Geometric description of the connecting homomorphism for Witt groups

GEOMETRIC DESCRIPTION OF THE CONNE CTING HOMOMO RPHISM F OR WITT GROUP S P A UL BALMER AND BAPTISTE CALM ` ES Abstract. W e give a geomet ric setup in which the connecting homomorphism in the localization long exact sequence for Witt gr oups decomp oses as the pull- bac k to the exceptional fib er of a suitable blo w-up foll ow ed by a push-for w ard. 1. Introduction Witt groups form a very in teresting cohomolo gy theory in algebra ic geometr y . (F or a surv ey , see [ 5 ].) Unlik e the better known K -theory and C how theory , Witt theory is not oriente d in the sense o f Levine-Mor el [ 17 ] o r Panin [ 22 ], as a lready visi- ble on the non-s tandard pro jective bundle theorem, see Ar a son [ 2 ] and W alter [ 26 ]. Another wa y of expressing this is that push-forwards do not e xist in sufficient generality for Witt groups. This “non-o r ientabilit y” can ma ke co mputations unex- pec tedly tricky . Indeed, the Witt groups of s uch elementary schemes as Grass mann v arieties will app ear for the fir st time in the companion ar ticle [ 6 ], wher eas the cor- resp onding computations for oriented co homologies have b een ach ieved mo re tha n 35 years a g o in [ 16 ], using the well-kno wn cellular decompositio n of Grassmann v arieties. See also [ 2 1 ] for general statements on ce llula r v a rieties. In oriented t heories, ther e is a very useful computational technique, recalled in Theorem 1.3 b elow, which allows inductive computations for f amilies of cellular v arieties. Our paper originates in an attempt to extend this r esult t o the non- oriented s etting of Witt theo ry . Roughly sp ea king, such a n extension is p oss ible “half of the time”. In the remaining “ half ”, some sp ecific ideas must come in and reflect the truly non-or ient ed behavior of Witt groups. T o explain this r o ugh statement, let us fix the setup, whic h will remain v alid for the en tire paper . 1.1. Setup. W e denote b y S ch the category of s eparated connected no e ther ian Z [ 1 2 ]-scheme. Let X , Z ∈ S ch b e schemes and let ι : Z ֒ → X b e a regula r closed immersion o f co dimension c ≥ 2 . Let B l = B l Z X b e the blow-up of X along Z and E the exceptional fiber . Let U = X − Z ∼ = B l − E b e the una ltered op en complement. W e ha ve a co mm utative dia gram (1) Z   ι / / X U ? _ υ o o n N ˜ υ } } { { { { { { { { E   ˜ ι / / ˜ π O O B l π O O with the usual mor phisms. Consider now a cohomology theor y with suppor ts, say H ∗ (2) · · · ∂ − → H ∗ Z ( X ) − → H ∗ ( X ) υ ∗ − → H ∗ ( U ) ∂ − → H ∗ +1 Z ( X ) − → · · · Date : Octob er 28, 2018. 1991 Mathematics Subje ct Classific ation. 19G12, 11E81. Researc h supported b y SNSF grant PP002-112579 and NSF gr an t 0654397. 1 2 P . BALMER AND B. CALM ` ES In this paper w e shall fo cus on the case of Witt gro ups H ∗ = W ∗ but we take inspiration from H ∗ being an orie nted cohomo logy theory . Ideally , we would like conditions for the v a nis hing of the co nnec ting homomorphism ∂ = 0 in the ab ov e lo calization long exact sequence. Even b etter would be conditions fo r the restrictio n υ ∗ to b e split surjective. When H ∗ is an oriented theor y , there is a well-known hypothesis und er which suc h a s plitting a ctually e x ists, namely : 1.2. Hypothesi s. Assume that there exists an auxiliary mor phism ˜ α : B l → Y (3) Z   ι / / X U ? _ υ o o N n ˜ υ } } { { { { { { { { α   E   ˜ ι / / ˜ π O O B l π O O ˜ α / / Y such that α := ˜ α ◦ ˜ υ : U → Y is a n A ∗ -bund le , i.e. every point of Y has a Zariski neighborho o d over which α is is o morphic to a trivia l A r -bundle, for so me r ≥ 0. See Ex. 1.5 for an explicit example with X , Y and Z b eing Grassmann v arieties . 1.3. Theorem (The o riented technique) . Under Setup 1.1 and Hyp othesis 1.2 , as- sume X , Y and Z r e gular. Assum e the c ohomolo gy t he ory H ∗ is homotopy invariant for r e gu lar schemes and oriented , in that it admits push-forwar ds along pr op er mor- phisms satisfying flat b ase-change. Then, t he r estriction υ ∗ : H ∗ ( X ) → H ∗ ( U ) is split surje ctive with explicit se ct ion π ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 , wher e π ∗ : H ∗ ( B l ) → H ∗ ( X ) is t he push-forwar d. Henc e the c onne cting homomorphism ∂ : H ∗ ( U ) → H ∗ +1 Z ( X ) vanishes and the ab ove lo c alization long exact se qu en c e ( 2 ) r e duc es to split short exact se quenc es 0 → H ∗ Z ( X ) → H ∗ ( X ) → H ∗ ( U ) → 0 . Pr o of. By homotopy in v ariance, we hav e α ∗ : H ∗ ( Y ) ∼ → H ∗ ( U ). By bas e-change, υ ∗ ◦ π ∗ = ˜ υ ∗ and since ˜ υ ∗ ◦ ˜ α ∗ = α ∗ , w e ha ve υ ∗ ◦ π ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 = id.  The dichotomy b etw e e n the cas es where the ab ov e technique extends to Witt groups and the ca s es where is do es no t, comes from the duality . T o understand this, recall that one can consider Witt groups W ∗ ( X, L ) with duality twisted by a line bundle L o n the scheme X . A ctually only the cla ss o f the twist L in P ic( X ) / 2 really matter s since w e have square-p er io dicity iso morphisms for all M ∈ Pic( X ) (4) W ∗ ( X, L ) ∼ = W ∗ ( X, L ⊗ M ⊗ 2 ) . Here is a condensed form of our Theorem 2.3 and Ma in Theo rem 2.6 below : 1.4. Theorem . U nder Hyp othesis 1.2 , assume X , Y and Z r e gular. L et L ∈ Pic( X ) . Then ther e exists an inte ger λ ( L ) ∈ Z (define d by ( 8 ) b elow) such that : (A) If λ ( L ) ≡ c − 1 mo d 2 then the r estriction υ ∗ : W ∗ ( X, L ) → W ∗ ( U, L | U ) is split surje ctive with a se ction given by the c omp osition π ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 . Henc e t he c onne ct ing homomorphism W ∗ ( U, L | U ) ∂ − → W ∗ +1 Z ( X, L ) vanishes and the lo c alization long ex act se qu enc e r e duc es to split short exact s e quenc es 0 − → W ∗ Z ( X, L ) − → W ∗ ( X, L ) − → W ∗ ( U, L | U ) − → 0 . (B) If λ ( L ) ≡ c mo d 2 then the c onn e cting homomorphism ∂ is e qual t o a c om- p osition of pul l-b acks and push-forwar ds : ∂ = ι ∗ ◦ ˜ π ∗ ◦ ˜ ι ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 . This statement r equires some explanations. First of all, note that w e ha ve used push-forwards for Witt groups, a long π : B l → X in (A) and along ˜ π : E → Z and ι : Z → X in (B). T o explain this, reca ll that the push-forward in Witt theory is only conditionally defined. Indeed, giv en a proper morphism f : X ′ → X b etw ee n (connected) regular s chemes a nd given a line bundle L ∈ Pic( X ), the push-forward homomorphism do es n ot map W ∗ ( X ′ , f ∗ L ) into W ∗ ( X, L ), a s one could naiv ely CONNECTING HOMOMORPHISM F OR WITT GR OUPS 3 exp ect, but the seco nd author and Hor nbo stel [ 8 ] show ed that Grothendieck-V erdier duality yie lds a twist by the relative canonic a l line bundle ω f ∈ Pic( X ′ ) : (5) W i +dim( f )  X ′ , ω f ⊗ f ∗ L  f ∗ − → W i ( X , L ) . Also note the shift by t he relativ e dimension, dim( f ) := dim X ′ − dim X , whic h is not problematic, s ince w e can a lwa ys replace i ∈ Z b y i − dim( f ). More trickily , if you ar e given a line bundle M ∈ Pic( X ′ ) a nd if y ou need a push-forward W ∗ ( X ′ , M ) → W ∗− dim( f ) ( X, ?) along f : X ′ → X , you first need to check that M is of the form ω f ⊗ f ∗ L for some L ∈ Pic( X ), at lea st mo dule squares. Otherwise, you simply do not k now h ow to push-forwar d . This is precis ely the sour ce of the dichotom y of Theorem 1.4 , as expla ined in Prop os ition 2.1 b elow. A t the end o f the day , it is only p ossible to transp ose to Witt groups the o r i- ent ed tec hniq ue of Theorem 1.3 when th e push-forward π ∗ exists for Witt g roups. But actually , the remark able part of Theore m 1.4 is Case (B), that is o ur Main Theorem 2.6 b e low, which gives a desc r iption of the connecting homomorphism ∂ when w e cannot prove it zero b y the oriented metho d. This is the par t where the non-oriented b ehavior r e ally app ears . See more in Remark 2.7 . Ma in Theor em 2.6 is especially striking since the orig inal definition of the connecting homomorphism given in [ 3 , § 4] do es no t have suc h a geometric flavor of pull-backs and push- forwards but rather in volv es abstra ct tech niques of tria ngulated catego ries, like symmetric co nes, a nd the like. Our new g eometric descr iption is also r e mark ably simple to us e in applica tions, see [ 6 ]. Here is the example in question. 1.5. Exa mple. Let k b e a field of characteristic not 2. (W e describe flag v arie ties ov er k b y giving their k -p oints, a s is customary .) Let 1 ≤ d ≤ n . Fix a codi- mension o ne s ubs pa ce k n − 1 of k n . Let X = Gr d ( n ) b e the Gra ssmann v ariety of d -dimensional subspaces V d ⊂ k n and let Z ⊂ X b e the closed sub v ariety of those subspaces V d contained in k n − 1 . The op en complement U = X − Z consists of thos e V d 6⊂ k n − 1 . F or such V d ∈ U , the subspace V d ∩ k n − 1 ⊂ k n − 1 has dimension d − 1 . This construction defines an A n − d -bundle α : U → Y := Gr d − 1 ( n − 1), mapping V d to V d ∩ k n − 1 . This situation rela tes the Grassmann v ariety X = Gr d ( n ) to th e smaller ones Z = Gr d ( n − 1) and Y = Gr d − 1 ( n − 1). Diagram ( 1 ) here b ecomes Gr d ( n − 1)   ι / / Gr d ( n ) U ? _ υ o o k K ˜ υ y y r r r r r r r r r r r r α   E   ˜ ι / / ˜ π O O B l π O O ˜ α / / Gr d − 1 ( n − 1) . The blow-up B l is t he v ar ie t y o f pairs of subspaces V d − 1 ⊂ V d in k n , s uch that V d − 1 ⊂ k n − 1 . The morphis ms π : B l → X a nd ˜ α : B l → Y forget V d − 1 and V d resp ectively . The morphism ˜ υ maps V d 6⊂ k n − 1 to the pair ( V d ∩ k n − 1 ) ⊂ V d . Applying Theorem 1.3 to this situation, Laksov [ 16 ] co mputes the Chow groups of Gras smann v a rieties by induction. F or Witt g r oups though, ther e ar e c ases where the r estriction W ∗ ( X, L ) → W ∗ ( U, L | U ) is not surjective (see [ 6 , Co r . 6.7]). Nevertheless, thank to our geometric description of the connecting homomo r phism, we hav e obtained a complete descriptio n of the Witt g roups of Grassma nn v arieties, for a ll shifts and a ll twists, to app e ar in [ 6 ]. In addition to the pre sent techniques, our computations involv e other ideas, sp ecific to Gr assmann v arieties , like Sch ub ert cells and desingular isations thereo f, plus so me co mbinatorial b o o kkeeping by means of special Y oung diagra ms. Including all this here w o uld mislea dingly hide the simplicity and g enerality of the present pap e r . W e therefore chose to publish the computation o f the Witt gr oups o f Gras s mann v arieties separately in [ 6 ]. 4 P . BALMER AND B. CALM ` ES The pap er is or ganized as fo llows. Section 2 is dedicated to the detailed ex plana- tion of the ab ov e dichotom y and the pro of of the a bove Cas e (A), see Theorem 2.3 . W e also e x plain Case (B) in our Main Theorem 2.6 but it s proof is deferred to Section 5 . The whole Section 2 is wr itten, a s ab ov e , under the a ssumption that a ll schemes ar e regular. This assumption s implifies the s tatement s but can b e removed at the price of intro ducing dualizing complexes a nd co herent Witt gro ups, which provide the na tural framew ork ov er non-re gular schemes. This gener alization is the purp os e o f Section 3 . There, we even dro p the a uxiliary Hypothes is 1 .2 , i.e. the do tted pa rt of Diagram ( 3 ). Indeed, our Main Lemma 3 .5 gives a v ery gener al description of the connecting homomor phism applied to a Witt class over U , if that class comes from the blow-up B l via res triction ˜ υ ∗ . The pro o f of Main L e mma 3.5 o ccupies Section 4 . Finally , Hyp othesis 1.2 re-enters the game in Section 5 , where we prov e our Main Theo rem 2.6 as a corolla ry of a non- regular genera liz ation given in Theor em 5.1 . F or the co nv enience o f the reader, we g athered in App endix A the needed res ults about Picard groups, canonical bundles and dualizing complex e s, which are s ometimes difficult to find in the literatur e . The conscient ious reader might wan t to start with that app endix. 2. The regular case W e keep notatio n as in Setup 1.1 and we assume a ll schemes to b e r egular. This section ca n als o be considered as an ex panded introductio n. As expla ined after Theor em 1 .4 a bove, we hav e to decide w he n the push-forward along π : B l → X a nd along ˜ π : E → Z exist. By ( 5 ), we ne e d to det ermine the canonical line bundles ω π ∈ Pic( B l ) and ω ˜ π ∈ Pic( E ). This is class ical and is recalled in App endix A . Fir st of a ll, Propos ition A.6 g ives Pic      Z   ι / / X U ? _ υ o o N n ˜ υ } } { { { { { { { { E   ˜ ι / / ˜ π O O B l π O O      ∼ = Pic( Z )  1 0    Pic( X ) ι ∗ o o  1 0    Pic( X ) . Pic( Z ) ⊕ Z Pic( X ) ⊕ Z  ι ∗ 0 0 1  o o (1 0) 8 8 q q q q q q q q q q The Z summands in Pic( B l ) and Pic( E ) are g e nerated by O ( E ) = O B l ( − 1) and O ( E ) | E = O E ( − 1) r esp ectively . Then Propos ition A.11 g ives the w a n ted (6) ω π = (0 , c − 1) in Pic( X ) ⊕ Z ∼ = Pic( B l ) and ω ˜ π = ( − ω ι , c ) in Pic( Z ) ⊕ Z ∼ = Pic( E ) . So, statistically , picking a line bundle M ∈ Pic( B l ) at random, there is a 50% chance of being a ble to push-forward W ∗ ( B l , M ) → W ∗ ( X, L ) along π f or some suitable line bundle L ∈ Pic( X ). T o justify this, obser ve that coker  Pic( X ) π ∗ / / Pic( B l )  2 ∼ = Z / 2 and tensoring by ω π is a bijection, s o half of the elements of Pic( B l ) / 2 ar e of the form ω π ⊗ π ∗ ( L ). The same proba bilit y of 50% applies to the push forward along ˜ π : E → Z but interestingly in complementary ca ses, a s we summariz e no w. 2.1. P rop ositi on. With the notation of 1.1 , assume X and Z r e gular. R e c al l that c = co dim X ( Z ) . L et M ∈ Pic( B l ) . L et L ∈ Pic( X ) and ℓ ∈ Z b e such that M = ( L, ℓ ) in Pic( B l ) = Pic( X ) ⊕ Z , t hat is, M = π ∗ L ⊗ O ( E ) ⊗ ℓ . (A) If ℓ ≡ c − 1 mo d 2 , we c an push-forwar d along π : B l → X , as fol lows : W ∗ ( B l , M ) ∼ = W ∗ ( B l , ω π ⊗ π ∗ L ) π ∗ → W ∗ ( X, L ) . CONNECTING HOMOMORPHISM F OR WITT GR OUPS 5 (B) If ℓ ≡ c mo d 2 , we c an push-forwar d along ˜ π : E → Z , as fol lows : W ∗ ( E , M | E ) ∼ = W ∗  E , ω ˜ π ⊗ ˜ π ∗ ( ω ι ⊗ L | Z )  ˜ π ∗ → W ∗− c +1 ( Z, ω ι ⊗ L | Z ) . In e ach c ase, t he isomorphism ∼ = c omes fr om squar e-p erio dicity in the twist ( 4 ) and the subse quent homomorph ism is the push-forwar d ( 5 ) . Pr o of. W e o nly hav e to c heck the congruences in Pic / 2. By ( 6 ), when ℓ ≡ c − 1 mo d 2, we have [ ω π ⊗ π ∗ L ] = [( L, ℓ )] = [ M ] in Pic( B l ) / 2. When ℓ ≡ c mo d 2, we hav e [ ω ˜ π ⊗ ˜ π ∗ ( ω ι ⊗ L | Z )] = [( L | Z , ℓ )] = [ M | E ] in Pic( E ) / 2. T o apply ( 5 ), note that dim( π ) = 0 since π is bir ational and dim( ˜ π ) = c − 1 since E = P Z ( C Z/X ) is the pro jective bundle of the rank- c co no rmal bundle C Z/X ov er Z .  So far , we hav e only used Setup 1.1 . Now add Hypothesis 1.2 with Y r egular. 2.2. Remark. Since P icard gro ups of reg ular schemes are homotopy in v ariant, the A ∗ -bundle α : U → Y yields an isomorphism α ∗ : Pic( Y ) ∼ → Pic( U ). Let us ide ntify P ic( Y ) with Pic( U ), and hence with Pic( X ) a s we did ab ov e since c = co dim X ( Z ) ≥ 2 . W e also hav e O ( E ) | U ≃ O U . Putting a ll this tog ether, the right-hand pa r t of Diagram ( 3 ) yields the follo wing on Picard gro ups : Pic      X U ? _ υ o o α   N n ˜ υ } } { { { { { { { { B l π O O ˜ α / / Y      ∼ = Pic( X )  1 0    Pic( X ) Pic( X ) ⊕ Z (1 0) 8 8 q q q q q q q q q q Pic( X ) .  1 λ  o o Note that the low er right map Pic( X ) ∼ = Pic( Y ) ˜ α ∗ − → Pic( B l ) ∼ = Pic( X ) ⊕ Z m ust be of the form  1 λ  by commutativit y (i.e. s inc e  1 0  ·  1 λ  = 1) but ther e is no reason fo r its se c o nd comp onent λ : Pic( X ) → Z to v anish. This is indeed a key observ atio n. In other words, we have t wo homomorphisms from Pic( X ) to Pic( B l ), the dir ect one π ∗ and the circumv olant o ne ˜ α ∗ ◦ ( α ∗ ) − 1 ◦ υ ∗ going via U a nd Y (7) Pic( X ) υ ∗ ≃ / / π ∗   6 = Pic( U ) ( α ∗ ) − 1 ≃   Pic( B l ) Pic( Y ) ˜ α ∗ o o and they do not coincide in genera l. The difference is measure d b y λ , which dep ends on the c hoice o f Y and on the choice of ˜ α : B l → Y , in Hypo thesis 1.2 . So, for ev ery L ∈ Pic( X ), the in teger λ ( L ) ∈ Z is defined b y the eq uation (8) ˜ α ∗ ( α ∗ ) − 1 υ ∗ ( L ) = π ∗ ( L ) ⊗ O ( E ) ⊗ λ ( L ) in Pic( B l ). Under the isomo r phism Pic( B l ) ∼ = Pic( X ) ⊕ Z , the abov e equa tion can be reformulated a s ˜ α ∗ ( α ∗ ) − 1 υ ∗ ( L ) =  L, λ ( L )  . 2.3. Theorem (Partial analo gue of Theorem 1.3 ) . With the notation of 1.1 , assume Hyp othesis 1.2 and assume X , Y , Z r e gular. R e c al l that c = co dim X ( Z ) . L et L ∈ Pic( X ) and c onsider t he inte ger λ ( L ) ∈ Z define d in ( 8 ) ab ove. If λ ( L ) ≡ c − 1 mo d 2 then t he r estriction υ ∗ : W ∗ ( X, L ) → W ∗ ( U, L | U ) is spli t surje ctive, with an explicit se ction given by the c omp osition π ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 W ∗ ( X, L ) W ∗ ( U, L | U ) ( α ∗ ) − 1 ≃   W ∗ ( B l , ω π ⊗ π ∗ L ) ∼ = W ∗  B l , ˜ α ∗ ( α ∗ ) − 1 L | U  π ∗ O O W ∗  Y , ( α ∗ ) − 1 L | U  ˜ α ∗ o o 6 P . BALMER AND B. CALM ` ES Pr o of. The whole p oint is that π ∗ can b e applied after ˜ α ∗ ◦ ( α ∗ ) − 1 , that is , o n W ∗  B l , ˜ α ∗ ( α ∗ ) − 1 υ ∗ ( L )  . This holds b y P rop osition 2.1 ( A) applied to (9) M := ˜ α ∗ ( α ∗ ) − 1 υ ∗ ( L ) ( 8 ) = ( L, λ ( L )) ∈ Pic( X ) ⊕ Z = Pic( B l ) . The assumption λ ( L ) ≡ c − 1 mo d 2 expres ses the hypothesis of Pr op osition 2.1 (A). Checking that w e indeed ha ve a section go es a s in the or iented case, see Thm. 1.3 : υ ∗ ◦ π ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 = ˜ υ ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 = α ∗ ◦ ( α ∗ ) − 1 = id . The firs t equality uses base-change [ 8 , Thm. 6.9] on the le ft-ha nd car tesian square : X U ? _ υ o o B l π O O U ? _ ˜ υ o o id O O L L | U ω π ⊗ π ∗ L L | U with r esp ect to the r ight-hand line bundles. Note that ( ω π ) | U = O U by ( 6 ).  2.4. Remark. In the ab ov e pro of, see ( 9 ), we do not apply Prop osition 2.1 to M being π ∗ L , as one could fir st expect; see Remar k 2.2 . Consequently , our condition on L , namely λ ( L ) ≡ c − 1 mo d 2, do es not only dep end on the codimens ion c of Z in X but also involv es (hidden in the definition of λ ) the par ticula r choice of the auxiliary sc heme Y and of the morphism ˜ α : B l → Y of Hyp othesis 1.2 . 2.5. Rem ark. The legitimate question is no w to decide what to do in the remain- ing case, that is, when λ ( L ) ≡ c mo d 2. As announced, this is the ce ntral goal of our pap er (Thm. 2.6 below). So, let L ∈ Pic( X ) be a twist such tha t push- forward a long π : B l → X cannot be applied to define a section to the restric tion W ∗ ( X, L ) → W ∗ ( U, L | U ) as abov e. Actually , w e can find examples o f such line bundles for which this re s triction is simply not s urjective (see Ex. 1.5 ). The na tural problem then b ecomes to compute the p ossibly non-zer o connecting ho momorphism ∂ : W ∗ ( U, L | U ) → W ∗ +1 Z ( X, L ). Although no t abso lutely necessar y , it actually sim- plifies the for mulation o f Theorem 2.6 to us e d´ evissage fro m [ 9 , § 6], i.e. the fact that pu sh-forward a long a r egular clo sed immers ion is an iso morphism (10) ι ∗ : W ∗− c ( Z, ω ι ⊗ L | Z ) ∼ → W ∗ Z ( X, L ) . Using this iso morphism, w e ca n replace the Witt gr oups with supp or ts b y Witt groups of Z in the lo calization long exa ct sequence, and obta in a lo ng exact sequence · · · / / W ∗ ( X, L ) υ ∗ / / W ∗ ( U, L | U ) ∂ ( ( R R R R R R R R R R R R ∂ ′ / / W ∗ +1 − c ( Z, ω ι ⊗ L | Z ) / / ≃ ι ∗   W ∗ +1 ( X, L ) / / · · · ( 11 ) W ∗ +1 Z ( X, L ) 6 6 l l l l l l l l l l l l W e now wan t to describ e ∂ ′ when λ ( L ) ≡ c mo d 2 (otherwise ∂ ′ = 0 by Thm. 2.3 ). By the c o mplete dichotomy of Prop os ition 2.1 , we know that when the pus h- forward π ∗ : W ∗ ( B l , M ) → W ∗ ( X, ?) does no t exist, her e for M = ˜ α ∗ ( α ∗ ) − 1 υ ∗ ( L ) by ( 9 ), then the following comp ositio n ˜ π ∗ ◦ ˜ ι ∗ exists and sta rts from the very gr oup where π ∗ cannot b e defined a nd a rrives in the very gro up where ∂ ′ itself a rrives : W ∗ ( B l , M ) ˜ ι ∗ − → W ∗ ( E , M | E ) ˜ π ∗ − → W ∗− c +1 ( Z, ω ι ⊗ L | Z ) . Hence, in a mo ment of exaltatio n, if we blindly a pply this obser v ation at the precise po int where the oriented technique fails for Witt g roups, w e se e tha t whe n we cannot define a section to restriction by the for mu la π ∗ ◦  ˜ α ∗ ◦ ( α ∗ ) − 1  we can instead define a mysterious homo mo rphism ( ˜ π ∗ ◦ ˜ ι ∗ ) ◦  ˜ α ∗ ◦ ( α ∗ ) − 1  . CONNECTING HOMOMORPHISM F OR WITT GR OUPS 7 2.6. Main Theorem. With the notation of 1.1 , assum e Hyp othesis 1.2 and assum e X , Y , Z r e gular. L et L ∈ P ic( X ) and r e c al l the inte ger λ ( L ) ∈ Z define d by ( 8 ) . If λ ( L ) ≡ c mo d 2 the n the c omp osition ˜ π ∗ ◦ ˜ ι ∗ ◦ ˜ α ∗ ◦ ( α ∗ ) − 1 is e qual to the c onne cting homomorphi sm ∂ ′ of ( 11 ) , that is, the fol lowing diagr am c ommutes : W ∗ +1 − c ( Z, ω ι ⊗ L | Z ) W ∗ ( U, L | U ) ( α ∗ ) − 1 ≃   ∂ ′ o o W ∗  E , ω ˜ π ⊗ ˜ π ∗ ( ω ι ⊗ L | Z )  ∼ = W ∗  E , ˜ ι ∗ ˜ α ∗ ( α ∗ ) − 1 L | U  ˜ π ∗ O O W ∗  Y , ( α ∗ ) − 1 L | U  ˜ ι ∗ ˜ α ∗ o o This statement implies Thm. 1.4 (B) since ∂ = ι ∗ ∂ ′ by ( 11 ). Its pro of will be given after g eneralization t o the non-regular se tting, at the end o f Section 5 . 2.7. Remark. Let us stress the pec ulia r combination of Theo rem 2.3 and Theo- rem 2.6 . Start with a Witt clas s w U ov er the op en U ⊂ X , for the duality twisted by so me L ∈ Pic( U ) = Pic( X ), and try to extend w U to a Witt class w X ov er X : ∂ ′ ( w U ) ← 50%   50% → w X  υ ∗ / / w U _ ( α ∗ ) − 1   w E _ ˜ π ∗ O O w B l  ˜ ι ∗ o o _ π ∗ O O w Y  ˜ α ∗ o o Then, either we can apply the same constr uction as for o riented theories , i.e. push- forward the cla ss w B l := ˜ α ∗ ◦ ( α ∗ ) − 1 ( w U ) fro m B l to X along π , constructing in this wa y a n extension w X := π ∗ ( w B l ) as wanted, or this la st push-forward π ∗ is forbidden on w B l bec ause of the twist, in which case the Witt class w U might simply not belong to the image of restr iction υ ∗ . The latter means that w U might hav e a non-zero b oundary ∂ ′ ( w U ) ov er Z , which then deserves to be computed. The little miracle precisely is that in order to compute this ∂ ′ ( w U ), it suffices to resume the ab ov e pro cess where it failed, i.e. with w B l , and, since we canno t push it forw ard along π , w e can consider the bifurca tion of Pr op osition 2.1 a nd res tr ict this class w B l to the e x ceptional fib er E , s ay w E := ˜ ι ∗ w B l , and then push it for ward along ˜ π . Of course, this do es not constr uct an e x tension o f w U anymore, since this new class ˜ π ∗ ( w E ) lives ov e r Z , not ov er X . Indeed, ther e is no reaso n a priori fo r this new class to g ive a nything sensible at all. Our Main Theo r em is that this construction in f act giv es a formula for the bo undary ∂ ′ ( w U ). Bottom line : Essent ial ly the same ge ometric r e cip e o f pu l l-b ack and push-forwar d either spl its the r estriction or c onstructs the c onne ct ing homomorph ism. In p artic- ular, the c onne cting homomorphism is explicitly describ e d in b oth c ases. 3. The non-regular case In Sectio n 2 , we res tricted o ur attention to the regular case in or der to grasp the main ideas. How ever, most results can b e sta ted in the grea ter g enerality of separated and no etheria n Z [ 1 2 ]-schemes admitting a dua lizing complex. The goal of this section is to pro v ide the relev ant background and to extend Theore m 1.4 to this non-regular setting, see Main Lemma 3.5 . 3.1. R emark. The c oher ent Witt gr oups ˜ W ∗ ( X, K X ) of a scheme X ∈ S ch (see 1.1 ) are defined using the de r ived catego ry D b coh ( X ) of complexes of O X -mo dules whose cohomolog y is coher ent a nd b ounded. Since X is no etherian and separated, D b coh ( X ) is eq uiv a lent to its sub categor y D b (coh( X )) of b ounded complexes of coher ent O X - mo dules; see for instance [ 8 , Prop. A.4]. The duality is defined using the derived functor RHom( − , K X ) where K X ∈ D b coh ( X ) is a dualizing c omplex (see [ 19 , § 3] 8 P . BALMER AND B. CALM ` ES or [ 8 , § 2]), meaning that the functor RHom( − , K X ) defines a dua lit y on D b coh ( X ). F or example, a s cheme is Gorenstein if and only if O X itself is an injectiv ely b ounded dualizing complex and, in that cas e , all other dualizing complexes are shifted line bundles (see Lemma A.7 ). Regular schemes ar e Gor enstein, and for them, coherent Witt groups coincide with the usual “lo cally free” W itt groups W ∗ ( X, L ) (i.e. t he ones defined using bo unded co mplexes of lo cally free sheav es instead of co herent ones). F o r a ny line bundle L , w e still ha ve a square-p erio dic ity isomorphism (12) ˜ W( X , K X ) ∼ = ˜ W( X , K X ⊗ L ⊗ 2 ) given b y the m ultiplication b y th e class in W 0 ( X, L ⊗ 2 ) o f the canonical form L → L ∨ ⊗ L ⊗ 2 , using the pairing betw een loca lly free and coherent Witt groups. F or an y c lo sed em b edding Z ֒ → X with open complement υ : U ֒ → X , the restriction K U := υ ∗ K X is a dualizing complex [ 19 , Thm. 3 .12] and the general triangulated framew o rk of [ 3 ] g ives a loca lization long exact sequence (13) · · · ∂ − → ˜ W ∗ Z ( X, K X ) − → ˜ W ∗ ( X, K X ) − → ˜ W ∗ ( U, K U ) ∂ − → ˜ W ∗ +1 Z ( X, K X ) − → · · · As for K -theory , no such sequence holds in gener al fo r s ingular schemes and lo ca lly free Witt groups. 3.2. Remark. F or coherent Witt groups, the push-forward alo ng a prop er mor- phism f : X ′ → X takes the following very round form : If K X is a dualiz ing complex o n X then f ! K X is a dua liz ing complex on X ′ ([ 8 , Prop. 3.9]) and the functor R f ∗ : D b coh ( X ′ ) → D b coh ( X ) induces a pus h-forwar d ([ 8 , Thm. 4.4]) (14) f ∗ : ˜ W i ( X ′ , f ! K X ) → ˜ W i ( X, K X ) . Recall that f ! : D Qcoh ( X ) → D Qcoh ( X ′ ) is the right a djoint of R f ∗ . If we twist the chosen dua lizing complex K X by a line bundle L ∈ P ic( X ), this is tra nspo rted to X ′ via the following formula (see [ 8 , Thm. 3.7 ]) (15) f ! ( K X ⊗ L ) ≃ f ! ( K X ) ⊗ f ∗ L . In t he regular ca se, push-fo rward ma ps are a ls o describ ed in Nenas he v [ 20 ]. 3.3. Remark. Let us also recall fr om [ 8 , Thm. 4.1] that the pull-back f ∗ : ˜ W i ( X, K X ) → ˜ W i ( X ′ , L f ∗ K X ) along a finite T or-dimension morphism f : X ′ → X is defined if L f ∗ ( K X ) is a dualizing complex (this is not a utomatically tr ue). T ogether with the push-forward, this pu ll-back satisfies the usual flat base-change formula (see [ 8 , Thm. 5.5 ]). A regular immer s ion f : X ′ ֒ → X has finite T o r-dimension since it is even p er fect (see [ 1 , p. 2 50]). Moreov er, in that c a se, L f ∗ is the same as f ! up to a t wist and a shift (see Prop os ition A.8 ), hence it pres erves dualiz ing co mplex es. 3.4. Prop ositio n. In Setup 1.1 , let K X b e a dualizing c omplex on X . L et L ∈ Pic( X ) and ℓ ∈ Z . Then K = π ! ( K X ) ⊗ π ∗ L ⊗ O ( E ) ⊗ ℓ is a dualizing c omplex on B l and any dualizing c omplex has t his form, for some L ∈ P ic( X ) and ℓ ∈ Z . Mor e over, the dichotomy of Pr op osition 2.1 her e b e c omes : (A) If ℓ ≡ 0 mo d 2 , we c an push-forwar d along π : B l → X , as fol lows : ˜ W ∗ ( B l , K ) ∼ = ˜ W ∗  B l , π ! ( K X ⊗ L )) π ∗ → ˜ W ∗  X , K X ⊗ L ) . (B) If ℓ ≡ 1 mo d 2 , we c an push-forwar d along ˜ π : E → Z , as fol lows : ˜ W ∗ ( E , L˜ ι ∗ K ) ∼ = ˜ W ∗ +1  E , ˜ π ! ι ! ( K X ⊗ L ))  ˜ π ∗ → ˜ W ∗ +1  Z, ι ! ( K X ⊗ L )  . As b efor e, in b oth c ases, t he fi rst isomorphism ∼ = c omes fr om s quar e-p erio dicity ( 12 ) and the se c ond morphism is push-forwar d ( 14 ) . CONNECTING HOMOMORPHISM F OR WITT GR OUPS 9 Pr o of. The complex K B l := π ! K X is a dualizing complex o n B l by Remark 3.2 . By Lemma A.7 and Pr op osition A.6 (i), all dualizing complexes on B l a r e of the form K = π ! ( K X ) ⊗ π ∗ L ⊗ O ( E ) ⊗ ℓ , for unique L ∈ Pic( X ) a nd ℓ ∈ Z . W e only need to chec k the re le v ant parit y for applying ( 12 ). Case (A) follows easily fr o m ( 15 ) by definition of K and parity of ℓ . In (B), we need to compare L˜ ι ∗ K and ˜ π ! ι ! ( K X ⊗ L )[1 ]. B y Pro po sition A.11 (iv), we know that ˜ ι ! ( − ) ∼ = ˜ ι ∗ O ( E )[ − 1 ] ⊗ L˜ ι ∗ ( − ). W e apply this and ( 15 ) in the second equality be low, the fir st one using simply that ι ˜ π = π ˜ ι : ˜ π ! ι ! ( K X ⊗ L )[1] ∼ = ˜ ι ! π ! ( K X ⊗ L )[1] ∼ = ˜ ι ∗ O ( E )[ − 1 ] ⊗ L˜ ι ∗  π ! ( K X ) ⊗ π ∗ L  [1] ∼ = ∼ = ˜ ι ∗ O ( E ) ⊗ L˜ ι ∗ ( K ⊗ O ( E ) ⊗− ℓ ) ∼ = ˜ ι ∗ O ( E ) ⊗ (1 − ℓ ) ⊗ L˜ ι ∗ K. Since 1 − ℓ is even, ˜ ι ∗ O ( E ) ⊗ (1 − ℓ ) is a square, a s desir ed.  W e now wan t to g ive the key technical result of the paper, which is an analogue of Theorem 1.4 in the no n-regular setting. The idea is to descr ib e the connecting homomorphism on Witt class es ov er U which admit an extension to the blow-up B l . The key fact is the exis tence of a n additional twist on B l , namely the twist by O ( E ), which disa pp ea rs o n U (see A.1 ) and hence allows Ca se (B) below. 3.5. Main Lemma. In Setup 1.1 , assume that X has a dualizing c omplex K X and let K U = υ ∗ ( K X ) and K B l = π ! ( K X ) ; se e R emarks 3.1 and 3.2 . L et i ∈ Z . (A) The fol lowing c omp osition vanishes : ˜ W i ( B l , K B l ) ˜ υ ∗ / / ˜ W i ( U, K U ) ∂ / / ˜ W i +1 Z ( X, K X ) . (B) The fol lowing c omp osition (wel l-define d sinc e ˜ υ ∗ O ( E ) ≃ O U ) ˜ W i  B l , K B l ⊗ O ( E )  ˜ υ ∗ / / ˜ W i  U, K U ⊗ ˜ υ ∗ O ( E )  ∼ = ˜ W i ( U, K U ) ∂ / / ˜ W i +1 Z ( X, K X ) c oincides with the c omp osition ˜ W i ( B l , O ( E ) ⊗ K B l ) ˜ ι ∗   ˜ W i +1 Z ( X, K X ) ˜ W i  E , L˜ ι ∗ ( O ( E ) ⊗ K B l )  ∼ = ˜ W i +1  E , ˜ π ! ι ! K X  ˜ π ∗ / / ˜ W i +1  Z, ι ! K X  ι ∗ O O wher e the latter isomorphi sm ∼ = is induc e d by the c omp osition (16) L˜ ι ∗ ( O ( E ) ⊗ K B l ) ∼ = ˜ ι ∗ ( O ( E )) ⊗ L˜ ι ∗ ( K B l ) ∼ = ˜ ι ! K B l [1] ∼ = ˜ π ! ι ! K X [1] . The proof of this result o ccupies Section 4 . Here a re j ust a couple of co mmen ts on the statemen t. Le t us first of a ll explain the announced sequenc e of isomor- phisms ( 16 ). The first one holds since L ˜ ι ∗ is a tensor functor and since O ( E ) is a line bun dle (hence is flat). The s econd o ne holds b y Propo sition A.9 (v). The last one follows by definition of K B l and the fact that ι ˜ π = π ˜ ι . Finally , note that w e use the pull-back ˜ ι ∗ on coheren t Witt groups as recalled in Remark 3.3 . 4. The main argument Surprisingly eno ugh for a problem in volving the blow-up B l = B l Z ( X ) of X along Z , see ( 1 ), the ca se wher e co dim X ( Z ) = 1 is als o interesting, even though, o f course, in that case B l = X and E = Z . In fa c t, this cas e is crucia l for the pro o f of Main Lemma 3.5 and this is w hy we deal with it first. In the “g eneral” pro of where co dim X ( Z ) is arbitra ry , w e will apply the case of c o dimension one to ˜ ι : E ֒ → B l . Therefore, we use the following notation to discuss co dimensio n one. 10 P . BALMER AND B. CALM ` ES 4.1. No tation. Let B ∈ S ch b e a scheme with a dualizing complex K B and ˜ ι : E ֒ → B b e a prime diviso r, that is, a reg ular clos ed immersio n of co dimension one, of a subscheme E ∈ S ch . Let O ( E ) b e the line bundle on B asso ciated to E (see Definition A.1 ). L e t ˜ υ : U ֒ → B b e the op en immersion of the op en complemen t E   ˜ ι / / B U ? _ ˜ υ o o U = B − E and let K U be the du alizing complex ˜ υ ∗ ( K B ) o n U . 4.2. Main Lemma in co di mension one. With Notation 4.1 , let i ∈ Z . Then : (A) The c omp osition ˜ W i ( B , K B ) ˜ υ ∗ / / ˜ W i ( U, K U ) ∂ / / ˜ W i +1 E ( B , K B ) is zer o. (B) The c omp osition ˜ W i  B , K B ⊗ O ( E )  ˜ υ ∗ / / ˜ W i  U, K U ⊗ ˜ υ ∗ O ( E )  ∼ = ˜ W i ( U, K U ) ∂ / / ˜ W i +1 E ( B , K B ) c oincides with the c omp osition ˜ W i ( B , O ( E ) ⊗ K B ) ˜ ι ∗   ˜ W i +1 E ( B , K B ) ˜ W i  E , L ˜ ι ∗ ( O ( E ) ⊗ K B )  ∼ = ˜ W i  E , ˜ ι ! K B [1]  ∼ = ˜ W i +1  E , ˜ ι ! K B  ˜ ι ∗ O O wher e the first isomorphi sm ∼ = is induc e d by the fol lowing isomorphism (17) L˜ ι ∗ ( O ( E ) ⊗ K B ) ∼ = ˜ ι ∗ ( O ( E )) ⊗ L˜ ι ∗ ( K B ) ∼ = ˜ ι ! ( K B )[1] . Pr o of. Cas e (A) is s imple : The comp osition of tw o consecutive morphisms in the lo calization lo ng exact sequence ( 13 ) is zero. Case (B) is the nontrivial one. The isomorphisms ( 1 7 ) ar e the same as in ( 16 ). A t this stage, we uploa d the definition of the connecting homomorphism for Witt gr oups ∂ : ˜ W i ( U, K U ) → ˜ W i +1 E ( B , K B ), which go es as follows : T a ke a non- degenerate s y mmetric space ( P , φ ) o ver U for the i th -shifted duality with v a lues in K U ; t here exists a possibly degenerate sy mmetr ic pair ( Q, ψ ) o ver B for the same duality (with v alues in K B ) which restr icts to ( P , φ ) over U ; compute its symmetric cone d ( Q, ψ ), which is essen tia lly the co ne of ψ equipped with a natura l metab olic form; see [ 3 , § 4 ] or [ 4 , Def. 2.3] for instance ; for any choice o f such a pair ( Q, ψ ), the b o unda ry ∂ ( P , φ ) ∈ ˜ W i +1 E ( B , K B ) is the Witt class of d ( Q, ψ ). There is nothing really sp ecific to dualizing complexes here. The abov e con- struction is a purely triangular one, as long a s one use s the same dualit y for the ambien t scheme B , for the op en U ⊂ B and for the Witt gr oup of B with supp orts in the clo sed complement E . The subtlety of statement (B) is that we start with a twisted dualit y o n the scheme B which is not the dualit y used for ∂ , but which agrees with it o n U b y the first isomor phism ∼ = in statemen t (B). Now, take a n elemen t in ˜ W i ( B , O ( E ) ⊗ K B ). It is the W itt-equiv alence class o f a symmetric spa ce ( P, φ ) o ver B with r e spe ct to the i th -shifted duality with v alues in O ( E ) ⊗ K B . The claim of the statement is tha t, mo dulo the ab ov e identifications of dualizing complexes, w e should hav e (18) ∂ ( ˜ υ ∗ ( P, φ )) = ˜ ι ∗ (˜ ι ∗ ( P, φ )) in ˜ W i +1 E ( B , K B ). By the abov e discussion, in order to compute ∂ ( ˜ υ ∗ ( P, φ )), we need to find a s ymmetric pair ( Q, ψ ) over B , for the duality given by K B , and such that ˜ υ ∗ ( Q, ψ ) = ˜ υ ∗ ( P, φ ). Note that we ca nnot take for ( Q, ψ ) the pair ( P, φ ) itself CONNECTING HOMOMORPHISM F OR WITT GR OUPS 11 bec ause ( P, φ ) is sy mmetric for the twisted duality O ( E ) ⊗ K B on B . Nevertheless, it is easy to “corr ect” ( P , φ ) as follows. As in Definition A.1 , w e ha ve a canonical homomorphism of line bundles : σ E : O ( E ) ∨ → O B . The pair ( O ( E ) ∨ , σ E ) is symmetr ic in the derived category D b (VB( B )) of vector bundles o ver B , with resp ect to the 0 th -shifted dualit y t wisted b y O ( E ) ∨ , because the ta rget of σ E is the dual o f its source : ( O ( E ) ∨ ) ∨ [0] ⊗ O ( E ) ∨ ∼ = O B . Let us define the w anted symmetric pair ( Q, ψ ) in D b coh ( B ) for the i th -shifted duality with v a lues in K B as the following pro duct : ( Q, ψ ) := ( O ( E ) ∨ , σ E ) ⊗ ( P , φ ) . Note that we tensor a complex of vector bundles with a cohe r ent one to get a coherent o ne, following the formalism of [ 4 , § 4] where such exter nal pro ducts are denoted b y ⋆ . W e claim that th e restriction of ( Q, ψ ) to U is nothing but ˜ υ ∗ ( P, φ ). This is easy to chec k since O ( E ) | U = O U via σ E (see A.1 ), whic h means ( O ( E ) ∨ , σ E ) | U = 1 U . So, by the construction of the connec ting homomorphism ∂ recalled at the beg inning of the pro of, w e know that ∂ ( ˜ υ ∗ ( P, φ )) can be computed as d ( Q, ψ ). This reads : ∂ ( ˜ υ ∗ ( P, φ )) = d  ( O ( E ) ∨ , σ E ) ⊗ ( P , φ )  . Now, we use tha t ( P , φ ) is non-deg enerate and that therefore (see [ 4 , Rem. 5.4] if necessary ) we can tak e ( P , φ ) out of the ab ove symmetric cone d ( ... ), i.e. (19) ∂ ( ˜ υ ∗ ( P, φ )) = d ( O ( E ) ∨ , σ E ) ⊗ ( P , φ ) . Let us c o mpute the symmetric cone d ( O ( E ) ∨ , σ E ) =: ( C, χ ). Note that this only inv olves vector bundles. W e define C to be the cone of σ E and w e equip it with a symmetric form χ : C ∼ → C ∨ [1] ⊗ O ( E ) ∨ for the duality used for ( O ( E ) , σ E ) but shifted by one, that is, for the 1 st shifted duality with v a lues in O ( E ) ∨ . One chec k s that ( C, χ ) is given by the follo wing explicit for mula : (20) C = χ    · · · / / 0 / /   0 / /   O ( E ) ∨ σ E / / − 1   O B / / 1   0 / /   0 / /   · · ·  C ∨ [1] ⊗ O ( E ) ∨ =  · · · / / 0 / / 0 / / O ( E ) ∨ − ( σ E ) ∨ / / O B / / 0 / / 0 / / · · ·  where the complexes ha ve O B in d egree zero. Now, observe that the complex C is a reso lution of ˜ ι ∗ ( O E ) over B , by Definition A.1 , that is , C ≃ ˜ ι ∗ ( O E ) in the derived category of B . Moreov er, by Pro po sitions A.8 a nd A.9 (ii), we have ˜ ι ! ( O ( E ) ∨ [1]) = ω ˜ ι [ − 1] ⊗ ˜ ι ∗ ( O ( E ) ∨ [1]) ∼ = O E . Using this, one c hecks the conceptually ob vious fact that χ is also the push-forward alo ng the p erfect mo r phism ˜ ι o f the unit for m on O E . See Remark 4.3 b elow for more details. This means that we hav e an isometry in D b E (VB( B )) d  O ( E ) ∨ , σ E  = ˜ ι ∗ (1 E ) of symmetric space s with resp ect to the 1 st shifted duality with v alues in O ( E ) ∨ . Plugging this last eq uality in ( 19 ), and using the pro jectio n formula (see Re- mark 4.3 ) we o bta in ∂ ( ˜ υ ∗ ( P, φ )) = ˜ ι ∗ (1 E ) ⊗ ( P, φ ) = ˜ ι ∗  1 E ⊗ ˜ ι ∗ ( P, φ )  = ˜ ι ∗  ˜ ι ∗ ( P, φ )  . This is the cla imed equality ( 18 ).  12 P . BALMER AND B. CALM ` ES 4.3. Remark. In the a bove pro o f, we use t he “conceptually obvious fac t” that the push-forward of the unit form o n O E is indeed the χ of ( 20 ). This fact is ob vious to the exp ert but w e cannot pr ovide a direct r eference for this exa ct statement. How ever, if the r eader do es not want to do this lengthy verification directly , the computation of [ 8 , § 7.2] can b e applied essentially v e r batim. The main difference is that here, we are c o nsidering a push-forward of lo cally free instead of coher ent Witt groups along a regular em b edding. Suc h a push-for ward is constructed using the same tensor formalis m a s the pro p e r push-for wards for coherent Witt gro ups considered in lo c. cit. a lo ng mor phis ms that are prop er, perfect and Gorens tein, which is true of a regula r em bedding . In lo c. cit. there is an as s umption that the schemes ar e Gor enstein, ensuring that the line bundles a re dualizing complexes. But here, the dualizing o b jects for our ca tegory o f complexe s o f lo cally free sheav es are line bundles anyw ay and the extra Go renstein a ssumption is irrelev ant. Moreov er, the pro jection formula used in the ab ov e pro of is e s tablished in com- plete gener ality for non neces sarily regula r schemes b y the same metho d as in [ 8 , § 5.7] using the pa iring betw een the lo cally free deriv e d categ o ry and the coherent one to the coheren t one. More precise ly , this pairing is just a restriction of the quasi-coher ent pair ing D Qcoh × D Qcoh ⊗ − → D Qcoh of loc. cit. to these subcatego ries. By the genera l tensor formalis m of [ 1 0 ], for any morphism f : X → Y as ab ove, for any ob ject A (resp. B ) in the quasi-coherent de r ived ca tegory of X (resp. Y ), w e obtain a pro jection mor phism in D Qcoh ( Y ) R f ∗ ( A ) ⊗ B − → R f ∗ ( A ⊗ L f ∗ ( B )) , see [ 10 , Pr op. 4.2.5]. It is an iso morphism by [ 8 , Thm. 3.7]. W e ac tua lly only use it for A a complex of lo cally free sheav es and B a complex with co herent and b ounded cohomolog y . The pr o jection for mula is implied by [ 10 , Thm. 5.5.1]. Pr o of of Main L emma 3.5 . Case (A) follows f rom the co dimension one ca se and the compatibilit y of p ush-forwards with connecting homomorphisms (here along the identit y of U ). Case (B) follo ws f rom t he outer comm utativity o f the fo llowing diagram : (21) ˜ W i ( U, K U ) ∂ / / ˜ W i +1 Z ( X, K X ) ˜ W i +1 ( Z, ι ! K X ) ι ∗ o o ˜ W i ( U, K U ) ∂ / / ˜ W i +1 E ( B l , K B l ) π ∗ O O ˜ W i +1 ( E , ˜ π ! ι ! K X ) ˜ ι ∗ o o ˜ π ∗ O O ˜ W i ( B l , O ( E ) ⊗ K B l ) ˜ υ ∗ O O ˜ ι ∗ / / ˜ W i  B l , L˜ ι ∗ ( O ( E ) ⊗ K B l )  ∼ = ˜ W i +1 ( E , ˜ ι ! K B l ) ∼ = O O W e shall now verify the inner commutativit y of this diag ram. The upp er left square of ( 21 ) co mm utes b y compatibility of push-forward with connecting ho- momorphisms. The upp er r ight square of ( 21 ) simply commutes by functoria lity of push-forward a pplied to ι ◦ ˜ π = π ◦ ˜ ι . Mo st in terestingly , the lower part of ( 2 1 ) commutes by Lemma 4.2 applied to the co dimensio n one inclusion ˜ ι : E ֒ → B l .  5. The Main Theorem in the no n -regular case Without regularity a s sumptions, we ha ve sho wn in Main Lemma 3.5 how to compute the connecting homomorphism ∂ : ˜ W ∗ ( U, K U ) → ˜ W ∗ +1 Z ( X, K X ) o n t hose Witt clas ses ov er U which come from B l = B l Z ( X ) b y res triction ˜ υ ∗ . The whole po int of adding Hyp othesis 1.2 is pr ecisely to split ˜ υ ∗ , that is, to construct for each Witt c lass o n U an extension on B l . In the regular case, this follows from CONNECTING HOMOMORPHISM F OR WITT GR OUPS 13 homotopy inv ar ia nce of Pica rd groups and Witt gr oups. In the non-reg ular setting, things are a little more complica ted. Let us give the statement and comment on the hyp o theses a fterwards (se e Remark 5.2 ). 5.1. Main Theorem in the non-regular c ase. In Setup 1.1 , assume that X has a dualizing c omplex K X and e quip U with the r estricte d c omplex K U = υ ∗ ( K X ) . Assume Hyp othesis 1.2 and further make the following hyp otheses : (a) Ther e exists a dualizing c omplex K Y on Y such that α ∗ K Y = K U . (b) The A ∗ -bund le α induc es an isomo rphism ˜ W ∗ ( Y , K Y ) ∼ → ˜ W ∗ ( U, K U ) . (c) The morphism ˜ α is of finite T or dimension and L ˜ α ∗ ( K Y ) is dualizing. (d) Se quenc e ( 25 ) is exact : Z → Pic( B l ) → Pic( U ) . (Se e Pr op osition A.3 .) Then L ˜ α ∗ ( K Y ) ≃ π ! K X ⊗ O ( E ) ⊗ n for some n ∈ Z , and the fol lowing holds true : (A) If n c an b e chosen even, the c omp osition π ∗ ˜ α ∗ ( α ∗ ) − 1 is a se ction of υ ∗ . (B) If n c an b e chosen o dd, the c omp osition ι ∗ ˜ π ∗ ˜ ι ∗ ˜ α ∗ ( α ∗ ) − 1 c oincides with the c onne cting homomorphi sm ∂ : ˜ W ∗ ( U, K U ) → ˜ W ∗ +1 E ( X, K X ) . Pr o of. By (c) and Remar k 3.2 respectively , b oth L ˜ α ∗ ( K Y ) and π ! K X are du al- izing complexes on B l . By Lemma A.7 (i), they differ by a shifted line bundle : L ˜ α ∗ ( K Y ) ≃ π ! K X ⊗ L [ m ] with L ∈ Pic( B l ) a nd m ∈ Z . Restricting to U , w e get K U ⊗ ˜ υ ∗ L [ m ] ≃ ˜ υ ∗ π ! K X ⊗ ˜ υ ∗ L [ m ] ≃ ˜ υ ∗ ( π ! K X ⊗ L [ m ]) ≃ ˜ υ ∗ L ˜ α ∗ ( K Y ) ≃ α ∗ K Y ≃ K U where the first equa lit y holds b y flat base-change ([ 8 , Thm. 5.5]). Thus, ˜ υ ∗ L [ m ] is the trivia l line bundle on U by L emma A.7 (ii). So m = 0 a nd, by (d), L ≃ O ( E ) ⊗ n for s o me n ∈ Z . This gives L ˜ α ∗ ( K Y ) ≃ π ! K X ⊗ O ( E ) ⊗ n as cla imed. W e now c o nsider coherent Witt groups. By (c) and Remark 3.3 , ˜ α induces a morphism ˜ α ∗ : ˜ W ∗ ( Y , K Y ) → ˜ W( B l, L ˜ α ∗ K Y ). By Lemma A.12 , the flat mor phism α induces a homomorphism α ∗ : ˜ W ∗ ( Y , K Y ) → ˜ W ∗ ( U, K U ) which is a ssumed to b e an iso morphism in (b). So, we ca n use ( α ∗ ) − 1 . When n is ev en, we hav e υ ∗ π ∗ ˜ α ∗ ( α ∗ ) − 1 = ˜ υ ∗ ˜ α ∗ ( α ∗ ) − 1 = α ∗ ( α ∗ ) − 1 = id where the first equalit y holds by flat base-c hange ([ 8 , Thm. 5.5]). This prov es (A ). On the other hand, when n is o dd, w e hav e ι ∗ ˜ π ∗ ˜ ι ∗ ˜ α ∗ ( α ∗ ) − 1 = ∂ ˜ υ ∗ ˜ α ∗ ( α ∗ ) − 1 = ∂ α ∗ ( α ∗ ) − 1 = ∂ where the first equa lit y holds by Ma in Lemma 3.5 (B).  5.2. Remark. Hyp othesis (a) in Theorem 5.1 is alwa ys true when Y admits a dualizing complex and homoto py inv ar ia nce holds ov er Y for the P icard g roup (e.g. Y regular ). Homotopy inv ar iance for co herent Witt gro ups sho uld hold in general but only app ears in the literature when Y is Gor enstein, se e Gille [ 12 ]. This means that H yp othesis (b) is a mild one. Hypo thesis (d) is discus sed in P r op osition A.3 . 5.3. Remark. In Theo rem 5.1 , the equatio n L ˜ α ∗ ( K Y ) ≃ π ! K X ⊗ O ( E ) ⊗ n , for n ∈ Z , s hould b e considered as a no n-regular a nalogue o f Equa tio n ( 8 ). In Remar k 2.2 , we discussed the compatibility of the v arious lines bundles on the s chemes X , U , Y and B l . Her e, w e need to con tro l the relationship b etw een dualizing complexes instead and we do so b y r estricting to U and b y using the exact sequence ( 25 ). Alternatively , one can remove Hypothes is (d) a nd dir ectly ass ume the relation L ˜ α ∗ ( K Y ) ≃ π ! ( K X ) ⊗ O ( E ) ⊗ n for so me n ∈ Z . This migh t hold in some pa r- ticular e xamples e ven if ( 25 ) is not exa ct. F or the conv enience of the reader, w e include the proofs of the follo wing facts. 5.4. Lemma. If X i s Gor enst ein, then Z and B l ar e Gor enstein. If X is r e gular, B l is r e gular. 14 P . BALMER AND B. CALM ` ES Pr o of. By Prop. A.8 , π ! ( O X ) is the line bundle ω π . Since π is prop er, π ! preserves injectiv ely b ounded dualizing complexes and ω π is dualizing and since it is a line bundle, B l is Gorenstein. The same proof holds for Z , since ι ! ( O X ) is ω ι (shifted) which is also a line bundle b y Pr op. A.11 . F or regula rity , see [ 18 , Thm. 1.19 ].  Pr o of of The or em 2.6 . Note that all the a s sumptions of Theor e m 5 .1 are fulfilled in the regular case, tha t is, in the setting of Section 2 . Indeed, if X and Y are regular , B l and U a re reg ular, and the dualizing complexes on X , Y , B l a nd U are simply shifted line bundles. The morphism α ∗ : Pic( Y ) → Pic( U ) is then an isomorphism (homotopy inv aria nce) and ˜ α is a uto matically of finite T or dimension, as any morphism to a regular scheme. Finally , the seque nce on Picard g r oups is exact by Pr op osition A.3 . Let K X = L be the chosen line bundle on X . Then set L U := K U = υ ∗ L and c ho os e L Y = K Y to be the unique line bundle (up to is o morphism) suc h that α ∗ L Y = L U . By ( 8 ), we hav e ˜ α ∗ L Y = π ∗ L ⊗ O ( E ) ⊗ λ ( L ) = π ! L ⊗ O ( E ) ⊗ ( λ ( L ) − c +1) , where the la st equa lity holds since π ! L = O ( E ) ⊗ ( c − 1) ⊗ π ∗ L by Pr op osition A.11 (v i). In o ther words, we hav e prov ed that ˜ α ∗ K Y = π ! K X ⊗ O ( E ) ⊗ ( λ ( L ) − c +1) . In The- orem 5.1 , we can then take n = λ ( L ) − c + 1 and the parity condition b ecomes λ ( L ) ≡ c − 1 mod 2 for Case (A) a nd λ ( L ) ≡ c mo d 2 for Case (B). So , Case (A) is the triv ial one and corres p o nds to Theorem 2.3 . Case (B) exactly gives Theorem 2.6 up to the identifications of line bundles explained in Appendix A .  Appendix A. Line bundles and d ualizing compl exes W e use Hartshor ne [ 15 ] or Liu [ 18 ] as gener al r eferences for algebra ic geo metry . W e still denote by S ch the categor y o f no etherian separ a ted co nnected schemes (we do not need “o ver Z [ 1 2 ]” in this app endix). A.1. Defini tion. Let ˜ ι : E ֒ → B be a regular clo sed immer sion of co dimension one, with B ∈ S ch . Consider the ideal I E ⊂ O B defining E (22) 0 / / I E σ E / / O B / / ˜ ι ∗ O E / / 0 . By assumption, I E is an inv er tible ideal, i.e. a line bundle. The line bund le asso- ciate d to E is defined as its dual O ( E ) := ( I E ) ∨ , se e [ 15 , I I.6.1 8]. W e th us hav e by construction a global section σ E : O ( E ) ∨ → O B , which v a nis hes exactly on E . This gives an explicit trivialization of O ( E ) outside E . On t he other hand, the restriction of O ( E ) to E is the normal bundle O ( E ) | E ∼ = N E /B . A.2. Example. Let B l = B l Z ( X ) be the blo w-up of X along a regula r clos ed immersion Z ֒ → X as in Setup 1.1 . Let I = I Z ⊂ O X be the shea f of idea ls defining Z . By construction of the blow-up, w e hav e B l = Pro j( S ) where S is the sheaf o f g raded O X -algebra s S := O X ⊕ I ⊕ I 2 ⊕ I 3 ⊕ · · · Similarly , E = Pr o j( S/ J ) where J := I · S ⊂ S is the sheaf o f homogeneous ideals J = I ⊕ I 2 ⊕ I 3 ⊕ I 4 ⊕ · · · So, E = P Z ( C Z/X ) is a pro jective bundle ov er Z asso ciated to the vector bundle C Z/X = I /I 2 which is the conormal bundle o f Z in X . Asso ciating O B l -sheav es to graded S -mo dules, the obvious exact sequence 0 → J → S → S / J → 0 yields (23) 0 / / e J σ E / / O B l / / ˜ ι ∗ O E / / 0 . Compare ( 22 ). This means that here I E = e J . But now, J is ob viously S (1) truncated in non-negative degrees. Since t wo graded S -mo dules which c o incide CONNECTING HOMOMORPHISM F OR WITT GR OUPS 15 ab ov e some degree hav e the s ame asso ciated sheav es, we have I E = e J = g S (1) = O B l (1). Consequent ly , O ( E ) = ( I E ) ∨ = O B l ( − 1). In particular , we g et (24) O ( E ) | E = O B l ( − 1) | E = O E ( − 1) . A.3. Prop os ition (Pica rd group in co dimension o ne) . L et B ∈ S ch b e a scheme and ˜ ι : E ֒ → B b e a r e gular close d immersion of c o dimension one of an irr e ducible subscheme E ∈ S ch with op en c omplement ˜ υ : U ֒ → B . We then have a c omplex (25) Z / / Pic( B ) ˜ υ ∗ / / Pic( U ) wher e the first map sends 1 to the line bu n d le O ( E ) asso ciate d to E . This c omplex is exact if B is n ormal, and ˜ υ ∗ is surje ctive when B is furthermor e r e gular. It is also exact when B is the blow-up of a normal scheme X along a r e gu lar emb e dding. Pr o of. ( 25 ) is a complex since O ( E ) is trivia l on U . When B is nor mal, Pic( B ) injects in the gr o up Cl( B ) of W eil divisors (see [ 18 , 7.1 .19 a nd 7.2.14 (c)]), for which the sa me sequence holds by [ 15 , Prop. I I.6.5]. Exactnes s of ( 25 ) then follows by diagram c hase. The surjectivity of ˜ υ ∗ when B is regular follows from [ 15 , Prop. I I.6 .7 (c)]. When B is the blow-up of X along Z , we can as s ume that co dim X ( Z ) ≥ 2 by the previous po int. Then, the result again follows by diagr am chase, using that Pic( B ) = Pic( X ) ⊕ Z , as prov ed in Prop osition A.6 (i) be low.  A.4. Remark. Note that the blow-up of a norma l scheme along a regular clo sed embedding isn’t necessarily normal if the subscheme is not reduced. F or example, take X = A 2 = Spec( k [ x, y ]) and Z defined by the equations x 2 = y 2 = 0 . Then, B l is the subscheme of A 2 × P 1 defined by the equations x 2 v = y 2 u where [ u : v ] ar e homogeneous co o r dinates for P 1 and it is easy t o check that the whole exceptional fiber is s ingular. Th us B l is no t norma l (not even regula r in codimension one). A.5. Propo sition (Picard group of a pro jectiv e bundle) . L et X ∈ S ch b e a (c on- ne cte d) scheme and F a ve ctor bund le over X . We c onsider the pr oje ctive bund le P X ( F ) asso ciate d to F . Its Pic ar d gr oup is Pic( X ) ⊕ Z wher e Z is gener ate d by O ( − 1) and Pic( X ) c omes fr om t he pul l-b ack fr om X . Pr o of. Surjectivity o f Pic( X ) ⊕ Z → Pic( P X ( F )) is a formal consequence of Quillen’s formula [ 23 , Prop. 4.3] for the K-theory of a pro jective bundle. Indeed, the deter- minant m ap K 0 → Pic is surjective with an ob vious set theoretic section a nd can easily b e co mputed o n each comp onent of Q uillen’s formula. Injectivit y is obtained by pulling back to the fib er of a point for the Z component, and b y the pro jection formula for the remaining Pic ( X ) component.  A.6. Prop o s ition (Picard gr oup o f a blo w-up) . U n der Setup 1.1 , we have : (i) The Pic ar d gr ou p of B l = B l Z ( X ) is isomorphi c to P ic( X ) ⊕ Z wher e the dir e ct summand Pic( X ) c omes fr om the pul l-b ack π ∗ and Z is gener ate d by the class of the exc eptional divi sor O ( E ) = O B l ( − 1) . (ii) If X is n ormal, the map υ ∗ : Pic( X ) → P ic ( U ) is inje ctive. If X is r e gular it is an isomo rphism. (iii) The exc eptional fib er E is the pr oje ctive bu nd le P ( C Z/X ) ove r Z and its Pic ar d gr oup is ther efor e Pic( Z ) ⊕ Z wher e Z is gener ate d by O E ( − 1) . (iv) The pul l-b ack ˜ ι ∗ : Pic ( B l ) → Pic( E ) maps [ O ( E )] ∈ Pic( B l ) to [ O E ( − 1)] . 16 P . BALMER AND B. CALM ` ES Under t hese identific ations, Diagr am ( 1 ) induc es the fol lowing pul l-b ack maps on Pic ar d gr oups : Pic( Z )  id 0    Pic( X ) ι ∗ o o υ ∗ / /  id 0    Pic( U ) . Pic( Z ) ⊕ Z P ic( X ) ⊕ Z  ι ∗ 0 0 id  o o ( υ ∗ 0) 6 6 n n n n n n n n n n n n Pr o of. By Example A.2 , we get (iv) and we can deduce (iii) from P rop osition A.5 . T o prov e (ii), use that fo r X normal (resp. reg ular) P ic ( X ) injects in to (resp. is isomorphic to) the gro up Cl( X ) of W eil diviso r s clas ses (see [ 18 , 7.1 .1 9 and 7.2.14 (c), r esp. 7.2.16]), and that Cl( X ) = Cl( U ) since co dim X ( Z ) ≥ 2. Finally , for (i), consider t he comm utative diagr am (26) D p erf ( Z ) L ˜ π ∗   D p erf ( X ) L ι ∗ o o L π ∗   D p erf ( E ) D p erf ( B l ) L˜ ι ∗ o o of induced functors on the derived categorie s of per fect c o mplexes. W e will use : F act 1 : The tensor triangulated functors L π ∗ and L ˜ π ∗ are fully faithful with left inv erse R π ∗ and R ˜ π ∗ resp ectively , see Thomason [ 24 , Lemme 2.3]. F act 2 : If M ∈ D p erf ( B l ) is such t hat L˜ ι ∗ ( M ) ≃ L ˜ π ∗ ( N ) for so me N ∈ D p erf ( Z ), then M ≃ L π ∗ ( L ) for some L ∈ D p erf ( X ), w hich must then be R π ∗ ( M ) by F act 1 . This follows from [ 11 , Pr op. 1.5]. (In their nota tion, o ur assumption implies that M is zero in all successive quotients D i +1 p erf ( B l ) / D i p erf ( B l ) hence b elongs to D 0 p erf ( B l ).) Hence Pic( X ) ⊕ Z → Pic( B l ) is injective : If L is a line bundle on X and n ∈ Z are such that L π ∗ ( L ) ⊗ O B l ( n ) is tr ivial then we get n = 0 by r estricting to E and a pplying (iii), and we get L ≃ R π ∗ L π ∗ L ≃ R π ∗ O B l ≃ R π ∗ L π ∗ O X ≃ O X by F act 1. So, let us chec k surjectivity of Pic( X ) ⊕ Z → P ic( B l ). Let M be a line bundle on B l . Using (iii) again and twisting with O B l ( n ) if neces sary , w e can assume that L˜ ι ∗ ( M ) is is o morphic to L ˜ π ∗ N = ˜ π ∗ N for some line bundle N on Z . By F act 2, there exists L ∈ D p erf ( X ) suc h that L π ∗ ( L ) ≃ M . It now suffices to check that this L ∈ D p erf ( X ) is a line bundle. The natural (ev aluation) map L ∨ ⊗ L → O X is an isomorphism, since it is so a fter applying the fully faithful tensor functor L π ∗ : D p erf ( X ) → D p erf ( B l ). So L ∈ D p erf ( X ) is an invertible ob ject, hence it is the m th susp ension of a line bundle for m ∈ Z , see [ 7 , Prop. 6.4]. Using ( 26 ), one checks by res tricting to Z that m = 0 , i.e. L is a line bund le.  * * * W e now discuss dualizing complexes a nd relative c anonical bundles. Firs t of all, we mention the essential uniqueness of dualizing complexes o n a scheme. A.7. Lemm a. L et X ∈ S ch b e a scheme admitting a dualizing c omplex K X . Then : (i) F or any line bun d le L and any inte ger i , the c omplex K X ⊗ L [ i ] is also a dualizing c omplex and any dualizing c omplex on X is of t his form. (ii) If K X ⊗ L [ i ] ≃ K X in the derive d c ate gory of X , for some line bund le L and some inte ger i , then L ≃ O X and i = 0 . In other wor ds, the s et of isomorphism classes of dualizing c omplexes on X is a princip al homo gene ous sp ac e under the action of Pic( X ) ⊕ Z . Pr o of. F or (i), see [ 19 , Lemma 3.9]. Let us prove (ii). W e have the isomorphisms O X ∼ → RHom( K X , K X ) ≃ RHom( K X , K X ⊗ L [ n ]) ≃ RHom( K X , K X ) ⊗ L [ n ] ∼ ← L [ n ] CONNECTING HOMOMORPHISM F OR WITT GR OUPS 17 in the co herent derived ca tegory . The fir st and las t ones hold by [ 19 , Pr op. 3.6]. W e th us o btain an isomorphism O X ≃ L [ n ] in the derived categ o ry of p erfect complexes (it is a full sub categor y of the coherent one). This forces n = 0 a nd the existence of an honest iso morphism o f sheaves O X ≃ L , see [ 7 , Prop. 6.4] if necessary .  W e no w use the notion of local complete in ter section (l.c.i.) mor phis m, that is , a mor phism which is lo cally a regular embedding followed by a smo o th morphis m, see [ 18 , § 6.3.2 ]. The adv antage of such morphisms f : X ′ → X is that f ! is just L f ∗ t wisted b y a line bun dle ω f and shifted by the relative dimension dim( f ). A.8. Prop ositi o n. L et f : X ′ → X b e an l.c.i. morphism with X , X ′ ∈ S ch . Assume that f is pr op er. Then f ! ( O X ) is a shif te d line bund le ω f [dim( f )] and ther e exists a natur al isomorph ism f ! ( O X ) ⊗ L f ∗ ( − ) ∼ → f ! ( − ) . In p articular, f ! pr eserves the sub c ate gory D p erf of D b coh . Pr o of. There is a lways a natura l mor phism f ! ( O X ) ⊗ L f ∗ ( − ) → f ! ( − ). O ne shows that it is an isomorphism and tha t f ! ( O X ) is a line bundle dir ectly from the defi- nition, since b oth these facts can b e chec ked lo cally , a re stable by comp osition and are true for (closed) regula r immersions a nd smo oth morphisms by Hartshor ne [ 14 , Ch. II I]. The sub categ ory D p erf is then preserved since bo th L f ∗ and tenso r ing by a line bundle pr eserve it.  The ab ov e prop ositio n r educes the descr iption of f ! to that of the line bundle ω f . A.9. Prop o s ition. In the fol lowing c ases, we have c oncr ete descriptions of ω f . (i) When f : X ′ → X i s smo oth and pr op er, ω f ≃ det(Ω 1 X ′ /X ) is the determi- nant of the she af of differ entials. In p articular, when f is the pr oje ction of a pr oje ctive bund le P ( F ) to its b ase, wher e F is a ve ctor bund le of r ank r , then ω f ≃ f ∗ (det F ) ⊗ O P ( F ) ( − r ) . (ii) When f : X ′ ֒ → X is a r e gular close d immersion, ω f ≃ det( N X ′ /X ) is the determinant of the normal bun d le. In p articular when f : E ֒ → B is the inclusion of a prime divisor (Def. A.1 ), we have ω f ≃ O ( E ) | E . Pr o of. See [ 25 , Pr op. 1 and Thm. 3]. See alternatively [ 18 , § 6.4.2].  A.10. Re mark. All morphisms along which we co nsider push-forward in this article are l.c.i. It might not be ob vious for π : B l → X bu t this follo ws from [ 1 , VI I 1.8 p. 424] (it is lo cally o f the form mentioned there). So, ω π is also a line bundle. Let us now describ e the r elative cano nical line bundles in terms of ω ι = det( N Z/X ). A.11. Prop osition. W ith the notation of Setup 1.1 , we have (i) ω ˜ ι = O ( E ) | E = O E ( − 1) (ii) ω ˜ π = ˜ π ∗ ω ∨ ι ⊗ O ( E ) ⊗ c | E = ˜ π ∗ ω ∨ ι ⊗ O E ( − c ) (iii) ω π = O ( E ) ⊗ ( c − 1) = O B l (1 − c ) . By Pr op osition A.8 , it implie s that we hav e (iv) ˜ ι ! ( − ) = O ( E ) | E ⊗ L˜ ι ∗ ( − )[ − 1] = O E ( − 1) ⊗ L˜ ι ∗ ( − )[ − 1] (v) ˜ π ! ( − ) = ˜ π ∗ ω ∨ ι ⊗ O ( E ) ⊗ c | E ⊗ L ˜ π ∗ ( − )[ c − 1] = ˜ π ∗ ω ∨ ι ⊗ O E ( − c ) ⊗ L ˜ π ∗ ( − )[ c − 1] (vi) π ! ( − ) = O ( E ) ⊗ ( c − 1) ⊗ L π ∗ ( − ) = O B l (1 − c ) ⊗ L π ∗ ( − ) . Pr o of. Points (i) a nd (ii) follow from Prop ositio n A.9 (ii) and (i), res pectively . They imply (iv) and (v). T o prove po int (iii) let us firs t observe that t he exa ct sequence ( 22 ) g ives rise to an exact triangle O B l ( l + 1) → O B l ( l ) → R ˜ ι ∗ ( O E ( l )) → O B l ( l + 1)[1] 18 P . BALMER AND B. CALM ` ES in D p erf ( B l ) for any l ∈ Z . Applying R π ∗ to this triangle a nd using that R π ∗ R ˜ ι ∗ O E ( l ) = R ι ∗ R ˜ π ∗ O E ( l ) = 0 for − c < l < 0 (b y [ 13 , 2.1.15]), w e o btain by induction that R π ∗ O B l ( l ) = R π ∗ O B l = O X for − c < l ≤ 0. In particular R π ∗ O B l (1 − c ) = O X . W e now use the filtration of [ 11 , Prop. 1.5]. Let us show that π ! ( O X ) ⊗ O B l ( c − 1) is in D 0 p erf ( B l ). By lo c. cit. it suffices to sho w that L ˜ ι ∗ ( π ! ( O X ) ⊗ O B l ( c − 1)) is in D 0 p erf ( E ). It follows from the sequence of isomorphis ms L˜ ι ∗ ( π ! ( O X ) ⊗ O B l ( c − 1)) ≃ L˜ ι ∗ π ! ( O X ) ⊗ O E ( c − 1) (iv) ≃ ˜ ι ! π ! ( O X ) ⊗ O E ( c )[1] ≃ ˜ π ! ι ! ( O X ) ⊗ O E ( c )[1] A.8 ≃ ˜ π ! ( ω ι ) ⊗ O E ( c )[ − c + 1 ] (v) ≃ O E . Since L := π ! ( O X ) ⊗O B l ( c − 1) is in D 0 p erf ( B l ), it is o f the form L π ∗ M for M = R π ∗ L (b y F act 1 in the pr o of of Prop. A.6 ) whic h w e compute by duality: R π ∗ ( π ! ( O X ) ⊗ O B l ( c − 1)) ≃ R π ∗ RHom( O B l , π ! ( O X ) ⊗ O B l ( c − 1)) ≃ ≃ R π ∗ RHom( O B l (1 − c ) , π ! ( O X )) ( † ) ≃ RHom(R π ∗ O B l (1 − c ) , O X ) ( ⋆ ) ≃ ≃ RHom( O X , O X ) ≃ O X where ( ⋆ ) is b y the co mputatio n a t the b eginning of the pro of, ( † ) is the dual- it y is omorphism a nd a ll o ther iso morphisms a re obtained as consequence s of the monoidal structure o n the D b coh inv olved (see [ 10 ] and [ 8 ] fo r details). Hence, π ! ( O X ) ≃ O B l (1 − c ) as a nnounced. This prov es (iii) and thus (vi).  Finally , we also use dualizing complexes in the con tex t of an A ∗ -bundle U → Y , i.e. a mo rphism that is lo cally of the form A n Y → Y (and is in par ticular flat). A.12. Lemma. L et α : U → Y b e an A ∗ -bund le. A s sume t hat Y admits a dualizing c omplex K Y . Then L α ∗ ( K Y ) = α ∗ ( K Y ) is a dualizing c omplex on U . Pr o of. This can b e chec ked lo cally , so we can ass ume α decomp oses as α = f ◦ u for an o pe n immersio n u : A n Y ֒ → P n Y follow ed by the s tr uctural pro jection f : P n Y → Y . Note that α , u and f are all flat. W e ha ve by Pr op ositions A.8 and A.9 (i) u ∗ f ! K Y [ n ] = u ∗ ( O ( − n − 1) ⊗ f ∗ K Y ) ≃ u ∗ f ∗ K Y = α ∗ K Y where the second equality comes fro m the tr iviality of O ( − n − 1 ) on A n Y . Now u ∗ f ! K Y [ n ] is dualizing be c a use pro per morphisms, op en immersions a nd shifting preserve dualizing complexes.  Ac knowledgmen ts : W e thank Ma rc Levine for precious discussions o n Theo- rem 1.3 and Example 1.5 , Jean F asel for a florilegium of relativ e bundles and Burt T otaro f or sev er al discussions on algebr aic geometr y . References 1. 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P aul Ba lm er, Dep a r tment of Ma thema tics, UCLA, Los Ang eles, CA 90095-15 55, USA E-mail addr ess : balmer@m ath.ucla. edu URL : http://www.mat h.ucla.e du/ ∼ balmer Baptiste Calm ` es, Laborar toire de Ma th ´ ema tiques de Lens, F acul t ´ e des Sciences J ean Perrin, Universit ´ e d’Ar tois, 62 307 Lens, France E-mail addr ess : calmes@m ath.jussi eu.fr URL : http://www.peo ple.math .jussieu.fr/~calmes