Pandharipande–Thomas stable-pair virtual class

We construct the named moduli object before using it. Choose a very ample A and put d=β(A). If the pure-dimension-one convention makes the moduli empty, the zero virtual class proves the assertion. For the relative calculations below write Y→B for a smooth projective threefold family, $Y_S=Y{\times }_B$ S, and $\pi :Y_S\to S$; the absolute claim is $Y=X$, B=Spec C. Work on affine Noetherian test bases S. The relative construction is included to supply the next family's compatibility, not as an extra hypothesis on the absolute claim. Choose a very ample A on X and put d=β(A). For a stable pair the image of $O_X\to F$ is $O_C$ for a Cohen–Macaulay curve C, and $0\to O_C\to F\to Q\to 0$

has Q of finite length. Here C has degree d: away from the finite support of Q, $F=O_C$, with their full scheme multiplicities. To check the support assertion, if a function annihilates the section but not F, its image aF is a nonzero finite-support subsheaf, contradicting purity. The image $O_C$ is pure as a subsheaf of F, so C has no embedded zero-dimensional associated point and is Cohen–Macaulay of dimension one.

There is a coarse uniform lower bound $\chi (O_C)\ge d(2-d)$. Pick a pencil of hyperplanes with centre disjoint from C. It gives q:C→P¹ with q*O(1)=A|_C. No component is contracted, since A has positive degree on every component; thus q is proper quasi-finite and finite. The pullback of a parameter on P¹ is regular on $O_C$, since it contains no component. Therefore $E=q_{\ast }{O}_C$ is torsion-free on P¹ and locally free. Its rank is d: the leading coefficient of $\chi (C,A^t)=\chi (P^1,E(t))$ is rank E.

Coordinate sections $x_j$ give algebra generators of E after dividing by either nonvanishing pencil coordinate. In any d-dimensional fibre algebra, the subspace generated by monomials of total degree at most m is all the algebra by $m=d-1$: it starts with dimension one, and if two successive subspaces agree, multiplication by every generator preserves it and it is already the whole generated algebra. Otherwise its dimension increases at each step. Consequently the global monomial maps ${\oplus }_{m=0}^{d-1}$ $O_{P^1}(-m{)}^{N_m}$ → E

are fibrewise surjective, hence surjective by Nakayama. Thus E(d−1) is globally generated, so its determinant has a nonzero section and nonnegative degree. It follows that deg E≥−d(d−1). Riemann–Roch for E on P¹ (also directly GRR in degree one) gives $\chi (O_C)=\chi (E)=d+\mathrm{deg}$ E≥d(2−d).

Since $\chi (O_C)\le \chi (F)=n$, only finitely many support Hilbert polynomials dt+c occur, with d(2−d)≤c≤n. If this interval is empty there are no pairs. Otherwise the actual projective Hilbert schemes for these polynomials form a finite union with a universal flat support family. Relative Serre vanishing and global generation, on finitely many Noetherian affine charts of that projective parameter, give ONE m such that every such $O_C(mA)$ is globally generated and $H^1(C,O_C(mA))=0$. Increase m if necessary. The sequence above gives H¹(X,F(mA))=0 and surjectivity H⁰(F(mA))→H⁰(Q). Lifting generators of Q and combining them with the generating sections of $O_C(mA)$ proves F(mA) globally generated. Its h⁰ is r=dm+n. Thus every stable-pair fibre has the same uniform generation and H¹ vanishing. This bound uses no semistable-sheaf or Quot boundedness theorem.

For families these conditions give π_*F(mA) locally free of rank r with every base change by the actual nonnegative cohomology complex and finite-free base-change criterion. Evaluation is surjective by fibrewise surjectivity and Nakayama. Empty/zero-support cases are treated according to the stated pure-dimension-one convention; if no pair exists, the presentation is empty. Put $V=O_X(-mA{)}^r$ and form the finite square-zero scheme $Y={\mathrm{Spec}}_X(O_X\oplus V)$, with $V\cdot V=0$.

The actual published finite-over-projective lemma makes Y projective. A quotient V→F with kernel K defines the ideal 0⊕K in $O_Y$ and a closed subscheme Z⊂Y containing the canonical X⊂Y. Conversely X⊂Z⊂Y means its ideal is contained in V, and that ideal is exactly an $O_X$-submodule K⊂V. The quotient algebra is $O_X\oplus (V\mathrm{/}K)$, with its indicated square-zero multiplication. These constructions are inverse on ALL test schemes, not merely on closed points.

Use the actual Hilbert scheme of Y for polynomial $P_{O_X}(t)+(dt+n)$, with the finite-pullback ample polarization. The condition X⊂Z is closed: it is vanishing of the universal ideal map $I_Z\to O_X$. Its target is flat over the Hilbert base, so the exact all-test lem-hilbert-family-vanishing-locus applies. On that closed locus $O_Z=O_X\oplus F$ as $O_X$-modules. Therefore $O_Z$ is flat over a test T if and only if F is flat, and its polynomial has the required value if and only if F has polynomial dt+n. This CLOSED Hilbert locus consequently represents the full required Quot functor of V, including nonreduced and non-Noetherian tests. It carries its universal quotient. A separate representability theorem for Quot has not been assumed.

Add a section $O_X\to F$ as follows. On affine charts the nonnegative finite-free complex K of the universal F represents its global sections by Spec Sym(coker((K¹)∨→(K⁰)∨)). These affine section schemes glue by their all-test functor. Thus there is a separated finite-type scheme of quotients together with a section.

Purity is an actual open on this scheme. For any flat family F on the fixed smooth threefold, the fibre modules have projective dimension at most three. A partial free resolution has base-flat syzygies; the third syzygy is fibrewise free and hence locally free by the actual flat/fibre-free criterion. Thus relative sheaf Ext³(F,ω_X) is the cokernel at the top of a local length-three vector-bundle dual resolution and commutes with every fibre base change. Its support has proper closed image in the parameter. On a regular threefold, a one-dimensional sheaf is pure exactly when its local projective dimension is at most two at closed points, by Auslander–Buchsbaum and the associated-prime criterion. A minimal resolution shows that a finite module of projective dimension three has nonzero top Ext³. Therefore removing that support image is exactly the geometric purity open.

The cokernel of the universal section is coherent. Its support is proper over the parameter, and the locus where its fibres have dimension at most zero is open by the actual proper fibre-dimension upper-semicontinuity theorem. Support commutes topologically with the fibre by Nakayama. Intersect with this open. The resulting families are precisely stable pairs. On this locus the uniform m calculation in uniform boundedness calculation applies, so π_*F(mA) is the universally compatible rank-r bundle. The quotient's coordinate restriction $O^r\to {\pi }_{\ast }F(mA)$ is a frame on its determinant-nonvanishing open. Call that separated finite-type scheme H.

Its functor is a stable pair together with a FULL frame of π_*F(mA): a frame gives the evaluation quotient V→F, and the Quot point plus section recover the pair and its frame. These are inverse on all tests. Changing the frame gives a $G{L}_r$ action. A pair has no nonidentity automorphism: if $\phi s=s$, φ−id factors through Q into F, and Hom(Q,F)=0 by relative purity. For Noetherian tests the exact argument is the one in the installed pair-deformation lemma: choose an annihilator element regular on the fibre F, lift it, and use the fibrewise-exact-flat-complex criterion to make it regular on F. For arbitrary tests, the finitely presented family, its morphism, inverse and equalities descend to a Noetherian stage; flatness descends by the actual approximation lemma, and purity/cokernel conditions are the just-constructed opens. The same vanishing descends back. Thus the $G{L}_r$ action has trivial scheme-theoretic stabilizer, including on nilpotent tests.

Faithfully flat module descent of F and its section is effective, with the quotient/frame data descending in the usual associated bundle. Hence $[H\mathrm{/}G{L}_r]$ represents the stable-pair groupoid on all tests. Its atlas H is a $G{L}_r$ torsor over any specified family, and its diagonal is affine finitely presented: after frames of both families an isomorphism is the CLOSED transporter in $G{L}_r$, the pullback of the diagonal of separated H. Affine descent gives this relative Isom scheme before frames. All automorphisms are trivial by purity/frame construction.

For completeness an étale presentation can be extracted without assuming that a smooth quotient sheaf is a scheme. The vertical cotangent of the atlas $H\to [H\mathrm{/}G{L}_r]$ has rank r² and is the dual Lie-algebra bundle. On an affine neighbourhood choose r² regular functions whose relative differentials form a basis at the chosen point; the orbit differential is injective because its kernel is the Lie algebra of the already proved trivial stabilizer, so its dual makes those relative cotangent generators restrictions of actual functions on H. The subscheme cutting their fixed values is, after shrinking, étale over the quotient: on every scheme test the atlas is a smooth torsor and the relative Jacobian is invertible. These transverse slices cover its geometric points. Their fibre-product relation is an étale equivalence relation represented by schemes, by the transporter computation, and is a monomorphism into the product because stabilizers are trivial. The actual published étale-equivalence-relation algebraic-space theorem gives a finite-type algebraic space M. The quotient and M have identical all-test functors. The universal quotient pair on H descends to X×M; the stabilizer vanishing makes its descent isomorphisms unique. Thus M is the needed fine moduli object, constructed rather than assumed. A fine PROJECTIVE SCHEME or GIT quotient was unnecessary.

The A-degree is already fixed. For any L∈Pic(X), GRR in codimension two gives χ(F⊗L)−χ(F)=β(L), since F has rank zero, c₁=0 and leading character β. The left side is locally constant in flat proper families by the finite cohomology complex. Thus a full specified Pic-degree functional selects a union of connected components of finite-type H.

For the source's integral complex homology convention there is an integral argument, not just a rational Chern-character assertion. A one-dimensional sheaf on the smooth threefold has canonically trivial determinant off its codimension-two support, and Hartogs gives the trivialization globally. Thus c₁=0 and the integral topological identity ch₂=−c₂ applies to its leading cycle. Explicitly −c₂(F) is the integral fundamental one-cycle class: away from finitely many exceptional points on its support, intersect with a transverse smooth complex surface disk. The restriction is a finite-length sheaf with its generic multiplicity. A composition series reduces it to residue-point sheaves; their two-coordinate Koszul resolution gives −c₂ equal to the positively oriented point. The coefficient is therefore precisely that generic length. The difference class is supported at the finitely many exceptional points, and local cohomology of a real six-ball at its centre has degree six, not four, so it contributes nothing in H⁴. This proves the integral equality, including torsion.

The universal F is perfect on X×H: its base-flatness and the relative three-dimensional free-resolution argument in purity/frame construction provide the actual perfect complex. Its integral topological c₂ is a class on X×H(C). Along a path in H(C) the fibre inclusions are homotopic, so $c_2(F_t)$ is constant. Connected finite-type complex algebraic components are analytically path connected; this also follows from their finite Whitney triangulations and normalized curve arcs. Poincaré duality on the smooth projective threefold identifies these integral H⁴ classes with the required integral H₂ cycle classes. Consequently a specified β selects a union of finitely many connected components, open and closed. No identification of Pic-degree with integral homology was used.

For a perfect complex E on $Y_S$ choose a bounded vector-bundle representative; in the projective-over-affine case it is obtained by successive negative ample-power surjections and a final locally free syzygy, as in the installed pair-deformation lemma. Choose a finite affine cover trivializing its terms. Lift this cover across a nilpotent base extension by principal affine opens. Intersections are affine, and restriction of rings across the thickening is surjective. On each intersection all morphisms are therefore matrices whose entries lift.

For reference the complex can be encoded by a Čech twisting cochain a: its Čech-degree-zero component is the local differential, its degree-one component is the change of basis, and a component of Čech degree p has internal degree 1−p. The equation is δa+a²=0, with the Alexander–Whitney multiplication sign (−1)^{internal degree of the first factor times Čech degree of the second}. In this equation δ deletes internal vertices; consequently its degree-two equation for degree-zero transition matrices is $g_{ij}$ $g_{jk}-g_{ik}=0$. Initially the components of Čech degree at least two vanish, since the original vector-bundle complex has strict transitions. There are finitely many Čech degrees and finitely many internal degrees. Endomorphism cochains have differential $d_a=\delta +[a,-]$. For the initial strict presentation this is the ordinary Čech complex of the sheaf complex Hom(E,E), so its cohomology is Ext(E,E), by affine Čech comparison.

We spell out why higher twisting data are effective, rather than assuming a moduli theorem. From local complexes with a twisting cochain form the total complex of the finite sums of $j_{i_0\dots i_p,\ast }{E}_{i_p}$, using δ and the maps $a_p$ for its differential. The twisting equation is exactly the assertion that its square is zero. After restriction to $U_i$, the augmented Čech complex has the extra-vertex contraction inserting i. For strict data that contraction identifies this total complex with $E_i$. For lifted data use the same contraction and the finite perturbation formula: successive corrections either raise Čech degree or acquire a factor from the square-zero ideal. Hence the geometric series for the corrected contraction terminates. It identifies the reconstructed complex locally with the lifted finite-free $E_i$. It is therefore perfect, and the maps and isomorphisms reconstructed by the same totalization are the actual morphisms and isomorphisms. Conversely, a perfect deformation, after adding split contractible pairs, is locally a lift of these graded free modules; its local identifications and their homotopies give precisely such twisting data. The finite Čech construction and its contraction show that adding contractible pairs or refining the cover does not change its quasi-isomorphism class or the Ext cochain model. No non-effective formal gluing is used. Let $S_0\subset S$ have square-zero ideal J, and let $E_0$ be perfect on $Y_{S_0}$. Lift its finite matrix data $a_0$ arbitrarily to a on $Y_S$. All entries lift by affineness of the intersections. Its curvature c=δa+a²

reduces to zero, so belongs to the total endomorphism complex of $E_0$ with coefficients π*J, in total degree two. The identity $d_a$ $c=0$ is obtained by expanding δ(δa+a²)+a(δa+a²)−(δa+a²)a and cancelling the terms in pairs; thus c is a cocycle. Replacing the lift by a+b, with b of total degree one and coefficients in J, changes c to $c+d_{a_0}b$ because b²=0. Changes of presentations and lifted bases induce the same cohomology class: their matrix conjugation carries curvature to conjugate curvature, and a change reducing to the identity contributes an exact term. Thus c defines $o(E_0,S)\in {\mathrm{Ext}}^2(E_0,E_0\otimes {\pi }^{\ast }J)$.

Its vanishing is equivalent to the existence of a deformation. If it vanishes, choose b with $d_{a_0}b=-c$; the corrected twisting equation holds and Čech reconstruction reconstructs an actual perfect deformation. Conversely a deformation supplies a lift with zero curvature. When zero, differences of corrected lifts are degree-one cocycles. A change of identification reducing to the identity is 1+h with h of degree zero in J and changes that cocycle by $d_{a_0}h$. Their classes consequently form a torsor under ${\mathrm{Ext}}^1(E_0,E_0\otimes {\pi }^{\ast }J)$; automorphisms reducing to the identity are Ext⁰. Higher homotopies, when present, are the negative Ext groups. In the stable-pair fixed-determinant situation these negative groups vanish, so the relevant groupoids have no higher-automorphism ambiguity.

Here is a calculation of the universal class, including the relations in a singular base. On an affine base $S_0=\mathrm{Spec}$ R over B, take a free graded-commutative DG B-algebra P→R in nonpositive degrees. It is constructed by polynomial generators in degree zero, degree−1 generators killing the relations, and successive negative generators killing the remaining negative cohomology. At every stage adjoining generators kills the specified homology classes; the union is a resolution. Its cotangent model is Ω_{P/B}⊗_P R. This is the actual semi-free presentation of the cotangent complex, not an assumption of smoothness of $S_0$.

A bounded local free graded complex over R lifts to a finite free graded DG module over P with the same basis degrees. To see this, first lift the entries of its degree-one differential to P⁰. Its square is a matrix in the acyclic kernel of P→R. Correct it by coefficients of degree−1 which raise the basis degree by two. At the next stage the defect is closed by the identity D(D²)=D²D and lies in the same acyclic kernel; correct it with coefficients of the next negative degree, raising basis degree once further. This terminates because there are only finitely many basis degrees. The identical filtration performs the correction on the finite twisting-cochain presentation: Čech degree is another bounded degree. Thus the construction also applies to the global descent model of Čech reconstruction.

Lift the degree-zero polynomial generators of P to R' for the square-zero extension R'→R; send the negative generators to zero. This gives a graded algebra map σ, whose failure to commute with differentials is the derivation $\kappa =\sigma d_P$ with values in J[1]. Since $d_P^2=0$ this is a cocycle, and changing the lifts changes it by a homotopy. On degree−1 relation generators its value is exactly the lifted relation in J. Quotienting these values by changes of polynomial-generator lifts is the elementary presentation of the extension class in ${\mathrm{Ext}}^1(L_{R\mathrm{/}B},J)$: this identifies κ with the Kodaira–Spencer connecting map, with the corresponding chosen shift sign.

Choose the connection ∇ which differentiates coefficients in the P-free graded basis and is zero on that basis, relative to the fixed Y directions. The commutator with the DG differential represents the relative Atiyah class: it is the connecting map for universal first principal parts, since a splitting is a connection and its failure to be a chain map is precisely this commutator. Changing a connection changes it by a differential, so its class is independent of the connection. In the basis it is the universal differential of the matrix A of D; with the connecting convention for the obstruction it is represented by $-d_{dR}A$. Evaluating A by σ gives a lifted differential whose square is $\sigma (A{)}^2=-\sigma (d_{P}A)=-\kappa (A)$. The global representative is the TOTAL Čech connection calculation, not a collection of equal local Ext classes. On a trivializing cover use the coefficient connection on each graded free term. On an overlap with transition $g$, its connection difference is $g^{-1}{d}_{dR}g$; include this as the Čech-degree-one component. For higher twisting components include their coefficient derivatives as well, with the same Alexander–Whitney signs. Equivalently apply the universal coefficient derivation entrywise to the entire twisting matrix $a$, not just to its degree-zero local differential. The total Atiyah cocycle is $-d_{dR}a$. On the coefficient resolution the equation is $d_{P}a+\delta a+a^2=0$; the differential on coefficient forms is the one induced by $d_P$. Applying $d_{dR}$ gives $(d_{\mathrm{\Omega }}+\delta +[a,-])(d_{dR}a)=0$, with the total graded signs. Thus the cocycle assertion includes the coefficient-resolution differential and remains valid on a singular base. A change of local connection by the degree-zero form $\eta$ changes this representative by $(d_{\mathrm{\Omega }}+\delta +[a,-])\eta$; conjugating the twisting matrix by a change of bases gives exactly this connection-difference term. Thus the total cocycle is independent of the local splittings, including their overlaps. It is the first-principal-parts boundary: in each free basis the principal-parts extension is split as a graded module and its off-diagonal differential is this commutator; the overlap off-diagonal entries are precisely the displayed connection differences. This identifies the TOTAL cocycle, and not only its local restrictions, with the canonical Atiyah arrow.

Contracting the displayed Atiyah matrix with κ gives exactly −κ(A). The same identity for total Čech matrices includes transition and higher-homotopy entries with their Alexander–Whitney signs; it is a single computation of the square of their total differential. Hence the curvature class in curvature construction is the product $o(E_0,S)=(id\otimes \kappa )\circ At(E_0\mathrm{/}Y)$. For the lifted total differential write $D=d_P+\delta +a$. The single equality $D^2=0$ over the coefficient resolution says $d_{P}a+\delta a+a^2=0$. Evaluating coefficients by $\sigma$ kills $d_P$ on the target but does not kill its matrix values: ${\tilde{D}}^2=\delta \sigma (a)+\sigma (a{)}^2=-\sigma (d_{P}a)=-\kappa (a)$. Contracting the TOTAL Atiyah cocycle of step27.1 with $\kappa$ is the same matrix $-\kappa (a)$, including transition and higher twisting entries. Consequently the equality holds in global hyper-Ext with coefficients in every coherent $J$, without an inference from equality after restriction to affine opens. Refining the cover or adding a contractible pair carries both sides by the same totalization comparison; the contraction identities of step21.1 carry their boundary homotopies as well.

One may reverse both conventional connecting signs; the resulting POT map is conjugated by multiplication by −1, an isomorphism of its source complex. Thus this calculation fixes the convention without an unproved sign assertion. Flatness of Y/B identifies the ambient base-extension Kodaira–Spencer class with ${\pi }^{\ast }\kappa (S_0\mathrm{/}S\mathrm{/}B)$; this also follows by tensoring the same free DG resolution with its base-flat affine coordinate rings.

The determinant line of a finite graded free complex is the alternating tensor product of its determinant lines. On a square-zero change of transition matrix, det(1+h)=1+tr(h); the graded alternating product gives supertrace. On a connection, d log $\det (g)=tr(g^{-1}dg)$. Thus its first-principal-parts class is tr At(E), and the scalar curvature of its determinant gluing is tr c. The preceding matrix identities prove both assertions, including transition entries; no independent trace-compatibility theorem is assumed. For a rank-one complex, identity followed by trace is multiplication by one, so RHom(E,E)=O⊕RHom(E,E)_0.

If its determinant is the specified trivial line, the determinant obstruction vanishes and o lies in Ext²_0. Vanishing of that trace-free class is sufficient: after a perfect lift exists, tensor it by the inverse of its determinant. Since its virtual rank is one, the new determinant is det(E)⊗det(E)^{-1}=O and its reduction has the required fixed determinant. The variation of determinant on the Ext¹ torsor is trace by the same first-order determinant calculation, so fixed-determinant lift classes form exactly the Ext¹_0 torsor. The fixed-determinant automorphisms are Ext⁰_0. In particular the universal Atiyah class of a family with fixed trivial determinant is trace free.

Let M be the fine stable-pair algebraic space just constructed and I its descended universal complex. All cotangent/POT tests are made on affine étale scheme charts of M, where its cotangent complex pulls back to the chart cotangent complex. The installed pair/complex deformation lemma identifies all its nilpotent deformations with the fixed-determinant deformations above. Put $Q=R{\pi }_{\ast }\mathrm{RHom}(I,I{)}_0$. Relative Serre duality gives the adjoint of the trace-free base Atiyah class as $E=R{\pi }_{\ast }(\mathrm{RHom}(I,I{)}_0\otimes {\omega }_{\pi })[2]$ → $L_{M\mathrm{/}B}$. The universal Atiyah arrow is defined globally by the DERIVED principal-parts boundary: resolve the coefficient structure algebra simplicially by free algebras, take its diagonal ideal modulo its square levelwise, and totalize. Its coefficient module is the cotangent complex, not merely the sheaf of differentials. Restriction maps and the simplicial face and degeneracy maps induce the comparison and coherence maps of this construction. The semi-free DG calculation above is its normalized characteristic-zero model. This defines the arrow globally, and the total cocycles above compute that same arrow on every affine étale chart. It therefore descends as a morphism, rather than by an attempted gluing of unrelated local Ext classes. The determinant calculation identifies its trace with the principal-parts boundary of $\det I$; the specified trivialization makes that boundary nullhomotopic. The rank-one scalar/trace splitting consequently gives its trace-free factor. Duality transports this GLOBAL morphism to the displayed $E\to L_{M\mathrm{/}B}$. On an étale overlap the universal pair, principal-parts boundary, trace splitting and duality trace all pull back to the same constructions. Their descent is inherited from these global maps and their specified determinant nullhomotopy.

Under this duality, Ext⁰(E,J)=Ext¹(I,I⊗πJ)_0 and Ext¹(E,J)=Ext²(I,I⊗πJ)_0. The product of this map with a square-zero Kodaira–Spencer class is therefore exactly the obstruction just proved, and its lift differences give exactly the same tangent torsor.

For clarity the cotangent criterion here need not be invoked just by name. It suffices throughout this criterion to test coherent modules J on Noetherian affine charts, so their square-zero extensions are again Noetherian and the installed pair-deformation lemma applies. Tangent maps for all these finite square-zero modules identify Hom(h⁰L,J) and Hom(h⁰E,J), hence h⁰(E)→h⁰(L) is an isomorphism. On an affine étale chart of M, every class of ${\mathrm{Ext}}^1(L_{M\mathrm{/}B},J)$ represents a square-zero B-algebra extension of M by J: in the free DG presentation in free-DG calculation its relation values are exactly a cocycle κ, and changes of generator lifts are the coboundaries. If such a class maps to zero in Ext¹(E,J), curvature construction and the pair-equivalence lemma extend the universal pair to that extension; the fine-moduli classifying map extends the original chart map to M. Formal étaleness of that chart lifts this map uniquely back to the chart, yielding a retraction of its square-zero extension. The algebra extension consequently splits, so the class is zero. Thus Ext¹(L,J)→Ext¹(E,J) is injective for every such J. For K=Cone(E→L), $h^{0}K=0$ and ${\mathrm{Ext}}^1(K,J)=\mathrm{Hom}(h^{-1}K,J)$. The long exact Hom sequence and the tangent isomorphism embed this group into the kernel just proved zero. Taking $J=h^{-1}K$ forces $h^{-1}K=0$. Therefore $h^{-1}(E)\to h^{-1}(L)$ is surjective. the relative-duality and Ext-amplitude calculations supply the amplitude calculation and relative duality interface, so this is exactly a POT, without citing an obstruction-compatibility theorem as an unproved premise.

Here is the needed extension of the published regular-local-base residue proof; we do not apply that published Statement outside its hypotheses. Let A be ANY Noetherian local coefficient ring and $P=P_A^N$. For a vector bundle F on P, take n₀ sufficiently large that $H^q(P,F(n))=0$ for $q>0$ and n≥n₀, by the actual projective coherent Serre-vanishing supplier. The published proper-flat finite-projective complex computes each such F(n) in nonnegative degrees. A bounded nonnegative projective complex with no positive cohomology splits from its highest degree downward: the last surjection onto its projective top term splits, and induction cancels it. Consequently H⁰(P,F(n)) is finite projective over A and commutes with every coefficient change. This uses actual vanishing plus the nonnegative complex, rather than assuming base change from fibrewise ranks.

The truncated graded section module M=⊕_{n≥n₀}H⁰(P,F(n)) is finite over $S=A[x_0,\dots ,x_N]$, by the existing graded-section finiteness theorem, and is flat over A by the preceding calculation. Resolve it by finite graded free S-modules. All syzygies remain A-flat. After N+1 steps the syzygy K has projective fibres over every $\kappa (q)[x_0,\dots ,x_N]$, since that polynomial ring over a field has global dimension N+1. At every prime of S, the pinned flat/fibre-free local criterion makes K locally free. Thus it is finite projective over S. The quotient K/S₊K is a finite graded projective A-module; each of its finitely many homogeneous pieces is free because A is local. Lift their homogeneous bases. The resulting finite graded free cover G→K is an isomorphism: graded Nakayama makes it surjective, projectivity of K splits its kernel sequence, and reduction modulo S₊ followed by graded Nakayama makes the kernel zero. Sheafification therefore gives a finite resolution of F by finite sums of twists $O_P(m)$, without regularity of A. Every perfect complex on P consequently belongs to the triangulated subcategory generated by those twists: use its bounded vector-bundle representative and the finite resolutions just constructed.

The Laurent Čech residue calculation for a twist works over EVERY coefficient ring. Its standard-cover complex decomposes by exponent vectors: mixed-sign vectors give contractible summands, all-nonnegative vectors give H⁰, and all-negative vectors give $H^N$. Multiplication pairs $x^a$ with $x^{-a-1}$, whose product has residue one. Thus for a twist the natural evaluation map gives $R\mathrm{\Gamma }(P,\mathrm{RHom}(K,O_P(-N-1)[N]))$ ≅ ${\mathrm{RHom}}_A(R\mathrm{\Gamma }(P,K),A)$.

It respects multiplication by every homogeneous section, hence is the actual natural chain pairing, not an arbitrarily chosen duality of cohomology groups. Finite sums, shifts, retracts and cones preserve this natural isomorphism, so the preceding finite-twist construction proves it for every perfect K. This is precisely the needed all-coefficient-ring replacement of the regular-base restriction in lem-relative-projective-space-derived-duality-regular-local-base; it does not assert that arbitrary coherent sheaves over singular A are perfect.

Embed the smooth projective relative threefold $Y_A$ in $P_A^N$. This immersion is regular of codimension N−3, including for singular A. In each fibre its ideal is generated by a regular sequence of that length, since both fibre schemes are smooth and its conormal rank is N−3. Lift these generators. Flatness of $Y_A$ and $P_A^N$ gives the exact ideal fibre sequence; Nakayama makes the lifts generate the total ideal. Their Koszul complex is exact on fibres, and its finite free terms are A-flat. The actual fibrewise-exact-flat-complex lemma makes it exact in the total local ring. Thus this is a regular sequence. Its dual Koszul complex computes $i!O_P(-N-1)[N]$ ≅ ${\omega }_{Y_A\mathrm{/}A}[3]$.

The identification of the line is the determinant of the conormal exact sequence. Koszul comparison for two generator sets is their determinant transition and composes correctly, so it glues and commutes with every base change. For a perfect K on $Y_A$, $i_{\ast }K$ is perfect on $P_A^N$ by the same Koszul resolution. Apply the actual finite-map derived coinduction adjunction and the just-proved projective-space pairing to get $R\mathrm{\Gamma }(Y_A,\mathrm{RHom}(K,\omega [3]))$ ≅ ${\mathrm{RHom}}_A(R\mathrm{\Gamma }(Y_A,K),A)$.

This is the relative duality and trace for the supplied embedding. Every complex in question is perfect over A: represent K by finitely many vector bundles and use the pinned proper-flat nonnegative complex on each one. Tensoring the pairing with an arbitrary A-module J, and using the finite affine Čech complexes and their flat terms to move derived tensor through finite totalizations, proves the same formula with π*J. In particular it proves exactly the Hom/Ext identifications in cotangent/POT test on the singular moduli base. The trace and evaluation are natural, and Koszul/conormal comparison identifies embeddings through their common product, so no choice of projective embedding changes the resulting adjunction map.

For a stable-pair fibre I, the triangle I→O→F gives Hom(I,O)=O and ${\mathrm{Ext}}^{<0}(I,O)=0$ as SHEAVES: Hom(F,O) and Ext¹(F,O) vanish by codimension two on the regular threefold. To verify these codimension vanishings directly, at each support prime choose a two-element O-regular sequence in its support ideal by prime avoidance; powers remain regular. For any sheaf annihilated by such a power, the two successive multiplication exact sequences make Ext⁰ and Ext¹ into O zero. The triangle H⁰I→I→Q[−1] likewise identifies ${\mathrm{Ext}}^{-1}(I,F)$ with Hom(Q,F)=0 by purity, and gives lower negative vanishing. Applying Hom(I,−) to I→O→F therefore gives sheaf Hom(I,I)=O and ${\mathrm{Ext}}^{<0}(I,I)=0$. Tensoring by the line ω gives the same assertions with target I⊗ω.

The local-to-global spectral sequence gives global negative Ext vanishing and Hom(I,I⊗ω)=H⁰(ω). The just-proved threefold duality identifies Ext³(I,I) with the dual of H⁰(ω); its trace is dual to the scalar inclusion, hence an isomorphism onto H³(O). Higher Ext vanish by the same duality and negative Ext vanishing for the ω twist. Trace-free Ext⁰ also vanishes since scalar identity followed by trace is rank(I)=1. Thus ${\mathrm{Ext}}^i(I,I{)}_0=0$ outside $i=1,2$.

Proper pushforward of the perfect endomorphism complex is perfect by the vector-bundle/cohomology construction in relative-duality proof. Locally a finite-free representative for Q can be trimmed to degrees one and two: on residue-field fibres the extraneous cohomology groups vanish, so upper endpoint maps are surjective with bundle kernels and lower endpoint maps are injective with bundle cokernels; cancel them finitely. Relative duality consequently identifies E with (Q[1])∨, of amplitude [−1,0]. cotangent/POT test now proves both POT axioms. This establishes the actual pair-specific RELATIVE perfect obstruction theory from its universal Atiyah class, retaining the nilpotent-deformation equivalence and trace compatibility rather than assuming them. The absolute case is B=Spec C. For a smooth parameter base B, form the homotopy pullback $E_{abs}=E{\times }_{L_{M\mathrm{/}B}}^h{L}_M$, using the cotangent triangle ${\pi }^{\ast }{\mathrm{\Omega }}_B\to L_M\to L_{M\mathrm{/}B}$. Its triangle ${\pi }^{\ast }{\mathrm{\Omega }}_B\to E_{abs}\to E$ shows directly that it is perfect of amplitude [−1,0]. The identity on π*Ω_B and the relative $h^0$-isomorphism/$h^{-1}$-surjection show the same two properties for $E_{abs}\to L_M$ by the long cohomology sequences. This is the absolute obstruction theory on the total moduli, with virtual rank increased by dim B.

The parameter H is quasi-projective with a $G{L}_r$-linearized ample line. The square-zero Hilbert construction has the coordinate $G{L}_r$-action; its Grassmannian/Plücker embedding has its natural linearization. The section scheme is affine of finite type over that projective scheme, so the pullback of an ample line is ample, with its induced linearization; restrict to H. For $U_N\subset \mathrm{Hom}(C^r,C^N)$ the full-rank open, codimension of its complement is N−r+1. The quotient $M_N=(H\times U_N)\mathrm{/}G{L}_r$

is a quasi-projective SCHEME: it is associated to the frame torsor over Gr(r,N), and the equivariant immersion of H gives its locally closed immersion into the associated projective bundle. It is the full-rank open in the rank-Nr vector bundle on the fine space M associated to its genuine frame bundle. The map $M_N\to M$ is smooth of relative dimension Nr.

On an algebraic space with trivial stabilizers the integral cycle/rational-equivalence convention has the same localization and vector-bundle homotopy as on schemes, with no division. Here is the needed reduction. The actual generic quotient theorem applied to an étale presentation supplies a dense open scheme in every integral closed subspace. On that dense open the ordinary scheme homotopy proof moves a vector-bundle cycle to a multiple of its generic affine-space fibre. All its finitely many rational functions spread over a smaller dense open. Localization leaves a cycle with lower-dimensional image; induction proves surjectivity of bundle pullback. Zero-section normalization supplies injectivity because its global normal cone is the identity bundle, with multiplicity one. No finite residual gerbe or stabilizer cover occurs, so this is integral. Localization follows directly by closure of cycles and rational functions. It gives, when N−r+1>dim M−i, $A_i(M,Z)$ ≅ $A_{i+Nr}(M_N,Z)$.

For two N use the common open $(H\times U_N\times U_L)\mathrm{/}G{L}_r$ in the corresponding associated vector bundles. Localization outside the increasing-codimension complements and bundle homotopy give the canonical comparison and its triple-product cocycle. All Chern, zero-section and regular Gysin operations used here are the scheme operations transported through that comparison.

Choose a two-affine-torsor replacement $T\to M_N$ and let p:T→M be its smooth composite. On affine T its projective module Ω_p has no morphism into $p^{\ast }{L}_M[1]$, since $L_M$ has no positive cohomology. Thus the cotangent triangle splits, and $E_T=p^{\ast }E\oplus {\mathrm{\Omega }}_p\to L_T$ is a POT. Two splittings differ by a map ${\mathrm{\Omega }}_p\to p^{\ast }{L}_M$, which lifts through p*E because $Cone(E\to L_M)$ has cohomology at most degree −2; the corresponding triangular source automorphism identifies the two POTs. This gives the actual well-defined smooth theory on T without assuming a global splitting on $M_N$. It has a strict conormal model and a global two-term projective resolution on this affine scheme. The actual replacement is explicit: first pull back the nonincidence affine bundle in $P^m\times (P^m{)}^{\vee }$; its pullback to the locally closed $M_N$ is quasi-affine. If its affine closure has complement $V(f_1,\dots ,f_t)$, the scheme $\sum f_j$ $z_j=1$ over that closure is an affine torsor over the quasi-affine open. Projective compactification P(W)\P(V) for each bundle torsor, together with projective-bundle formula and localization, proves its integral Chow pullback is an isomorphism.

The cotangent sequence of a smooth affine replacement splits because its relative cotangent is projective and the derived target has no positive cohomology. A two-term projective POT [E⁻¹→E⁰] on the affine replacement can be made strict [F→Ω_P] with Ω-degree map the identity: lift Ω_P→h⁰E to k:Ω_P→E⁰, set F=ker(E⁻¹⊕Ω_P→E⁰), (u,w)↦du+kw, and correct its map to [I/I²→Ω_P] by a homotopy h with $d_{I}h={\phi }^{0}k-id$. The conormal map is (u,w)↦−φ⁻¹u−hw and is surjective by the POT axioms. Thus the normal cone embeds in F∨ and its integral zero-section class is defined on that actual affine scheme.

Independence is explicit. For two global bundle resolutions F,H of the same obstruction stack ℰ, $P=F_1{\times }_{\mathcal{E}}$ $H_1$ is a vector-bundle torsor over each; inverse pullback for the SAME composite P→base applied to the common flat-pulled cone gives the same zero-section class. On affine replacements strict quasi-isomorphisms and their homotopies represent all specified derived comparisons, so this also compares the smooth cotangent splittings and ambient choices. The intrinsic cone under a smooth replacement is obtained from the pulled-back normal-cone chart by the added relative tangent directions; this is an equality of cone algebras and tangent actions on their common product charts. It consequently gives exactly the smooth pullback of the virtual class. The common matrix replacements therefore yield compatible classes in $A_{c\beta +Nr}(M_N,Z)$, hence an actual integral class in $A_{c\beta }(M,Z)$. This removes the need to assume either projectivity of M or a global POT resolution on M itself. Its virtual rank is the installed trace-free GRR lemma. The identical good-pair/affine construction transports the published scheme refined-Gysin identities for the relative-family proof. To make the first comparison precise, let $r_F=\mathrm{rk}{F}_1$, $r_H=\mathrm{rk}{H}_1$, and let $s_F,s_H$ be the ranks of the degree-zero bundles in the two dual presentations. The 2-fibre product $P=F_1{\times }_{\mathcal{E}}{H}_1$ is an affine-bundle torsor of rank $s_H$ over $F_1$ and of rank $s_F$ over $H_1$; the equality of virtual ranks gives $r_F+s_H=r_H+s_F$. Write $q_F,q_H$ for these maps and $p:P\to M$ for their common composite. Bundle homotopy and affine-torsor homotopy make $p^{\ast }$ an isomorphism. If ${\beta }_F=0_{F_1}^{!}[C_F]$, then bundle homotopy gives $p^{\ast }{\beta }_F=q_F^{\ast }[C_F]$. The same equation holds for $H$. These two cone cycles are equal: each is the flat pullback of the SAME intrinsic cone along $P\to \mathcal{E}$. Hence $p^{\ast }{\beta }_F=p^{\ast }{\beta }_H$, and injectivity gives ${\beta }_F={\beta }_H$, integrally. For a smooth replacement $t:T\to M$, the local cotangent splitting presents the new obstruction stack as the old pulled-back stack with the relative tangent in its quotient term; the intrinsic cone has that same extra tangent quotient. Pulling both to the chosen degree-one bundle therefore gives exactly $t^{\ast }{C}_F$, with its scheme-theoretic cone algebra and multiplicities. This proves $[T,E_T{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=t^{\ast }[M,E{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$ in the scheme comparisons, and explains why the relevant pullback is smooth/flat rather than an arbitrary nonflat cone base change.

For the relative theorem take the source's smooth separated finite-type complex parameter B. All preceding boundedness and square-zero Hilbert constructions are relative over B: the support degree bound is fibrewise and independent of the embedding dimension, the finite union of relative Hilbert schemes is projective over B, and finitely many affine base charts give a uniform m. This constructs the relative finite-type fine algebraic space with universal pair. We do not assume that a smooth separated B is quasi-projective.

Here is an explicit additional framing which makes the same good-pair Chow construction available globally. Work on each of the finitely many open-and-closed integral components of regular B. For an affine open U⊂B its complement is pure codimension one. Indeed if p were the generic point of a higher-codimension complement component, shrink an affine neighbourhood V of p to remove the other complement components. Then V∖(B∖U)=V∩U is affine by separatedness. Normal Hartogs across this codimension-at-least-two complement identifies its functions with Γ(V,O); both affine schemes would therefore be equal, a contradiction. Since regular local rings are factorial, the reduced codimension-one complement is an effective Cartier divisor D. Thus $U=B_s$ for the canonical section s of O(D).

Choose a finite affine cover $U_i=B_{s_i}$ of this kind and put $L_i=O(D_i)$. Let B'=∏_B $L_i^{\times }$ be the scheme of simultaneous nonzero frames of these lines, a torus $T=(G_m{)}^a$ torsor. Pullback turns the $s_i$ into global functions $f_i$ on B'. Their principal opens cover B', and each is affine: it is a product of line-frame affine torsors over the affine $U_i$. A quasi-compact separated scheme with a finite affine principal-open cover by global functions is quasi-affine. Explicitly global-section localization identifies $\mathrm{\Gamma }(B^{\mathrm{\prime }},O{)}_{f_i}$ with the coordinate ring of that principal open. Choose finitely many numerators generating these rings and the $f_i$, and let A₀ be the resulting finite-type C-subalgebra of Γ(B',O). Its localization at each $f_i$ is exactly that coordinate ring. Hence B' is the open union of these $D(f_i)$ in Spec A₀. This proves the required finite-type quasi-affine model, not just a formal ample-family assertion.

Pull the relative framed parameter H to B'. It is quasi-projective over this quasi-affine scheme, hence quasi-projective over C, with an ample line linearized for $G{L}_r\times T$. One can see the linearization directly: its relative Hilbert Plücker line and the square-zero coordinate action are canonical; multiply by the pullbacks of the framed base lines as needed. On the quasi-affine base their global generator sections have finite-dimensional $G{L}_r\times T$ spans, by the finite coaction formula for rational representations, giving the equivariant projective embedding used for the good pairs. The resulting fine relative moduli is $[H_{B^{\mathrm{\prime }}}\mathrm{/}(G{L}_r\times T)]$.

The action is free: a stabilizer first fixes all the base line frames, so its torus coordinate is one; its remaining $G{L}_r$ coordinate is a pair automorphism and is one. Its universal pair descends from the same construction. For T take the product good pair $(C^N\setminus \{0\}{)}^a$, with quotient $(P^{N-1}{)}^a$ and complement codimension N. Combine it with the $G{L}_r$ full-rank pair. All replacements are quasi-projective schemes associated over $\mathrm{Gr}(r,N)\times (P^{N-1}{)}^a$, with complement codimension tending to infinity. The same integral bundle-homotopy/localization comparison and affine-torsor cone construction of integral Chow construction therefore apply to the GLOBAL relative moduli over B. This constructs the integral relative virtual class without assuming B quasi-projective or a global two-term resolution on the algebraic space itself. Smooth-family pullback and the compatible base-square Gysin calculation can then be proved on common scheme replacements and transported through their injective integral Chow comparisons.

The additional actual published interfaces used here are regular-local factoriality, normal height-one Hartogs, Cartier–Weil comparison, section extension from nonvanishing opens, localization of quasicoherent global sections on quasi-compact separated schemes, and the existing finite field/group-scheme affine descent. Every occurrence of quasi-projectivity in the good-pair argument is now on the explicitly constructed B' replacement, not imposed on the original smooth parameter B.

We have therefore constructed the fine finite-type stable-pair algebraic space, its universal fixed-determinant complex, the actual relative and absolute perfect obstruction theories, and a canonical integral virtual class by compatible scheme cone replacements. The good-pair/affine-torsor comparisons and common-cone argument prove independence of every chosen frame, replacement, embedding and two-term resolution. The installed trace-free Euler-characteristic lemma [F1] gives its dimension $c_{\beta }={\int }_{\beta }{c}_1(T_X)=-\chi (\mathrm{RHom}(I,I{)}_0)$, so the absolute class belongs to $A_{c_{\beta }}(P_n(X,\beta ),Z)$, as asserted. No weighted Euler-index or external pair-specific POT theorem is a premise.