<p>Artin, Tate and Van den Bergh initiated the field of noncommutative projective algebraic geometry by fruitfully studying geometric data associated to noncommutative graded algebras. More specifically, given a field <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation> and a graded <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </math></EquationSource> </InlineEquation>, they defined an inverse system of projective schemes <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varvec{\Upsilon }_{\varvec{A}} \varvec{=} \varvec{\{}{\varvec{\Upsilon }_{\varvec{d}}\varvec{(A)}}\varvec{\}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </msub> <mrow> <mo mathvariant="bold">=</mo> </mrow> <mrow> <mo mathvariant="bold" stretchy="false">{</mo> </mrow> <mrow> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">d</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> <mrow> <mo mathvariant="bold" stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This system affords an algebra, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">B</mi> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, built out of global sections, and a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>-algebra morphism <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varvec{\tau }\varvec{:} \varvec{A} \varvec{\rightarrow } {\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">τ</mi> </mrow> <mrow> <mo mathvariant="bold">:</mo> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> <mrow> <mo mathvariant="bold">→</mo> </mrow> <mi mathvariant="bold">B</mi> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We study and extend this construction. We define, for any natural number <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varvec{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">n</mi> </mrow> </math></EquationSource> </InlineEquation>, a category <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\texttt {PSys}^{\varvec{n}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="monospace">PSys</mi> <mrow> <mi mathvariant="bold-italic">n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> of <i>projective systems of schemes</i> and a contravariant functor <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\textbf{B}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">B</mi> </math></EquationSource> </InlineEquation> from <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\texttt {PSys}^{\varvec{n}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="monospace">PSys</mi> <mrow> <mi mathvariant="bold-italic">n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> to the category of associative <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>-algebras. We realize the schemes <InlineEquation ID="IEq13"> <EquationSource Format="TEX">${\varvec{\Upsilon }_{\varvec{d}}\varvec{(A)}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">d</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\varvec{\hbox {Proj}\ } {\textbf{U}}_{\varvec{d}}\varvec{(A)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mtext>Proj</mtext> <mspace width="4pt" /> </mrow> <msub> <mi mathvariant="bold">U</mi> <mrow> <mi mathvariant="bold-italic">d</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq15"> <EquationSource Format="TEX">${\textbf{U}}_{\varvec{d}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">U</mi> <mrow> <mi mathvariant="bold-italic">d</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is a functor from associative algebras to commutative algebras. We characterize when the morphism <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\varvec{\tau }\varvec{:} \varvec{A} \varvec{\rightarrow } {\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">τ</mi> </mrow> <mrow> <mo mathvariant="bold">:</mo> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> <mrow> <mo mathvariant="bold">→</mo> </mrow> <mi mathvariant="bold">B</mi> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is injective or surjective in terms of local cohomology modules of the <InlineEquation ID="IEq17"> <EquationSource Format="TEX">${\textbf{U}}_{\varvec{d}}\varvec{(A)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">U</mi> <mrow> <mi mathvariant="bold-italic">d</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Motivated by work of Walton, when <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\varvec{\Upsilon }_{\varvec{A}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> consists of well-behaved schemes, we prove a geometric result that computes the Hilbert series of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">${\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">B</mi> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <msub> <mrow> <mi mathvariant="bold">Υ</mi> </mrow> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We provide many detailed examples that illustrate our results. For example, we prove that for some non-AS-regular algebras constructed as twisted tensor products of polynomial rings, <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\varvec{\tau }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">τ</mi> </mrow> </math></EquationSource> </InlineEquation> is surjective or an isomorphism.</p>

Algebras Associated to Inverse Systems of Projective Schemes

期刊论文
发表日期：

2026年5月14日

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Andrew Conner,
Peter Goetz

摘要

Artin, Tate and Van den Bergh initiated the field of noncommutative projective algebraic geometry by fruitfully studying geometric data associated to noncommutative graded algebras. More specifically, given a field $\mathbb {K}$ $K$ and a graded $\mathbb {K}$ $K$ -algebra $\varvec{A}$ $A$ , they defined an inverse system of projective schemes $\varvec{\Upsilon }_{\varvec{A}} \varvec{=} \varvec{\{}{\varvec{\Upsilon }_{\varvec{d}}\varvec{(A)}}\varvec{\}}$ $Υ_{A} = {Υ_{d} (A)}$ . This system affords an algebra, ${\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$ $B (Υ_{A})$ , built out of global sections, and a $\mathbb {K}$ $K$ -algebra morphism $\varvec{\tau }\varvec{:} \varvec{A} \varvec{\rightarrow } {\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$ $τ : A \to B (Υ_{A})$ . We study and extend this construction. We define, for any natural number $\varvec{n}$ $n$ , a category $\texttt {PSys}^{\varvec{n}}$ ${PSys}^{n}$ of projective systems of schemes and a contravariant functor ${\textbf{B}}$ $B$ from $\texttt {PSys}^{\varvec{n}}$ ${PSys}^{n}$ to the category of associative $\mathbb {K}$ $K$ -algebras. We realize the schemes ${\varvec{\Upsilon }_{\varvec{d}}\varvec{(A)}}$ $Υ_{d} (A)$ as $\varvec{\hbox {Proj}\ } {\textbf{U}}_{\varvec{d}}\varvec{(A)}$ $Proj U_{d} (A)$ , where ${\textbf{U}}_{\varvec{d}}$ $U_{d}$ is a functor from associative algebras to commutative algebras. We characterize when the morphism $\varvec{\tau }\varvec{:} \varvec{A} \varvec{\rightarrow } {\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$ $τ : A \to B (Υ_{A})$ is injective or surjective in terms of local cohomology modules of the ${\textbf{U}}_{\varvec{d}}\varvec{(A)}$ $U_{d} (A)$ . Motivated by work of Walton, when $\varvec{\Upsilon }_{\varvec{A}}$ $Υ_{A}$ consists of well-behaved schemes, we prove a geometric result that computes the Hilbert series of ${\textbf{B}}\varvec{(}\varvec{\Upsilon }_{\varvec{A}}\varvec{)}$ $B (Υ_{A})$ . We provide many detailed examples that illustrate our results. For example, we prove that for some non-AS-regular algebras constructed as twisted tensor products of polynomial rings, $\varvec{\tau }$ $τ$ is surjective or an isomorphism.