This entry is about the formal dual topological space of a commutative ring. For the very different notion of a similar name in higher algebra see at ring spectrum. For more see at spectrum - disambiguation.
Given a commutative ring , its spectrum is the topological space whose points are the prime ideals of and whose topology is the Zariski topology on these prime ideals. This topological case is also called the prime spectrum of , the latter terminology however applies to noncommutative rings as well.
However, usually by one means more: the locally ringed space which is obtained by equipping the above topological space by a unique sheaf of commutative local rings such that for every principal localization of commutative rings we have and the restrictions and where divides are the corresponding localizations of rings. Global sections functor assigns to every ringed space the ring of global sections. Restrict this functor to the functor of global sections from the subcategory of commutative locally ringed spaces to the category of commutative rings. This functor has its right adjoint and this is precisely the Spec-functor.
If the prime spectrum is taken with a structure of a locally ringed space then one usually says the affine spectrum (this terminology never used just for the underlying topological space or a set).
Every locally ringed space isomorphic to an affine spectrum is said to be an affine scheme.
projective spectrum?
modules over a ring are equivalent to quasicoherent sheaves over its spectrum
Last revised on August 30, 2024 at 11:11:12. See the history of this page for a list of all contributions to it.