Finite functorial semi-norms and representability

Functorial semi-norms are semi-normed refinements of functors such as singular (co)homology. We investigate how different types of representability affect the (non-)triviality of finite functorial semi-norms on certain functors or classes. In particular, we consider representable functors, generalised cohomology theories, and so-called weakly flexible homology classes in singular homology and l1-homology.Comment: 18 pages; v3: small changes as suggested by the referee; v2: clarified Example 4.3, added referenc

On group gradings on PI-algebras

We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] \leq exp(W)^K$. A G-grading $W = \bigoplus_{g \in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} \neq 0$ for any $r \geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.Comment: 17 page

Cohomology operations and algebraic geometry

The manuscript is an overview of the motivations and foundations lying behind Voevodsky's ideas of constructing categories similar to the ordinary topological homotopy categories. The objects of these categories are strictly related to algebraic varieties and preserve some of their algebraic invariants.Comment: This is the version published by Geometry & Topology Monographs on 29 January 200