Reconstruction of projective curves from the derived category
aa r X i v : . [ m a t h . AG ] F e b RECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVEDCATEGORY
DYLAN SPENCE
Abstract.
We prove that integral projective curves with Cohen-Macaulay singularitiesover an algebraically closed field of characteristic zero are determined by their triangulatedcategory of perfect complexes. This partially extends a theorem of Bondal and Orlov tothe case of singular projective varieties, and in particular shows that all integral complexprojective curves are determined by their category of perfect complexes.
Contents
1. Introduction 12. Derived categories of sheaves 43. Reconstruction for projective curves 9References 181.
Introduction
Derived categories of sheaves are a relatively recent development in algebraic geometry.Besides intrinsic interest, they present a fruitful approach to noncommutative and homo-logical methods in algebraic geometry, and since their initial development, many interestingconnections to more classical questions have been discovered. For any variety X we canassociate to it the bounded derived category of coherent sheaves D b ( X ), which is the Verdierquotient of bounded chain complexes of objects in coh X by the subcategory of boundedacyclic complexes. The resulting category is no longer abelian, but rather admits a differentstructure known as a triangulation.The first question one might ask is how much information about X is lost in this process.It is a well-known theorem of Gabriel that the abelian category coh X completely determines X , but the analogous question for D b ( X ) is more interesting. If we consider D b ( X ) with itstensor product as a symmetric monoidal triangulated category, then it remains a completeinvariant [Bal05]. However there are many interesting equivalences that do not respect thetensor product. So instead considering D b ( X ) purely as a triangulated category, we ask thesame question. In the case of a smooth projective variety, the answer is tied to the intrinsicgeometry of X , namely if ω X is (anti) ample, then the famous “reconstruction theorem” ofBondal and Orlov in [BO01] states that the derived category is still a complete invariant.However in the case where ω X is neither (in particular, trivial), then there are examples ofnon-isomorphic varieties with equivalent derived categories. Historically the first exampleswere found by Mukai [Muk81], and the specific functor employed in the equivalence is Mathematics Subject Classification.
Key words and phrases.
Derived categories, Reconstruction. now referred much more generally as an “integral functor”, with the term “Fourier-Mukaifunctor” typically reserved for those integral functors which are equivalences. Integralfunctors have taken center stage as the geometric functors of primary consideration, dueboth to utility and necessity, as a seminal result of Orlov [Orl97] showed that any derivedequivalence between two smooth projective varieties is given by a Fourier-Mukai functor.An aspect of the theory which has, up until somewhat recently (see [Bal11, HRLMdS07,SdSSdS12, HRLMSdS09, KKS18]), been mostly ignored is the case when X is singular.However many of the basic tools are now in place for its detailed study. For example, thework of Lunts and Orlov [LO10] has shown that fully faithful functors between projectivevarieties with arbitrary singularities are integral functors. In addition, many details aboutthe structure of arbitrary triangulated categories are now known. In this paper we areinterested in the question of reconstruction, more precisely, to what extent is X determinedby D b ( X ) (or Perf X ) when X is a singular variety. To give some context for our result, it isknown that the reconstruction theorem of Bondal and Orlov holds for Gorenstein varietieswith ample or anti-ample dualizing bundle, proven by Ballard [Bal11] and de Salas and deSalas [SdSSdS12]. We provide an answer to the reconstruction problem for the next mostreasonable type of singularity, those of Cohen-Macaulay type. However due to complicationsin higher dimensions, we make the additional simplification to consider only dim X = 1,namely, projective curves. Since any integral projective curve is automatically Cohen-Macaulay, we lose no generality by our assumptions on the singularities, with arbitrarysingularities (necessarily in higher dimensions) left for future work.To give more context for out result, let us remind the reader about what is knownfor dimension one. As a special case of the results cited above, any curve X which isregular or Gorenstein with ample or anti-ample dualizing bundle is completely determinedby Perf X . The simplicity of curves allows us to reformulate conditions on the canonicalbundle to conditions on the arithmetic genus p a ( X ). It is well known that any regular curvewith p a ( X ) = 1 has anti-ample (= 0) or ample ( ≥
2) canonical bundle ω X . When X isGorenstein, the situation is only marginally more complicated. As is well-known, p a ( X ) = 0can only occur when X ∼ = P ; on the other hand by Riemann-Roch, the dualizing bundleon a Gorenstein curve with p a ( X ) ≥ C , there are few possibilities; the curve is either regular (and hence an ellipticcurve) or the curve is Gorenstein, and must be either a cuspidal or nodal cubic. The formercase can be handled in several ways, originally in [HVdB07] it was shown that any twoderived equivalent elliptic curves must be isomorphic (see also [Ber07]), and the analogousresult in the latter cases are proven in [LM14, LMTP17]. If we are not over C , see [War14,Section 2.3] for a discussion when X is regular. Assembling the results in this and theprevious paragraph, it follows that if X is a regular or Gorenstein complex projective curve,it is completely determined by Perf X .Our result is the following: Theorem A.
Let X and Y be integral projective varieties such that dim X = 1 and X isstrictly Cohen-Macaulay (i.e. contains a singular point which is not smooth or Gorenstein).Then Perf X ∼ = Perf Y if and only if X ∼ = Y . From our result, it then follows that all integral complex projective varieties of dimensionone are completely determined by their triangulated category of perfect complexes.
ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 3
We remark here that since derived equivalences in the singular case are all Fourier-Mukai functors, the work of Ruip´erez et al. [HRLMSdS09, Theorem 4.4] implies thatdim Y = dim X and that Y is Cohen-Macaulay. In our proof we will comment on a differentway to reach the same conclusion.Let us now comment on the structure of the paper. We have broken the discussioninto two parts, the first part serving as a reminder on some relevant aspects of derivedcategories. While the original proof of the reconstruction theorem in [BO01] works for anyequivalence, our proof relies upon, in an essential way, the work of Lunts and Orlov [LO10]who showed that equivalences between derived categories of projective varieties are givenby Fourier-Mukai functors. As such the first subsection is devoted to a review of this veryimportant class of functors. The fundamental result of this subsection is Lemma 2.5, whichshows that under our conventions, a Fourier-Mukai equivalence between any of D(Qcoh( − )),D b ( − ), or Perf( − ) (see the conventions below for notation) automatically induces a Fourier-Mukai equivalence between the other two pairs. This result is certainly already known tothe experts, but we included a proof due to lack of an adequate reference. The followingsubsection is devoted to a review of a replacement for the Serre functor on the boundedderived category, which following Ballard in [Bal11], we label a Rouquier functor.The only other part of the paper is devoted to the proof. We first take the opportunityto establish a special class of objects of Perf X , the perfect zero-cycles. They are a class ofskyscraper sheaves on our curve X which are structure sheaves of thickened points; theirmost useful property is their ability to detect the support of a complex in D b ( X ). Aftersome minor comments about the Rouquier functor in the case of projective curves, the proofthen begins in earnest. The proof consists of two major steps. The first is to classify, ina manner very similar to previous work in [BO01] and [SdSSdS12], the perfect zero-cycleson our curves. We then use these objects to show that the Fourier-Mukai functor preservesthe structure sheaves of closed points, and well known techniques will finish the proof. Acknowledgements.
The author wishes to thank Valery Lunts for his valuable input,encouragement, and continual support while advising him on this problem. He also wishesto thank Alexander Polishchuk, Fernando Sancho de Salas, and especially Xuqiang Qinfor encouragement and helpful discussions. The author is also indebted to Noah Riggen-bach, whose careful reading and criticisms of the initial drafts considerably improved theexposition.In addition the author and this project benefitted from the workshops “Derived Cate-gories, Moduli Spaces and Deformation Theory” and “The Geometry of Derived Categories”held in Cetraro and Liverpool respectively in 2019, and we thank the organizers and theparticipants of the events for providing a stimulating research environment.
Conventions.
Throughout we work over a fixed algebraically closed field k of characteristiczero. In addition, unless explicitly stated otherwise, any scheme X is assumed to be aprojective variety; in particular for us this means an integral separated scheme of finitetype over Spec k , such that X admits a closed immersion into some projective space. A(projective) curve refers to a (projective) variety of dimension one.Unadorned products of varieties and unadorned tensor products of sheaves on X alwaysmean the fiber product over Spec k and the tensor product over O X respectively. Weindicate the right derived functor of a left exact functor F by R F , and similarly the leftderived functor of a right exact functor G by L G . DYLAN SPENCE
By D ∗ (Qcoh X ) and D ∗ ( X ) we always mean the derived category of complexes of quasi-coherent sheaves and coherent sheaves respectively, with appropriate decoration ∗ ∈ {− , + , b } meaning bounded above, below, and bounded respectively. In particular, we identify D b ( X )with the full subcategory of D(Qcoh X ) consisting of complexes of quasi-coherent sheaveswith bounded coherent cohomology sheaves. By Perf X we mean the full triangulated sub-category of D(Qcoh X ) consisting of complexes which are Zariski-locally isomorphic (inD(Qcoh X )) to a bounded complex of vector bundles. It follows from our assumptions thatevery perfect complex on X is strictly perfect, that is, globally isomorphic to a boundedcomplex of vector bundles, as any separated scheme with an ample family of line bundlessatisfies this [TT90, Theorem 2.4.3].2. Derived categories of sheaves
Here we review a few auxiliary facts about derived categories of sheaves which will beof use. The first section is dedicated to the study of integral functors and the focus ismainly on when they can be extended or restricted between subcategories of D(Qcoh X ).The second section focuses on a more general version of the Serre functor on D b ( X ), whichwas originally studied in [Rou08]. Fourier-Mukai Partners.
Two schemes X and Y are said to be derived equivalent ifthere is a triangulated equivalence between the two categories D(Qcoh X ) and D(Qcoh Y ).It is a desirable property for any functor F : D(Qcoh X ) → D(Qcoh Y ) between the twocategories to be represented by an object on the product, namely, consider the product: X × YX Y, π X π Y with π X and π Y the projections. Given some object K in D(Qcoh( X × Y )), called thekernel, we ask if F is isomorphic to the triangulated functorΦ X → Y K ( − ) = R π Y ∗ ( π ∗ X ( − ) L ⊗ K ) : D(Qcoh X ) → D(Qcoh Y ) . If so, we say that F is an integral functor with kernel K . If it is also an equivalence, we saythat it is a Fourier-Mukai functor.Fourier-Mukai functors also make sense when, instead of considering the unbounded de-rived category of quasi-coherent sheaves, we consider other suitable subcategories, namelyPerf X and D b ( X ). For both of these subcategories, we make a nearly identical definitionas above.It is a theorem of Orlov [Orl97, Theorem 2.2] that for X and Y smooth projective varieties,then any triangulated fully faithful functor admitting adjoints between D b ( X ) and D b ( Y )is an integral functor. The question then is to extend this to the singular case, and usingthe formalism of dg-enhancements, Lunts and Orlov were able to show the following. Lemma 2.1 ([LO10], Corollary 9.12) . Let X and Y be quasi-compact separated schemesover a field k . Assume that X has enough locally free sheaves. Let F : D(Qcoh X ) → D(Qcoh Y ) be a fully faithful functor that commutes with coproducts. Then there is an object K ∈
D(Qcoh( X × Y )) such that the functor Φ X → Y K is fully faithful and Φ X → Y K ( C ) ∼ = F ( C ) for any C ∈ D(Qcoh X ) . ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 5
We emphasize that it may not be true that there is an isomorphism of functors F ∼ = Φ X → Y K .From elementary arguments however, it is true that if F is an equivalence, then Φ X → Y K willbe as well.However the main object of study, for many technical reasons, is typically not D(Qcoh X ).Instead one works with the better behaved subcategories D b ( X ) and Perf X . Again, Luntsand Orlov were able to give a very general result, dependent on the geometry of X . Lemma 2.2 ([LO10], Corollary 9.17) . Let X be a projective scheme with maximal torsionsubsheaf of dimension zero T ( O X ) trivial, and Y a noetherian scheme. Let F : D b ( X ) → D b ( Y ) be a fully faithful functor that has a right adjoint. Then there is an object K ∈ D b ( X × Y ) such that F ∼ = Φ X → Y K | D b ( X ) . For the category of perfect complexes, they were also able to conclude:
Lemma 2.3 ([LO10], Corollary 9.13) . Let X be a projective scheme with maximal torsionsubsheaf of dimension zero T ( O X ) trivial, and Y a quasi-compact and separated scheme.If F : Perf X → Perf Y is a fully faithful functor, then there is an object K in D b ( X × Y ) such that F ∼ = Φ X → Y K | Perf X .Remark . In Theorem A, the assumption that X was integral allows us to conclude that O X is a torsion-free sheaf, and so the hypotheses of Lemmas 2.2 and 2.3 are satisfied. Remark . We record here for the reader that integral functors in great generality willcommute with all coproducts. The only difficulty is the derived pushforward, but so longas f : X → Y is quasi-compact and quasi-separated, R f ∗ will commute with arbitrarycoproducts, see [Sta19, Lemma 08DZ]. Alternatively one may use the existence of adjoints[Riz17]. Remark . It is worthwhile to point out that working with D b ( X ) can be troublesome inthe singular case. If the kernel of an integral functor is not nice enough the derived tensorproduct can escape to the bounded above derived category. For some criteria for an objectto define a functor between bounded derived categories, see [HRLMSdS09, Proposition 2.7].Alternatively, one could work directly with D − ( X ).Although in our assumptions in Theorem A are Perf X ∼ = Perf Y , it so happens that wewill need to pass to D b ( X ) eventually. This is a delicate question since we are in the singularsetting. Before we providing an answer we should first recall the notion of a compact objectin a triangulated category and the notion of a compactly generated category. Definition 2.1.
Given a triangulated category T , an object A is called compact if thefunctor Hom T ( A, − ) commutes with all coproducts. The full thick subcategory of compactobjects is denoted T c . T is said to be compactly generated if ( T c ) ⊥ = 0 . It is well known (see [BvdB03, Theorem 3.1.1]) that if X is quasi-compact and quasi-separated, D(Qcoh X ) is a compactly generated triangulated category, generated by a singleperfect complex, and D(Qcoh X ) c = Perf X . Compact generation also provides a veryconvenient characterization of the category T in question. The following is due to Neemanin [Nee96, Lemma 3.1]. Lemma 2.4.
Let
S ⊂ T be a full triangulated subcategory of a compactly generated trian-gulated category T . If S contains T c and is closed under all coproducts of its objects, then S ∼ = T . DYLAN SPENCE
Our primary example is T c = Perf X and T = D(Qcoh X ). In particular, any full subcate-gory of D(Qcoh X ) which contains Perf X and all coproducts of objects in Perf X must beeverything. Lemma 2.5.
Let X and Y be two projective varieties over the field k . Then given K ∈
D(Qcoh( X × Y )) , the following are equivalent: (1) Φ X → Y K : D(Qcoh X ) ∼ → D(Qcoh Y ) , (2) Φ X → Y K : D b ( X ) ∼ → D b ( Y ) , (3) Φ X → Y K : Perf X ∼ → Perf Y ,Furthermore, if any one of the three above occurs, then K ∈ D b ( X × Y ) . One should also compare this result to [CS18, Proposition 7.4].
Proof.
We first show that (1) implies (3). Indeed, Perf X ⊂ D(Qcoh X ) are precisely thecompact objects, namely those objects A such that Hom( A, − ) commutes with all coprod-ucts. It is then immediate that Φ X → Y K restricts to an equivalence Perf X ∼ → Perf Y . Thisshows (1) = ⇒ (3).Again assume that (1) holds. By [Rou08, Proposition 7.46], an object B ∈ D(Qcoh X ) isin D b ( X ) if and only if for all A ∈ Perf X we havedim ⊕ i Hom(
A, B [ i ]) < ∞ . Since, by the previous paragraph, the integral functor preserves compact objects, applyingthis fact shows that Φ X → Y K restricts to an equivalenceΦ X → Y K | D b ( X ) : D b ( X ) ∼ → D b ( Y ) . This shows that (1) = ⇒ (2).Let us now assume (2). By [Rou08, Lemma 7.49], an object A ∈ D(Qcoh X ) belongs toPerf X if and only if dim ⊕ i Hom(
A, B [ i ]) < ∞ for all B ∈ D b ( X ). The case when A ∈ D b ( X ) implies that Φ X → Y K restricts to an equivalenceΦ X → Y K | Perf X : Perf X ∼ → Perf Y. This shows (2) = ⇒ (3).To complete the proof of the first claim, it is enough to show that (3) implies (1). Con-sider the naive extension of this functor to Φ X → Y K : D(Qcoh X ) → D(Qcoh Y ) by simplyallowing the Fourier-Mukai functor to take inputs from D(Qcoh X ). We claim that thisfunctor is fully faithful on all of D(Qcoh X ). Let h Perf X i ⊕ ⊂ D(Qcoh X ) be the mini-mal full triangulated subcategory containing Perf X and all coproducts of perfect objects.By Lemma 2.4, h Perf( X ) i ⊕ = D(Qcoh X ). Consider the full triangulated subcategory ofD(Qcoh X ) defined by T = { Q | Φ X → Y K : Hom( E, Q ) ∼ = Hom(Φ X → Y K ( E ) , Φ X → Y K ( Q )) , for all E ∈ Perf X } . By assumption, this category contains Perf X , and since E is compact and Fourier-Mukaifunctors commute with all coproducts, it also contains h Perf X i ⊕ . Thus T = D(Qcoh X ).Similarly, consider the full triangulated subcategory of D(Qcoh X ) defined by S = { Q | Φ X → Y K : Hom( Q, F ) ∼ = Hom(Φ X → Y K ( Q ) , Φ X → Y K ( F )) , for all F ∈ D(Qcoh X ) } . ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 7
Since we have shown that T = D(Qcoh X ), the same logic again implies h Perf X i ⊕ ⊂ S ,and so must be all of D(Qcoh X ). This shows that the extension to the unbounded derivedcategory is fully faithful.Now to see that the extension is essentially surjective, it again is enough by Lemma 2.4that the essential image be a full triangulated subcategory and is closed under coproductsof objects in Perf Y . The essential image is full as we have shown above that the functoris fully faithful, and the second condition follows as the Fourier-Mukai functor preserves allcoproducts. This shows that (3) implies (1), and completes the proof of the first claim. Letus now turn to the second statement.By restricting to Perf X , by the work of Lunts and Orlov in Lemma 2.3, we concludethat there is an object E ∈ D b ( X × Y ) such that Φ X → Y E ∼ = Φ X → Y K . Now by [CS14, Remark5.7], the kernel is unique, and so E ∼ = K . Since this kernel realizes all three equivalences,the result follows. (cid:4) To conclude this subsection, we recall a well-known fact, namely if an integral functorbetween two projective varieties preserves structure sheaves of points, then the kernel ofthe functor is supported on the graph of a morphism between the varieties in question. Inparticular, if the integral functor is a Fourier-Mukai functor, the two varieties are isomor-phic. This was originally known for smooth projective varieties (see for example, [Huy06,Corollary 5.23 and 6.14]), but the crux of the proof is the knowledge that the functor isrepresented by an object on the product. Since this is known for singular varieties, the samestatement holds.
Lemma 2.6.
Let Φ X → Y K : D b ( X ) → D b ( Y ) be an integral functor between projective vari-eties such that for all closed points x ∈ X , Φ X → Y K ( k ( x )) ∼ = k ( y )[ r ] for some closed point y ∈ Y and r ∈ Z . Then K ∼ = σ ∗ L [ m ] for a section σ : X → X × Y of π X , L ∈
Pic X , and m ∈ Z , and Φ X → Y K ( − ) ∼ = R f ∗ (( − ) ⊗ L )[ m ] for a morphism f : X → Y . Further, f is an isomorphism if and only if Φ X → Y K is anequivalence of categories. Recall that the support of a complex
K ∈ D b ( X ), denoted supp K , is the union of thesupports of its cohomology sheaves. In this scenario the support is a closed subscheme. Proof.
The proof of the first statement can be found in several places, for example, [HVdB07,Proposition 4.2], [BBHR09, Corollary 1.12], or originally, in steps 2 and 3 of the proof of[Orl02, Theorem 2.10] where it is stated in the setting of abelian varieties. We note herethat as a byproduct of the proof, the section σ : X → X × Y is an isomorphism between X and supp K .By letting f = π Y ◦ σ , the projection formula easily shows thatΦ X → Y K ( − ) ∼ = R f ∗ (( − ) ⊗ L )[ m ] , and it is clear that this functor is an equivalence if and only if f is an isomorphism. Indeed,the “if” direction is obvious, for the converse, the inverse functor Φ Y → X E must satisfy thesame hypotheses, and hence there is a section ζ : Y → X × Y realizing an isomorphismbetween Y and supp K . Setting g = π X ◦ ζ , it follows that g is the inverse to f . (cid:4) Remark . There are other ways to see that an equivalence satisfying Φ X → Y K ( k ( x )) ∼ = k ( y )[ r ] implies that X ∼ = Y . For example, the reader should see [Fav12, Lemma 3.5] for aproof which is closer in spirit to the original proof of Bondal and Orlov in [BO01]. DYLAN SPENCE
Rouquier Functors.
All results in this section are contained in [Rou08, Section 4] and[Bal11, Section 5].Given a smooth projective variety X , the bounded derived category D b ( X ) possesses adistinguished autoequivalence given by S X ( − ) = ( − ) ⊗ ω X [dim X ] : D b ( X ) → D b ( X ) . This functor captures the notion of Serre duality, and as such is called a Serre functor.More generally given a k -linear triangulated category T , a Serre functor is a triangulatedautoequivalence S : T → T for which there are natural isomorphisms η A,B : Hom T ( B, S ( A )) → Hom T ( A, B ) ∗ , where “ ∗ ” indicates the vector space dual. Typically the presence and properties of a Serrefunctor on a triangulated category T has significant consequences for the structure of T .The first question is that of existence. Fixing A ∈ T , we see that if the functorHom T ( A, − ) ∗ is a representable functor for any such A , with representing object in T ,then T has a so-called weak Serre functor, that is, a Serre functor that is not necessarilyan autoequivalence. The question of representability can be difficult, but if T is compactlygenerated and cocomplete, a convenient criterion exists known as Brown representability: Theorem 2.1 ([Nee96], Theorem 3.1) . Let T be a k -linear, compactly generated, and co-complete triangulated category. If H : T →
Vect k is a contravariant functor sending alldistinguished triangles to long exact sequences, and H sends all coproducts to products,then H is representable. It is well-known that the functors Hom T ( A, − ) are always cohomological, that is, theysatisfy the hypotheses of Brown representability; so to conclude that T has a weak Serrefunctor by this approach, it follows that the functor Hom T ( A, − ) must commute with allcoproducts. This is the same as saying that A is compact in T , and so it follows that if thisfunctor is represented by an object in T c , the category T c will have a weak Serre functor.Let us now specialize to the geometric case, where T = D(Qcoh X ). It is well-known that X is regular if and only if Perf X ∼ = D b ( X ), and so in the singular case we no longer expectD b ( X ) to possess a Serre functor. However it is still possible for Perf X ; if X is Gorenstein,this is precisely what occurs. Proposition 2.1 ([Bal11, SdSSdS12]) . If X is a proper variety over a field k , the category Perf X has a Serre functor if and only if X is Gorenstein. In particular, the functor hasthe formula: S X ( − ) = ( − ) ⊗ ω X [dim X ] . The failure of Perf X to have a Serre functor in the non-Gorenstein case is not due toa failure of existence, but rather the representing objects for the functors Hom( A, − ) nolonger lie in Perf X . This observation is due to Rouquier [Rou08, Section 4], where thefollowing definition was given. Definition 2.2.
Let S and T be arbitrary k -linear triangulated categories, and F : S → T bea k -linear triangulated functor. A Rouquier functor associated to F is a k -linear triangulatedfunctor R F : S → T for which there are natural isomorphisms η A,B : Hom T ( B, R F ( A )) → Hom T ( F ( A ) , B ) ∗ . In particular, if S = T and F is the identity functor, then a Rouquier functor is just aweak Serre functor. As we had discussed above, the existence of a Serre functor is closely ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 9 related to the representability of the functors Hom T ( A, − ) ∗ , and in the case of a Rouquierfunctor, a similar result also holds. Lemma 2.7 ([Bal11], Lemma 5.7) . A necessary and sufficient condition for the existenceof R F is the representability of the functors Hom T ( F ( A ) , − ) ∗ for A an object of S . If itexists, its unique. In particular, if the triangulated category is compactly generated, then the previous discus-sion implies the following.
Lemma 2.8 ([Rou08], Corollary 4.23) . Let T be a k -linear, compactly generated, and cocom-plete triangulated category. Then the Rouquier functor associated to the inclusion T c → T exists. Furthermore, if T c has finite dimensional Hom spaces, then the Rouquier functor isfaithful. Our main example is the case of projective schemes. For the following, see Proposition7.47 and Remark 7.48 in [Rou08], see also [Bal11, Example 5.12] for more details.
Lemma 2.9.
Let f : X → Spec k be a projective scheme over the field and ι : Perf X ֒ → D (Qcoh X ) the inclusion functor. Then the Rouquier functor R X := R ι exists and isisomorphic to ( − ) L ⊗ f ! O Spec k . Furthermore, it maps
Perf X into D b ( X ) . Another extremely useful feature of a Serre functor is that it commutes with all equiva-lences of triangulated categories. In our situation, there is an analogous statement.
Lemma 2.10 ([Bal11], Lemma 5.15) . Assume that there are equivalences of triangulatedcategories φ : T → T ′ and ψ : S → S ′ and functors F : S → T and F ′ : S ′ → T ′ makingthe diagram S S ′ T T ′ ψF F ′ φ commute. Then if R F exists, so does R F ′ , and moreover R F ′ ◦ ψ ∼ = φ ◦ R F . Reconstruction for projective curves
In this section, after some more preliminaries, we give the proof of Theorem A. Wefirst establish a class of objects on projective curves which we call perfect zero-cycles; onemay think of these as “test objects” against which we will find the support of particularcomplexes. Following this, we exploit the perfect zero-cycles to show that the Fourier-Mukaitransform which realizes the equivalence Perf X ∼ = Perf Y preserves the structure sheavesof closed points. Then the theorem will follow from Lemma 2.6. Preliminaries: Perfect zero-cycles.
In the original proof, due to Bondal and Orlov in[BO01], point objects in D b ( X ) were characterized as the structure sheaves of closed pointsby using the Serre functor. In the singular case, since structure sheaves of closed point areno longer perfect, we work primarily with Perf X for the reasons mentioned in Remark 2.3.Let us first recall from [HRLMdS07, Section 1.3] a construction of what we will refer toas Koszul zero-cycles . Let X be a projective curve (which by our conventions, is integral). Then for any closed point x ∈ X , the local ring ( O X,x , m x ) is an integral noetherian ringof dimension one. Choosing any element of the maximal ideal yields a non-zero divisor f ∈ m x ⊂ O X,x , and letting ( f ) be the ideal sheaf determined locally by this nonzerodivisor, this determines a closed subscheme Z x , supported only at x ∈ X . Since f is anonzero divisor, multiplication by a local section yields a resolution of the structure sheaf O Z x = O X / ( f ) by locally free sheaves of length 1 (= dim X ), hence O Z x is perfect as anobject of D b ( X ). It is worthwhile to point out that this is just the Koszul complex associatedto a regular sequence of length one, which provides some justification for the name. Remark . Much more generally, a sequence of elements ( x , ..., x n ) ⊂ m in a noetherianlocal ring R is called a regular sequence if x i is a nonzero divisor on R/ ( x , ..., x i − ) forall i ≤ n and R/ ( x , ..., x n ) = 0. The length of any maximal regular sequence is constantand is called the depth of R , denoted by depth R . The depth should be thought of as analgebraic definition of the notion of dimension; in general the depth and dimension do notagree, but they can be related via depth R ≤ dim R . Further, if I is an ideal such that R/I has finite projective dimension as an R -module, there is a close relationship between thedepth of a ring and the projective dimension of R/I (more generally, any R -module of finiteprojective dimension), proj . dim . R/I + depth R/I = depth R, known as the Auslander-Buchsbaum formula [BH93, Therem 1.3.3]. Finally, a ring is saidto be a Cohen-Macaulay ring if depth R = dim R .We can also globalize this, namely a locally noetherian scheme X is said to be a Cohen-Macaulay scheme if O X,x is a Cohen-Macaulay ring for all x ∈ X . While we will only needthe most basic facts about Cohen-Macaulay rings, a more thorough treatment can be foundin [BH93].Our primary example is when R is a reduced local ring of dimension one, where such ringsare automatically Cohen-Macaulay (this is [BH93, Exercise 2.1.20(a)]). Since the definitionof a Cohen-Macaulay scheme is local, it follows that any integral curve is Cohen-Macaulay.A maximal regular sequence in such a ring R is simply a nonzero divisor f ∈ R as we foundabove, and it follows that the quotient satisfies dim R/ ( f ) = 0. As indicated, we also havea finite resolution by free R -modules of finite rank given by the Koszul resolution:0 → R · f → R → R/I → . Remark . Considering again the geometric situation where X is a projective curve, forany P ∈ Perf X , End Perf X ( P ) is a finite dimensional (noncommutative) k -algebra withmultiplication given by composition. Now for any other object Q ∈ Perf X , the spaceHom Perf X ( P, Q ) has the structure of a right End
Perf X ( P )-module, with the action givenby pre-composition. Choosing some closed point x ∈ X , if we consider a Koszul zero-cycle P = O Z x , since Z x is affine, we can identify O Z x with its global sections and thusHom O X ( O Z x , O Z x ) ∼ = Hom Perf X ( O Z x , O Z x ) ∼ = O Z x , which shows that in this case the endomorphism algebra is commutative. Further, sinceExt is local, the Koszul resolution yieldsHom Perf X ( O Z x , O Z x [1]) ∼ = O Z x . Thus the right O Z x -module Hom Perf X ( O Z x , O Z x [1]) is a cyclic module. ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 11
These Koszul zero-cycles will be the prototypical example of our point-like objects. Moregenerally however, there is, to the author’s knowledge, no good categorical way to test ifan ideal is generated by a regular sequence. So we give below the definition of a perfectzero-cycle, whose definition was inspired by [SdSSdS12, Definition 1.6] in the context ofGorenstein schemes.
Definition 3.1.
Let X be projective curve and let x ∈ X be a closed point. A perfectzero-cycle at x is a closed subscheme Z x ֒ → X supported at x such that (1) Z x is a zero dimensional scheme, (2) O Z x is an object of Perf X , and (3) Hom Perf X ( O Z x , O Z x [1]) is a cyclic right O Z x -module. Here and from now on, we will always denote a perfect zero-cycle supported at x ∈ X by Z x . Remark . By Remark 3.2, it is clear that every Koszul zero-cycle is a perfect zero-cycle,and since we can construct a Koszul zero-cycle for every closed point x ∈ X , this showsthat the category Perf X has “enough perfect zero-cycles”. Remark . Clearly any regular closed point x ∈ X with its structure sheaf k ( x ) is aKoszul zero-cycle. Conversely, if a closed point x ∈ X with structure sheaf k ( x ) is a perfectzero-cycle, it follows by assumption that k ( x ) is perfect as an object of D b ( X ). The well-known result of Auslander, Buchsbaum, and Serre (see for example, [BH93, Theorem 2.2.7]),which states that a noetherian local ring R is regular if and only if its residue field has finiteprojective dimension as an R -module. Thus x ∈ X must be regular.One of the most important uses of the set of perfect zero-cycles is their ability to probe thesupport of a complex. Recall that the support of a complex Q ∈ D b ( X ), denoted supp Q , isthe union of the supports of its cohomology sheaves. Since there are finitely many non-zerocohomology sheaves, and each cohomology sheaf is coherent, this gives a closed subschemeof X . A priori this subscheme could be non-reduced, but we will always implicitly take thereduced subscheme structure when we refer to the support of a complex.For the reader’s convenience, we include the following elementary lemma. Lemma 3.1.
Let ( R, m , k ) be a local noetherian ring and M a finitely generated module.If m ∈ supp M , M admits a surjection M → k . Further if { m } = supp M , then M alsoadmits an injection k → M .Proof. If M is finitely generated and supported at the maximal ideal, then Nakayama’slemma implies that M/ m M = 0, and further is a finite dimensional k -vector space. Choosingany further projection to a one-dimensional subspace, we have a surjection M → k .Consider a prime ideal p ⊂ R . Recall that p is said to be an associated to M if and onlyif R/ p is isomorphic to a submodule of M . Further, since R is noetherian, standard resultsin commutative algebra imply that the set of associated primes is always nonempty and isa subset of the support of M . Now if { m } = supp M , it follows that m is an associatedprime to M , and hence there is a submodule of M which is isomorphic to k . The inclusionof submodules k → M yields the claim. (cid:4) The reader should compare the following result to [BBHR09, Proposition A.91] and[HRLMSdS09, Lemma 3.5].
Lemma 3.2.
Let X be a Cohen-Macaulay variety and Q ∈ D b ( X ) . Given a closed point x ∈ X , x ∈ supp Q if and only if Hom D b ( X ) ( Q, O Z x [ m ]) = 0 for some m ∈ Z and all perfectzero-cycles Z x supported at x . We note here that we only need the property that a the structure sheaf of a perfectzero-cycle is a supported at only a single closed point.
Proof.
Since X is assumed Cohen-Macaulay, for any closed point there is a perfect zero-cyclesupported on it (see Remark 3.3). Suppose a closed point x ∈ X belongs to the support of Q and denote the cohomology sheaves of Q by H i . We consider the spectral sequence(3.1) E p,q = Hom D b ( X ) ( H − q , O Z x [ p ]) = ⇒ Hom D b ( X ) ( Q, O Z x [ p + q ]) . Since x belongs to the support of Q , there is at least one cohomology sheaf H q supportedat x . Let ι x : { x } ֒ → X be the inclusion of the closed point x , and using the adjunctionbetween the inverse image and pushforward, there is an isomorphism:Hom O X,x ( H qx , O Z x ) ∼ = Hom O X ( H q , O Z x ) , where we identify O Z x with its pushforward since it is a skyscraper sheaf on X .To show that these spaces are nonzero it is clearly enough to see that the left-hand side isnonzero. In particular the stalk H qx is a finitely generated O X,x -module which is supportedon the maximal ideal m x by assumption, and O Z x is a finitely generated O X,x module whosesupport is precisely the maximal ideal. Hence by Lemma 3.1 there is a nonzero morphism H qx → k ( x ) → O Z x , which shows that the morphism space is nonzero.So now taking the maximal q such that H q x = 0 at x , we see that the term E , − q survivesto the E ∞ page as there are no negative Ext groups between two sheaves, so the differentialending at E , − q starts from zero, and the differential leaving E , − q hits terms which arezero simply because we had chosen q maximal, and all Ext groups between sheaves withdisjoint support are trivial (using the local-to-global spectral sequence). Hence this termsurvives, and since it is nonzero, we see Hom D b ( X ) ( Q, O Z x [ − q ]) = 0.Conversely, suppose that Hom D b ( X ) ( Q, O Z x [ m ]) = 0 for some m ∈ Z . Suppose by way ofcontradiction that x / ∈ supp Q , equivalently, x / ∈ supp H i for all i ∈ Z . But since sheaveswith disjoint support have no nontrivial Ext groups, the spectral sequence 3.1 yields acontradiction. (cid:4) Preliminaries: The Rouquier functor.
Before we move on to the proof, there is onemore piece of the puzzle which we should elaborate on. In Lemma 2.9 we recalled from[Rou08, Remark 7.48] that the Rouquier functor R X associated to any projective scheme isgiven by R X ( − ) = ( − ) L ⊗ f ! O Spec k , and has essential image in D b ( X ). We would now like to take the opportunity to remindthe reader of the relevant features of the dualizing complex f ! O Spec k in the setting where X is a projective curve. By assumption we have a closed immersion i : X ֒ → P n , and so thedualizing complex may be taken to be [Sta19, Lemma 0AA2, Lemma 0AA3] f ! O Spec k = R H om ( i ∗ O X , ω P n [ n ]) | X . ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 13
Since X is Cohen-Macaulay by Remark 3.1, this complex has exactly one non-zero coho-mology sheaf in degree − f ! O Spec k ∼ = H − ( f ! O Spec k )[1] . We denote this cohomology sheaf by ω X . This is nothing more then the classical dualizingsheaf appearing in Serre duality [Sta19, Lemma 0BS3]. Putting this all together, if X is aprojective curve, the Rouquier functor can be given explicitly as R X ( − ) = ( − ) L ⊗ ω X [1] . It is well-known that ω X is a divisorial sheaf (reflexive and generically locally free ofrank 1) and in particular torsion-free. Indeed on the regular locus U ⊂ X , this definitionreduces to give an isomorphism of coherent sheaves ω X | U ∼ = det Ω U/k , where the latter is thecanonical bundle on U (see for example, [Kle80, Proposition 22] or [Kov17, Lemma 3.7.5]).Further, the restriction of ω X to the Gorenstein locus is a line bundle [Sta19, Lemma 0BFQ].With this description of the dualizing sheaf ω X , let us introduce some convenient notation.Given a projective curve X , we define the non-Gorenstein locus X nG to be the set consistingof all points x ∈ X such that ω X,x is not cyclic as a O X,x -module. By definition on X \ X nG the stalk ω X,x of the canonical sheaf is generated by a single element and since it is torsion-free, it must be free of rank 1, hence ω X | U ∼ = O U for some open neighborhood containing x . Using this notation, we may reformulate our assumptions in Theorem A and say that aprojective curve X is strictly Cohen-Macaulay if and only if X nG = ∅ . The Proof of Theorem A.
The first step is to characterize the perfect zero-cycles asobjects of Perf X . Recall from our conventions that all varieties are assumed integral.As with the definition of a perfect zero-cycle, the following is a direct generalization of[SdSSdS12]. Definition 3.2.
Let P be an object of Perf X , ι, R X : Perf X → D b ( X ) the inclusion andRouquier functor respectively. Then we say that P is a strong point object of codimension d ≥ if it satisfies (1) R X ( P ) ∼ = ι ( P )[ d ](2) Hom( P, P [ j ]) = 0 for j < , (3) Hom( P, P ) is a commutative local k -algebra, (4) Hom( P, P [ d ]) , is a cyclic right Hom(
P, P ) -module.If P only satisfies (2) through (4), we say that it is a weak point object (where the codimen-sion is taken by default to be dim X = 1 ). Its clear from the definition that the structure sheaf of a perfect zero-cycle supported in X \ X nG is a strong point object. However the perfect zero-cycles which are supported in X nG are weak point objects which are not strong; see Lemma 3.3 below. In addition, wewish to point out that there may be weak point objects which are not structure sheaves ofperfect zero-cycles, but see Lemma 3.5 for a partial answer to this. Lemma 3.3.
Let P be a strong point object of codimension d on a projective curve X .Then the following hold: (1) d = 1 , and (2) supp P ∩ X nG = ∅ . Remark . Before we begin the proof, we remind the reader that a straightforward applica-tion of Nakayama’s Lemma gives a convenient criterion for when a closed point ι x : { x } ֒ → X belongs to the support of a complex C , namely this occurs if and only if L ι ∗ x C = 0. Proof.
Taking cohomology of both sides of the isomorphism R X ( P ) ∼ = i ( P )[ d ] in Definition3.2 we see that(3.2) H i +1 ( P L ⊗ ω X ) ∼ = H i + d ( P ) . Now using that both complexes are bounded above, set j = max { i | H i ( P ) := H i = 0 } . Itfollows from right exactness of the tensor product that H j ( P L ⊗ ω X ) ∼ = H j ⊗ ω X and so j is also maximal with respect to H j ( P L ⊗ ω X ) = 0, as the right hand side vanishesif and only if H j vanishes. Now in (3.2), it follows that i + 1 is maximal if and only if i + d is maximal, and further that i + 1 = i + d . Hence d = 1.Now fixing some closed point ι x : { x } ֒ → X , passing to stalks via ι − x is an exact func-tor, and so it commutes with taking cohomology of the complex P . Let H ix = ι − x H i = H i ( ι − x ( P )) be the cohomology sheaves of the complex P x = ι − x ( P ). Then let j x be maxi-mal such that H j x x = 0. Then again by right-exactness of the tensor product and exactnessof the inverse image, taking right-most cohomology (which is in degree j x by assumption)of L ι ∗ x ( P L ⊗ ω X ) ∼ = L ι ∗ x P yields an isomorphism of k ( x )-vector spaces H j x x ⊗ O X,x ω X,x ⊗ O X,x k ( x ) ∼ = H j x x ⊗ O X,x k ( x ) . However if x ∈ X nG then ω X,x is generated by at least two elements as a O X,x -module andby Nakayama’s lemma the left hand side has strictly larger dimension then the right handside. This is a contradiction, so we conclude that the cohomology sheaves of P have supportdisjoint from X nG . (cid:4) Lemma 3.4.
Let X be a projective curve which is strictly Cohen-Macaulay ( X nG = ∅ ).Then every point object P in Perf X is isomorphic to O Z x [ r ] for some x ∈ X \ X nG and r ∈ Z , where Z x is a perfect zero-cycle supported at x ∈ X . The following proof is almost completely identical to the analogous result in [SdSSdS12].We include it here for the reader’s convenience.
Proof.
Let P = O Z x be the structure sheaf of a perfect zero-cycle supported on some closedpoint x ∈ X \ X nG . It follows from the fact that x ∈ X \ X nG that P satisfies (1) inDefinition 3.2. P obviously satisfies (2) in Definition 3.2, as does any sheaf concentrated insome degree, and since O Z x is a cyclic O X -module, it follows thatHom Perf X ( P, P ) ∼ = Hom O X,x ( O Z x , O Z x ) ∼ = O Z x and this is (3) in Definition 3.2. The last property is immediate from the definition of aperfect zero-cycle. This shows that perfect zero-cycles supported on X \ X nG are strongpoint objects.Conversely, the first step is to show that P is supported at a single closed point. Lemma3.3 implies that the support is proper. Since a closed proper subset of an integral curveis a finite set of points, we see dim(supp P ) = 0. If P were supported on at least twodistinct points, it follows that P can be decomposed into a nontrivial direct sum. Yet the ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 15 assumption that Hom(
P, P ) be a local k -algebra implies that P is indecomposable. Alltogether this implies that P is supported at a single point.Next, we show that property (2) of Definition 3.2 implies that P has a single cohomologysheaf. Suppose that P has two nontrivial cohomology sheaves, H i , H j , i < j minimal andmaximal respectively. Since we chose i minimal, respectively j maximal, we have a nonzeromorphism H i → P [ i ], respectively P [ j ] → H j . Now since H j and H i are supported at asingle common point, Lemma 3.1 implies the existence of a nonzero morphism H j → H i .Hence the composition P [ j ] → H j → H i → P [ i ]is nontrivial, but this gives a nonzero element in Hom( P [ j ] , P [ i ]) = Hom( P, P [ i − j ]) whichis a contradiction as i − j <
0. Hence P ∼ = F [ r ] ∈ Perf X for some r ∈ Z and F ∈ coh X .Now we have an exact sequence0 → G → F → F / m x F → X , where G is the kernel of the projection F → F / m x F ∼ = F ⊗ O X,x k ( x ). This givesa distinguished triangle in D b ( X ), and applying Hom( F , − ) gives a long exact sequenceof abelian groups. Now since F is a coherent sheaf with zero-dimensional support, it hasdepth F = 0 as a module over the local ring. The Auslander-Buchsbaum formula implies,since the local ring is Cohen-Macaulay, that the projective dimension of F is dim X = 1.Thus, in the long exact sequence of abelian groups, we have a surjection · · · → Hom( F , F [1]) → Hom( F , F / m x F [1]) → , and since the former is a cyclic Hom( F , F )-module, the latter must be as well. In par-ticular this implies that F / m x F is a one dimensional k ( x )-vector space, as otherwise,Hom( F , F / m x F [1]) would split into a direct sum of Hom spaces, and hence not be cyclic.This then implies that F is cyclic by Nakayama’s lemma, and so F ∼ = O X / I with I theannihilator of F in O X which is m x -primary. Taking the closed subscheme Z x defined by I ,we see that Z x is supported at a single closed point x ∈ X \ X nG , and P ∼ = O Z x [ r ]. Hencethe claim. (cid:4) The same method of proof also yields the following.
Lemma 3.5.
Let X be projective curve and P ∈ Perf X be a weak point object with zero-dimensional support. Then P ∼ = O Z x [ r ] for some x ∈ X and r ∈ Z . As presented in Lemma 2.3, any equivalence Φ : Perf X → Perf Y is isomorphic to aFourier-Mukai functor Φ X → Y K ( − ) = R π Y ∗ ( π ∗ X ( − ) L ⊗ K ) , where X π X ← X × Y π Y → Y are the projections and K ∈ D b ( X × Y ). Here and from now on, fix one such Fourier-Mukai functor. Using the perfect zero-cycles, we aim to show that (extending Φ X → Y K to anequivalence D b ( X ) ∼ = D b ( Y )) the functor preserves the structure sheaves of closed points. Remark . From the definition of a point object it is clear that an equivalence Perf X ∼ =Perf Y must preserve the weak point objects, however since the Rouquier functor is notan endofunctor of Perf X , one may worry that the strong point objects are not preserved.Recall that from Lemma 2.5, the Fourier-Mukai functor Φ X → Y K : Perf X ∼ = Perf Y extendsto an equivalence D b ( X ) ∼ = D b ( Y ), and since the Rouquier functor is valued in the bounded derived category of coherent sheaves by Lemma 2.9 and commutes with equivalences as inLemma 2.10, we get a commutative squarePerf X Perf Y D b ( X ) D b ( Y ) . Φ X → Y K R X R Y Φ X → Y K Now from Definition 3.2, strong point objects remain perfect complexes after an applica-tion of the Rouquier functor, so by this observation and the commutativity of the above, itfollows that if P is a strong point object in Perf X , then so is Φ X → Y K ( P ) in Perf Y .Note that this immediately implies that dim X = dim Y = 1 by considering the codimen-sions of strong points objects in both categories. Since then Y is assumed to be integral,dim Y = 1, and so Y is Cohen-Macaulay for dimension reasons.We now see that the assumption that we have an equivalence implies that although Y was not assumed to be strictly Cohen-Macaulay, all strong point objects are shifts of perfectzero-cycles. Lemma 3.6.
Let X and Y be projective curves with X strictly Cohen-Macaulay, and Φ X → Y K : Perf X → Perf Y an equivalence of categories. Then all strong point objects on Y are of the form O Z y [ r ] for a perfect zero-cycle Z y and r ∈ Z . Further, given any closedpoint x ∈ X , there exists a closed point y ∈ Y and r ∈ Z such that Φ X → Y K ( O Z x ) ∼ = O Z y [ r ] .Proof. We first show that the existence of the equivalence Φ X → Y K implies that every strongpoint object of Perf Y is of the same form. Denote by ˜ P ( − ) the strong point objects inPerf( − ), and by P ( − ) the objects in Perf( − ) isomorphic to a shift of a structure sheaf of aperfect zero-cycle supported off the non-Gorenstein locus. Then by definition of the objectsin ˜ P ( X ) and Remark 3.6, they are preserved by equivalences, and so Lemma 3.4 and thatΦ X → Y K is an equivalence gives us the following inclusions of sets P ( X ) = ˜ P ( X ) ∼ = ˜ P ( Y ) ⊃ P ( Y ) . We would like to show that ˜ P ( Y ) = P ( Y ). Consider a strong point object Q ∈ ˜ P ( Y ). ByLemma 3.5, it is enough to show that this strong point object has zero dimensional support.Note that by the inclusion of sets above, Q is in the essential image of the Fourier-Mukaifunctor, that is Φ X → Y K ( O Z x ′ [ r ]) ∼ = Q for some closed point x ′ ∈ X , r ∈ Z , and perfectzero cycle Z x ′ . Similarly, given any strong point object O Z y supported on a closed point y ∈ Y \ Y nG , we can find x ∈ X \ X nG such that Φ X → Y K ( O Z x ) = O Z y [ s ] for some s ∈ Z . Bychoosing some other closed point in Y \ Y nG if necessary, we can arrange that x = x ′ . SinceΦ X → Y K is an equivalence of categories, and strong point objects with disjoint support arehomologically orthogonal, it follows from fully faithfulness thatHom( O Z x ′ , O Z x ) ∼ = Hom( Q, O Z y [ t ]) = 0for all t ∈ Z , but by Lemma 3.2 this means Q is supported on a proper (zero-dimensional)subset. Hence by Lemma 3.5 Q ∼ = O Z x [ r ] for some x ∈ X \ X nG and r ∈ Z . This shows thefirst statement.Now we want to show that perfect zero-cycles (regardless of where they are supported)are preserved by the Fourier-Mukai functor. If they are supported in X \ X nG , the previousparagraph implies the claim, so let x ∈ X nG . Again consider the complex Q = Φ X → Y K ( O Z x ). ECONSTRUCTION OF PROJECTIVE CURVES FROM THE DERIVED CATEGORY 17
Choosing some x ′ ∈ X \ X nG , the conclusion of the previous paragraph says Φ X → Y K ( O Z x ′ ) ∼ = O Z y ′ [ r ] for some r ∈ Z . Fully faithfulness implies thatHom( O Z x , O Z x ′ ) ∼ = Hom( Q, O Z y ′ [ r ]) = 0 , for all r ∈ Z , which by Lemma 3.2 tells us that supp Q is a proper subset of Y , and beingclosed, must be zero-dimensional since Y is an integral curve. Now again Lemma 3.5 impliesthat Q ∼ = O Z y [ s ] for some y ∈ Y nG and s ∈ Z . This completes the proof. (cid:4) Remark . Note that Lemma 3.6, together with Remark 3.6, implies for all closed points x ∈ X , there is a closed point y ∈ Y such that F ( O Z x ) = O Z y [ r ] for some r ∈ Z . It followsform this statement that if x ∈ X is a regular point, so is y ∈ Y . Indeed by 3.4 it is enoughto see that if O Z x ∼ = k ( x ), then O Z y ∼ = k ( y ). Since Φ X → Y K is fully faithful,Hom( k ( x ) , k ( x )) ∼ = Hom( O Z y , O Z y ) . Since the left hand side is the residue field k ( x ), so must the right hand side. But theonly module over O Y,y with this endomorphism algebra is the residue field. This shows O Z y ∼ = k ( y ) and in particular, y ∈ Y is a smooth point.We are now in a position to prove Theorem A. Proof of Theorem A.
We show that the Fourier-Mukai functor Φ X → Y K preserves the struc-ture sheaves of closed points, and rest of the proof follows from Lemma 2.6.Using Lemma 2.5, first extend the equivalence Φ X → Y K to an equivalence D b ( X ) ∼ → D b ( Y ).Fix an arbitrary closed point x ∈ X , and denote by Q the complex Φ X → Y K ( k ( x )). If weconsider any other closed point x ′ ∈ X which is distinct from x , it follows via Lemma 3.2that Hom D b ( X ) ( k ( x ) , O Z x ′ [ r ]) = 0 for all r ∈ Z . Since Φ X → Y K is fully faithful,Hom D b ( X ) ( k ( x ) , O Z x ′ [ r ]) ∼ = Hom D b ( Y ) ( Q, Φ X → Y K ( O Z x ′ )[ r ]) = 0for all r ∈ Z as well. By Lemma 3.6, the Fourier-Mukai preserves the perfect zero-cycles,and so there exists some y ′ ∈ Y and s ∈ Z such that Φ X → Y K ( O Z x ′ )[ r ] ∼ = O Z y ′ [ s ]. Finally,applying once again Lemma 3.2, it follows that supp Q is a proper closed subset of Y , hencezero-dimensional.The rest of the proof proceeds similarly to the proof of Lemma 3.4. Noting thatHom D b ( X ) ( k ( x ) , k ( x )[ i ]) = ( i < k ( x ) i = 0 , and once again using that Φ X → Y K is fully faithful, it follows thatHom( Q, Q [ i ]) = ( i < k ( x ) i = 0 . First consider when i = 0. Since the endomorphisms are a field, the complex Q must besupported at a single point. Now consider when i <
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