A toy example in Minimal Model Program ◮ In minimal model program for 3-folds, Mori connected minimal models with flops. ◮ A flop is a pair of birational proper surjections: X Y Z of 3-folds with certain properties. In particular, X and Y are similar to being smooth (terminal singularity) and we will pretend they are smooth. ◮ The morphisms f , g are small contractions; outside a few curves on X and Y and a few points on Z they are isomorphic, and the preimage of a point in Z is at worst curves. ◮ In general, a proper morphism f : X → Z to an equi-dimensional variety Z is called small (to representation theorists and some algebraic geometers) if codim { z ∈ Z | dim f − 1 ( z ) ≥ i } ≥ 2 i + 1. ◮ Do you agree this looks like a relative version of allowable chains?
A toy example in Minimal Model Program, cont. codim { y ∈ Y | dim f − 1 ( y ) ≥ i } ≥ 2 i + 1. Do you agree this looks like a relative version of allowable chains? ◮ It is indeed the case that the machinery of perverse sheaves is able to treat small proper morphisms as if they are smooth of dimension 0, i.e. ´ etale. ◮ Birational ´ etale morphisms are isomorphisms. For us this means we have Rf ∗ IC X = IC Z = Rg ∗ IC Y . But X and Y are (almost) smooth! We have H ∗ ( X ; Q ) = H ∗ ( Z ; Rf ∗ Q X ) = H ∗ ( Z ; Rf ∗ IC X [ − 3]) = H ∗ ( Z ; IC Z [ − 3]) = IH ∗ ( Z ; Q ) . ◮ Same for Y , so H ∗ ( X ; Q ) = IH ∗ ( Z ; Q ) = H ∗ ( Y ; Q ). ◮ For Mori, this proved that birationally equivalent minimal models in 3d have isomorphic (co)homology groups in Q -coefficients. Sweet?
Review, I ◮ Let A be your favorite abelian category. We form C ( A ), the abelian category of complexes of objects in A . That is, objects are X : X n → X n +1 } n ∈ Z with X n ∈ A and d n X ◦ d n − 1 X = { X n , d n = 0. X Morphisms are Hom( X , Y ) = { f = { f n : X n → Y n } n ∈ Z | f n ◦ d n − 1 = d n − 1 ◦ f n − 1 } . X Y ◮ We also have K ( A ) as the quotient category of C ( A ) by modding out homotopy, i.e. same objects as C ( A ) but two morphisms are identified if they are homotopic , i.e. f , g : X → Y are identified if there exists h = { h n : X n → Y n − 1 } n ∈ Z such that f n − g n = h n +1 ◦ d n X + d n − 1 ◦ h n . Y ◮ For any complex X ∈ A one defines the cohomology X ) / Im( d n − 1 H n ( X ) := ker( d n ) ∈ A . Any morphism f : X → Y in X C ( A ) induces a morphism of cohomology H n ( f ) : H n ( X ) → H n ( Y ). Homotopic morphisms induce the same map on cohomology, so this extends to K ( A ).
Review, II ◮ A quasi-isomorphism is a morphism in C ( A ) or K ( A ) which induces isomorphisms on cohomology in all degrees. ◮ The derived category D ( A ) is the additive category whose objects are the same in C ( A ) and K ( A ), and morphisms are “ X ← Y → Z ” where Y → X is a quasi-isomorphism and Y → Z is a general morphism in K ( A ). We refer the reader to [KS, § 1.6] for detail. ◮ We again have cohomology functors H n ( − ) : D ( A ) → A for any degree n . ◮ Some useful facts: 1. D ( A ) is abelian iff A is semisimple. 2. Not every object X in D ( A ) is isomorphic to the complex of its cohomology, that is ( ... → H n − 1 ( X ) → H n ( X ) → H n +1 ( X ) → ... ) where all maps are trivial. This is the case if and only if A has homological dimension ≤ 1. In particular this is true for Ab , the category of abelian groups, but not for Sh( CP 1 ), the sheaf of abelian groups on CP 1 .
Review, 2.5 Not every object X in D ( A ) is isomorphic to the complex of its cohomology, that is ( ... → H n − 1 ( X ) → H n ( X ) → H n +1 ( X ) → ... ) where all maps are trivial. This is the case if and only if A has homological dimension ≤ 1. In particular this is true for Ab , the category of abelian groups, but not for Sh( CP 1 ), the sheaf of abelian groups on CP 1 . (This slide is not needed from the rest and may be skipped.) In fact, If X 0 , Y 0 ∈ A and X , Y ∈ D ( A ) are the associated complexes supported only at degree 0, then Hom D ( A ) ( X , Y [ n ]) = Ext n A ( X 0 , Y 0 ); one may prove this via an injective resolution of Y 0 . In particular, if A = Sh( CP 1 ), and X 0 = Y 0 = Q CP 1 , then Ext 2 Sh( CP 1 ) ( X 0 , Y 0 ) = H 2 ( CP 1 ; Q ) = Q and we may pick a non-trivial morphism in α ∈ Hom D ( A ) ( X , Y [2]). Let Z = M ( α ) be the mapping cone (see latter slides). The cohomology of Z is supported at degree − 2 and − 1 thanks to the long exact sequence associated to the distinguished triangle X → Y → Z → X [1], but Z is not isomorphic to its cohomology; if it is, then one may rotate the triangle to show that α = 0 and arrive at a contradiction.
Review, III ◮ For every 1 X ∈ D ( A ), we may have truncation τ ≥ 0 ( X ) = Y with Y 0 = X 0 / Im d − 1 X ( X − 1 ), Y n = X n for n ≥ 1 and Y n = 0 for n ≤ − 1. Likewise τ ≤ 0 ( X ) = Y with Y 0 = ker( d 0 X : X 0 → X 1 ), Y n = X n for n ≤ − 1 and Y n = 0 for n ≥ 0. ◮ We have X isomorphic to τ ≥ 0 ( X ) iff H n ( X ) = 0 for all n < 0 and likewise for τ ≤ 0 ( X ). Hence if X has its cohomology supported in degree from [ a , b ], we have X ∼ = τ ≥ a ( τ ≤ b ( X )) = τ ≤ b ( τ ≥ a ( X )) is represented by a bounded complex. ◮ This suggests a good notion of D + ( A ) (resp. D − ( A ) and D b ( A )) the full subcategory of bounded below (resp. bounded above, bounded.) complexes. ◮ Also, A is the full subcategory of D ( A ) of those whose cohomology is supported at degree 0. 1 It’s really a bad idea to write X ∈ C for some category C , as objects in a category really form groupoid rather than a set. Cheng-Chiang is just being horribly lazy here.
Triangulated category ◮ For X ∈ C ( A ), the shift X [ m ] is the complex given by X [ m ] n = X m + n and d n X [ m ] = ( − 1) m d n + m . X ◮ The mapping cone of a morphism f : X → Y in C ( A ) is the complex associated to the double complex X → Y by having X n at degree ( n , − 1) and Y n at degree ( n , 0). They come with natural morphisms Y → M ( f ) and M ( f ) → X [1] so that 0 → Y → M ( f ) → X [1] → 0 is a short exact sequence in C ( A ). ◮ A triangle in C ( A ) is X → Y → Z → X [1] where X , Y , Z ∈ C ( A ) and the arrows are morphisms in C ( A ). This induces the same definition for K ( A ) and D ( A ). ◮ A distinguished triangle in K ( A ) is a triangle in K ( A ) that is f isomorphic to some ( X − → Y → M ( f ) → X [1]) in K ( A ) Likewise, a distinguished triangle is a triangle in D ( A ) that is isomorphic to f some X − → Y → M ( f ) → X [1] in D ( A ).
Short exact sequence in the derived category? ◮ Any distinguished triangle X → Y → Z in D ( A ) induces a long exact sequence ... → H n ( X ) → H n ( Y ) → H n ( Z ) → H n +1 ( X ) → ... → Y 0 → Z 0 → 0 is a short f ◮ On the other hand, suppose 0 → X 0 − exact sequence in A . Let X ∈ D ( A ) be the complex only at degree 0 given by X 0 and likewise for Y , Z ∈ D ( A ). Then there is a natural i quasi-isomorphism M ( f ) − → Z so that by putting δ ( f ) i f δ ( f ) := ( Z ← − M ( f ) → X [1]) we have X − → Y → Z − − → X [1] is a distinguished triangle. ◮ The Grothendieck group K 0 ( A ) of A as the free abelian group generated by objects on A mod out [ X 0 ] + [ Z 0 ] − [ Y 0 ] for any short exact sequence 0 → X 0 → Y 0 → Z 0 → 0 in A , and define χ ( X ) = � ( − 1) n [ H n ( X )] ∈ K 0 ( A ) for any X ∈ D b ( A ). The one proves that a distinguished triangle X → Y → Z → X [1] again satisfy χ ( X ) + χ ( Z ) = χ ( Y ).
Triangulated category ◮ We have a list of properties for distinguished triangles in K ( A ) and the same for D ( A ) (see [KS, § 1.4]). f 1. X − → Y → Z → X [1] is a distinguished triangle iff − f [1] Y → Z → X [1] − − − → Y [1] is. id 2. If X − → X → 0 → X [1] is a distinguished triangle. 3. Any commutative diagram X Y X ′ Y ′ in K ( A ) (resp. D ( A )) can be completed to a morphism of distinguished triangles X Y Z X [1] X ′ Y ′ Z ′ X ′ [1]
Triangulated category, II We have a list of properties for distinguished triangles in K ( A ) and the same for D ( A ) (see [KS, § 1.4]). g f ◮ Given morphisms X − → Y and Y − → Z . We have distinguished triangles X → Y → M ( f ) → X [1], Y → Z → M ( g ) → Y [1] and X → Z → M ( g ◦ f ) → X [1]. They should be related. The last property states: 4. Writing Z ′ = M ( f ), X ′ = M ( g ) and Y ′ = M ( g ◦ f ), there exists a distinguished triangle Z ′ → Y ′ → X ′ → Z ′ [1], so that these morphisms make the following diagrams commute: Z ′ Y ′ X Y Y Z Y ′ X ′ Z ′ Y ′ X [1] X ′ Y [1] Z Z ′ [1] Y ′ X ′
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