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2 votes
1 answer
326 views

Extension by zero operation

Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$. What are some examples and situations which ...
24 votes
1 answer
837 views

Is there a useful theory of D-modules on smooth (non-analytic) manifolds?

D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
5 votes
1 answer
286 views

Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf

Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
3 votes
1 answer
467 views

Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?

Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
2 votes
0 answers
111 views

Canonicity in split sequence in cotangent spaces

Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence $$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$ where $\mathfrak{m}_p$ is the maximal ...
4 votes
0 answers
278 views

Are manifolds "naturally" ringed or locally ringed spaces?

My question is about whether it's more natural to see manifolds as ringed spaces or as locally ringed spaces. I think I have arguments for both points of view. On the one hand, it's reasonable to ...
2 votes
1 answer
178 views

Are vector bundles acyclic for $\Gamma_c$?

Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}...
2 votes
1 answer
399 views

Locally free sheaves and vector bundles over smooth connected projective curve

Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
3 votes
1 answer
747 views

Are cohomology functors sheaves?

Question is the following: Is the functor $H^n_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$? More generally, are cohomology functors sheaves in ...
9 votes
1 answer
447 views

Results in “generalised smooth spaces” that did not hold in the case of smooth manifolds

Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page: Manifolds are fantastic spaces. It’s a pity that there aren’t more of them. I understand that this category $\text{...
5 votes
1 answer
332 views

Is the sheaf associated to a differential structure of a specific type?

On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...
2 votes
0 answers
35 views

If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?

Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
16 votes
3 answers
3k views

Physical interpretations/meanings of the notion of a sheaf?

I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
9 votes
0 answers
378 views

Is there any notion of "smoothification" from $\mathbb{R}$-schemes to generalized smooth spaces?

I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{...
6 votes
1 answer
752 views

Sections of the conormal bundle

Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$. Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
24 votes
4 answers
6k views

What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...