Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological and scheme-theoretic properties of the $D$-graded Proj construction

Felix Goebler

TL;DR

The paper extends the classical Proj construction to multigraded rings graded by a finitely generated abelian group $D$ by introducing D-prime ideals and the multihomogeneous spectrum Proj^D_B(S) for conical rings (S,B). It proves a multigraded version of the Nullstellensatz and establishes a scheme structure on Proj^D_B(S) via affine opens D_+(f) that correspond to localizations S_{(f)}, yielding Proj^D_B(S) ≅ Proj^B_D(S). It develops a homogeneous Zariski topology on the D-prime spectrum, analyzes when the resulting Proj is separated through algebraic surjectivity of multiplication maps mu_{(fg)}, and explains non-separatedness via linear degree dependencies. The work further develops the theory of sheaves on multigraded spectra, including the construction of Serre twists O_X(d), their coherence, and an isomorphism S_d ≅ Γ(X, O_X(d)) in the noetherian factorial setting. Finally, it provides practical criteria for invertibility and reflexivity of Serre twists, linking invertibility to the intersection of degree-subgroups D^f and offering structural insight into the Picard group and potential ample-family constructions in the multigraded context.

Abstract

We generalize the topological description of the $\mathbb{N}$-graded Proj construction to the multigraded Proj construction for factorially graded rings that are graded by finitely generated abelian groups $D$. However, there is one big structural difference: While the classical description is given by the space of homogeneous prime ideals not containing the irrelevant ideal, we characterize the multigraded Proj setting using $D$-prime ideals, i.e.\ ideals that have the prime property, but only for homogeneous factorizations. In particular, we establish a multigraded version of the Nullstellensatz. Additionally, we present algebraic conditions for separability in terms of factorially graded rings, and observe that Proj$^D(S)$ is not separated in many cases. Finally, building on Mayeux-Riche's definition of Serre twists, we give a criterion for their invertibility.

Topological and scheme-theoretic properties of the $D$-graded Proj construction

TL;DR

The paper extends the classical Proj construction to multigraded rings graded by a finitely generated abelian group by introducing D-prime ideals and the multihomogeneous spectrum Proj^D_B(S) for conical rings (S,B). It proves a multigraded version of the Nullstellensatz and establishes a scheme structure on Proj^D_B(S) via affine opens D_+(f) that correspond to localizations S_{(f)}, yielding Proj^D_B(S) ≅ Proj^B_D(S). It develops a homogeneous Zariski topology on the D-prime spectrum, analyzes when the resulting Proj is separated through algebraic surjectivity of multiplication maps mu_{(fg)}, and explains non-separatedness via linear degree dependencies. The work further develops the theory of sheaves on multigraded spectra, including the construction of Serre twists O_X(d), their coherence, and an isomorphism S_d ≅ Γ(X, O_X(d)) in the noetherian factorial setting. Finally, it provides practical criteria for invertibility and reflexivity of Serre twists, linking invertibility to the intersection of degree-subgroups D^f and offering structural insight into the Picard group and potential ample-family constructions in the multigraded context.

Abstract

We generalize the topological description of the -graded Proj construction to the multigraded Proj construction for factorially graded rings that are graded by finitely generated abelian groups . However, there is one big structural difference: While the classical description is given by the space of homogeneous prime ideals not containing the irrelevant ideal, we characterize the multigraded Proj setting using -prime ideals, i.e.\ ideals that have the prime property, but only for homogeneous factorizations. In particular, we establish a multigraded version of the Nullstellensatz. Additionally, we present algebraic conditions for separability in terms of factorially graded rings, and observe that Proj is not separated in many cases. Finally, building on Mayeux-Riche's definition of Serre twists, we give a criterion for their invertibility.
Paper Structure (7 sections, 27 theorems, 46 equations)

This paper contains 7 sections, 27 theorems, 46 equations.

Table of Contents

  1. $D$-graded Proj
  2. Multihomogeneous D-prime Spectra
  3. Homogeneous Zariski Topology
  4. Topological description of Proj
  5. Separation Criteria
  6. Sheaves on Multigraded Spectra
  7. Invertibility and reflexivity of Serre twists

Key Result

Theorem A

Let $S$ be a factorially and effectively $D$-graded integral domain and $(S, B)$ a conical ring. Then we can equip its multihomogeneous $D$-prime spectrum with the structure of a scheme in such a way, that for relevant $f \in B$ the homeomorphisms $\psi_f\colon D_+(f) \to \mathop{\mathrm{Spec}}\nolimits(S_{(f)})$ of topological spaces become isomorphisms of schemes and thus give rise to an affine

Theorems & Definitions (78)

  • Theorem A: \ref{['thm:Proj_D=Proj^D']}
  • Theorem B: \ref{['thm:lin_dep_not_sep']}
  • Definition 1.1
  • Definition 1.2
  • Corollary 1.3
  • Definition 1.4: Multigraded spectrum of a multigraded ring
  • Example 1.5
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • ...and 68 more