Maps between non-commutative spaces.pdf

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 356, Number 7, Pages 2927–2944 S 0002-9947(03)03411-1 Article electronically published on November 18, 2003 MAPS BETWEEN NON-COMMUTATIVE SPACES S. PAUL SMITH Abstract. Let J be a graded ideal in a not necessarily commutative graded k-algebra A = A 0 ⊕A 1 ⊕· · · in which dim k A i ∞for all i. We show that the map A →A/J induces a closed immersion i : Proj nc A/J →Proj nc A between the non-co
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  TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 356, Number 7, Pages 2927–2944S 0002-9947(03)03411-1Article electronically published on November 18, 2003 MAPS BETWEEN NON-COMMUTATIVE SPACES S. PAUL SMITH Abstract.  Let  J   be a graded ideal in a not necessarily commutative graded k -algebra  A  =  A 0 ⊕ A 1 ⊕···  in which dim k  A i  < ∞ for all  i . We show that themap  A → A/J   induces a closed immersion  i  : Proj nc A/J   → Proj nc A  betweenthe non-commutative projective spaces with homogeneous coordinate rings  A and  A/J  . We also examine two other kinds of maps between non-commutativespaces. First, a homomorphism  φ  :  A  →  B  between not necessarily commu-tative  N -graded rings induces an affine map Proj nc  B  ⊃  U   →  Proj nc  A  froma non-empty open subspace  U   ⊂ Proj nc B . Second, if   A  is a right noetherianconnected graded algebra (not necessarily generated in degree one), and  A ( n ) is a Veronese subalgebra of   A , there is a map Proj nc A  →  Proj nc A ( n ) ; weidentify open subspaces on which this map is an isomorphism. Applying thesegeneral results when  A  is (a quotient of) a weighted polynomial ring producesa non-commutative resolution of (a closed subscheme of) a weighted projectivespace. 1.  Introduction This paper concerns maps between non-commutative projective spaces of theform Proj nc A . Before summarizing our main results we define the relevant terms.Following Rosenberg [8, p. 278] and Van den Bergh [13], a non-commutative space  X   is a Grothendieck category  Mod X  . A  map  g  :  Y   → X   between two spacesis an adjoint pair of functors ( g ∗ ,g ∗ ) with  g ∗  :  Mod Y   → Mod X   and  g ∗ left adjointto  g ∗ . The map  g  is  affine  [8, page 278] if   g ∗  is faithful and has a right adjoint.For example, a ring homomorphism  ϕ  :  R  →  S   induces an affine map  g  :  Y   →  X  between the affine spaces defined by  Mod Y   :=  Mod S   and  Mod X   :=  Mod R .Let  k  be a field. An  N -graded  k -algebra  A  is  locally finite  if dim k  A i  < ∞  for all i . The non-commutative projective space  X   with homogeneous coordinate ring  A is defined by Mod X   :=  GrMod A/ Fdim A (see Section 2), andProj nc A  := ( Mod X, O X ) , where  O X  is the image of   A  in  Mod X  . Thus Proj nc A  is an enriched quasi-schemein the language of [13]. Let  Y   be another non-commutative projective space withhomogeneous coordinate ring  B . A map  f   : Proj nc B  → Proj nc A  is a map  f   :  Y   → X   such that  f  ∗ O X  ∼ = O Y   . Received by the editors September 18, 2002 and, in revised form, April 29, 2003.2000  Mathematics Subject Classification.  Primary 14A22; Secondary 16S38.The author was supported by NSF grant DMS-0070560. c  2003 American Mathematical Society 2927  2928 S. PAUL SMITH When  A  is a commutative  N -graded  k -algebra we write Proj A  for the usualprojective scheme. We will always view a quasi-separated, quasi-compact scheme X   as a non-commutative space by associating to it the enriched space ( Qcoh X, O X ).The rule  X   → ( Qcoh X, O X ) is a faithful functor. Summary of results.  The main results in this paper are Theorems 3.2, 3.3, 4.1, and Proposition 4.8.A map  g  :  Y   → X   is a  closed immersion  if it is affine and the essential image of  Mod Y   in  Mod X   under  g ∗  is closed under submodules and quotients. Theorem 3.2shows that a surjective homomorphism  A → A/J   of graded rings induces a closedimmersion  i  : Proj nc A/J   → Proj nc A . The functors  i ∗ and  i ∗  are the obvious ones(see the proof of  3.2). It seems to be a folklore result that  i ∗ is left adjoint to  i ∗ , butwe could not find a proof in the literature so we provide one here. Several peoplehave been aware for some time that this is the appropriate intuitive picture, but,as far as I know, no formal definition of a closed immersion has been given and sono explicit proof has been given.If   A  is a graded subalgebra of   B , commutative results suggest there should bea closed subspace  Z   of   Y   = Proj nc B  and an affine map  g  :  Y  \ Z   →  Proj nc A .Theorem 3.3 establishes such a result under reasonable hypotheses on  A  and  B . Infact, that result is set in a more general context, namely a homomorphism  φ  :  A → B  of graded rings. Corollary 3.4 then says that if   φ  :  A  →  B  and  B  is a finitelypresented left  A -module, then there is an affine map  g  : Proj nc B  → Proj nc A . Thisis a (special case of a) non-commutative analogue of the commutative result that afinite morphism is affine.If   A  is a quotient of a commutative polynomial ring, and  A ( n ) is the gradedsubring with components ( A ( n ) ) i  =  A ni , then there is an isomorphism of schemesProj A  ∼ = Proj A ( n ) . Verevkin [12] proved that Proj nc A  ∼ = Proj nc A ( n ) when  A  isno longer commutative, but is connected and generated in degree one. Theorem4.1 shows that when  A  is not required to be generated in degree one, there is stilla map Proj nc A  →  Proj nc A ( n ) , and Proposition 4.8 describes open subspaces onwhich this map is an isomorphism.The results here are modelled on the commutative case, and none is a surprise.In large part the point of this paper is to make the appropriate definitions sothat results from commutative algebraic geometry carry over verbatim to the non-commutative setting. Thus we formalize and make precise some of the terminologyand intuition in papers like [2] and [7]. In Example 4.9 we show how our results apply to a quotient of a weighted poly-nomial ring to obtain a birational isomorphism  g  : Proj nc A → X   = Proj A , where X   is a commutative subscheme of a weighted projective space. It can happen that X   is singular whereas Proj nc A  is smooth. Thus we can view Proj nc A → Proj A  assomething like a non-commutative resolution of singularities. Furthermore, in thissituation  g ∗ g ∗  ∼ = id.We freely use basic notions and terminology for non-commutative spaces fromthe papers [9], [10], and [13]. 2.  Definitions and preliminaries Throughout this paper we assume that  A  is a  locally finite N -graded algebra  over afield  k . Thus  A  =  A 0 ⊕ A 1 ⊕··· , and dim k A i  < ∞ for all  i . The  augmentation ideal m of   A  is  A 1 ⊕ A 2 ⊕··· . If   A 0  is finite dimensional and  A  is right noetherian, then it  MAPS BETWEEN NON-COMMUTATIVE SPACES 2929 follows that dim k  A i  < ∞ for all  i  because  A ≥ i /A ≥ i +1  is a noetherian  A/ m -module.We write  GrMod A  for the category of   Z -graded right  A -modules, and define Tails A  :=  GrMod A/ Fdim A, where  Fdim A  is the full subcategory consisting of direct limits of finite dimensional A -modules. Equivalently,  Fdim A  consists of those modules in which every elementis annihilated by a suitably large power of   m . We write  π  for the quotient functor GrMod A → Tails A  and  ω  for its right adjoint.The  projective space with homogeneous coordinate ring  A  is the space  X   definedby  Mod X   :=  Tails A . We write Proj nc A  = ( Mod X, O X ), where  O X  denotes theimage of   A  in  Tails A .A  closed subspace  Z   of a space  X   is a full subcategory  Mod Z   of   Mod X   that isclosed under submodules and quotient modules in  Mod X   and such that the inclusionfunctor  i ∗  :  Mod Z   → Mod X   has both a left adjoint  i ∗ and a right adjoint  i ! .Two spaces are  isomorphic  if their module categories are equivalent. Hence amap  Y   → X   is a closed immersion if and only if it is an isomorphism from  Y   to aclosed subspace of   X  .The  complement  X  \ Z   to a closed subspace  Z   is defined by Mod X  \ Z   :=  Mod X/ T , the quotient category of   Mod X   by the localizing subcategory  T  consisting of those X  -modules  M   that are the direct limit of modules  N   with the property that  N   hasa finite filtration  N   =  N  n  ⊃  N  n − 1  ⊃ ··· ⊃  N  1  ⊃  N  0  = 0 such that each  N  i /N  i − 1 is in  Mod Z  . Because  T  is a localizing category, there is an exact quotient functor  j ∗ :  Mod X   →  Mod X  \ Z  , and its right adjoint  j ∗  :  Mod X  \ Z   →  Mod X  . The pair(  j ∗ ,j ∗ ) defines a map  j  :  X  \ Z   → X  . We call it an  open immersion .We sometimes write  Mod Z  X   for the category  T  and call it the category of   X  -modules  supported on  Z  .Let  f   :  Y   → X   be a map. If   f  ∗  is faithful, then the counit id Y   → f  ! f  ∗  is monicand the unit  f  ∗ f  ∗  → id Y   is epic. Watt’s Theorem for graded modules.  Let  A  and  B  be  Z -graded  k -algebras.We recall the analogue of Watt’s Theorem proved by Del Rio [3, Proposition 3]that describes the  k -linear functors  GrMod A → GrMod B  that have a right adjoint.A bigraded  A - B -bimodule is an  A - B -bimodule M   =  (  p,q ) ∈ Z 2  p M  q such that  A i .  p M  q .B j  ⊂  i +  p M  q + j  for all  i,j,p,q   ∈  Z . Write  ⊗  for  ⊗ k . If   L  is agraded right  A -module, we define L  ¯ ⊗ A M   :=  q ∈ Z ( L  ¯ ⊗ A M  ) q , where ( L  ¯ ⊗ A M  ) q  is the image of    p ( L −  p ⊗  p M  q ) under the canonical map  L ⊗ M   → L ⊗ A  M  . This gives  L  ¯ ⊗ A M   the structure of a graded right  B -module; it is a  B -module direct summand of the usual tensor product  L ⊗ A  M  .If   N   is a graded right  B -module, we defineHom B ( M,N  ) := { f   ∈ Hom Gr B ( M,N  ) | f  (  p M  ∗ ) = 0 for almost all  p } .  2930 S. PAUL SMITH This is made into a graded right A -module by declaring that deg f   =  p  if   f  ( i M  ∗ ) = 0for all  i  = −  p . Hence Hom B ( M,N  )  p  is naturally isomorphic to Hom Gr B ( −  p M  ∗ ,N  ),and there is a natural isomorphismHom B ( M,N  ) =   p Hom Gr B ( −  p M  ∗ ,N  ) . The usual adjoint isomorphism between Hom and  ⊗  then induces an isomorphismHom Gr B ( L  ¯ ⊗ A M,N  ) ∼ = Hom Gr A ( L, Hom B ( M,N  )) , (2-1)showing that  − ¯ ⊗ A M   :  GrMod A → GrMod B  is left adjoint to Hom B ( M, − ). Theorem 2.1  (Del Rio [3]) .  Let   A  and   B  be graded   k -algebras, and   F   :  GrMod A → GrMod B  a   k -linear functor having a right adjoint. Then   F   ∼ =  − ¯ ⊗ A M  , where   M  is the bigraded   A - B -bimodule  M   =   p ∈ Z F  ( A (  p )) with homogeneous components   p M  q  =  F  ( A (  p )) q .If   F   also commutes with the twists by degree, then   F   is given by tensoring with a graded   A - B -bimodule, say   V   =   n V  n . The corresponding   M   in this case is  M   =  V  (  p )  with   p M  q  =  V  (  p ) q .The left   A -action on   M   is given by declaring that   x ∈ A i  acts on   p M  ∗  as   F  ( λ x ) ,where   λ x  :  A (  p ) → A (  p  + i )  denotes left multiplication by   x . 3.  Maps induced by graded ring homomorphisms Throughout this section we assume that  A  and  B  are locally finite  N -gradedalgebras over a field  k .We consider the problem of when a homomorphism  φ  :  A  →  B  of graded ringsinduces a map  g  : Proj nc B  → Proj nc A  and, if it does, how the properties of   g  aredetermined by the properties of   φ .Associated to  φ  is an adjoint triple ( f  ∗ ,f  ∗ ,f  ! ) of functors between the categoriesof graded modules. Explicitly,  f  ∗ = −⊗ A B ,  f  ∗  = −⊗ B  B A  is the restriction map,and  f  ! =   p ∈ Z  Hom Gr B ( B ( −  p ) , − ). We wish to establish conditions on  φ  whichimply that these functors factor through the quotient categories in the followingdiagrams: GrMod B  f  ∗ −−−−→  GrMod A π   π Tails B  Tails A GrMod B  f  ∗ ,f  ! ←−−−−  GrMod A π   π Tails B  Tails A Lemma 3.1.  Let   A  and   B  be Grothendieck categories with localizing subcategories  S ⊂ A  and   T ⊂ B . Let   π  :  A → A / S  and   π  :  B → B / T  be the quotient functors, and let   ω  and   ω  be their right adjoints. Consider the following diagram of functors: A  F  −−−−→  B π  π  A / S B / T .
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