Notes on the Zariski Tangent Space
نویسندگان
چکیده
For perspective, recall how we define the tangent space of a differentiable manifold M . We coverM by open neighborhoods Ui which are identified with R , and then we transfer our understanding of the tangent space at a point of R to define the tangent space at a point in Ui. This can be shown to be independent of choices. This approach is not a good idea for an affine algebraic set X because X does not have an open cover by Zariski open sets that are identified with an open set in some C. First of all, if this were the case, then X would be smooth, so we would miss information about singularities, but secondly even smooth affine varieties are not necessarily locally isomorphic to some C.
منابع مشابه
Foliations with vanishing Chern classes
In this paper we aim at the description of foliations having tangent sheaf TF with c1(TF) = c2(TF) = 0 on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of F is an Abelian variety. It turns out that the analytic type of the Zariski closures of leaves may vary from leaf to leaf. W...
متن کاملThe Tangent Space at a Special Symplectic Instanton Bundle on IP
ABSTRACT :Let MISimp,IP2n+1(k) be the moduli space of stable symplectic instanton bundles on IP with second Chern class c2 = k (it is a closed subscheme of the moduli space MIIP2n+1(k)) We prove that the dimension of its Zariski tangent space at a special (symplectic) instanton bundle is 2k(5n − 1) + 4n − 10n + 3 , k ≥ 2. It follows that special symplectic instanton bundles are smooth points fo...
متن کاملThe Graded Classical Prime Spectrum with the Zariski Topology as a Notherian Topological Space
Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...
متن کاملOn two problems concerning the Zariski topology of modules
Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...
متن کامل