Regular schemeIn algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth. For an example of a regular scheme that is not smooth, see Geometrically regular ring#Examples.
Regular embeddingIn algebraic geometry, a closed immersion of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor. For example, if X and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding.
Ideal sheafIn algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces. Let X be a topological space and A a sheaf of rings on X. (In other words, (X, A) is a ringed space.) An ideal sheaf J in A is a subobject of A in the of sheaves of A-modules, i.e., a subsheaf of A viewed as a sheaf of abelian groups such that Γ(U, A) · Γ(U, J) ⊆ Γ(U, J) for all open subsets U of X.
Nagata's compactification theoremIn algebraic geometry, Nagata's compactification theorem, introduced by , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper morphism. Nagata's original proof used the older terminology of Zariski–Riemann spaces and valuation theory, which sometimes made it hard to follow.
Hilbert series and Hilbert polynomialIn commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes.
Crystalline cohomologyIn mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by and developed by . Crystalline cohomology is partly inspired by the p-adic proof in of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963).
Blowing upIn mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion. Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup.
