let me spell out the case $n = 1$, $d = 2$. Cover $\mathbb{P}^1$ …

The bare question: Let $\mathcal{C}$ be an $\infty$-topos, and let $\tau_{\leq 0}\mathcal{C}$ be the subcategory of 0-truncated objects (which is the nerve of an ordinary Grothendieck topos: see HTT 6 …

It might not be the best reference for a systematic study of stacks and some of the terminology is old, but Mumford's "Picard Groups of Moduli Problems" (1965) might be a nice complement. It explains …

There is a nice treatment of it in chapter 1 of Brylisnki's Loop Spaces, Characteristic Classes and Geometric Quantization. In the Stacks projects, look for section 19.19.

Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$. These statements ar …

If $\pi: E \to M$ is a bundle over a topological space $M$, you can define a sheaf on $M$ that associates to each open set $U \subseteq M$ the set of sections over it, i.e., maps $\sigma: U \to E$ suc …

On a complete nonsingular curve over an algebraically closed field, a line bundle of degree zero with a global section is necessarily the trivial bundle. This is lemma IV.1.2 in Hartshorne's Algebraic …