In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical...

elements, then the Proj of R in this sense is equivalent to the category of coherent sheaves on the usual Proj of R. Hence, the construction can be thought...

One of the basic constructions in commutative algebraic geometry is the Proj construction of a graded commutative ring. This construction builds a projective...

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homogeneous polynomial of the same degree) play an essential role in the Proj construction of projective schemes. When working over the real numbers, or more...

Nullstellensatz. More generally, for an arbitrary graded ring R, the Proj construction disregards all irrelevant ideals of R. Zariski & Samuel 1975, §VII...

they both make sense. Just as Spec replaces affine varieties, the Proj construction replaces projective varieties in modern algebraic geometry. Just as...

{\displaystyle k} . This can be easily generalized to any base ring, see Proj construction (in fact, we can define projective space for any base scheme). The...

manifolds can be built by gluing together open sets of Rn. The Proj construction is the construction of the scheme of a projective space, and, more generally...

^{n}(k)} in the usual sense. An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted...

that furthermore is complete. cf. Glossary of scheme theory The Proj construction in fact gives more than a mere scheme: a sheaf in graded modules over...

algebra of polynomials on a vector space V then Proj S is P(V). This Proj construction gives rise to a contravariant functor from the category of graded...

terms of the Proj construction, this subscheme can be written as Proj ⁡ R [ x 0 , … , x n ] / ( f ) . {\displaystyle \operatorname {Proj} R[x_{0},\ldots...

blow-up is constructed (see also Rees algebra). Here Proj is the Proj construction on graded sheaves of commutative rings. The exceptional divisor of...

line bundle determines Y by the Proj construction: Y = Proj  ⨁ a ≥ 0 H 0 ( X , L ⊗ a ) . {\displaystyle Y={\text{Proj }}\bigoplus _{a\geq 0}H^{0}(X,L^{\otimes...

, P ( V ) = Proj ⁡ ( k [ V ] ) = Proj ⁡ ( Sym ⁡ ( V ∗ ) ) {\displaystyle \mathbf {P} (V)=\operatorname {Proj} (k[V])=\operatorname {Proj} (\operatorname...

rings generated by elements of degree 1 produce projective schemes by Proj construction while finitely generated graded modules correspond to coherent sheaves...

an) is isomorphic to P(da0,da1,...,dan). This is a property of the Proj construction; geometrically it corresponds to the d-tuple Veronese embedding. So...

these cases may be dealt with on the same footing by means of the Proj construction. The irrelevant ideal J generated by all the Xi corresponds to the...

quantum deformation of the exterior algebra by a quadratic form Proj construction § Proj of a quasi-coherent sheaf, an application of symmetric algebras...

(2004), Theorem 2.1.33. The dimension of the Proj construction Proj ⁡ R ( K X ) {\displaystyle \operatorname {Proj} R(K_{X})} , a projective variety called...

algebraic construction is a method by which an algebraic entity is defined or derived from another. Instances include: Cayley–Dickson construction Proj construction...

_{B}^{n},\,x\mapsto (s_{0}(x):\dots :s_{n}(x))} . (See also Proj construction#Global Proj.) Remark: The above point of view (which goes under the name...

Evaluation Program, Aerospace Medical Laboratory, Wright-Patterson AFT, Proj. No. 7164, Task No 71832 Kubis, Joseph F.; McLaughlin, Edward J. (1966),...

space using the Proj construction, X ≅ Proj ⁡ ⨁ r ≥ 0 H 0 ( X , L k r ) ↪ P ( H 0 ( X , L k ) ∗ ) {\displaystyle X\cong \operatorname {Proj} \bigoplus _{r\geq...

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because that the fibers of E, the vector spaces, are contractible. Proj construction cone (algebraic geometry) ruled surface (an example of a projective...

affine cone of R. Similarly, the relative Proj P ( C ) = Proj X ⁡ R {\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R} is called the projective cone...

Chretienneau developed OpenProj at Projity in 2007. It moved out of beta with Version 1.0 on January 10, 2008. In late 2008, Projity was acquired by Silver...

finitely many degree-1 elements, and Proj ⁡ ( R ) {\displaystyle \operatorname {Proj} (R)} refers to the Proj construction) can be described as the Serre quotient...