Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over distributions. Are there any applications of the former which cannot be obtained using the latter? Any important examples? I was told one general property of hyperfunctions … Read more

When is a coherent subsheaf determined by its global sections

I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections. The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between separated $S$-schemes of finite type, over a Noetherian base $S$. Then, $\mathcal{A}$ is defined to be a subsheaf of the … Read more

Reference request: sheaf-theoretic operations in the classical topology?

Like many graduate students before trying to learn something about étale cohomology and Deligne’s proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing basic sheaf-theoretic operations in the classical topology. Here are some sources I found so far which discuss this, some of which by way of wise … Read more

Weierstrass model of an elliptic curve: a line bundle over the base

Let S be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map π:S→C where C is a compact Riemann surface. Disclaimer: Physicist here. Intuitively I think of this as a compact Riemann surface (e.g. CP1) where over each point z we … Read more

How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component?

Let f:X→Y be a proper morphism. In section 6.4. of Liu’s book he introduces the r-dualizing sheaf ωf for f which satisfies f∗HomOX(F,ωf)≅HomOY(Rrf∗F,OY) for all quasi-coherent sheaves F on X. In the special case of f being finite he proves that the 0-dualizing sheaf is given by f!OY=HomOY(f∗OX,OY) where this is considered an OX-module via … Read more

Examples of non-hypercomplete sheaves on affine schemes

Let A be a commutative ring and let O be a sheaf of E∞-ring spectra on SpecA such that π0O=OSpecA. Lurie provides a criterion when (SpecA,O) coincides with SpecO(SpecA), namely if the homotopy groups πnO are quasi-coherent sheaves on SpecA and O is hypercomplete (Spectral Algebraic Geometry, Proposition 1.6.1.1). To get a better understanding why … Read more

Infinity-categorical exceptional push-forward

Classically, if f:X→Y is a map of locally compact Hausdorff topological spaces, one can define the exceptional push-forward functor f!:Sh(X;k)→Sh(Y;k) among k-valued sheaves for, say, a field k. Then one can form the derived functor to obtain a functor amond derived categories, take the right adjoint and get Verdier duality out of this. Is there’s … Read more

Understanding a step in proof of sheaf version Verdier duality

Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss. So all proofs I can find factors through a particular statement, which goes to Kashiwara-Shapira, which leaves it to readers as exercise ((2.6.25) on page 114). The statement: Let f:X→Y be a map of “nice” spaces … Read more