We investigate pure monomorphisms and pure epimorphisms in varieties of universal algebras with applications to equationally compact (= pure injective) algebras, pure projective algebras and perfect varieties.

Let R be a regular local ring, K its field of fractions and A1, A2 two Azumaya algebras with involutions over R. We show that if A1 ⊗R K and A1 ⊗R K are isomorphic over K, then A1 and A2 are isomorphic over R. In particular, if two quadratic spaces over the ring R become similar over K then these two spaces are similar already over R. The results are consequences of a purity theorem for similar...

— GROTHENDIECK'S absolute cohomological purity conjecture is proved for Q7-etale cohomology, and for etale cohomology with 7.1 f coefficients if / is large. The proof depends on Quillen's localization theorem for algebraic K-thcory and my comparison of algebraic and topological X-thcory. In this note, I show how my theorem relating algebraic and topological K-theory yields an absolute cohomolog...

We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saorín. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it for...

OBJECTIVES Amphotericin B (AmB) is a drug of choice for treatment of disseminated fungal infections, but its use is often associated with severe adverse effects. Our observation that generic formulations of AmB contain multiple polyene components led us to propose that removal of other polyenes would yield a high purity AmB (AmBHP) with an improved therapeutic index. METHODS To test that prem...

Adámek and Sousa recently solved the problem of characterizing the subcategories K of a locally λ-presentable category C which are λ-orthogonal in C, using their concept of Kλ-pure morphism. We strenghten the latter definition, in order to obtain a characterization of the classes defined by orthogonality with respect to λ-presentable morphisms (where f :A B is called λ-presentable if it is a λ-...

We consider smooth completion of algebraic manifolds. Having some information about its singular completions or about completions of its images we prove purity of cohohomology of the set at infinity. We deduce also some topological properties. The work is based on the study of perverse direct images for algebraic maps.

Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them. I want to begin with a classical formulation of purity, due to Hilbert: In modern mathematics one strives to preserve the purity ...