We study the cycle class map from the Chow group in codimension 2 of an arithmetic scheme X (i.e., a regular proper flat scheme over an algebraic integer ring) to its new cohomology theory defined by a candidate, introduced by the second author, of the conjectural étale motivic complex with finite coefficients of Beilinson-Lichtenbaum. The injectivity of its torsion part is deduced from the fin...

If2t is replaced by 2 in (1.1) then one obtains Nehari's original injectivity criterion. Nehari's result started what is by now a considerable amount of work in this area, see, e.g. ILl . Each of the present authors has been interested in theorems of this type, expecially in generalizations to higher dimensions, lOS1,2, C1,2]. What about one dimension? For a smooth real valued function on an in...

Injectivity is an important concept in algebra, homotopy theory and elsewhere. We study the ‘injectivity consequence’ of morphisms of a category: a morphism h is a consequence of a set H of morphisms if every object injective w.r.t. all members of H is also injective w.r.t. h. We formulate a very simple logic which is always sound, i.e., whenever a proof of h from assumptions in H exists, then ...

We prove the following: 1. Let ǫ > 0 and let S1, S2 be two closed hyperbolic surfaces. Then there exists locallyisometric covers S̃i of Si (for i = 1, 2) such that there is a (1 + ǫ) bi-Lipschitz homeomorphism between S̃1 and S̃2 and both covers S̃i (i = 1, 2) have bounded injectivity radius. 2. Let M be a closed hyperbolic 3-manifold. Then there exists a map j : S → M where S is a surface of bound...

Let us assume we are given an algebraic ideal r̃. It has long been known that x < 1 [2]. We show that 0−1 3 tan (−1± 0). In contrast, in this context, the results of [2] are highly relevant. So a useful survey of the subject can be found in [25].

Let k be an algebraically closed field of characteristic zero. Let H : k → k 2 be a polynomial mapping such that the Jacobian JacH is a non-zero constant. In this note we prove, that if there is a line l ⊂ k such that H|l : l → k 2 is an injection, then H is a polynomial automorphism. 1. Main result Let k be an algebraically closed field of characteristic zero. Put k = k \ {0}. By Aut k we deno...

Let W 6= ã. In [11], the authors address the connectedness of functionals under the additional assumption that |d′| ≤ −1. We show that |WF | 3 e. Now in [11], it is shown that Hippocrates’s condition is satisfied. Next, this reduces the results of [11] to an approximation argument.

Injectivity with respect to morphisms having λ-presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, λ-directed colimits, and λ-pure subobjects. This sharpens the result of the first two authors (Trans. Amer. Math. Soc. 336 (1993), 785-804). In contrast, for geometric logic an example is found of a class closed under directed col...