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<p>We study the spectrum of closed subcategories in a quasi-scheme, i.e. a Grothendieck category <i>X</i>. The closed subcategories are the direct analogs of closed subschemes in the commutative case, in the sense that when <i>X</i> is the category of quasi-coherent sheaves on a quasi-projective scheme <i>S</i>, then the closed subschemes of <i>S</i> correspond bijectively to the closed subcategories of <i>X</i>. Many interesting quasi-schemes, such as the noncommutative projective scheme <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\operatorname {Qgr-}\hspace{-2.0pt}B = \operatorname {Gr-}\hspace{-2.0pt}B/\operatorname {Tors-}\hspace{-2.0pt}B\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Qgr-</mo> <mspace width="-2.0pt" /> <mi>B</mi> <mo>=</mo> <mo>Gr-</mo> <mspace width="-2.0pt" /> <mi>B</mi> <mo stretchy="false">/</mo> <mo>Tors-</mo> <mspace width="-2.0pt" /> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> associated to a graded algebra <i>B</i>, arise as quotient categories of simpler abelian categories. In this paper, we will show how to describe the closed subcategories of any quotient category <i>X</i>/<i>Y</i> in terms of closed subcategories of <i>X</i> with special properties, when <i>X</i> is a category with a set of compact projective generators.</p>

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Closed Subcategories of Quotient Categories

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  • 发表日期：

    2026年3月8日

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  • Daniel Rogalski

摘要

We study the spectrum of closed subcategories in a quasi-scheme, i.e. a Grothendieck category X. The closed subcategories are the direct analogs of closed subschemes in the commutative case, in the sense that when X is the category of quasi-coherent sheaves on a quasi-projective scheme S, then the closed subschemes of S correspond bijectively to the closed subcategories of X. Many interesting quasi-schemes, such as the noncommutative projective scheme \(\operatorname {Qgr-}\hspace{-2.0pt}B = \operatorname {Gr-}\hspace{-2.0pt}B/\operatorname {Tors-}\hspace{-2.0pt}B\) Qgr- B = Gr- B / Tors- B associated to a graded algebra B, arise as quotient categories of simpler abelian categories. In this paper, we will show how to describe the closed subcategories of any quotient category X/Y in terms of closed subcategories of X with special properties, when X is a category with a set of compact projective generators.