On the Chow theory of projectivizations

In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$. In this process, we establish the decomposition of Chow groups for the cases of Cayley's trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.


Introduction
Let X be a Cohen-Macaulay scheme of pure dimension, and G a coherent sheaf on X of rank r and homological dimension ≤ 1, that is, locally over X, there is a two-step resolution 0 → F → E → G → 0, where F and E are finite locally free sheaves. (If X is regular, this condition on G is equivalent to E xt i X (G , O X ) = 0 for all i ≥ 2.) The projectivization π : P(G ) := Proj X Sym • O X G → X of G is generically a projective bundle with fiber P r−1 ; However, the dimension of the fiber of π jumps along the degeneracy loci (see §2.1) of G .
The derived category of P(G ) was studied in [JL18], where we prove (under certain regularity and dimension conditions) that there is a semiorthogonal decomposition: (For a space Y , D b coh (Y ) stands for its bounded derived category of coherent sheaves). The theorem states that the (right) orthogonal of the "projective bundle part" of D b coh (P(G )) is given by the derived category of another projectivization P(E xt 1 (G , O X )), which is a Springer type partial desingularization of the singular locus of G . See [JL18] for more details In this paper, we establish the Chow-theoretic version of the above formula: Theorem. (See Theorem 4.1) Let X and G be as above. Assume either (A) P(G ) and P(E xt 1 (G , O X )) are non-singular and quasi-projective, and the degeneracy loci of G satisfy a weak dimension condition (4.1); Or (B) All degeneracy loci of G (are either empty or) have expected dimensions.
(2) Nested Hilbert schemes of surfaces §5. 2. Let S be a smooth quasi-projective surface, and denote by Hilb n (S) the Hilbert scheme of n points on S, by convention Hilb 1 (S) = S, Hilb 0 = point. Denote Hilb n,n+1 (S) the nested Hilbert scheme. Then projectivization formula of derived categories [JL18] can be applied to obtain a semi-orthogonal decomposition of D(Hilb n,n+1 (S)), see Belmans-Krug [BK19]. In this paper, we show that for any k ≥ 0, there is an isomorphism of integral Chow groups: CH k (Hilb n,n+1 (S)) ≃ CH k−1 (Hilb n−1,n (S)) ⊕ CH k (Hilb n (S) × S) and a similar decomposition for Chow motives, see Corollary 5.4. (3) Voisin maps §5. 3. Let Y be a cubic fourfold not containing any plane, let F (Y ) be the Fano variety of lines on Y , and let Z(Y ) be the corresponding LLSvS eightfold [LLSvS17]. Voisin [Voi16] constructed a rational map v : F (Y ) × F (Y ) Z(Y ) of degree six, Chen [Chen18] showed that the Voisin map v can be resolved by blowing up the indeterminacy locus Z = {(L 1 , L 2 ) ∈ F (Y ) × F (Y ) | L 1 ∩ L 2 = ∅}, and the blowup variety is a natural relative Quot-scheme over Z(Y ) if Y is very general. The main theorem can be applied to this case, and implies that for any k ≥ 0, there is an isomorphism of Chow groups: where Z = P(ω Z ) is a Springer type (partial) resolution of the indeterminacy locus Z, which is an isomorphism over Z\∆ 2 , and a P 1 -bundle over the type II locus ∆ 2 = {L ∈ ∆ ≃ F (Y ) | N L/Y ≃ O(1) ⊕2 ⊕ O(−1)} which is an algebraic surface. See Corollary 5. 6.
The results of this paper could also be applied to many other situations of moduli spaces, for example, moduli of sheaves on surfaces [Ne17,Ne18], and the moduli spaces of extensions of stable objects in K3 categories, which are generalizations of the varieties resolving Voisin's maps [Voi16,Chen18]. Another such example is provided by the pair of Thaddeus moduli spaces [Tha] M C (2, L ) → N C (2, L ) and M C (2, L ∨ ⊗ ω C ) → N C (2, L ) studied by Koseki and Toda [KT]. (Here, L is a line bundle of odd degree d > 0, N C (2, L ) is the moduli space of rank 2 semistable vector bundles over a curve C, with determinant L , and M C (2, L ) is the space M ω of [Tha], where ω = [ d−1 2 ]). The results of this paper on flips §3.2 and projectivizations Theorem 4.1 would shed light on the study of Chow theory of N C (2, L ). 1 Convention. Throughout this paper, X is a Noetherian scheme of pure dimension, and G is a coherent sheaf over X. We say that G has rank r if the rank of G (η) := G ⊗ κ(η) is r at the generic point η of each irreducible component of X. Assume all schemes in consideration are defined over some fixed ground field k. The terms "locally free sheaves" and "vector bundles" will be used interchangeably. We use Grothendieck's notations: for a coherent sheaf F on a scheme X, denote by P X (F ) = Proj X Sym • O X F its projectivization; we will write P(F ) if the base scheme is clear from context. For a vector bundle V , we also use P sub (V ) := P(V ∨ ) to denote the moduli space of 1-dimensional linear subbundles of V .
For motives, we use the covariant convention of [KMP,Mu1,Mu2,Vi13,Vi15]. In particular, [KMP] contains a dictionary for translating between covariant and contravariant conventions. For a smooth projective variety X over a field k, denote by h(X) its class (X, Id X , 0) in Grothendieck's category of integral Chow motives of smooth projective varieties over k. Notice that under the covariant convention, for a morphism f : X → Y of smooth projective varieties, Γ f induces the pushforward map f * : h(X) → h(Y ) and [Γ t f ] induces the pullback map f * : h(Y ) → h(X)(dim Y − dim X). Moreover, h(P 1 ) = 1 ⊕ L = 1 ⊕ 1(1), where 1 = h(Spec k), and L = 1(1) is the Lefschetz motive. In particular, the covariant Tate twist coincide with tensoring with L, i.e., h(X)(i) = h(X) ⊗ L i for all i ∈ Z. Furthermore, CH ℓ (h(X)(n)) = CH ℓ−n (X) and CH k (h(X)(n)) = CH k−n (X). We will use h to denote the action c 1 (O(1)) ∩ ( ) on motives when the line bundle O(1) is clear from the context. Acknowledgement. The author would like to thank Arend Bayer for many helpful discussions, thank especially Huachen Chen for bringing his attention to this problem and many helpful discussions on Voisin maps and his work [Chen18], and thank Dougal Davis for helpful conversations. This project started during a workshop at Liverpool, for which the author thanks the organisers Alice Rizzardo and Theo Raedschelders for hospitality. The author also thank the referee for the careful reading and many helpful suggestions which greatly improve the exposition of the paper. This work is supported by the Engineering and Physical Sciences Research Council (EPSRC) [EP/R034826/1]. (1) Let G be a coherent sheaf of (generic) rank r over a scheme X. For an integer k ∈ Z, the degeneracy locus of G of rank ≥ k is defined to be

Preliminaries
where G (x) := G x ⊗ O X,x κ(x) is the fiber of G at x ∈ X. Notice that X ≥k (G ) = X if k ≤ r; X sg (G ) := X ≥r+1 (G ) is called first degeneracy locus (or the singular locus) of G . (2) Let σ : F → E a morphism of O X -modules between locally free sheaves F and E on X. For an integer ℓ, the degeneracy locus of σ of rank ℓ is defined to be is the map induced by σ on the fibers.
The degeneracy loci X ≥k (G ) and D ℓ (σ) have natural closed subscheme structures given by Fitting ideals, see [Laz,§7,2]. The two notions are related as follows: let σ : F → E be an O X -module map between finite locally free sheaves, and let G := Coker(σ) be the cokernel, then X ≥k (G ) = D rank E −k (σ) as closed subschemes of X.
The expected codimension of . If G has homological dimension ≤ 1 and rank r, for example if G = Coker(F σ − → E ) is the cokernel of an injective map of O X -modules between finite locally free sheaves, then for any i ≥ 0, the expected codimension of X ≥r+i (G ) ⊂ X is i(r + i).
In the universal local situation where X = Hom k (W, V ) is the total space of maps between two vector spaces W and V over a field k, there is a tautological map τ : GKZ, GG]). Let X = Hom k (W, V ), denote D ℓ = D ℓ (τ ) ⊆ X the degeneracy locus of the tautological map τ of rank ℓ. Then for any 0 ≤ ℓ ≤ min{rank W, rank V }, the singular locus of D ℓ is D ℓ−1 . Furthermore, for any regular point A ∈ D ℓ \D ℓ−1 : In general, let σ : F → E be a map between vector bundles over a scheme X. For a fixed integer ℓ, regarding the open degeneracy locus D := D ℓ (σ)\D ℓ−1 (σ), we have the following: 3. Assume X is a Cohen-Macaulay k-scheme, and D := D ℓ (σ)\D ℓ−1 (σ) ⊂ X has the expected codimension (rank E − ℓ)(rank F − ℓ). Then σ| D : E | D → F | D has constant rank ℓ over D; K := Ker σ| D and C := Coker σ| D are locally free sheaves over D of ranks rank E − ℓ and rank F − ℓ respectively. Moreover, D ⊂ X is a locally complete intersection subscheme with normal bundle N D/X ≃ K ∨ ⊗ C.
Proof. First, we prove the lemma for the total Hom space H = | Hom X (F , E )|. Denote π : H = | Hom X (F , E )| → X the projection, and let D ℓ := D ℓ (τ H ) ⊂ H be the degeneracy locus for the tautological map τ H : π * F → π * E . As the statement is local, we may assume The desired result holds for H and D by Lemma 2. 2. In general, the map σ : F → E induces a section map s σ : X → H, such that σ = s * σ τ H and D = D × X H. Since s is the section of a smooth separated morphism, it is a regular closed immersion. Since H and X are Cohen-Macaulay, D ֒→ H is a regular immersion, and the intersection D = D× X H ֒→ X has the expected codimension, therefore the inclusion D ֒→ X is also a regular immersion, with normal bundle N D/X = s * σ N D/H . Finally, s * σ N D/H = K ∨ ⊗C holds since K ∨ = s * σ Coker(τ ∨ H ) and C = s * σ Coker(τ H ).

2.2.
Chow groups of projective bundles. Let X be a scheme, and E a locally free sheaf of rank r on X. Denote π : P(E ) := Proj(Sym • E ) → X the projection. Notice that our convention P(E ) = P sub (E ∨ ) is dual to Fulton's [Ful]. For simplicity, from now on we will denote ζ = c 1 (O P(E ) (1)), and use the notation ζ i ·β := c 1 (O P(E ) (1)) i ∩β, where β ∈ CH(P(E )), to denote the cap product. For each i ∈ [0, r − 1], we introduce the following notations: The following results are summarised and deduced from [Ful, Proposition 3.1, Theorem 3.3] but presented in a way that fits better into our current work. (1) (Duality) For any α ∈ CH(X), (2) For any k ∈ N, there is an isomorphism of Chow groups: (3) The projection to the i-th summand of the above isomorphism is given by Therefore for any i, j ∈ [0, r − 1], the following holds: Proof.
Notice that our maps π i * (resp. projectors π * i π i * ) are nothing but the explicit expressions of the correspondences g i (resp. orthogonal projectors p r−i ) that are inductively defined in [ Remark 2. 5. The projector π i * can be expressed via the universal quotient bundle as: This duality is explained for more general Grassmannian bundles in [J20].

Blowups.
Let Z ⊂ X be a codimension r ≥ 2 locally complete intersection subscheme. Denote π : X → X the blowup of X along Z, with exceptional divisor E ⊂ X. Then E = P(N ∨ Z/X ) is a projective bundle over Z. We have a Cartesian diagram: The excess bundle V for the diagram is defined by the short exact sequence: From the excess bundle formula [Ful,Theorem 6.3], one obtains the key formula for blowup: The following is summarised from [Ful, Proposition 6.7]: (1) The following holds: π * π * = Id CH(X) and p * (c r−1 (V ) ∩ p * ( )) = Id CH(Z) .
(2) For any k ≥ 0, there exists a split short exact sequence: where a left inverse of the first map is given by (ε, α) → p * ε.
. Note that the well-known formula of (3) follows from (2) by the identification where CH k (E) p * =0 denote the subgroup {γ ∈ CH k (E) | p * γ = 0} of CH k (E). A similar and more detailed argument is given later in the case of standard flips, see Theorem 3.6. There are similar results on Chow motives by Manin [Man]; see also Corollary 3.10 below.

Cayley's trick and standard flips
The projectivization can be viewed as a combination of the situation of Cayley's trick and flips. In this chapter we study the Chow theory of the latter two cases. 3 Let E be a locally free sheaf of rank r ≥ 2 on a scheme X, and s ∈ H 0 (X, E ) a regular section, and denote Z := Z(s) the zero locus of the section s. Denote the projectivization by q : P(E ) = Proj Sym • E → X. Then under the canonical identification the section s corresponds canonically to a section f s of O P(E ) (1) on P(E ). Denote the divisor defined by f s by: Thus H s is a P r−2 -bundle over X\Z, and a P r−1 -bundle over Z. It follows that H s | Z coincides with P Z (N i ), the projectivization of the normal bundle of inclusion i : Z ֒→ X. The situation is illustrated in the following commutative diagram, with maps as labeled: where the maps f and g are given by (where p i * is defined below similar to (2.1)). A left inverse of f is given by (⊕ r−2 i=0 α i , ε) → ⊕ r−2 i=0 p i+1 * ε. Furthermore, the above sequence induces an isomorphism i=0 ζ i · π * α i + j * p * γ, and in this decomposition the projection map to the first (r − 1)-summands CH k (H s ) → CH k−(r−2)+i (X), i = 0, 1, . . . , r − 2, is given by β → π i * β, where π i * is defined below by (3.4), and the projection to the last summand For simplicity, we introduce the following notations. For the projective bundles q : P(E ) → X and p : P(N i ) → Z, similar to (2.1), we denote the projections to the i-th factors by q i * : CH k (P(E )) → CH k−(r−1)+i (X), p i * : CH k (P(N i )) → CH k−(r−1)+i (Z), which are explicitly given as follows: for any i = 0, 1, . . . , r − 1, Furthermore, for any i ∈ [0, r − 1], α ∈ CH(X), γ ∈ CH(Z), we denote q * i α := ζ i · q * α and p * i γ := ζ i · p * γ.
Proof. (a). In fact, from [JL18, Remark. 2.5] the Euler sequence for P(N i ) is equivalent to: 3.5) where N i = E | Z and N j ≃ Ω P(E )/X (1)| P(N i ) . Therefore the excess bundle for the diagram 3.1 is given by O P(N i ) (1). Now from [Ful,Theorem 6.3] and [Ful, Proposition 6.2(1), Proposition 6.6], one has: From the Euler sequence (3.5) and N i = E | Z : is a divisor representing the class ζ = c 1 (O(1)). For any β ∈ CH(P(G | U )), by Theorem 2.4 applied to P(G U ), there exists a unique α i ∈ CH(U), such that β = r−2 i=0 (ι * ζ) i · π * α i . Therefore the following holds: where the last equality follows from the projection formula and ι * ι * ( ) = ζ · ( ). From the uniqueness statement of Theorem 2.4 applied to P(E ), we know that Therefore over U, the following holds: β = Since the ambient square of (3.1) is flat, by the flat base-change formula we have: Here the second equality follows from the key formula of Step (a). Now j * j * (p * 0 p 0 * ε) = j * j * ε = 0. By Step (c), p * 0 p 0 * ε = 0, hence β = j * ε = 0. Theorem 3.1 follows from the above Proposition 3.2 as follows: Proof of Theorem 3. 1. The fact gf = 0 follows from Step (a). The surjectivity of g is Step To show the exactness of (3.2), suppose for α i ∈ CH(X) and ε ∈ CH(P( Hence the sequence (3.2) is exact. To prove the last statement, we show that for any β ∈ CH(H s ), there exists a unique ε ∈ CH(P(N i )), such that p 1 * ε = . . . = p r−1 * ε = 0, and In fact, for any expression β = r−2 i=0 π * i α i + j * ε, by replacing ε by ε − r−2 i=0 p * i+1 p i+1 * ε and α i by α i + i * (p i+1 * ε), we may assume p 1 * ε = . . . = p r−1 * ε = 0. Hence by the projective bundle formula, ε = p * γ for a unique γ ∈ CH(Z). Now by the flat base-change formula, Therefore Hence we have established the identification: .) Moreover, the projection maps to first (r − 1)-summands are respectively given by β → α i = π i * β, for i = 0, 1 . . . , r − 2. For the formula of the projection to the last summand, it suffices to notice that In the above proof, we have actually shown that the relations , and that the isomorphism (3.3) is given by: Corollary 3.4. If X, H s and Z are smooth and projective varieties over some ground field k, then there is an isomorphism of Chow motives: Proof. By Manin's identity principle, it suffices to notice that for any smooth T , the schemes Z × T ⊂ X × T and H s × T are also in the same situation of Cayley's trick Theorem 3.4. Hence the identities of Remark 3.3 hold for the Chow motives.
Example 3.5. . Let Y ⊂ P n be any complete intersection subvariety over a field k of codimension c ≥ 1, say cut out by a regular section of the vector bundle , if we fix a positive integer r ≥ max{ d i − n − c, 1 − c}, then Y ⊂ P n ⊂ P n+r = X is the zero subscheme of a regular section s of the ample vector bundle and similarly for Chow motives if we assume Y is smooth. Hence the Chow group (resp. motive, rational Hodge structure if k ⊂ C and Y smooth) of every complete intersection Y can be split embedded into that of a Fano variety F Y , with complement given by copies of the Chow group (resp. motive, rational Hodge structure) of a projective space P n+r .
By (the same argument of) [LLW10,§1], there exists a vector bundle F ′ of rank m + 1 such that If we blow up X along P , we get π : X → X with exceptional divisor E = P sub (N P/X ) = P S, sub (F ) × S P S, sub (F ′ ). Furthermore, one can blow down E along fibres of P S, sub (F ) and get π ′ : X → X ′ and π ′ (E) =: The above birational map f : X X ′ is called a standard (or ordinary) flip of type (n, m). Note that X > K X ′ (resp. X ≃ K X ′ ) if and only if n > m (resp. n = m).
The geometry is illustrated in the following diagram, with maps as labeled: , the expected relations of derived categories for the flip (resp. flop) f : X X ′ are established by Bondal-Orlov [BO]. In this section we establish the corresponding relations on Chow groups, which complements the results of [LLW10,§3].
From now on we assume n ≥ m, i.e. X ≥ K X ′ . Denote Γ the graph closure of f in X ′ × X, which is nothing but X = X × X X ′ . Denote by Γ * : CH k (X) → CH k (X ′ ) and Denote by V and V ′ the respective excess bundles for the blow up π : X → X and π ′ : X → X ′ , i.e. they are defined by the short exact sequences: Theorem 3.6 (Standard flips). Let f : X X ′ be a standard flip as above, and assume X ′ is non-singular and quasi-projective. Then (1) The following holds: (2) There exists a split short exact sequence: where a left inverse of the first map is given by The above exact sequence induces an isomorphism of Chow groups Furthermore, in the above decomposition, the projection to the first summand is given by α → α ′ = Γ * α.
Notice that in the flop case m = n, this result recovers the invariance of Chow groups under flops in [LLW10]; and in the flip case m < n, this theorem completes the discussion of [LLW10,§2.3] by providing the complementary summands of the image of Γ * in the Chow group CH(X). Finally, as a blowup can be viewed as a standard flip of type (n, 0), above theorem recovers the blowup formula Theorem 2.8.
Proof of first part of (1). The equality Γ * Γ * = Id follows exactly the same line of proof of [ , where W is the image of W , and is also the proper transform of W ′ along the birational rational map f −1 . Now we have: A direct computation of dimensions shows that, for a general point s, the fiber E B,s over s has dimension: Now E B,s must contain positive fibers of of p ′ s : P n s × P m s → P m s , as n + 1 > n ≥ m. Hence Remark 3.7. Notice that the above argument does not work in the other direction for , then the above argument still works, i.e., the following holds: For the intermediate cases m for certain cycles Z ⊂ P supported on P ; These cycles will be precisely explained by the statements (2) and (3) of the theorem.
Proof of second part of (1). It follows from Lemma 2.7 that: and The map Φ * • Φ * is given by the convolution of correspondences where p ij are the obvious projections from P ′ × S P × S P ′ to the corresponding factors; The cohomological degree m is computed via m + n − dim(P/S) = m. To avoid confusion, we denote the product P ′ × S P × S P ′ by P ′ 1 × S P × S P ′ 2 , and denote the relative O(1)-classes of P ′ 1 and P ′ 2 by ζ ′ 1 and ζ ′ 2 respectively. Therefore Since p 13 * (ζ k ) = 0 for all 0 ≤ k ≤ n − 1, in the above expression, the only terms inside the parentheses that could survive p 13 * are the ones whose indices satisfy t − i − j ≥ 0. Thus we may assume the indices of the summation satisfy j ≤ t ≤ m and 0 ≤ i ≤ t − j. From the definition of the Segre class of F , we have p 13 * (ζ n+k ) = s k (F ), hence: From c(F )s(F ) = 1, we know that t−j i=0 c i (F ) · s t−i−j (F ) = 0 unless t = j, in which case c 0 (F )s 0 (F ) = 1. Hence above expression reduces to (For the second equality, we used Lemma 2.7.) On the other hand, the diagonal ∆ P ′ ⊂ P ′ × S P ′ is the zero locus of a regular section s of the rank m vector bundle O P ′ (1)⊠T P ′ /S (−1); The section s under the canonical identification Before proceeding the rest of the proof of Theorem 3.6, we study more about the maps Φ * and Φ * . First, notice that the projective bundle formula Theorem 2.4 can be regarded as equipping CH(P ) and CH(P ′ ) with natural "free module structures over CH(S)".
We could also write down explicitly the projectors to the last (n − m) summands; We omit the details here as we will not need them. As before, by Manin's identity principle: Corollary 3. 10. If X and X ′ are smooth and projective over some ground field k, then there is an isomorphism of Chow motives over k: As before, the blowup formula for Chow motives of [Man] could be viewed as the case m = 0 of the above corollary, as a blowup can be viewed as a standard flip of type (n, 0).

Main results
Let G be a coherent sheaf of homological dimension ≤ 1 on X (i.e., X is covered by open subschemes U ⊂ X over which there is a resolution F σ − − → E ։ G such that F and E are locally free of rank m and n respectively, and G = Coker(σ) is of rank r = n − m ≥ 0). Denote the projection by π : P(G ) → X. Similar to the projective bundle case, for any i ∈ [0, r − 1], denote by π * i : CH k−(r−1)+i (X) → CH k (P(G )) be map π * i ( ) = ζ i · π * ( ), where ζ = c 1 (O P(G ) (1)). Consider the fiber product Denote the projections by r + : Γ → P(G ) and r − : Γ → P(E xt 1 (G , O X )). As before, we denote Γ * : CH k−r (P(G )) → CH k (P(E xt 1 (G , O X ))) and Γ * : CH k (P(E xt 1 (G , O X ))) → CH k−r (P(G )) the maps induced by the correspondence [Γ] ∈ CH(P(G ) × P(E xt 1 (G , O X ))), i.e. Γ * ( ) = r − * r * + ( ) and Γ * ( ) = r + * r * − ( ). The main result of this paper is the following: Theorem 4. 1. Let X be a Cohen-Macaulay scheme of pure dimension, and let G be a coherent sheaf of rank r ≥ 0 on X of homological dimension ≤ 1. Assume either (A) P(G ) and P(E xt 1 (G , O X )) are non-singular and quasi-projective, and Then for any k ≥ 0, there is an isomorphism of Chow groups: The projection β → α i is given by the map π i * of Lemma 4.4, 0 ≤ i ≤ r − 1, and the projection β → γ is given by (−1) r Γ * .
Remark 4.2. (i) If X is irreducible, then the dimension condition (4.1) of (A) is equivalent to the requirement that P(E xt 1 (G , O X )) maps birationally to X ≥r+1 (G ), and P(G ), P(E xt 1 (G , O X )) and Γ are irreducible and have expected dimensions: (ii) The only place that we need P(G ) and P(E xt 1 (G , O X )) to be nonsingular and quasiprojective in (A) is to use Chow's moving lemma. Hence the result holds as long as Chow's moving lemma holds for P(G ) and P(E xt 1 (G , O X )). (iii) It follows from [JL18, Theorem 3.4] that if X is nonsingular, and P(G ), P(E xt 1 (G , O X )) have expected dimension. Then P(G ) is nonsingular if and only if P(E xt 1 (G , O X )) is.
(iv) The requirement r + 2i for codimension in (A) is much weaker than the expected one i(r + 1) if i >> 1, which is required by (B). On the other hand, (B) only requires very weak regularity conditions on the schemes -X being Cohen-Macaulay.
(In fact, the Cohen-Macaulay condition can be dropped, as long as each stratum X ≥i (G )\X ≥i+1 (G ) ⊂ X is a regular immersion of expected dimension.) Corollary 4. 3. If P(G ), P(E xt 1 (G , O X )) and X are smooth and projective over some ground field k, then there is an isomorphism of Chow motives: Proof. Similarly as Corollary 3.4, for any smooth T , the same constructions and the theorem applies to X × T and G ⊠ O T , hence in particular the identities Id = Γ * Γ * + π * i π i * , Γ * Γ * = Id, π i * π * i = Id, etc (see Lemma 4.4 and Lemma 4.9 below) hold for all X × T and G ⊠ O T . Then the result follows from Manin's identity principle.
Before proceeding with the proofs of the theorem, we first explore some general facts.

Proof.
For simplicity, we may assume G = Coker(F σ − → E ), then c(G ) = c(E )/c(F ) and ι : P(G ) ֒→ P(E ) is given by a regular section of the vector bundle F ∨ ⊗ O P(E ) (1). The for any a ∈ [0, r − 1], Now set µ := ν + j, notice that above terms which survives under q * have the indices range 0 ≤ µ ≤ i − 1, i = µ + a ≥ a and 0 ≤ ν ≤ µ, therefore: Hence (1) follows. In general, it suffices to notice that the maps π i * π * i and π i * π * j are globally defined and their values do not depend on local presentations.
For simplicity of notations, from now on we denote: Therefore we have a fibered diagram: Lemma 4. 5. Assume P(G ), P(K ) and Γ have expected dimensions (see Remark 4.2 (i)).
(3) The following holds: Proof. It suffices to prove in a local situation, i.e. 0 → F σ − → E → G → 0 for vector bundles F and E of rank m and n. Dually we have is zero, hence σ factorises through a map of vector bundles σ : T P(F ∨ )/X (−1) → π ′ * E . For the reason of ranks, it easy to see the following sequence is exact: Therefore π ′ * G has homological dimension ≤ 1, and P(π ′ * G ) ⊂ P(π ′ * E ) = P(K ) × X P(E ) is the zero scheme of a regular section of the vector bundle Ω P(F ∨ ) (1) ⊠ O P(E ) (1). The last equality follows directly from commutativity of projectivization and fiber products.
For (2), consider the following factorisation of the (transpose of) the diagram (4.3): (Here for simplicity we use q to denote both projections of projectivization of E .) The normal bundles are N ι = F ∨ ⊠ O P(E ) (1), and N ι ′ = Ω P(F ∨ ) (1) ⊠ O P(E ) (1). Since the right square of the diagram is a smooth and flat, the excess bundle is given by r * For (3), the first equality is Lemma 4.4. For any γ ∈ CH(P(K )), for i ∈ [0, r − 1], (The last equality holds since q m+i+j * has index range m + i + j ≥ m.) Similarly for any α ∈ CH(X) and i ∈ [0, r − 1], since q is the projection of a P n−1 -bundle, and m − 1 + i ≤ m + r − 2 ≤ n − 2.

First approach.
In this approach, we use Chow's moving lemma, hence need P(G ) and P(K ) to be nonsingular and quasi-projective. The idea is: over the open part of the first degeneracy locus, the theorem is almost the case of Cayley's trick. Then the "error" terms over higher degeneracy loci can be estimated by dimension counting. A similar strategy was used by Fu-Wang to show invariance of Chow groups under stratified Mukai flops [FW08].
We first need the following variant of Cayley's trick: Lemma 4.6 (Variant of Cayley's trick). Assume G is a coherent sheaf of homological dimension 1 over a variety X, and let i : Z ֒→ X be a locally complete intersection subscheme of codimension r + 1, such that G has constant rank r over X\Z, and constant rank r + 1 over Z. Denote Γ := P Z (i * G ) = P(G ) × X Z, and denote Γ * : CH(P(G )) → CH(Z) and Γ * : CH(Z) → CH(P(G )) the maps induced by [Γ] and [Γ] t . Then the following holds: . Furthermore, the following decomposition of identity holds: Proof. It suffices to notice that the argument of Theorem 3.1 for these statements only depends on the properties of the normal bundles, hence still works here. More precisely, we may assume G = Coker(F σ − → E ) for simplicity, then over Z there exits a line bundle L such that there is an exact sequence of vector bundles: Also we have a similar picture as Cayley's trick (3.1): Denote G Z := i * G , which is a vector bundle on Z. Then it is easy to compute the normal bundles are

and excess bundle for the left square is
.10 for the more general situation). Therefore )∩p * ( )) = (−1) r Id, and the rest of the orthogonal relations follow from Lemma 4. 5. Finally, for the last identity, it suffices to show the subjectivity of Γ * + π * i . For any β ∈ CH(P(G ), β ′ = β− π * i π i * β is supported on P Z (G Z ), hence can always be expressed in the form , and hence we are done.
Remark 4.7. If we modify the map f in Theorem 3.1 by f : , then the sequence of Theorem 3.1 is still exact. In fact, if we denote p i * the projectors with respect to O P(E ) (1), then p * i •i * = j * •π * i holds. Although for i ∈ [0, r −1], i * • p i+1 * and π i * • j * are no longer the same, due to the additional factor c 1 (L), but they differ by an invertible upper triangular change of basis as Remark. 2.6. Hence in Proposition 3.2, except from the key formula (a) now becomes π * i • i * = j * ((ζ − c 1 (L) ∩ p * ( )), the rest still holds. Similar for other the statements.
Lemma 4.8. If P(K ) = P(E xt 1 (G , O X )) is nonsingular and quasi-projective, and the dimension condition (4.1) of (A) holds, then the following holds: Proof. The following arguments follow the strategy of Fu-Wang [FW08] for stratified Mukai flops, which is itself a generalization of [LLW10]'s treatment for standard flops and flips (see also §3.2). For any class [W ] ∈ CH k (P(K )), by Chow's moving lemma, we may assume W intersects transversely with i≥1 P(K ) i . First, notice that over the open subsetX := X\X 1 ,P(K ) := P(K ) 0 \P(K ) 1 ≃Z := X 0 \X 1 i − →X is an inclusion of codimension r + 1, and G has constant rank r overX\Z, has constant rank r + 1 overZ, andΓ ≃ P(i * G ) ⊂P(G ) := P(G ) 0 \P(G ) 1 . Therefore we are in the situation of variant of Cayley's trick Lemma 4.6. Therefore by Lemma 4.6, if we setW = W ∩X, then the cycle r * + r + * r * − [W ] is represented by a k-cycle˚ W which maps generically one to one to a k-cycle which is rationally equivalent toW , and that Now back to the whole space, if we let W be the closure of˚ W in Γ. Then where W is the k-dimensional cycle as above, mapping generically one to one to a k-cycle that is rationally equivalent to (−1) r W , a C ∈ Z, and F C are k-dimensional irreducible schemes supported over π ′ (W ∩ i≥1 P(K ) i ). (More precisely, let C ′ be irreducible component of π −1 π ′ (W ∩ i≥1 P(K ) i ), then the fiber F C runs through the components {C = π(C ′ ) ⊂ π ′ (W ∩ i≥1 P(K ) i )}; Here different C ′ may have the same image C.) For any F C , take the largest i such that there is a component But since the general fiber of π ′ over s has dimension i, hence F C,s contains positive dimension fibers of r − . Therefore r − * [F C ] = 0. Hence Lemma 4.9. If P(G ) is nonsingular and quasi-projective, and the dimension condition (4.1) of (A) holds, then for every [V ] ∈ CH k (P(G )) tor. the following holds: . By moving lemma we may assume V intersects transversely with i≥1 P(G ) i . Similar to the proof of Lemma 4.8, by variant of Cayley's trick Lemma 4.6, overX : . Therefore there existsW representing Γ * [V ] ∈ CH k−r (P(K )) such that r −1 − (W ) is a k-dimensional cycle and r + * (r −1 − (W )), though supported on P(G ) 1 , is rationally equivalent to (−1) rV in P(G ).
Therefore over the whole space, we have: where V is the closure of r −1 − (W ) in Γ, and hence r + * V is rationally equivalent to (−1) r V , a C ∈ Z, and F C are irreducible k-dimensional cycles supported over π(V ∩ i≥1 P(G ) i ). Similar as before, for any F C , take the largest i ≥ 1 such that there is a component . For a general s ∈ B C , the fiber F C,s has dimension: Now since the general fiber of π over s has dimension r + i, hence F C,s contains positive dimension fibers of r + . Therefore r + * [F C ] = 0, and

4.2.
Second approach. The idea of this second approach is that: if we stratify the space X as before, then over each stratum the theorem reduces to a situation very similar to standard flips case §3. 2. Since we will argue over each stratum, we will need all strata to achieve the expected dimensions, but do not require regularity on the total space. Lemma 4. 10. Let G be a coherent sheaf on a Cohen Macaulay scheme X of homological dimension ≤ 1 and rank r. For a fixed integer i ≥ 0, denote Z = X ≥r+i+1 (G ), and assume X ≥r+i+2 (G ) = ∅. (That is, Z is the bottom degeneracy locus of G ; G has constant rank r + i + 1 over Z, and has rank ≤ r + i over X\Z.) Assume furthermore that Z ⊂ X has the expected codimension (i + 1)(r + i + 1). Denote K = E xt 1 (G , O), i : Z ֒→ X the inclusion. Then G Z := i * G , K Z := i * K are vector bundles over Z of rank r+i+1 and i+1 respectively. Consider the following base-change diagram for the fibered product Γ = P(G ) × X P(K ) along the base-change Z ֒→ X, with names of maps as indicated: (4.4) Then the normal bundles of the closed immersions i, j, k, ℓ are respectively given by (1), and N ℓ = Ω P(G Z )/Z (1) ⊠ Ω P(K Z )/Z (1).
The excess bundle for the front square is given by V = O P(G Z ) (1)⊠K Z , and the excess bundle for the back square is V ′ = O P(G Z ) (1) ⊠ Ω P(K Z )/Z (1). Therefore Similarly the excess bundle for the bottom square is given by W = G Z ⊠ O P(K Z ) (1), and for the top square is W ′ = Ω P(G Z )/Z (1) ⊠ O P(K Z ) (1). Therefore Proof. As the statements are local, we may assume G = Coker(F σ − → E), where E, F are vector bundles of rank n and m. Then by our assumption on Z and Lemma 2.3, Z ⊂ X is a closed locally compete intersection subscheme, and N i = G Z ⊗ K Z ; Moreover, the image im(σ| Z ) ⊂ E| Z is a vector sub-bundle; let us denote it by B Z . Therefore the map σ| Z induces two short exact sequences of vector bundles over Z: Next, over P(G ) ⊂ P(E), the composition π * F π * σ − − → π * E → O P(E) (1) is zero, hence π * σ factors through a map between vector bundles σ : π * F → Ω 1 P(E)/X (1). As the rank of σ at a point p ∈ P(G ) agrees with the rank of σ at π(p), therefore Z := π −1 (Z) = P(G Z ) is the bottom degeneracy locus of σ. We claim that there is an exact sequence of vector bundles: Then by Lemma 2.3, To prove the claim, it suffices to notice that over Z there is a commutative diagram: In the diagram, the three columns and the last two rows are exact, hence the first row is a short exact sequence. Combined with the short exact sequence of vector bundles 0 → π * Z (K ∨ Z ) → π * Z (F | Z ) → π * Z (B Z ) → 0, the claim follows. Notice that in above argument we do not use the condition n ≥ m, hence the same argument works for all the other cases. and furthermore for any a ∈ [0, r − 1], denote π * Z,a ( ) := c top (V ) · ζ a ∩ π * Z ( ). Then (1) Ψ * Ψ * = (−1) r Id; (2) Then for any k ≥ 0, there is an isomorphism of Chow groups: given by (⊕ r−1 a=0 α a , γ) → r−1 a=0 π * Z,a α a + Ψ * γ.
Proof. For the first two statements, notice that if we write F = G ∨ Z , F ′ = K ∨ Z , with rank n = r +i and m = i, and S = Z, then P = P(G Z ), P ′ = P(K Z ), E = Γ Z , and we are in a very similar situation as the standard flip case §3. 2. In fact, for (1), using the notation of the proof of Theorem 3.6, then Ψ * and Ψ * correspond to the correspondence given by (−1) n c n (V ′ ) and (−1) m c m (V ) respectively (instead of c m (V ) for Φ * and c n (V ′ ) for Φ * ). However the composition c n (V ′ ) * c m (V ) is still computed by the same formula as c m (V ) * c n (V ′ ) (with the role of first and third factor of the product P ′ × S P × S P ′ switched), by commutativity of intersection product. Hence c n (V ′ ) * c m (V ) = [∆ P ′ ], and Ψ * Ψ * = (−1) m+n Id = (−1) r Id.
For (3), it follows directly from Lemma 4.10 that for any γ ∈ CH(P(K Z )), and similarly Γ * j * = k * Ψ * . Also for any a ∈ [0, r − 1] and α ∈ CH(Z), Proof of theorem 4.1 under condition (B). Stratify the space X by the same way as in the first approach, namely X i := X ≥r+i+1 (G ) for i ≥ −1, and similarly for P(G ) i , P(K ) i and Γ i . For each i ≥ −1, we will denote the natural inclusions by: i i : X i ֒→ X, j i : P(G ) i ֒→ P(G ), k i : P(K ) i ֒→ P(K ) and ℓ i : Γ i ֒→ Γ. For i ≥ 0, we also denote i i,i−1 : X i ֒→ X i−1 the natural inclusion, and j i,i−1 , k i,i−1 and ℓ i,i−1 are defined similarly. Finally for each pair (i, j) with j > i ≥ −1, denote by X i\j := X i \X j ; P(G ) i\j , P(K ) i\j and Γ i\j are defined by the same manner. By abuse of notations, the inclusion i i : X i\j ֒→ X\X j = X −1\j is also denoted by i i , and similarly for other inclusions. For any fixed integer i ≥ 0, if we assume condition (B) of Theorem 4.1 is satisfied, then Z := X i\i+1 ⊂ X\X i+1 = X −1\i+1 is a locally complete intersection subscheme of codimension (i + 1)(r + i + 1), and G has constant rank r + i + 1 over Z. Therefore the conditions of Lemma 4.10 are satisfied by Z ⊂ X\X i+1 and G , with P(G Z ) = P(G ) i\i+1 , P(K Z ) = P(K ) i\i+1 and Γ Z = Γ i\i+1 , also i = i i , j = j i , k = k i and ℓ = ℓ i . Hence results of Lemma 4.11 can be applied. Now our goal is to show the isomorphism of Lemma 4.11 (2) over each stratum can indeed be integrated into an isomorphism of the map (4.2) of Theorem 4. 1.
Surjectivity of the map (4.2). For each i ≥ −1, there is an exact sequence: for which if i = i max + 1, then the middle term is the whole space, where i max is the largest number such that X imax = ∅. (Since X is locally Noetherian of pure dimension, there exists only finitely many strata and such an i max always exists.) Therefore inductively we see CH(P(G )) is generated by the images of j i * : CH(P(G ) i\i+1 ) → CH(P(G )) for all strata P(G ) i\i+1 , i ≥ −1, where i = −1 corresponds to the open stratum.
Hence we need only show that the image of the map (4.2) contains the image of the strata CH(P(G ) i\i+1 ) in CH(P(G )) for each i ≥ − 1. The open stratum case i = −1 follows from projective bundle formula. For other cases, i.e i ≥ 0, set Z := X i\i+1 ⊂ X\X i+1 as above, and for simplicity denote j * := j i * : CH(P(G ) i\i+1 ) → CH(P(G )), which agrees with notations of Lemma 4.10 and Lemma 4.11; Similarly for the maps i, k, ℓ. Then by Lemma 4.11 (2), any α ∈ CH(P(G ) i\i+1 ) = CH(P(G Z )) can be written as α = r−1 a=0 π * Z,a α a + Ψ * γ, for certain α a ∈ CH(Z) and γ ∈ P(K Z ) = P(K ) i\i+1 . Therefore by Lemma 4.11 (3), i.e. the image of j * is contained in the image of the map (4.2). Hence we are done.

Injectivity of the map (4.2). This part is a little tricky;
The key observation is that the above excision exact sequence becomes a short exact sequence if we take the image of first map. The injectivity of π * a follows from Lemma 4.4; It remains to show the injectivity of Γ * . For each i ≥ −1, there is a commutative diagram of short exact sequences: where recall the maps k i * and j i * are the inclusions to (an open subset of) the whole space: We want to show that for each i ≥ 0, the map Γ * | Im k i * is injective. Set Z := X i\i+1 ⊂ X\X i+1 as above, then the question reduces to show: in the following commutative diagram (which is commutative by Lemma 4.11(3)) the injection Ψ * induces an injection Γ * on the image. In fact, for any γ ∈ CH(P(K ) i\i+1 ), if Γ * k i * γ = j i * Ψ * γ = 0, then by Lemma 4.11 (1), we have γ = (−1) r Ψ * Ψ * γ. Therefore by Lemma 4.11 (3), Hence Γ * | Im k i * is injective. Now by induction, starting with the case i = 0, when the injectivity of Γ * | −1\1 follows from the following commutative diagram we can inductively show that Γ * | −1\i is injective for all i = 0, 1, 2, . . . , i max , i max + 1, where i max is the largest number i max such that X imax = ∅. Therefore Γ * = Γ * | −1\imax+1 is injective on the whole space. Notice that, from above argument, we also obtain that Γ * Γ * = (−1) r Id holds, since it is true on the image of each stratum. Together with Lemma 4.4 and Lemma 4.5, this completes the proof of Theorem 4. 1. 4. 3. First examples.

4.3.1.
Universal Hom spaces. Let S be a Cohen-Macaulay scheme, and let V and W be two vector bundles over S. Without loss of generality, we may assume rank W ≤ rank V . Consider the total space of maps between V and W : Then there are tautological maps over X: Let G = Coker(φ) and K = E xt 1 (G , O X ) = Coker(φ ∨ ). Then it is easy to see that the condition (B) of Theorem 4.1 is satisfied, and Theorem 4.1 holds for P(G ) = Tot P(V ) (W ∨ ⊗ S Ω P(V )/S (1)) and P(K ) = Tot P(W ∨ ) (Ω P(W ∨ )/S (1) ⊗ S V ).
Notice that any map σ : W → V over S determines a section s σ : S → X, such that s * σ φ = σ, s * σ φ ∨ = σ ∨ . Then Coker(σ) and Coker(σ ∨ ) (and their projectivizations) are just the pullbacks of G and K (and the projectivizations P(G ) and P(K )) along the section map s σ . Similarly, we can consider the projectivization version: Over Y there are tautological maps: Then One may also consider the linear sections of the space Y as in HPD theory [Kuz07,BBF16].

Flops and Springer resolutions.
In the situation of Theorem 4.1, if we take r = 0, then P(G ) and P(K ) = P(E xt 1 (G , O X )) are both Springer type partial desingularizations of the first degeneracy locus X sg (G ) = X ≥1 (G ) ⊂ X. They are related by a flop, and Γ = P(G )× X P(K ) is the graph closure for the rational map P(G ) P(K ). For simplicity, we assume X is irreducible. Then Theorem 4.1 states that if either (A) P(G ), P(K ) are smooth and quasi-projective (hence they are both resolutions of X sg (G )), Γ = P(G ) × X P(K ) is irreducible and dim Γ = dim X − 1; or (B) X is Cohen-Macaulay and codim X ≥i (G ) = i 2 for i ≥ 1.

4.3.3.
Cohen-Macaulay subschemes of codimension 2. Let X be an irreducible scheme, and Z ⊂ X a codimension 2 subscheme whose ideal I Z has homological dimension ≤ 1. This holds in particular for any codimension two Cohen-Macaulay subscheme Z ⊂ X inside a regular scheme X, by the Auslander-Buchsbaum theorem. (In fact, in this case X clearly has the resolution property, there always exist locally free sheaves F and E , and a short exact sequence 0 → F → E → I Z → 0, with rank F = rank E − 1; and by the Hilbert-Burch theorem, any Cohen-Macaulay codimension 2 subscheme of X arises in this way.) Consider the degeneracy X ≥1+i (I Z ) for i ≥ 0 as before (note rank I Z = 1), then X ≥1+i (I Z ) is the loci where the ideal I Z needs no less than i + 1 generators. It is known (e.g. see [ES]) that if codim X ≥1+i (I Z ) ≥ i + 1 for i ≥ 1, then π : P(I Z ) = Bl Z X → X is the blowup of X along Z and is irreducible, and Z := P(E xt 1 (I Z , O X )) is the Springer type desingularization of Z. Notice that if X is Goreinstein, then Z ≃ P(E xt 1 (I Z , ω X )) = P(ω Z ), where ω X and ω Z are the dualizing sheaves. Theorem 4.1 states that if whether (A) Bl Z X and Z are smooth and quasi-projective, Z maps birational to Z (hence Z is a resolution of Z), and codim X ≥1+i (I Z ) ≥ 1 + 2i for i ≥ 1 (or equivalently Γ := Bl Z X × X Z is irreducible and dim Γ = dim X − 1); or (B) X is Cohen-Macaulay and codim X ≥1+i (I Z ) = i(1 + i) for i ≥ 1.
Then for any k ≥ 0, there is an isomorphism of Chow groups:

Applications
If 0 ≤ d ≤ g − 1, then AJ maps birationally onto the Brill-Noether loci W 0 d ⊂ Pic d (C), which has codimension g − d, and the dimension jumps over W i d for i ≥ 1. The cases g − 1 ≤ d ≤ 2g − 2 and 0 ≤ d ≤ g − 1 are naturally related by the involution Following Toda [Tod18b], from now on we use the following notation: set an integer n ≥ 0, and set (We do not restrict ourselves to n ≤ g−1, though this is the most interesting case.) Therefore apart from the usual Abel-Jacobi map, we also have its involution version: The fiber of AJ ∨ over a point L ∈ Pic d (C) is the linear system |L ∨ (K C )| = P sub (H 1 (C, L ) * ) = P(H 1 (C, L )). Therefore we have the following fibered diagram: Pic g−1+n (C).
To prove the corollary, we show that the above situation fits into the picture of Theorem 4.2 and satisfies condition (A) (if C is not hyperelliptic).
Set X := Pic g−1+n (C), and let D be an effective divisor of large degree on C. ∀L ∈ Pic(X), the exact sequence 0 → L → L (D) → L (D)| D → 0 induces an exact sequence: Globalizing (the dual of) above sequence yields the desired picture: let L univ be the universal line bundle of degree g − 1 + n on C × X, and pr C , pr X be obvious projections, then E := (pr X * (pr * C O(D) ⊗ L univ )) ∨ and F := (pr X * (pr * C O D (D) ⊗ L univ )) ∨ are vector bundles on X of ranks deg(D) + n and deg(D), with a short exact sequence where G := Coker(σ) is the sheafification of H 0 (C, L ) ∨ , has homological dimension ≤ 1 and rank n, and K := E xt 1 (G , O X ) = Coker(σ ∨ ) is the sheafification of H 1 (C, L ). Therefore Then the stratification X i := X ≥n+i+1 (G ) for i ≥ −1 of the Theorem 4.2 corresponds to Brill-Noether loci as follows (recall d = g − 1 + n, d ′ = g − 1 − n):
The case when g ≥ 3 and C is not hyperelliptic. We show the condition (A) is satisfied, i.e.
The case g ≥ 3, and C is hyperelliptic. Take a disc D in the moduli space M g intersecting transversely to the hyperelliptic locus, with zero point [C], and consider the universal curve C over D. Then the general fiber of C is non-hyperelliptic, and by above estimates condition (A) is satisfied by the family C (with relative Hilbert schemes Hilb g−1±n (C /D) of zero dimensional subscheme on the fibres of length g − 1 ± n) as well as the generic fibre C η . Therefore the identities of the maps between Chow groups (e.g. Γ * Γ * = Id, decomposition of Id = Γ * Γ * + i π * i π i * , etc) of Theorem 4.1 for Hilb g−1±n (C /D) (or C (g−1±n) η ) specialize to the same identities for the central fiber C 0 = C (see [Ful,Ch. 10]), hence induces the isomorphism of Corollary 5.1 for the hyperelliptic curve C.
Combining these two formulae, one obtains the desired result for h Q . By using Baño's works [dB1, dB2] as above, [GL20, Proposition 1.6] also independently obtains the isomorphism of Chow motives of Corollary 5.1 with rational coefficients.
If we denote X = F (Y ) × F (Y ), and let E xt i f (F , P) be the sheafification of the group Ext i (F, P ) for the family f : X × Y → X, the following is proved in Chen's work [Chen18]: (1) E xt 1 f (F , P) = I Z (where I Z is the ideal sheaf of Z ⊂ X, and Z is the incident locus {L 1 ∩ L 2 = ∅} defined above), has homological dimension 1, and Z ⊂ X = F (Y ) × F (Y ) is Cohen-Macaulay of codimension 2.
of stable quotients of E inside A ⊂ Ku(Y ) over Z(Y ), where A is the heart of σ.
Therefore the sheaf I Z satisfies condition (B) of Theorem 4.2. If we consider π ′ : Z := P X (E xt 2 f (P, F )) = P Z (ω Z ) → X (2) This work is inspired by its counterpart in derived categories [JL18], where the projectivization formula was proved using the techniques developed in [JLX17,T15,Kuz07]. It is interesting whether or not one can "decategorify" other interesting semiorthogonal decompositions obtained by these techniques. Examples include various cases of homological projective duality and flops, see [JLX17,JL18,T15,Kuz07]. Note that usually, results of derived categories only imply ungraded results for rational Chow groups and motives; but see [BT16] where essential graded information of Chow groups is recovered from derived categories. (3) The projectivization formula for derived categories is closely related to the wall-crossing and d-critical flips studied by Toda [Tod18a,Tod18b]. It would be interesting to extend the results of this paper to the cases of Donaldson-Thomas type moduli spaces considered there. (4) The projectivization formula considered in this paper fits into a broad framework of the study of Quot schemes of locally free quotients [J20, J21]. (5) Since the resolution P(Ext 1 (G , O X )) → X sg (G ) is usually IH-small, it is reasonable to expect one may replace CH(P(Ext 1 (G , O X ))) by the intersection Chow group [CH] of X sg (G ). (6) The projectivization formula of Chow groups should hold for Deligne-Mumford stacks, with CH replaced by CH Q ; It would also be interesting to study the ring structure of CH(P(G )) in the case when X and P(G ) are smooth.