Nonnoetherian geometry (Q2816951)

A major principle in classical algebraic geometry is \textit{locality}, namely, properties which can be determined by considering an open neighborhood of a point rather than the global space structure. As an important example, the dimension of an algebraic variety is local. The purpose of this paper is to provide some tools for geometric understanding of affine schemes whose ring of global sections is nonnoetherian but of finite Krull dimension. In such situation, it may occur that some points have positive dimension.NEWLINENEWLINEThe author introduces the notion of \textit{depiction} of an algebra. This is a finitely generated algebra containing the original algebra, such that the induced morphism of affine schemes is surjective and noetherianity of closed points is encoded by the corresponding local rings at that point of the two algebras being equal. Using this notion, a \textit{geometric codimension} (resp. \textit{codimension}) is defined at a point, as the minimum height of prime ideals lying over that point in some depiction (resp.~the Krull dimension of the ring minus the geometric codimension).NEWLINENEWLINEThese notions allow the author to interpret nonnoetherian singularities as points of positive geometric dimension.NEWLINENEWLINEThe paper also provides applications for noncommutative rings. The author defines an \textit{impression} of a finitely generated (noncommutative) algebra as a representation into matrix ring over some finitely generated commutative algebra, which is surjective onto the matrix rings obtained by compositing it with the quotient map modulo a ageneric prime ideal (of the commutative algebra), and such that the induced morphism from the spectrum of the commutative underlying algebra to the spectrum of the center of the represented algebra is surjective too.NEWLINENEWLINEThe author applies the notion of impression to the context of quiver algebras, encoding noetherianity of the algebra and its center in terms of a depiction associated with the impression. An example of a quiver algebra is given, in which the geometric codimension of the vertex simple modules is equal to their projective dimension, thereby hinting that the geometric (co)dimension notion defined in this paper plays an important role in the geometric investigation of noncommutative algebras.