aa r X i v : . [ m a t h . A C ] J a n Convergence of two obstructions for projective modulesSatya Mandal ∗ University of Kansas, Lawrence, Kansas 66045, USA [email protected]
26 January 2020
Abstract:
Let X = Spec ( A ) denote a regular affine scheme, over a field k , with / ∈ k and dim X = d . Let P denote a projective A -module ofrank n ≥ . Let π ( L O ( P )) denote the (Nori) Homotopy Obstruction set,and g CH n ( X, Λ n P ) denote the Chow Witt group. In this article, we define anatural (set theoretic) map Θ P : π ( L O ( P )) −→ g CH n ( X, Λ n P ) In this article we establish a natural (set theoretic) map from (Nori) Homo-topy Obstructions to the Chow Witt group obstructions [BM], for projectivemodules to split off a free direct summand. We avoid repeating the extensivebackground comments given in [MM1, MM2], on this whole set of problemson obstructions for projective modules, sometimes referred to as the "ho-motopy program". To facilitate further discussions, in this introduction, let X = Spec ( A ) denote a regular affine scheme, over a field k , with / ∈ k and dim X = d . Let P denote a projective A -module of rank n . ∗ Partially supported by a General Research Grant (no 2301857) from U. of Kansas see [MM2, Mu, M2] and others for slightly varying versions of (1.1)).
Question 1.1 (Homotopy Question) . Suppose X = Spec ( A ) is a smoothaffine variety, with dim X = d . Let P be a projective A -module of rank n and f : P ։ I be a surjective homomorphism, onto an ideal I of A .Assume Y = V ( I ) is smooth with dim Y = d − n . Also suppose Z = V ( J ) ⊆ Spec ( A [ T ]) = X × A is a smooth subscheme, such that Z intersects X × transversally in Y × . Now, suppose that ϕ : P [ T ] ։ JJ is a surjective mapsuch that ϕ | T =0 = f ⊗ AI . Then the question is, whether there is a surjectivemap F : P [ T ] ։ J such that (i) F | T =0 = f and (ii) F | Z = ϕ . Assume n ≥ d + 3 . Euler class groups:
Further, Nori also gave a definition of Euler classgroups to house obstructions for splitting. Given invertible modules L ,Nori’s definition was expanded ([BS1, BS2, MY]) to define Euler class groups E n ( A, L ) for integers ≤ n ≤ d . This was defined ideal theoretically, usinggenerators and relations. When n = d , Euler class groups work fairly well.For projective A -modules P of rank d , a Euler class e ( P ) ∈ E d ( A, Λ d P ) isdefined. It was conjectured and proved [BS2, BS3] that e ( P ) = 0 ⇐⇒ P ∼ = Q ⊕ A. For projective A -modules P , with rank ( P ) = n < d , attempts to define Eulerclasses e ( P ) failed.On the other hand, note that there is a homotopy relation ingrained inthe Homotopy question (1.1). For this reason, a local P -orientation is definedto be a pair ( I, ω ) , where I ⊆ A is an ideal and ω : P ։ II is a surjectivemap. Let L O ( P ) denote the set of all local P -orientations. By substituting T = 0 , , one obtains two maps L O ( P ) L O ( P [ T ]) T =1 / / T =0 o o L O ( P ) ∼ on L O ( P ) , which is in deed anequivalence relation [MM2]. The homotopy obstruction set π ( L O ( P )) wasdefined to be the set of all equivalence classes. Further, an obstruction class ε H ( P ) ∈ π (cid:16)g L O ( P ) (cid:17) is defined. It was also established [MM2] that ε H ( P ) is neutral ⇐⇒ P ∼ = Q ⊕ A (1)when n ≥ d + 3 and A is essentially smooth over a perfect field. The Ho-motopy question (1.1) was settled affirmatively ( under the same hypothesis )by Bharwadekar-Keshari [BK], which was used to establish the above.These set of ideas of Nori ( along with the activities going on, e.g. [BS1,BS2, BS3, Mu, M2]) received added significance in 2000, with the intro-duction of Chow Witt groups, by Barge and Morel [BM, Mo], to housesuch a possible obstruction. The work of Jean Fasel [F1] added substan-tially to the literature. For integers ≤ n ≤ d , and line bundles L on X , a group g CH n ( X, L ) , to be called Chow Witt groups was introduced.For any projective A -module P , with rank ( P ) = n , an obstruction class ε CW ( P ) ∈ g CH n ( X, Λ n P ) was defined. As in the case of Euler class groups,when n = d , it was conjectured [BM] and proved that [Mo, Theorem 8.14] ε CW ( P ) = 0 ⇐⇒ P ∼ = Q ⊕ A By that time, it started appearing not so promising, that Euler class groups E n ( A, Λ n P ) may be able to house such obstructions, for splitting. The pos-sibility to use the homotopy relations ingrained in (1.1) to construct a house π ( L O ( P )) for obstructions ε H ( P ) was considered only recently [MM2].However, π ( L O ( P )) is an invariant of P itself. On the other hand, ChowWitt groups g CH n ( X, Λ n P ) were invariants of X (and of the determinant).Further, they were fully functorial, analogous to Chow groups CH n ( X ) .While obstruction class ε CW ( P ) ∈ g CH n ( X, Λ n P ) was defined [BM], for all n := rank ( P ) , attempts to define obstruction classes in E n ( A, Λ n P ) failed,for n < d . Other than continued efforts to answer some of the questionsor conjectures, alluded above, newer perspective evolved that these two ap-proaches aught to have some relationship. This is precisely what we respondto, in this article, by establishing a natural map (set theoretic) Θ P : π ( L O ( P )) −→ g CH n ( X, Λ n P ) with Θ P ( ε H ( P )) = ε CW ( P ) .
3e establish this for all projective modules P , with ≤ n = rank ( P ) ≤ d .While it was established in [MM2] that π ( L O ) ( P ) has a additive structure,when n ≥ d + 2 , there is no such well defined structures outside this rangeof n . Therefore, the map Θ P , defined above, would be a set theoretic maponly, in general, and respects additivity when n ≥ d + 2 . Further, recall thatthe obstruction class ε H ( P ) ∈ π ( L O ) ( P ) is defined, for all n := rank ( P ) .The obstruction class ε H ( P ) detects splitting properties of P , under theconditions stated above (1). With this result in mind, the following naturalquestion emerges. Question 1.2. [Agreement Question]
Whether the map Θ P is injective?If and when the answer to (1.2) is affirmative, Chow Witt obstructions ε CW ( P ) would detect splitting, under the same hypotheses above.This author took the liberty to refer to this whole set of problems, as the"Homotopy Program" (e. g. [MS]). More precise outline of the program wasgiven in [MM2]. The possibility of the existence of a map Θ P , as above, wasmentioned as a part of the program ([MM2, Part 2 , pp. 173]), which weaccomplish in this article.We comment on the organization of the article. In section 2, we providebackground on Homotopy obstruction, mainly from [MM2, MM1]. In section3, we associate a symmetric form Φ( I, ω ) to certain representatives ( I, ω ) ofthe elements [( I, ω )] ∈ π ( L O ( P )) . In section 4, we provide background onChow Witt groups. In section 5, we establish the map Θ P . Acknowledgement.
This author is thankful to Marco Schlichting for hiscontinued academic support and valuable discussions.
Throughout this article A will denote a noetherian commutative ring, with dim A = d , and A [ T ] will denote the polynomial ring in one variable T . Wewill assume / ∈ A . For an A -module M , denote M [ T ] := M ⊗ A [ T ] .Likewise, for a homomorphism f : M −→ N of A -modules, f [ T ] := f ⊗ A [ T ] .Further, P with denote a projective A -module with rank ( P ) = n , and ≤ n ≤ d . Our main results would assume A is a regular ring, containing a4eld k , with / ∈ k . For a such a projective A -module P , as above, the setof equivalence classes of homotopy obstructions π ( L O ( P )) was defined in[MM2]. We recall some of the the essential elements of the definition of, andalternative descriptions of π ( L O ( P )) from [MM1]. Definition 2.1.
Let A be a noetherian commutative ring, with dim A = d and P be a projective A -module with rank ( P ) = n . By a local P -orientation , we mean a pair ( I, ω ) where I is an ideal of A and ω : P ։ II is a surjective homomorphism. We will use the same notation ω for the map PIP ։ II , induced by ω . A local local P -orientation will simply be referredto as a local orientation , when P is understood. Denote L O ( P ) = { ( I, ω ) : (
I, ω ) is a local P orientation }L O n ( P ) = { ( I, ω ) ∈ L O ( P ) : height ( I ) = n } g L O ( P ) = { ( I, ω ) ∈ L O ( P ) : height ( I ) ≥ n } Note g L O ( P ) = L O n ( P ) ∪ { ( A, } (2)The (Nori) Homotopy obstruction set π ( L O ( P )) was defined, by thepush forward diagram: L O ( P [ T ]) T =0 / / T =1 (cid:15) (cid:15) L O ( P ) (cid:15) (cid:15) L O ( P ) / / π ( L O ( P )) in Sets. (3)While one would like to define π (cid:16)g L O ( P ) (cid:17) similarly, note that substitution T = 0 , would not yield any map from g L O ( P [ T ]) to g L O ( P ) . However, notethat the definition of π ( L O ( P )) by push forward diagram (3), is only analternate way of saying the following: For ( I , ω ) , ( I , ω ) ∈ L O ( P ) , we write ( I , ω ) ∼ ( I , ω ) , if there is an ( I, ω ) ∈ L O ( P [ T ]) such that ( I (0) , ω (0)) =( I , ω ) and ( I (1) , ω (1)) = ( I , ω ) . In this case, we say ( I , ω ) is homotopicto ( I , ω ) . Now ∼ generates a chain equivalence relation on L O ( P ) , which wecall the chain homotopy relation. The above definition (3) means, π ( L O ( P )) is the set of all equivalence classes in L O ( P ) .5he restriction of the relation ∼ on g L O ( P ) ⊆ L O ( P ) , generates a chainhomotopy relation on g L O ( P ) . Define π (cid:16)g L O ( P ) (cid:17) to be the set of all equiv-alence classes in g L O ( P ) . It follows, that there is natural map ϕ : π (cid:16)g L O ( P ) (cid:17) −→ π ( L O ( P )) The following is from [MM2].
Proposition 2.2.
Let A and P be as in (2.1). Then, The map ϕ is surjective.Assume further that A is a regular ring containing a field k , with / ∈ k .Then ∼ is an equivalence relation on g L O ( P ) . Moreover, ϕ is a bijection. Remark 2.3.
With notations as in (2.1) the following are some useful ob-servations.1. Suppose ( I , ω ) , ( I , ω ) ∈ g L O ( P ) and ( I , ω ) ∼ ( I , ω ) . By defintionthere is homotopy H ( T ) = ( I, ω ) ∈ L O ( P [ T ]) such that H (0) = ( I , ω ) and H (1) = ( I , ω ) . By moving Lemma arguments, similar to [MM2,Lemma 4.5], we can assume that H ( T ) ∈ g L O ( P [ T ]) .2. If A is Cohen Macaulay, then g L O ( P ) is in bijection with the set (cid:26) ( I, ω ) ∈ L O n ( P ) : ω : PIP ∼ −→ II is an isomorphism (cid:27) ∪ { ( A, } Moreover, for ( I, ω ) ∈ L O n ( P ) , I is a local complete intersection ideal. The essence of the arguments in the this section, can be traced back thefollowing theorem of Altman and Kleiman [AK, Theorem 4.5].6 heorem 3.1.
Suppose A is a commutative noetherian ring and I is a locallycomplete intersection ideal, with height ( I ) = n . Suppose L is an invertible A -module. Then, there is a natural isomorphism Ext n (cid:18) AI , L (cid:19) ∼ −→ Hom (cid:18) Λ n II , L I L (cid:19) (4)Now assume A is Cohen Macaulay ring. With the notations in abovesection (2), let ( I, ω ) ∈ L O n ( P ) , with rank ( P ) = n . Then, ω induces anisomorphism det( ω ) : Λ n PIP ∼ −→ Λ n II . With L = Λ n P , (4) induces anisomorphism Φ( I, ω ) , as in the following commutative diagram: Ext n (cid:0) AI , L (cid:1) / / Hom (cid:0) Λ n II , L I L (cid:1) λ det( ω ) λ (cid:15) (cid:15) AI Φ( I,ω ) h h ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ (5)In the rest of this section, we elaborate this, to associate ( I, ω ) Φ( I, ω ) , asymmetric form. C M n ( X ) Suppose X = Spec ( A ) is Cohen Macaulay scheme, with dim X = d . Forintegers ≤ n ≤ d , in [M1], the subcategory C M n ( X ) ⊆ Coh ( X ) , wasdefined to be the full subcategory of objects C M n ( X ) = {F ∈ Coh ( X ) : grade ( F ) = proj dim( F ) = n } where proj dim( F ) denotes the locally free dimension of F . Given an invert-ible sheaf L , on X , the association F 7→ F ∨ := E xt n ( F , L ) is a duality in C M n ( X ) . This endows C M n ( X ) with a structure of an exact category withduality, to be denoted by C M n ( X, L ) . Given ( I, ω ) ∈ g L O ( P ) , we constructa symmetric form Φ( I, ω ) in C M n ( X, L ) , for L ∼ = Λ n P . Definition 3.2.
Let X = Spec ( A ) be a Cohen Macaulay scheme with dim X = d . Let P be a projective A -module with rank ( P ) = n . Fix anisomorphism χ : Λ n P ∼ −→ L , for some invertible sheaf L on X . ( For our urpose, L = Λ n P ). Let ( I, ω ) ∈ L O n ( P ) . We have AI ∈ C M n ( X ) and ω : PIP ∼ −→ II is an isomorphism. Consider a surjective lift f of ω , as in thediagram P f / / / / (cid:15) (cid:15) (cid:15) (cid:15) I ∩ J (cid:15) (cid:15) (cid:15) (cid:15) PIP ω ∼ / / II with I + J = A, height ( J ) ≥ n Write P ∗ = Hom ( P, L ) . The map f has a Koszul complex extension andthere is map of complexes, as follows / / Λ n P ϕ n := χ (cid:15) (cid:15) d n / / · · · / / Λ P d := f / / ϕ (cid:15) (cid:15) A d / / ϕ (cid:15) (cid:15) AI ∩ J / / ϕ (cid:15) (cid:15) ✤✤✤ / / L / / · · · / / (Λ n − P ) ∗ d ∗ n / / (Λ n P ) ∗ / / Ext n (cid:0) AI ∩ J , L (cid:1) / / (6)where Ext n (cid:0) AI ∩ J , L (cid:1) := co ker( d ∗ k ) . Here1. Up to a sign, the maps ϕ r : Λ r P −→ (Λ n − r P ) ∗ are obtained by theperfect duality ( Λ r P ⊗ Λ n − r P −→ L sending [ H , Ex. . b ) , pp. p ∧ · · · ∧ p r ) ⊗ ( p r +1 ∧ · · · ∧ p n ) χ ( p ∧ · · · ∧ p r ∧ p r +1 ∧ · · · ∧ p n )
2. Both the sequences are exact, and ϕ is the induced map.3. Since all the maps ϕ r are isomorphisms, so is ϕ . Define ( Φ( I, ω ) := ϕ ( f ) ⊗ AI : AI ∼ −→ (cid:0) AI (cid:1) ∨ Also , for ( A, ∈ g L O ( P ) Φ( A,
0) := 0 (7)Clearly, Φ( I, ω ) is a symmetric form in C M n ( X ) . It appears that Φ( I, ω ) depends on the lift f , while such choices lead to naturally isometric forms.However, in our case, P is given and for each ( I, ω ) such choices of the lift f would not create any set theoretic issue.8 roposition 3.3. Use the notations as in Definition 3.2. Then, Φ( I, ω ) , asdefined in (7), is a well defined symmetric form in C M n ( X, L ) . Proof.
As explained above.
Before we proceed, we recall two different formulations of the Gersten Wittcomplex from [BW]. Let X be a regular scheme over a field k , with / ∈ k with dim X = d . Let L be an invertible sheaf on X and ≤ n ≤ d be an integer. For x ∈ X we denote X x := Spec ( O X,x ) , and X ( n ) := { x ∈ X : co dim( x ) = n } . The following on Gersten Witt complexes is from[BW], developed based on the work of Paul Balmer [B] on Witt groups oftriangulated categories. The Gersten Witt complex, on X , and with dualityinduced by P Hom X ( P, L ) , has the following two isomorphic descrip-tions: / / L x ∈ X (0) W (cid:16) D bfl ( V ( X x , L x ) (cid:17) / / ≀ (cid:15) (cid:15) · · · / / L x ∈ X ( n ) W n (cid:16) D bfl ( V ( X x , L x ) (cid:17) ∂nW / / ≀ (cid:15) (cid:15) · · · / / L x ∈ X (0) W (cid:16) C M ( X x , L x ) (cid:17) / / · · · / / L x ∈ X ( n ) W ( C M n ( X x , L x )) dnW / / · · · (8) terminating at the L x ∈ X ( d ) -term. Here C M n ( X x , L x ) turns out to be thecategory of finite length O X,x -modules, with duality M Ext n ( M, L x ) .Subsequently, we would also use the notation M od fl ( O X,x ) := C M n ( X x , L x ) .As in the diagram (8), the differentials in the first and the second lines would,respectively, be denoted by ∂ nW and d nW . The complex in the second line wouldbe denoted by C • ( X, L , W ) . Also, let I ( X x , L x ) ⊆ W ( C M n ( X x , L x )) de-note the fundamental ideal, I r ( X x , L x ) denote its exponents. For r ≤ ,denote I r ( X x , L x ) = W ( C M n ( X x , L x )) .Thus far, the work of Jean Fasel [F1] provides the most comprehensivefoundation available on Chow Witt groups. To recall the definition of ChowWitt groups, the following diagram would be helpful. The Chow Witt group9 CH n ( X, L ) of co dimension n cycles, is defined by the following diagram: G n ( X ) (cid:15) (cid:15) ζ / / d G ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ L x ∈ X ( n − I (cid:0) O X,x (cid:1) (cid:15) (cid:15) d I ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ G n ( X ) (cid:15) (cid:15) / / d G ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ L x ∈ X ( n ) I (cid:0) O X,x (cid:1) (cid:15) (cid:15) d I ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ G − n ( X ) L x ∈ X ( n +1) I − (cid:0) O X,x (cid:1)L x ∈ X ( n − u ( k ( x )) / / d K ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ L x ∈ X ( n − u ( k ( x )) u ( k ( x ))2 ∼ / / ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ L x ∈ X ( n − I (cid:16) O X,x (cid:17) I (cid:16) O X,x (cid:17) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ L x ∈ X ( n ) Z [ x ] / / ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ L x ∈ X ( n ) Z Z ∼ / / L x ∈ X ( n ) Z Z (9) We ignored to include the second coordinates, associated to L , which inducethe duality. In left lower diagonal, u ( k ( x )) = K ( k ( x )) denotes the group ofunits of the residue field k ( x ) , and Z [ x ] = K ( k ( x )) is the K -group of k ( x ) .The upper left diagonal of the diagram is obtained by cartesian product. TheChow Witt groups is defined by g CH n ( X, L ) := ker ( d G ) Image ( d G ) Remark 4.1.
The following are some useful information on the diagram (9):1. For all these complexes, diagonally, degree zero corresponds to L x ∈ X ( n ) .The upper right diagonal complex is denoted by C • n ( X, L , I ∗ ) . The dif-ferentials d rI are the restrictions of the differentials of the Gersten Wittcomplex (8). The lower left diagonal complex C • n ( X, K M ) is known asMilnor K -theory sequence [Mi].Since the Chow Witt group g CH n ( X, L ) is the homology, at degreezero, we are mainly concerned with deg = 1 , , − . In these cases, allthe complexes are fairly elementary.10. The upper left diagonal complex, denoted by G • n ( X, L ) is obtained bycartesian product.3. Two complexes G • n ( X, L ) , and C • n ( X, I ∗ ) extends in either direction,by taking L x ∈ X ( r ) with r = 0 , . . . , n, . . . , d . The lower left complex C • n ( X, K M ) terminates, as shown.4. There is a natural isomorphism G n ( X, L ) ∼ = L x ∈ X ( n ) GW ( X x , L x ) ,where GW denotes the Grothedieck Witt group.5. There are other descriptions of the complex G • n ( X, L ) , or the ChowWitt groups, as follows:(a) Given a field F , with / ∈ F , analogous of Milnor K -groups K r ( F ) , there are groups K MWr ( F ) , known as Milnor Witt groups[Mo] . Using the Milnor Witt groups K MWr ( k ( x )) , one can con-struct Gersten complexes C • n ( X, L , K MW ) , exactly as in the lowerright corner of the diagram (9). One can prove that G • n ( X, L ) ∼ = C • n ( X, K MW , L ) . See [Mo], [F2, Theorem 1.2, pp. 6] for furtherdetails.(b) Chow-Witt group g CH n ( X, L ) can also be defined as the homologyof the Gersten complexes of the Grothendieck Witt groups [FS, S,M1].6. Consider a regular local ring ( R, m ) , with dim R = d and / ∈ R . Let k = R/ m . Given these, we can associate a number of Witt groups, asfollows:(a) Most fundamental is the Witt group W ( k ) of quadratic forms [L].(b) The Witt group W ( V ( k ) , ∗ ) of the category V ( k ) , of finite dimen-sional k -vector spaces, with duality V Hom ( V, k ) .(c) The Witt group W (cid:0) M od fl ( R ) , Ext d (cid:1) of the category M od fl ( R ) ,of finite length R -modules, with duality M Ext d ( M, A ) .11d) The Witt group W (cid:0) D b (cid:0) M od fl ( R ) (cid:1) , Ext d (cid:1) of the bounded de-rived category D b (cid:0) M od fl ( R ) (cid:1) , of the category M od fl ( R ) , of finitelength R -modules, with duality induced by M Ext d ( M, A ) .This was mentioned above.(e) The Witt group W d (cid:0) D dfl ( P ( R )) , Hom ( − , A ) (cid:1) of the boundedderived category D dfl ( P ( R )) , of complexes finite rank free R -modules, with finite length homology, and duality induced by P Hom ( P, A ) . ( We would not consider skew dualities. )We tried to be consistent with the notations in [BW]. Readers are re-ferred to [BW] for further details on derived Witt groups. It turnsout that all these Witt groups are (naturally) isomorphic [B, BW,QSS, L]. The isomorphism of the two descriptions of Gersten Wittcomplexes (8), mentioned above, is a consequence of this fact. How-ever, some of the isomorphisms would depend on some choices to bemade. While the first one W ( k ) would be elementary [L], the last one W d (cid:0) D bfl ( P ( R )) , Hom ( − , A ) (cid:1) may also be nicer to work with. Thisis because the duality P Hom ( P, A ) is much more tangible than,dualities associated to "Ext".We restate [F1, Lemma 10.3.4], as follows. Lemma 4.2.
Let X = Spec ( A ) be a regular scheme, / ∈ A , and L be aninvertible sheaf on X . Consider the duality induced by P Hom ( P, L ) onthe bounded derived category D b ( P ( X )) of the category P ( X ) of projective A -modules of finite rank. For x := ℘ ∈ X denote X x := Spec ( A ℘ ) . Let ∂ nW : M x ∈ X ( n ) W n (cid:0) D bfl ( X x , L x ) (cid:1) −→ M x ∈ X ( n +1) W n +1 (cid:0) D bfl ( X y , L y ) (cid:1) (10)denote the differential of the Gersten Witt comples (8) ( which is a map be-tween Witt groups of two quotient categories ). Let f , . . . , f n ∈ A be a regularsequence and K ( f , . . . , f n ) be the Koszul complex. Let ϕ : K ( f , . . . , f n ) ∼ −→ ( f , . . . , f n ) ∗ be the corresponding symmetric form. Let t ∈ A be a nonzero divisor (or isomorphism) on A ( f ,...,f n ) . Then, ∂ nW : [( K ( f , . . . , f n ) , tϕ )] = [( K ( f , . . . , f n ; t ) , ϕ ∧ t )] (11)where ϕ ∧ t : K ( f , . . . , f n ; t ) ∼ −→ K ( f , . . . , f n ; t ) ∗ denotes the symmetricform. Proof. . We note that the cone of the multiplication map t : K ( f , . . . , f n ) −→ K ( f , . . . , f n ) is the Koszul complex K ( f , . . . , f n ; t ) . Now, the lemma followsfrom the following Lemma 4.3.The following is a more general formulation of Lemma 4.2. Lemma 4.3.
Let X = Spec ( A ) be a regular scheme, / ∈ A , and dim X = d . Let L be an invertible sheaf on X . Consider duality induced by P Hom ( P, L ) on D b ( P ( X )) , and on the other associated categories. Fix an in-teger n . Let ∂ nW and d nW denote the differentials in the Gersten Witt complex(8). Let D n ( P ( X )) ⊆ D b ( P ( X )) denote the subcategory of objects Q • , with co dim ( H i ( Q • )) ≥ n, ∀ i . In D n ( P ( X )) , consider a complex P • :=0 / / P n / / P n − / / · · · P / / / / P / / H i ( P • ) = 0 ∀ i = 0 , and M := H ( P • ) Then M ∈ C M n ( X ) . Let ϕ : P • ∼ −→ P ∗• be a symmetric form . Then ,ϕ induces a symmetric form ϕ : M ∼ −→ M ∨ in C M n ( X, L )where M ∨ := Ext n ( M, L )Let t ∈ A is a non zero divisor (or isomorphism) on M The map P • tϕ (cid:15) (cid:15) / / P n / / tϕ n (cid:15) (cid:15) P n − / / tϕ n − (cid:15) (cid:15) · · · P / / / / tϕ (cid:15) (cid:15) P / / tϕ (cid:15) (cid:15) P ∗• / / P ∗ / / P ∗ / / · · · P ∗ n − / / / / P ∗ n / /
13s a symmetric morphism, which is also a lift of tϕ . By the work of Balmer[B, Definition 5.16], [BW, Thm 2.1, 3.1], we have ∂ nW [( P • , tϕ )] = [( Cone ( P • , tϕ ) , ψ )] where ψ : Cone ( P • , tϕ ) −→ Cone ( P • , tϕ ) ∗ denotes the induced duality map.By ( T R ) axiom (see [B]), we have a map of the triangles: P • a / / P • / / ϕ (cid:15) (cid:15) Cone ( t ) / / (cid:15) (cid:15) T P • P • tϕ / / P ∗• / / Cone ( tϕ ) / / T P • Since ϕ is a quasi isomorphism, Cone ( t ) ∼ = Cone ( tϕ ) . Therefore, ∂ nW [( P • , tϕ )] = [( Cone ( P • , tϕ ) , ψ )] = [(( Cone ( P • , t ) , ϕ ∧ t )] (12)where we use the notation ϕ ∧ t : ( Cone ( P • , t ) −→ ( Cone ( P • , t ) ∗ to denotethe duality induced by ϕ and t . ( We remark that cones are very similarto Koszul complexes. For this reason the degree r term of the cone can bewritten as ( Cone ( P • , t )) r = P r ⊕ P r − = P r ⊕ P r − ∧ Ae . )Further note, since t is a non zero divisor on M = H ( P • ) , the sequence ( / / H ( P • ) t / / H ( P • ) / / H ( Cone ( P • , t )) / / , and H i ( Cone ( P • , t )) = 0 ∀ i = 0 We obtain the commutative diagram / / M t / / ϕ (cid:15) (cid:15) M / / ϕ (cid:15) (cid:15) MtM / / ψ (cid:15) (cid:15) / / Ext n ( M, L ) t / / Ext n ( M, L ) / / Ext n +1 (cid:0) MtM , L (cid:1) / / where ψ is the induced isometry in C M n +1 ( X, L ) . Consequently, d nW [( M, tϕ )] = (cid:20)(cid:18)
MtM , ψ (cid:19)(cid:21) (13)
Proof.
As described! 14
The convergence of two obstructions
In this final section, we prove our main result by establishing a natural map Θ P : π ( L O ( P )) −→ g CH n ( X ) . Definition 5.1.
Let X = Spec ( A ) be a Cohen-Macaulay affine scheme,with dim X = d and / ∈ A . Let P be a projective A -module with rank ( P ) = n and χ : Λ n P ∼ −→ L be an isomorphism. ( Without lossof generality L = Λ n P ). Refer to the diagram (9) of definition of ChowWitt groups g CH n ( X, L ) . For ( I, ω ) ∈ g L O ( P ) , consider the symmetricform Φ( I, ω ) ∈ C M n ( X, L ) , as in Proposition 3.3. This defines an ele-ment Φ( I, ω ) I ∈ C n ( X, L , I ∗ ) , and an element in Φ( I, ω ) K ∈ C n (cid:0) X, K M (cid:1) ,which patch to define an element in Φ( I, ω ) G ∈ G n ( X, L ) . It follows that d I (Φ( I, ω ) I ) = 0 (see [BW]), and hence d G (Φ( I, ω ) G ) = 0 . Therefore, Φ( I, ω ) G represents an element in g CH n ( X, L ) . Define, a set theoretic map Ω : g L O ( P ) −→ g CH n ( X, L ) by Ω( I, ω ) := Φ(
I, ω ) G ∈ g CH n ( X, L ) (14)The following commutative diagram would be helpful, g L O ( P ) Φ / / Ω , , C M n ( X, L ) ι n / / Z ( G n ( X, L )) q (cid:15) (cid:15) g CH n ( X, L ) where Z ( − ) = ker( d G ) . (15)We remark that there is a direct GW -way to look at the same. Readersare referred to [M1, Remark 4.14(1)], and the GW analogues of the diagramcorresponding to [M1, Remark 4.5(1)]. Theorem 5.2.
Suppose A is a regular ring, containing a field k with / ∈ k ,and dim A = d ≥ . Let P be a projective A -module of rank n , and L = Λ n P .Then, the map Ω in (14) factors through a set theoretic map ̺ , as in the15ommutative diagram: g L O ( P ) Ω & & ▼▼▼▼▼▼▼▼▼▼ β / / π ( L O ( P )) Θ P (cid:15) (cid:15) ✤✤✤ g CH n ( X, L ) (16) Proof.
By (2.2), π ( L O ( P )) = π (cid:16)g L O ( P ) (cid:17) . Let ( I , ω ) , ( I , ω ) ∈ g L O ( P ) be such that β (( I , ω )) = β (( I , ω )) ∈ π (cid:16)g L O ( P ) (cid:17) . By Remark 2.3,there is ( I, ω ) ∈ L O n ( P [ T ]) such that ( I, ω ) | T =0 = ( I , ω ) and ( I, ω ) | T =1 =( I , ω ) . The projection map p : X × A −→ X induces a pull back map p ∗ : g CH k ( X, L ) ∼ −→ g CH k ( X × A , p ∗ L ) , which is an isomorphism [F1, Cor11.3.2], [F2, Theorem 2.15], [Mo]. Consider the commutative diagram g CH n ( X, L ) p ∗ ≀ (cid:15) (cid:15) g L O ( P ) Ω Φ / / (cid:15) (cid:15) C M n ( X, L ) qι n ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ p ∗ (cid:15) (cid:15) g L O ( P [ T ]) Ω - - Φ / / C M n ( X × A , p ∗ L ) qι n ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ g CH n ( X × A , p ∗ L ) (17)We are required to prove that Ω( I , ω ) = Ω( I , ω ) . Since p ∗ (the last verticalarrow) is an isomorphism, it is enough to prove that p ∗ (Ω( I i , ω i ))) = Ω( I, ω ) for i = 0 , (18)Since T and T − are interchangeable, it would suffice to establish the case i = 0 , which is done subsequently in Lemma 5.3. The proof is complete. T = 0 We establish the equation (18), in this subsection.16 emma 5.3.
Let X = Spec ( A ) be a regular affine scheme, over Spec ( k ) ,where k is a field with / ∈ k and dim X = d . Let P be a projective A -module with rank ( P ) = n , with ≤ n ≤ d . Let ( I, ω ) ∈ L O n ( P [ T ]) be suchthat ( I , ω ) ∈ g L O ( P ) . Let p : X × A −→ X denote the projection map.Then, p ∗ (Ω( I , ω ))) = Ω( I, ω ) . Proof.
We compute d I ( T · Φ (
I, ω ) I ) ∈ C n +1 ( X, L , I ∗ ) . Consider a resolu-tion of AI and its dual: / / P n / / ϕ n (cid:15) (cid:15) · · · / / P / / ϕ (cid:15) (cid:15) A / / ϕ (cid:15) (cid:15) AI / / Φ( I,ω ) (cid:15) (cid:15) / / P ∗ / / · · · / / P ∗ n − / / P ∗ n / / Ext n (cid:0) AI , L (cid:1) / / So, ϕ is a symmetric form in D n ( P ( X )) , induced by Φ( I, ω ) . Note T is anonzero divisor on AI . By Lemma 4.3, we have ( see diagram (8) for notations ) ∂ nW ( P • , T ϕ ) = [(( P • , Cone ( T )) , ϕ ∧ T )] . Therefore , d nW ( T · Φ (
I, ω ) I ) = h(cid:16) A [ T ]( I,T ) , ψ (cid:17)i d I (( h T i − h i ) · Φ (
I, ω ) I ) = h(cid:16) A [ T ]( I,T ) , ψ (cid:17)i ∈ C n +1 ( X, L , I ∗ ) (19)where ψ : ( A [ T ]( I,T ) ∼ −→ Ext n +1 (cid:16) A [ T ]( I,T ) , L (cid:17) is the isometry in C M n +1 ( X, L ) in-duced by ϕ ∧ T , ( then we take image in the quotient category, which is pointwise localizations ). Note, ψ is independent of the resolution P • chosen above.Likewise, d nW ( T · p ∗ Φ ( I (0) , ω ) I ) = h(cid:16) A [ T ]( I (0) ,T ) , ψ (cid:17)i d I (( h T i − h i ) · p ∗ Φ ( I (0) , ω ) I ) = h(cid:16) A [ T ]( I (0) ,T ) , ψ (cid:17)i ∈ C n +1 ( X, L , I ∗ ) (20)where ψ is like wise induced, but independent of choice of resolution. Claim : d nW ( T · Φ (
I, ω ) I ) = d nW ( T · p ∗ Φ ( I (0) , ω ) I ) ∈ C n +1 ( X, L , I ∗ ) (21)Let ℘ , . . . , ℘ r be the minimal primes over I (0) . Then, both sides of thisequation are supported on { ( ℘ , T ) , . . . , ( ℘ r , T ) } which turns out to be the set17f all minimal primes over ( I (0) , T ) . We semi localize. Write S = A \ ( ∪ ri =1 ) , B = S − A and F = S − P .Fix a lift ϕ : P [ T ] ։ I ∩ J , of ω , where I + J = A [ T ] , heght ( J ) ≥ n . Itfollows J (0) ℘ i and J ( ℘ i , T ) for all i . Fix a basis e , . . . , e k of F , whichinduces a basis of F [ T ] . Write f i ( T ) := ϕ ( e i ) . So, (cid:26) ( S − I, T ) = ( f ( T ) , . . . , f n ( T ); T )= ( S − I (0) , T ) = ( f (0) , . . . , f n (0); T ) are regular sequences in B [ T ] Since ( f ( T ) , . . . , f k ( T ); T ) and ( f (0) , . . . , f k (0); T ) differ by an elementarymatrix, the Koszul complexes K ( f ( T ) , . . . , f k ( T ); T ) ∼ = K ( f (0) , . . . , f k (0); T ) are isometric. To see this, write f ( T ) f ( T ) · · · f n ( T ) T = · · · ∗ · · · ∗ · · · ∗ · · · ∗ · · · f (0) f (0) · · · f n (0) T = ∆ f (0) f (0) · · · f n (0) T . So , det ∆ = 1With M = B [ T ]( f ( T ) ,f ( T ) ,...,f n ( T ) ,T ) = B [ T ]( f (0) ,f (0) ,...,f n (0) ,T ) M ∨ = Ext n +1 ( M, L ⊗ B [ T ])and e F = F [ T ] ⊕ B [ T ] e we have the composition of the maps of complexes: / / Λ n +1 e F / / Λ n e F / / Λ n ∆ (cid:15) (cid:15) · · · / / Λ e F / / (cid:15) (cid:15) e F ( f i (0); T ) / / ∆ (cid:15) (cid:15) B [ T ] / / M / / / / Λ n +1 e F / / can (cid:15) (cid:15) Λ n e F / / can (cid:15) (cid:15) · · · / / Λ e F / / can (cid:15) (cid:15) e F ( f i ( T ); T ) / / can (cid:15) (cid:15) B [ T ] / / can =1 (cid:15) (cid:15) M / / ψ (cid:15) (cid:15) / / B [ T ] ∗ / / e F ∗ / / ∆ ∗ (cid:15) (cid:15) · · · / / Λ n − F ∗ / / (cid:15) (cid:15) Λ n e F ∗ / / Λ n ∆ ∗ (cid:15) (cid:15) Λ n +1 e F ∗ / / M ∨ / / / / B [ T ] ∗ / / e F ∗ / / · · · / / Λ n − e F ∗ / / Λ n e F ∗ / / Λ n +1 e F ∗ / / M ∨ / / The composition agrees with the symmetric form of K ( f ( T ) , . . . , f n ( T ) , T ) ,at degree zero. This, in fact, establishes that ψ = ψ and hence the symmetricforms in claim (21) are isometries. ( For clarity, we point out two things: (1) wo forms ψ and ψ can differ by unit modulo the ideal. (2) The composition (Λ • ∆) ∗ ( K ( f ( T ) , . . . , f n ( T ) , T ))(∆) • ) agrees with ( K ( f (0) , . . . , f n ( o ) , T )) inthe derived category, because they agree in the homotopy category. )This completes the proof of the Claim (21). So, it follows (cid:26) d I (( h T i − h i ) · Φ (
I, ω ) I ) = d I (( h T i − h i ) · p ∗ Φ ( I (0) , ω ) I ) d K ( T · Φ (
I, ω ) K ) = d K ( T · p ∗ Φ ( I (0) , ω ) K ) where the first identity follows from Claim (21), and the second identityfollows by similar, and more basic, arguments on the K -theory complex.Patching this two identities, it follows d G ( T · Φ (
I, ω ) G ) = d G ( T · p ∗ Φ ( I (0) , ω ) G ) ∈ G n +1 ( X, L ) (22)Consider the commutative diagram / / G n ( A, L ) p ∗ / / (cid:127) _ (cid:15) (cid:15) G n ( A [ T ] , L [ T ]) · T (cid:15) (cid:15) G n +1 ( A [ T ] , L [ T ]) G n +1 ( A [ T ] , L [ T ]) d G o o In deed, (22) establishes that the retraction of Φ( I, ω ) G is equal to Φ ( I (0) , ω ) G . Since p ∗ : g CH n ( X, L ) ∼ −→ g CH n (cid:0) X × A , p ∗ L (cid:1) is an isomorphism ., pre-image of Ω (
I, ω ) is Ω ( I (0) , ω (0)) . Therefore, p ∗ (Ω( I , ω ))) = Ω( I, ω ) .The proof is complete. References [AK] Altman, Allen; Kleiman, Steven Introduction to Grothendieck dual-ity theory. Lecture Notes in Mathematics, Vol. 146
Springer-Verlag,Berlin-New York
Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 1,127?152.[BM] Barge, Jean; Morel, Fabien Groupe de Chow des cycles orientés etclasse d’Euler des fibrés vectoriels. (French) [The Chow group of ori-ented cycles and the Euler class of vector bundles]
C. R. Acad. Sci.Paris Sér. I Math.
330 (2000), no. 4, 287-290.[BK] Bhatwadekar, S. M.; Keshari, Manoj Kumar A question of Nori: pro-jective generation of ideals. K -Theory 28 (2003), no. 4, 329-351.[BS1] Bhatwadekar, S. M.; Sridharan, Raja On Euler classes and stably freeprojective modules. Algebra, arithmetic and geometry,
Part I, II (Mum-bai, 2000), 139-158, Tata Inst. Fund. Res. Stud. Math., 16,
Tata Inst.Fund. Res., Bombay,
Compositio Math . 122 (2000), no. 2, 183-222.[BS3] Bhatwadekar, S. M.; Sridharan, Raja Projective generation of curvesin polynomial extensions of an affine domain and a question of Nori.
Invent. Math.
133 (1998), no. 1, 161-192.[F1] Fasel, Jean Groupes de Chow-Witt. (French) [Chow-Witt groups]M’em. Soc. Math. Fr. (N.S.) No. 113 (2008)[F2] Fasel, Jean Lectures on Chow-Witt groups, arXiv:1911.08152[FS] Fasel, J.; Srinivas, V. Chow-Witt groups and Grothendieck-Wittgroups of regular schemes. Adv. Math. 221 (2009), no. 1, 302-329.[H] Hartshorne, Robin Algebraic geometry. Graduate Texts in Mathemat-ics, No. 52.
Springer-Verlag, New York-Heidelberg , 1977. xvi+496 pp.[G] Gille, Stefan A graded Gersten-Witt complex for schemes with a dual-izing complex and the Chow group.
J. Pure Appl. Algebra
208 (2007),no. 2, 391-419.[L] Lam, T. Y. Introduction to quadratic forms over fields. Graduate Stud-ies in Mathematics, 67.
American Mathematical Society , Providence,RI, 2005. 20Mo] Morel, Fabien A -algebraic topology over a field. Lecture Notes inMathematics, 2052. Springer, Heidelberg, 2012.[MM1] Satya Mandal and Bibekananda Mishra, Some perspective on Homo-topy obstructions, arXiv:1811.04510[MM2] Mandal, Satya; Mishra, Bibekananda The monoid structure on ho-motopy obstructions. J. Algebra
540 (2019), 168-205.[MM3] Mandal, Satya; Mishra, Bibekananda The homotopy obstructions incomplete intersections.
J. Ramanujan Math. Soc.
34 (2019), no. 1, 109-132.[M1] Mandal, Satya Witt, GW, K -theory of quasi-projective schemes. J.Pure Appl. Algebra 221 (2017), no. 2, 286-303.[M2] Mandal, Satya Homotopy of sections of projective modules. With anappendix by Madhav V. Nori. J. Algebraic Geom . 1 (1992), no. 4, 639-646.[MY] Mandal, Satya; Yang, Yong Intersection theory of algebraic obstruc-tions. J. Pure Appl. Algebra 214 (2010), no. 12, 2279-2293.[MS] Mandal, Satya; Sheu, Albert J. L. Local coefficients and Euler classgroups.
J. Algebra
322 (2009), no. 12, 4295-4330.[Mi] Milnor, John Algebraic K -theory and quadratic forms. Invent. Math.
Algebra, K -theory, groups, and education (NewYork, 1997) , 153-174, Contemp. Math., 243, Amer. Math. Soc., Prov-idence, RI,
J. Algebra