Zariski Density of Associated Graded Primes and Degree--1 Torsion in Special Fibers of Rees Algebras

Résumé fourni par la source

We study the geometric and homological conditions under which the set of associated primes of the associated graded ring $\gr_I(R)$ of a Noetherian ring $R$ along an ideal $I$ is Zariski dense in $\Spec R$. A unifying criterion is established: density occurs precisely when the special fibers of the Rees algebra exhibit generic degree--1 torsion. Equivalently, on a dense open subset, every minimal reduction $J\subset I$ fails to induce an injective map on degree--1 components $(\gr_J(R))_1 \to (\gr_I(R))_1$, or, equivalently, the reduction number $r_J(I)$ is positive. This correspondence links topological density on $\Spec R$ to algebraic data of reductions, analytic spread, and Rees valuations. The framework remains stable under localization, completion, integral closure, and flat base change, and persists through Veronese and symbolic filtrations. Quantitative bounds are given in standard graded settings via Castelnuovo--Mumford regularity, and explicit examples---monomial, determinantal, and almost complete intersection ideals---demonstrate the criterion’s sharpness. The results provide a cohesive view of how degree--1 behavior in the special fiber governs the global distribution of associated graded primes.

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