Let F be a field, let D be a subring of F , and let X be the Zariski-Riemann space of valuation rings containing D and having quotient field F . We consider the Zariski, inverse and patch topologies on X when viewed as a projective limit of projective integral schemes having function field contained in F , and we characterize the locally ringed subspaces of X that are affine schemes.

Let Γ2 ⊆ Γ1 be finitely generated subgroups of GLn0 (Z[1/q0]). For i = 1 or 2, let Gi be the Zariski-closure of Γi in (GLn0 )Q, Gi be the Zariski-connected component of Gi, and let Gi be the closure of Γi in ∏ p-q0 GLn0 (Zp). In this article we prove that, if G1 is the smallest closed normal subgroup of G1 which contains G2 and Γ2 y G2 has spectral gap, then Γ1 y G1 has spectral gap.

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type A10 + A9 that are defined over Q( √ 5) and are conjugate to each other by the action of Gal(Q( √

Let $f \colon X \to X$ be a surjective endomorphism of normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure $X$. Using this, extend second author's result to singular surfaces extent that either $X$ has $f$-invariant non-constant rational function, or $f$ infinitely many Zaris...

Let L be a complete lattice. We introduce and characterize the prime L-submodules of a unitary module over a commutative ring with identity. Finally, we investigate the Zariski topology on the prime L-Spectrum of a unitary module, consisting of the collection of all prime L-submodules, and prove that for L-top modules the Zariski topology on L-Spec(M) exists. © 2007 Elsevier B.V. All rights res...