topological stable rank one and AF-algebra construction on Cantor set

ag.algebraic geometry – Finite decision by semi-stable bundles Answer

Hello pricey customer to our community We will proffer you an answer to this query ag.algebraic geometry – Finite decision by semi-stable bundles ,and the respond will breathe typical via documented info sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning in regards to the respond to this query.

ag.algebraic geometry – Finite decision by semi-stable bundles

Devlin Mallory’s strategy is actually rectify, and in reality the status is plane a miniature higher than he urged.

Let $V$ breathe an capricious vector bundle of rank $n$. Let $W$ breathe a steady bundle of rank $n+1$ (if one exists) or a semistable bundle. Fix an copious bundle $mathcal O(1)$.

I pretense that for $m$ sufficiently sizable, there’s a map $W(-m) to V$ that’s surjective and whose kernel is a line bundle (and thus is mechanically steady).

It suffices to take $m$ sizable sufficient that $Votimes W^vee otimes mathcal O(m)$ is globally generated. We can perceive sections of this line bundle as maps $W(-m) to V$, and the fiber of the part at a degree corresponds to the fiber of the map at a degree.

Maps from an $m+1$-dimensional vector area to an $m$-dimensional vector area have complete rank outdoors of a codimension $2$ locus. Thus amongst world sections of $V otimes W^vee otimes mathcal O(m)^vee$, these with out complete rank at a given level have codimension $2$. So these which fail to have complete rank at a number of factors of $C$ have codimension $1$. Thus we are able to discover a part which has complete rank at each level, which suggests it’s surjective and its kernel is a line bundle.

Over finite fields, this codimension dispute would not labor, however a similar counting dispute does.

we are going to proffer you the answer to ag.algebraic geometry – Finite decision by semi-stable bundles query by way of our community which brings all of the solutions from a number of dependable sources.

Add comment