Closed Subcategories of Quotient Categories

Abstract 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 $$\operatorname {Qgr-}\hspace{-2.0pt}B = \operatorname {Gr-}\hspace{-2.0pt}B/\operatorname {Tors-}\hspace{-2.0pt}B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>Qgr-</mml:mo> <mml:mspace/> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mo>Gr-</mml:mo> <mml:mspace/> <mml:mi>B</mml:mi> <mml:mo>/</mml:mo> <mml:mo>Tors-</mml:mo> <mml:mspace/> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> 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.