Local to global equivalence for homomorphisms

Examples of the failure of the local-global principle for similarity of matrices

“…Our first theorem generalizes the Mayer-Vietoris sequence that was given in [3]. The proof is a long and relatively straightforward calculation available from the authors.…”

“…For a complete discussion of non-abelian cohomology see [4]. The same analysis for the relation equivalence for homomorphisms was given in [3] so we state most results without proof since the arguments are analogous to those in [3].…”

“…The object of study is GS(f) which is the set of similarity classes of endomorphisms locally similar to f. For an open set U c X we say f is similar to g over U when 1 €3 f is similar to 1 €3 g over Ox(U). When the modules are finitely ~resented, local similarity is equivalent to similarity over every open set in a cover U of X [3].…”

“…The "Drozd condition", given by Guralnick and Levy in [6], is then used to give a sufficient condition for some of the terms in the Mayer-Vietoris sequence to be abelian groups. Some results from [3] are then restated in the context of similarity. …”