# Contents

## Definition

### Of submanifolds

In differential geometry a conormal bundle of an embedded submanifold is the (fiberwise linear) dual of the normal bundle.

### Of locally ringed subspaces

The phrase conormal bundle is also used for more general conormal sheaf in the study of locally ringed spaces, especially of analytic spaces and algebraic schemes.

### Of abelian subcategories

Even more generally, Alexander Rosenberg defines a conormal bundle of a topologizing subcategory $S$ of an abelian category $A$ as follows.

He first modifies the notion of the defining sheaf of ideals of a closed subscheme to the notion of defining ideal of a topologizing subcategory as the endofunctor $\mathcal{I}=\mathcal{I}_S\in End(A)$ which is the subfunctor of identity $Id_A$ assigning to any $M\in A$ the intersection of kernels $Ker(f)$ of all morphisms $f: M\to N$ where $N\in Ob(S)$. One can show that if $T\subset S$ is an inclusion of topologizing subcategories, then $\mathcal{I}_{S}\subset \mathcal{I}_{T}$. In particular, for Gabriel multiplication of topologizing subcategories we have $\mathcal{I}_{S\circ S} \subset \mathcal{I}_S$.

Then the conormal bundle is simply $\Omega_S = \mathcal{I}_S/\mathcal{I}_{S\circ S}$, similarly to the sheaf case.

Revised on May 12, 2011 17:00:07 by Urs Schreiber (131.211.238.109)