Displaying similar documents to “Normal forms of matrices in topoi”

The Rank+Nullity Theorem

Jesse Alama (2007)

Formalized Mathematics

Similarity:

The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to |B - A|, and that T is one-to-one on B - A.

Homological projective duality

Alexander Kuznetsov (2007)

Publications Mathématiques de l'IHÉS

Similarity:

We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these...

Families of differential forms on complex spaces

Vincenzo Ancona, Bernard Gaveau (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

On every reduced complex space X we construct a family of complexes of soft sheaves Λ X ; each of them is a resolution of the constant sheaf ℂ X and induces the ordinary De Rham complex of differential forms on a dense open analytic subset of X . The construction is functorial (in a suitable sense). Moreover each of the above complexes can fully describe the mixed Hodge structure of Deligne on a compact algebraic variety.