Singular projective bases and the affine Bol operator

Gröbner bases for families of affine or projective schemes

Let I be an ideal of the polynomial ring A[x] = A[x1, . . . , xn] over the commutative, noetherian ring A. Geometrically I defines a family of affine schemes over Spec(A): For p ∈ Spec(A), the fibre over p is the closed subscheme of affine space over the residue field k(p), which is determined by the extension of I under the canonical map σp : A[x] → k(p)[x]. If I is homogeneous there is an ana...

Affine and projective planes

Affine and Projective Universal Geometry

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyp...

Operator-valued bases on Hilbert spaces

In this paper we develop a natural generalization of Schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. We prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. We prove that the operators of a dual ov-basis are continuous. We also dene the concepts of Bessel, Hilbert ov-basis and obta...

Projective Dimension and the Singular Locus

For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.