Let (M,ω) be a closed symplectic manifold and Ham(M,ω) the group of Hamiltonian diffeomorphisms of (M,ω). Then the Seidel homomorphism is a map from the fundamental group of Ham(M,ω) to the quantum homology ring QH∗(M ; Λ). Using this homomorphism we give a sufficient condition for when a nontrivial loop ψ in Ham(M,ω) determines a nontrivial loop ψ × idN in Ham(M ×N, ω ⊕ η), where (N, η) is a c...

Let F be a field. We shall let W(F) denote the Wit t ring of equivalence classes of nondegenerate symmetric bilinear forms over F. There are many questions one migh t ask about the extent to which one can prescribe the Wit t ring of an extension field of F and the natural homomorphism between these Wit t rings induced by the field inclusion. In this note we consider a formally real field F and ...

For any ring homomorphism R −→ R′ of noetherian rings, the “extension of scalars” morphism from (Sch/R) to (Sch/R′) which sends X/R to X ⊗R R′/R′, extends to a group homomorphism from K0(Sch/R) to K0(Sch/R). Suppose A is an abelian group, and ρ is an “additive function” from (Sch/R) to A, i.e., a rule which assigns to each X ∈ (Sch/R) an element ρ(X) ∈ A, such that ρ(X) depends only on the isom...

Extending ideas of twisted equivariant K-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective Z2-graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups....

By a topological ring we mean a ring X with a Hausdorff topology such that (x, y) 7→ x + y, x 7→ −x, (x, y) 7→ x · y are continuous. A morphism of topological rings is a continuous homomorphism of rings. An inverse system of topological rings is a family of topological rings Xi and a family of morphisms πi,j : Xi → Xj for i, j ∈ I with i ≥ j, such that when i ≥ j ≥ k, πi,k = πj,k ◦ πi,j . If Y ...

Problem 2. (a) For any ring R, define a ring structure on the abelian group R̃ = R×Z such that (0, 1) is the identity of R̃ and the inclusion map R ↪→ R̃, r 7→ (r, 0) is a rng homomorphism. (b) Let C be the category of rings without identity. Show that the functor F : C → Rings such that F (R) = R̃ is a left adjoint to the forgetful functor G : Rings→ C taking a ring with identity R to R considered...

Let p be a prime number, and let A ring in which is nilpotent . In this paper, we consider the maps K q + 1 ( [ x ] / m ) , → n induced by homomorphism ↦ We evaluate these maps, up to extension, for general terms of topological Hochschild homology, regular F -algebras groups de Rham-Witt forms. After evaluation, give calculation relative -group O certain perfectoid fields

For an arbitrary ring R, the largest strong left quotient ring Ql (R) of R and the strong left localization radical lR are introduced and their properties are studied in detail. In particular, it is proved that Ql (Q s l (R)) ≃ Q s l (R), l s R/ls R = 0 and a criterion is given for the ring Ql (R) to be a semisimple ring. There is a canonical homomorphism from the classical left quotient ring Q...