Equivariant K-theoretic virtual structure sheaf and determinant twist

The intrinsic cone and its inverse image are equivariant without a choice of unrelated cone actions. The action isomorphism $a^{\ast }{E}^{\bullet }\simeq {\mathrm{pr}}_2^{\ast }{E}^{\bullet }$ over $T\times M$ is an isomorphism of obstruction theories over the pulled cotangent complex. The intrinsic-cone charts use associated graded ideals, which commute with flat base change; applying that isomorphism identifies the two pulled cone charts. The group-action cocycle identifies these maps on $T\times T\times M$. Thus the inverse-image cone ideal descends with an equivariant structure in the chosen equivariant bundle $\mathrm{O}\mathrm{b}\mathrm{s}$. The zero section, tautological Koszul differential and resulting homology are equivariant. Coherence and finite Koszul length consequently give the asserted class in $G_0^T(M)$ for the chosen presentation.

The following stabilization calculation verifies one necessary presentation comparison explicitly. Adding an equivariant contractible summand $[V\stackrel{1}{\to }V]$ adds $V^{\vee }$ to both dual terms. The inverse-image cone becomes $C{\times }_M{V}^{\vee }$ in $\mathrm{O}\mathrm{b}\mathrm{s}\oplus V^{\vee }$: the added denominator acts transitively by translations on the added numerator. Its zero-section Koszul complex is the tensor product of the old complex with the coordinate Koszul complex on $V^{\vee }$. The latter resolves its zero section and stays exact after tensoring with ${\mathcal{O}}_C$ because its polynomial variables form a regular sequence over every base ring. Its derived restriction is therefore ${\mathcal{O}}_C$ in degree zero. The two alternating homology classes agree. The determinant factors also cancel, since the same $\det V$ occurs once with each exponent. This proves invariance under equivariant contractible stabilization. The following common-atlas argument proves the arbitrary-presentation comparison as well; DM virtual localization is proved in the subsequent constructions.

For arbitrary two equivariant presentations of the same obstruction theory over $L_M$, write the canonically identified linear Picard stack as $\mathcal{E}=[F_1\mathrm{/}{F}_0]=[G_1\mathrm{/}{G}_0]$. Form the fibre product $P=F_1{\times }_{\mathcal{E}}{G}_1$. As a sheaf this consists of pairs of numerator sections together with an isomorphism between their images. Addition and multiplication by local functions are induced by the linear Picard structure, so $P$ is a sheaf of ${\mathcal{O}}_M$-modules. The first projection has kernel $G_0$: an object of its zero fibre is a section $g$ of $G_1$ and an isomorphism from zero to $g$, exactly the section of $G_0$ giving that translation. The projection is locally surjective because $G_1\to \mathcal{E}$ is an atlas. Thus $0\to G_0\to P\to F_1\to 0$ is an exact sequence of sheaves of modules. Locally lift a finite basis of $F_1$ through this epimorphism; extending the lifts linearly splits the sequence. Hence $P$ is a vector bundle. The other projection gives $0\to F_0\to P\to G_1\to 0$. Both constructions are intrinsic and equivariant; no equivariant splitting is required.

Pull back the same intrinsic cone $\mathcal{C}\hookrightarrow \mathcal{E}$ to $P$, obtaining one closed cone $C_P$. Its inverse-image descriptions along the two projections are equal scheme-theoretically by their common fibre-product definition. If $q:P\to F_1$ is the first vector-bundle epimorphism, it is flat and ${\mathcal{O}}_{C_P}=q^{\ast }{\mathcal{O}}_{C_F}$. The zero sections satisfy $q\circ 0_P=0_{F_1}$. Associativity of derived tensor therefore gives the natural global equivariant isomorphism $L{0}_P^{\ast }{\mathcal{O}}_{C_P}=L{0}_P^{\ast }L{q}^{\ast }{\mathcal{O}}_{C_F}\simeq L{0}_{F_1}^{\ast }{\mathcal{O}}_{C_F}$. Every zero-section pullback here has finite Tor amplitude, computed by its finite coordinate Koszul complex. The same argument for $P\to G_1$ identifies this object with $L{0}_{G_1}^{\ast }{\mathcal{O}}_{C_G}$. Taking alternating homology classes proves equality in $G_0^T(M)$ for arbitrary equivariant global presentations, rather than gluing an equality of $G_0$ classes from an atlas. The compatible isomorphism of obstruction theories over $L_M$ is essential: the argument compares the same embedded intrinsic cone, not unrelated POTs. The two exact bundle sequences also give $\det F_1\otimes (\det F_0{)}^{-1}\cong \det G_1\otimes (\det G_0{)}^{-1}$; this is the same virtual determinant comparison, with duals as prescribed in the Definition.

Saturation and independence of the fixed datum. Here is the exact finite-isogeny argument. At a geometric coarse-fixed object $x$, the group of pairs $(t,x\simeq t\cdot x)$ is an algebraic group extension of $T$ by the finite inertia group $G_x$. Its connected component $H$ maps onto $T$ with finite kernel of order at most $D$. Conjugation of connected $H$ on finite $G_x$ is constant, hence trivial. The commutator in $H$ is consequently a map into a finite group and is trivial; there is no positive-dimensional unipotent part because its image in $T$ would be trivial and the kernel is finite. Thus $H$ is a torus and $H\to T$ is an isogeny. Its cocharacter-lattice cokernel has exponent dividing its kernel order $e\le D$. Since $e\mid N$, the power map $[N]:T_N\to T$ factors through $H$, uniquely because a connected torus has no nontrivial map to a finite group. This gives a unique coherent $T_N$-fixed structure at every geometric coarse-fixed object. Conversely a fixed structure makes its coarse orbit a point. Thus the geometric image is exactly the coarse fixed set, and the complement is the geometric moving complement required below.

For families, evaluation is fully faithful. The discrepancy between two fixed structures, or between an underlying isomorphism and its transported one, is a section of the finite unramified inertia over the connected torus times the base. It is the identity at the torus identity. On geometric fibres it is therefore constant and equal to the identity. Two sections of an unramified separated space with equal geometric values are equal: their equalizer is an open-and-closed immersion containing every geometric point. Consequently the discrepancy is identically trivial, including over nonreduced bases. Evaluation is a monomorphism. It is proper because $F_N$ is proper and $M$ separated, by its graph and the proper projection, so it is a closed immersion.

If $N$ is replaced by $Nk$, the unique geometric lift through $H$ has cocharacter exponent $Nk\mathrm{/}e$. Its restriction to ${\mu }_k=\ker (T_{Nk}\to T_N)$ is trivial, since $e\mid N$. Hence the kernel of the further isogeny acts trivially on every fixed family: it is trivial on all geometric fibres, and finite unramified inertia gives equality on the family as above. Fppf descent of the fixed coherence along that isogeny gives a unique $T_N$-fixed structure. Thus $F_{Nk}\simeq F_N$ as functors on all schemes, rather than only as sets of geometric points. A common divisible refinement compares any two allowed integers.

Finite-isogeny comparison: actual equivariant coherent categories. Let $q:T^{\mathrm{\prime }}\to T$ be an isogeny of complex algebraic tori with finite diagonalizable kernel $K$, and pull the specified $T$ action on $M$ back along $q$. The kernel acts through the identity of $T$ with the canonical coherence supplied by this pullback. Thus a $T^{\mathrm{\prime }}$-equivariant coherent sheaf has a canonical $K$ action as an ${\mathcal{O}}_M$-module. Its character projectors split it as a finite direct sum indexed by $X^{\ast }(K)$. The projectors are exact: after adjoining the roots of unity in $\mathbb{C}$ they are the finite sums $\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\sum }_{k\in K}\kappa (k{)}^{-1}k$. All summands are coherent and preserved by $T^{\mathrm{\prime }}$ because the tori commute.

The trivial-$K$ summand category is equivalent to ${\mathrm{Coh}}^T(M)$. To see this directly, a linearization over $T^{\mathrm{\prime }}\times M$ with trivial $K$ action identifies its two pullbacks to $(T^{\mathrm{\prime }}{\times }_T{T}^{\mathrm{\prime }})\times M=K\times T^{\mathrm{\prime }}\times M$ by the identity on the kernel factor. Faithfully flat descent along the finite flat $T^{\mathrm{\prime }}\to T$ therefore gives a unique linearization over $T\times M$; the cocycle descends because it holds after the same faithfully flat pullback. Conversely every $T$ linearization pulls back to one with trivial $K$ action. These functors are exact inverses.

Every character $\kappa$ of $K$ extends to a character $\tilde{\kappa }$ of $T^{\mathrm{\prime }}$ by the exact sequence of character lattices. Tensoring by its inverse identifies the $\kappa$-summand category with the trivial summand category. Choose one lattice representative for each character of $K$. The corresponding Laurent monomials form a basis of $R(T^{\mathrm{\prime }})$ over $R(T)$; the preceding exact category decompositions consequently prove $R(T^{\mathrm{\prime }}){\otimes }_{R(T)}{G}_0^T(M)\stackrel{\sim }{\to }{G}_0^{T^{\mathrm{\prime }}}(M)$, where $a\otimes [F]$ maps to the character twist of the pulled-back sheaf. In particular this is a finite free, faithfully flat scalar extension, and restriction on $G_0$ is injective.

Character localization is compatible with this comparison. If $\rho \mathrm{\ne }1$ is a character of $T^{\mathrm{\prime }}$, a positive multiple ${\rho }^n$ descends to $T$. Inverting $1-{\rho }^n$ makes $1-\rho$ invertible, since $(1-\rho )(1+\rho +\cdots +{\rho }^{n-1})=1-{\rho }^n$. Thus localizing the two character rings at all nontrivial characters gives the same scalar extension as first extending and then localizing at the $T^{\mathrm{\prime }}$ characters. It remains faithfully flat. Any equality between specified $T$-equivariant classes can therefore be checked after the isogeny.

Character torsion on the geometric moving complement. Let $M$ be the specified separated finite-type complex DM stack with torus $T$ action, and let $U$ contain no point whose coarse $T$ orbit is a point. This is the geometric moving complement, independent of whether an honest fixed structure exists before an isogeny. Let $S\subset R(T)$ be generated by all $1-\chi$, $\chi \mathrm{\ne }1$. We prove $S^{-1}{G}_0^T(U)=0$.

Write $\mathcal{Y}=[U\mathrm{/}T]$ and take any nonempty reduced integral closed support $W\subset \mathcal{Y}$. Denote the successive dense restrictions below by $W^{\circ }$. The following construction supplies the two quotient steps at exactly this characteristic-zero DM/torus scope. A gerbe over a sheaf means that objects exist fppf locally and that any two objects with the same sheaf image are isomorphic fppf locally. Every quotient below is fppf sheafification of the orbit presheaf. We use the ordinary scheme and étale-space framework of DM stacks: an actual étale equivalence-relation presentation gives its algebraic space; algebraic-space overlaps may be replaced by scheme étale charts in the same presentation. This foundational étale construction, the Jacobian criterion and generic flatness are retained. Neither the general flat-equivalence quotient Bootstrap 04S6 nor the general flat free-action quotient Bootstrap 06PH is an assumed supplier; both needed instances are constructed here by explicit smooth transverse charts. (a) The diagonal of $\mathcal{Y}$ is affine of finite type. Over two objects, its isomorphisms are pairs of a torus element and an isomorphism in $U$; the latter is a base change of the finite diagonal of $U$. The transporter is therefore finite over the corresponding torus, and affine. These assertions descend over local trivializations of the torus torsors defining the two objects. The same applies to $W$. Choose a finite affine smooth atlas $A_0\to W$. Then $R_0=A_0{\times }_W{A}_0$ is a scheme, since it is affine over the scheme $A_0\times A_0$. Its projections are smooth. The pullback of the reduced stack $W$ is reduced, because smooth morphisms have geometrically reduced fibres. (b) Shrink $W$ to its nonempty smooth locus. Here is why that locus is well defined and dense in this setting, without assuming the integral support was initially smooth. The reduced finite-type scheme $A_0$ has a dense open smooth locus over $\mathbb{C}$. This locus is groupoid invariant: for an arrow with source $a$ and target $b$, smoothness of both projections identifies regularity at $a$ and $b$. Equivalently, smooth local completions are formal power-series extensions after separable residue-field extension; both dimension and embedding dimension increase by the same number, so regularity holds at one end exactly when it holds on the arrow and at the other end. In characteristic zero regular finite-type local rings are smooth. Thus the smooth locus descends to a dense open of $W$. Restrict the atlas and choose finitely many affine opens covering it. From now on $A$ and $R=A{\times }_{W}A$ are smooth finite-type schemes over $\mathbb{C}$. (c) Let $G\to A$ be the stabilizer group scheme, the pullback of the inertia. It is affine and of finite presentation. There is a dense open of $A$ on which it is flat by generic flatness. Take the union of all such opens. It is invariant: on $R$ the source and target pullbacks of $G$ are isomorphic by conjugation. If $O\subset A$ is a flat open, then $t(s^{-1}(O))$ is open because $t$ is smooth, and the group scheme is flat there by faithfully flat descent along $s^{-1}(O)\to t(s^{-1}(O))$. This map is smooth and surjective onto that open. Hence that image is contained in the same maximal flat open. Restrict $W$ once more to the corresponding dense open, and restrict $A,R,G$. (d) Every geometric fibre $J$ of $G$ is smooth; the special form $[U\mathrm{/}T]$ gives a proof without invoking the general characteristic-zero smoothness theorem for group schemes. Over an algebraically closed characteristic-zero field $k$, $J\to T_k$ is the finite transporter map described in the preceding construction, with kernel $K$ the finite étale inertia of the corresponding object of $U$. Let $H\subset T_k$ be its scheme-theoretic image. Its coordinate Hopf algebra is a quotient of the Laurent group algebra $k[\mathrm{\Lambda }]$, $\mathrm{\Lambda }=X^{\ast }(T)$, hence is spanned by images of characters, all group-like. Distinct group-like elements in a coalgebra are linearly independent: choose a shortest dependence, apply comultiplication, subtract the first group-like element tensored with that dependence, and use independence of the remaining terms to force every group-like element in the dependence to equal the first. The only identifications are therefore a subgroup $L\subset \mathrm{\Lambda }$, and $\mathcal{O}(H)=k[\mathrm{\Lambda }\mathrm{/}L]$. The finitely generated abelian group $\mathrm{\Lambda }\mathrm{/}L$ is a product of a free group and finite cyclic groups; its group algebra is a product of Laurent polynomial algebras because characteristic zero makes the torsion polynomials have distinct roots. Thus $H$ is smooth. (e) The finite map $J\to H$ is surjective by the lying-over property of integral extensions. It is flat on a nonempty open of each component of $H$ by generic flatness. Translations spread a flat open to all of $H$: every geometric $h\in H(k)$ lifts to $j\in J(k)$, and translation by $j$ over translation by $h$ is an isomorphism of this finite map. Translates of a flat open therefore cover every closed point, and hence all of the finite-type Jacobson scheme $H$. Thus $J\to H$ is finite faithfully flat. The group identity gives $J{\times }_{H}J\cong J\times K$, $(j,k)\mapsto (j,jk)$. After that faithfully flat base change it is the projection with finite étale factor $K$, so it is finite étale; this may also be checked by flat descent and vanishing of its module of differentials. Therefore $J$ is smooth over $k$. Since $G\to A$ is flat of finite presentation with geometrically smooth fibres, it is smooth. (f) We now construct the first quotient in the required general relative smooth form. Let $C$ and $X$ be finite-type $\mathbb{C}$-schemes, let $X\to C$ be smooth, and let a smooth group scheme $H\to C$ of finite presentation act freely on $X$ over $C$. Fix a closed point $x\in X(\mathbb{C})$ and shrink about its image so the relative dimensions are constant. Freeness injects $\mathrm{Lie}(H{)}_x$ into $T_{X\mathrm{/}C,x}$: a tangent vector in its kernel would give a nonidentity infinitesimal stabilizer, contrary to the action map being a monomorphism. Write $d=\dim H_c$. Choose $d$ regular functions $f_1,\dots ,f_d$ on an affine neighbourhood of $x$ whose relative differentials restrict to a basis on this orbit tangent subspace. Replace $f_i$ by $f_i-f_i(x)$, using the constants $f_i(x)\in \mathbb{C}$. Their common zero locus, after shrinking about $x$, is therefore an actual $\mathbb{C}$-scheme $V$ containing $x$, smooth over $C$ of relative codimension $d$. Its relative tangent space is complementary to the orbit tangent space at all its points. (g) The action map $H{\times }_{C}V\to X$ is étale. At $(1,v)$ its relative derivative is the direct-sum map from the orbit tangent space and $T_{V\mathrm{/}C,v}$, an isomorphism. Both source and target are smooth over $C$, so the relative Jacobian criterion makes the action map étale there; translation in $H$ and the action on $X$ make it étale at every $(h,v)$. Its image is open and $H$-saturated. Take the collection of these transverse affine slices through closed $\mathbb{C}$-points. Their saturated open images contain every closed point of $X$ and therefore cover all of $X$: a nonempty closed complement in a finite-type $\mathbb{C}$-scheme would have a closed point. Quasi-compactness gives a finite subcover. Let $V$ denote their disjoint union. The map $H{\times }_{C}V\to X$ is étale and surjective. (h) On $V$ the relation “lie in the same $H$ orbit” is represented by the scheme $E=V{\times }_X(H{\times }_{C}V)$. One projection $E\to V$ is the base change of the étale map in the preceding construction, so it is étale; inversion gives the other projection étale as well. The map $E\to V\times V$ is a monomorphism because the action is free. The groupoid identities come from multiplication and inversion in $H$. Thus $(V,E)$ is an actual étale equivalence relation and presents an algebraic space $Z=V\mathrm{/}E$ in the étale-space sense under this ordinary étale-presentation convention. Since every orbit of $X$ meets $V$ étale locally, its sheaf quotient is exactly $X\mathrm{/}H=Z$, not just a bijection on geometric points. Indeed local lifts exist by the preceding construction and two lifts are equal precisely through $E$; sheafification gives inverse maps. (i) Freeness identifies $H{\times }_{C}X\to X{\times }_{Z}X$ as an isomorphism of fppf sheaves: an orbit pair has a group element locally, and that element is unique and so glues. In particular $X{\times }_{Z}V\cong H{\times }_{C}V$. Thus the chart base change of $X\to Z$ is the smooth surjective projection $H{\times }_{C}V\to V$. It follows directly on the étale chart $V\to Z$ that $X\to Z$ is smooth, surjective and locally of finite presentation, and is an fppf $H$-torsor. This finishes the required free-action quotient construction; no quotient theorem for a general flat group scheme was used. (j) Apply this construction to $X=R$, $C=A$ via the target $t$, and $H=G$. The action $G{\times }_{A,t}R\to R$, $(g,r)\mapsto gr$, is free by cancellation of arrows. It preserves both endpoints. The hypotheses hold because $R\to A$ and $G\to A$ are smooth. Hence $Q=R\mathrm{/}G$ is the algebraic space just constructed, and $R\to Q$ is a smooth surjective torsor. The source and target descend to $s_Q,t_Q:Q\to A$. The transverse slice charts used to construct $Q$ are smooth over $A$ by the preceding construction, so $t_Q$ is smooth. Inversion descends and interchanges the two maps, proving $s_Q$ smooth as well. (k) The map $j_Q:Q\to A\times A$ is a monomorphism on all test schemes. Two local lifts $r,r^{\mathrm{\prime }}$ with the same endpoints differ by the unique target stabilizer element $r^{\mathrm{\prime }}{r}^{-1}$, so have the same class. This equality descends from the fppf cover on which they lift. Composition of classes is well defined: replacing one arrow by a target stabilizer translate changes its composite by another target stabilizer translate, with conjugation moving a source stabilizer across an intervening arrow. This also proves that inversion and identity descend. Thus $Q\rightrightarrows A$ is a smooth equivalence relation in algebraic spaces. It remains to construct its quotient; its existence is not inferred merely from that description. (l) For this second construction let $E=e^{\ast }{T}_{Q\mathrm{/}A,s_Q}$ on the smooth scheme $A$, and let $D\subset T_A$ be its image under $d{t}_Q$. The map into $T_A$ is injective: a vector in both endpoint kernels would be a nontrivial arrow over the dual numbers with fixed endpoints, prohibited by the monomorphism $j_Q$. Both its domain rank and its fibrewise rank are locally constant, so its image is a vector subbundle. At an arrow $r:a\to b$, multiplication by $r$ identifies $\ker (d{s}_Q{)}_r$ with $E_b$ and its target derivative image with $D_b$. Similarly $\ker (d{t}_Q{)}_r$ maps isomorphically onto $D_a$ under the source derivative. As both endpoint derivatives are surjective, their joint tangent image induces an isomorphism $T_{a}A\mathrm{/}{D}_a\cong T_{b}A\mathrm{/}{D}_b$: lift a vector by one endpoint and take its class at the other, with independence precisely supplied by the stated kernel images. (m) Through each closed point $a\in A(\mathbb{C})$ choose a locally closed smooth affine transverse slice $V_i$, defined by $d_i=\mathrm{rank}D$ functions whose differentials restrict to a basis on $D$, after subtracting their values at $a$ as constants in $\mathbb{C}$. Shrink the slice so $T_{V_i}\oplus D{\mathrm{\mid }}_{V_i}=T_A{\mathrm{\mid }}_{V_i}$. Consider $Q{\times }_{s_Q,A}{V}_i\to A$ by the target map. Its source is smooth over $\mathbb{C}$. At any arrow from $v\in V_i$ its derivative is an isomorphism: the source-normal part is the isomorphism in the preceding construction, while its vertical part maps isomorphically onto the target orbit tangent space. Therefore this map is étale. Its image is an open saturated neighbourhood of $V_i$. These open images contain every closed point and hence cover all of the finite-type Jacobson scheme $A$; quasi-compactness gives a finite subcover and we put $V={\coprod }_i{V}_i$. In particular $Q{\times }_{s_Q,A}V\to A$ is étale and surjective. (n) Restrict the relation to $V$, obtaining $Q_V=Q{\times }_{A\times A}(V\times V)$. Its projections to $V$ are étale. For example, on $s_Q^{-1}(V_i)$ impose the equations defining $V_j$ at the target. Their differentials are a basis on $\ker (d{s}_Q)$ by the preceding construction and transversality; the relative Jacobian criterion makes the resulting source projection étale. The other projection follows by inversion. Thus $(V,Q_V)$ is an explicit étale equivalence relation presenting an algebraic space $B=V\mathrm{/}{Q}_V$ in the étale-space convention under this ordinary étale-presentation convention. Equivalently take scheme étale presentations of the overlap spaces to read it in scheme-chart form. No general smooth-equivalence-relation quotient theorem is an input: substeps (l)–(n) have actually produced its transverse étale presentation. (o) This $B$ represents the sheaf quotient $A\mathrm{/}Q$. Every section of $A$ is étale locally related to $V$ by the surjection in the preceding construction; the choice of a lift changes it precisely by $Q_V$. These give inverse maps of fppf sheaves between $A\mathrm{/}Q$ and $V\mathrm{/}{Q}_V$. The formula $A{\times }_{B}V\cong Q{\times }_{s_Q,A}V$ follows by the same unique-arrow argument as the monomorphism in the preceding construction. Hence $A\to B$ is smooth, surjective and locally of finite presentation, as checked after the étale cover $V\to B$. The construction is finite type because all the charts and their finite union are finite type. (p) Finally $[A\mathrm{/}R]=W^{\circ }\to B$ is a gerbe. Any section of $B$ lifts fppf locally to $A$ by the preceding construction, and hence to an object of $W^{\circ }$. Two objects with the same section of $B$ lift fppf locally to $a,b\in A$; their equality in $B$ is the unique arrow class in $Q$ between them. The smooth surjective torsor $R\to Q$ lifts that class fppf locally to an actual arrow of $R$, which is an isomorphism of those objects in $W^{\circ }$. These are the two defining gerbe conditions. They also show that the gerbe and the sheaf of isomorphism classes constructed here agree with the original stack on every test scheme, including nonreduced tests. (q) The integral stack $W^{\circ }$ has irreducible sheaf of isomorphism classes, and $B$ is reduced: the transverse charts are smooth schemes and étale over $B$. The generic residual space of this reduced irreducible algebraic space is a spectrum of a field. Concretely an étale scheme chart gives its generic field extension; the two generic overlap maps impose the field equalizer, equivalently ordinary finite separable field descent. Pulling the constructed gerbe back to this generic field gives the required generic gerbe. Since the construction is made on a dense open of every integral support, the Noetherian support-lowering argument that consumes it retains the entire original separated finite-type DM/torus scope; singularities and changing stabilizers remain in the lower-dimensional closed complements, where the same construction is repeated. No general positive-characteristic or arbitrary-flat-group quotient statement has been asserted. Thus a dense open of every integral support has the constructed generic gerbe, with no free-flat-action or flat-equivalence quotient left as an unproved input. Since the consuming argument lowers supports, its full original $[U\mathrm{/}T]$ scope is preserved. The exact published quotient uses motivating this reconstruction are Stacks 06QJ and Bootstrap 04S6/06PH; the construction here is the sufficient smooth characteristic-zero route, not a claim of the general positive-characteristic flat theorem.

At that generic gerbe, the stabilizer $J$ has a homomorphism to $T$. Its kernel is the finite inertia inherited from $U$, and its image $H$ is a proper closed subgroup of $T$. Indeed $H=T$ would say that every torus element carries the corresponding geometric object to an isomorphic one, exactly a point with zero-dimensional coarse torus orbit, excluded from $U$. Choose a nonzero character $\chi$ vanishing on $H^0$; this is possible because $H^0$ is a proper subtorus. Some positive power ${\chi }^n$ vanishes on all of $H$, since $H\mathrm{/}{H}^0$ is finite. Therefore the equivariant character line $L_{{\chi }^n}$ is trivial on the entire stabilizer $J$ of the generic gerbe.

A line with trivial stabilizer action on a gerbe descends to the base field: choose an fppf object, use its stabilizer-triviality to make the descent identification independent of the chosen isomorphism of two lifts, and descend the resulting one-dimensional vector space. A line over a field has a nonzero basis, so $L_{{\chi }^n}$ is isomorphic to the trivial line on that generic gerbe. For any coherent sheaf $F$ with this integral support, this gives a generic isomorphism $F\simeq F\otimes L_{{\chi }^n}$. It spreads to a dense open by finite presentation. This spreading may be checked after the smooth atlas by clearing finitely many denominators in the matrix and its inverse; the equality with the groupoid descent matrices holds after further shrinking because it holds generically. Thus it is an isomorphism of sheaves on the stack open, rather than an equality of $G_0$ classes checked on an atlas.

There is an explicit support-lowering relation. Take the closure of the graph of that open isomorphism inside the coherent sheaf $F\oplus (F\otimes L_{{\chi }^n})$. Concretely this is the inverse image of the graph under restriction to the open; it is a quasi-coherent subsheaf and is coherent because the ambient sheaf is Noetherian, as seen on the atlas. Each graph projection has kernel and cokernel supported in the lower-dimensional closed complement. Its two exact-sequence formulas therefore express $(1-{\chi }^n)[F]$ as a finite sum of coherent-sheaf classes of strictly smaller support dimension. With finitely many top-dimensional integral support components, take the product of their character factors; it lowers all top-dimensional supports. Noetherian induction terminates. Nilpotent structure is retained by the finite filtration by powers of the support ideal. Every coherent class is consequently annihilated by a product of elements of $S$, proving the asserted vanishing.

The same graph construction on $[M\mathrm{/}T]$, stopped when a support is contained in the geometric fixed substack, proves surjectivity of its closed pushforward after character localization. A sheaf supported on a finite thickening of that substack has a finite ideal-adic filtration, whose graded terms are actual sheaves on the substack. Thus it comes from closed pushforward in $G_0$. Injectivity still requires a left inverse, supplied in the intended virtual-localization argument by the relative virtual pullback and its self-intersection formula; vanishing alone does not prove it.

Moving Euler factors on a fixed DM stack. Let $F$ be a separated finite-type complex DM stack with coherently trivial torus action, and let $B$ be an equivariant vector bundle with only nonzero torus weights. The moving Euler factor $a={\lambda }_{-1}(B)$ acts invertibly on $S^{-1}{G}_0^T(F)$, where $S$ is generated by $1-\chi$ for nontrivial characters. The assertion concerns its action on coherent $G$-theory, not an unsupported rank-nilpotence assertion in the representation ring of finite inertia.

On a geometric generic gerbe of an integral support, let $G$ be its finite stabilizer. After a finite field extension, $G$ and its representations split. Write the fibre of $B$ as a direct sum of its nonzero torus weights tensored with finite-dimensional $G$ representations. Multiplication by $a$ on the finite free abelian group $R(G)$ has a characteristic polynomial $p_G(z)$ over $R(T)$. Over $\mathbb{C}$ the character table diagonalizes this multiplication: its eigenvalue at a conjugacy class $g$ is $\det (1-g\mid B)$, with the torus weights retained, namely a product of terms $1-\chi \zeta$ where $\chi \mathrm{\ne }1$ and $\zeta$ is a root of unity of order dividing $\mathrm{\mid }G\mathrm{\mid }$. The constant coefficient of $p_G$ is the product of these eigenvalues up to sign. It is a Laurent polynomial over $\mathbb{Z}$ and becomes a unit after $S$ localization: each root-of-unity factor divides $1-{\chi }^N$ for a suitable positive $N$, and the total product has unit Laurent leading coefficient. Equivalently, use all Galois conjugates to group the factors into cyclotomic factors in the Laurent character monomials. Cayley–Hamilton gives $p_G(a)=0$ on the generic representation group, with unit constant coefficient after localization.

This also applies to a nonneutral generic gerbe. Coherent representations there form a semisimple category in characteristic zero: after a finite splitting extension they are modules over finite-dimensional semisimple twisted group algebras. Its Grothendieck group is free on the simple modules. Scalar extension sends distinct simple classes to nonzero combinations of disjoint Galois orbits of split simples, hence is injective on this group. Thus the same polynomial identity, verified after splitting, holds before that extension. No finite scheme-cover pushforward surjectivity on $G_0(BG)$ is being assumed.

One can choose a single polynomial for all integral supports. Since $F$ is separated DM of finite type, its inertia is finite with uniformly bounded geometric degree $b$. The weights and rank of $B$ range over a finite set, after separating its finitely many open-and-closed rank/weight loci. There are finitely many finite groups of order at most $b$ (encode a group law by its finite multiplication table), and finitely many representations of dimension bounded by each weight rank (finite irreducible lists with bounded multiplicities). Take the product $p(z)$ of the finitely many corresponding characteristic polynomials. Its constant coefficient is a unit after localization, and $p(a)$ vanishes on every generic support representation.

Apply the graph/support-lowering construction of the support construction below to the generic representation identities. They express $p(a)[H]$, for any coherent sheaf $H$, as a sum of classes supported in strictly lower dimension; the relations are actual exact-sequence relations obtained by extending generic stable isomorphisms, not a gluing of $G_0$ equalities. If $d=\dim F$, the operator $p(a{)}^{d+1}$ is therefore zero on $G_0^T(F)$. The polynomial $q(z)=p(z{)}^{d+1}$ has unit constant term in $S^{-1}R(T)$. Writing $q(z)=q(0)+zr(z)$ gives $a^{-1}=-q(0{)}^{-1}r(a)$ as an operator. This proves the claimed moving-Euler invertibility, including finite-inertia representations, singular supports and nonneutral generic gerbes.

The induced fixed perfect obstruction theory. The fixed summand really is a POT, rather than merely a complex of the right rank. The map $i^{\ast }{L}_M\to L_F$ annihilates nonzero weights, so gives $E_{fix}\to L_F$. To check its cotangent conditions work over an affine test $S\to F$ and a square-zero extension by a module $J$ with trivial torus weight. The POT on $M$ assigns its underlying lifting obstruction in ${\mathrm{Ext}}^1(E,J)$ and its lift torsor in ${\mathrm{Ext}}^0(E,J)$, functorially for the torus action. Invariants in these groups are precisely the groups computed from $E_{fix}$: weight decomposition is exact. If the fixed obstruction vanishes, the underlying lift torsor is nonempty. An algebraic torus acting on an affine torsor under a rational vector module has a fixed point: applying the weight-zero projector to its affine coordinate-linear extension gives an invariant splitting. This is the degree-one cocycle calculation of exactness of invariants; the finite-dimensional submodules containing the finitely many data suffice. Thus a lift may be chosen fixed. Its coherence is unique because the DM diagonal is unramified, so two isomorphisms of infinitesimal lifts agreeing on the closed fibre agree; the action cocycle then holds by that same uniqueness. Conversely a fixed lift is an underlying lift, so its obstruction vanishes. Differences of fixed lifts are exactly the invariant lift differences. These statements for every $J$ identify the degree-zero cotangent map as an isomorphism and its degree-minus-one map as surjective: applying the square-zero derivation and extension interpretation of the cotangent complex, its Hom and Ext sequences have exactly those kernels and obstruction images. Hence $E_{fix}\to L_F$ satisfies the stated POT axioms. All constructions are local on the test base and natural under restriction, so supply the global induced theory.

Global relative fixed-embedding pullback and self-intersection. Fix an isogeny and a closed geometric fixed substack $i:F\hookrightarrow M$ on which the pulled-back torus action is coherently trivial; the complement is the geometric moving complement in the preceding construction. Let the given global equivariant POT restrict as $E{\mathrm{\mid }}_F=E_{fix}\oplus E_{mov}$, with $E_{mov}=[A\to B]$ in degrees $-1,0$. Put $V=B^{\vee }$ and $W=A^{\vee }$, so $N^{vir}=V-W$.

The fixed POT is $E_{fix}\to L_F$. The modified POT is ${\tilde{E}}_F=E_{fix}\oplus A[1]\to L_F$, projection followed by the fixed map. The projection $[A\to B]\to A[1]$ and the fixed identity give a distinguished triangle $E{\mathrm{\mid }}_F\to {\tilde{E}}_F\to B[1]\to E{\mathrm{\mid }}_F[1]$. The cotangent triangle is $i^{\ast }{L}_M\to L_F\to L_{F\mathrm{/}M}\to i^{\ast }{L}_M[1]$. Their compatible map is obtained by splitting into weights: $L_F$ has weight zero, so the moving map to it is canonically zero, and the induced cone map on the moving summand is the required $B[1]\to L_{F\mathrm{/}M}$. This is a global map of derived complexes, not separately chosen lifts of conormal generators.

It is a relative POT. Indeed $h^0(L_{F\mathrm{/}M})=0$. In the long exact cohomology diagrams for the two triangles, the maps at $h^0(E{\mathrm{\mid }}_F)$ and $h^0({\tilde{E}}_F)$ are isomorphisms, while those at $h^{-1}(E{\mathrm{\mid }}_F)$ and $h^{-1}({\tilde{E}}_F)$ are surjective. A direct diagram chase gives surjectivity $B\to h^{-1}(L_{F\mathrm{/}M})=I\mathrm{/}{I}^2$: lift the image of a conormal class to $h^0(E{\mathrm{\mid }}_F)$, correct its image in $h^0({\tilde{E}}_F)$ using the isomorphism there, then lift its remaining predecessor at $h^{-1}(L_F)$ through the stated surjection. This is the relative closed-immersion POT axiom, and the global cone map embeds $C_{F\mathrm{/}M}$ into $V$.

There is a global specialization on coherent sheaves. The extended Rees algebra ${\mathcal{O}}_M[t,I\mathrm{/}t]\subset {\mathcal{O}}_M[t,t^{-1}]$ defines $D_{F\mathrm{/}M}$, with ordinary part $M\times {\mathbb{G}}_m$ and Cartier special fibre $C_{F\mathrm{/}M}$. For a coherent sheaf $H$ on $M$, take the closure of $H⊠{\mathcal{O}}_{{\mathbb{G}}_m}$ in this deformation (equivalently its Rees module for the induced ideal filtration), then take its derived restriction at $t=0$. Differences between coherent extensions are supported on the special fibre. Their derived restriction is zero in $G_0$ because its self-intersection complex is $[\mathcal{O}\stackrel{0}{\to }\mathcal{O}]$, the normal line of the principal divisor being trivial. Thus this defines the global operation ${\sigma }_i$ independently of extension. Its construction uses the full ideal algebra and descends as an equivariant algebra/module, so no equality of $G_0$ classes is inferred from an atlas.

Compose with the closed cone inclusion and the finite Koszul zero-section pullback: $i^{!}=0_V^{!}{j}_{\ast }{\sigma }_i:G_0^T(M)\to G_0^T(F)$. All three operations are now explicitly constructed. For a coherent sheaf $H$ on $F$, the closure of $i_{\ast }H⊠{\mathcal{O}}_{{\mathbb{G}}_m}$ is the sheaf on $F\times {\mathbb{A}}^1$ on which $I\mathrm{/}t$ acts by zero. It is $t$-flat, and its special fibre is the zero-section pushforward of $H$ into $C_{F\mathrm{/}M}$. Consequently $i^{!}{i}_{\ast }[H]=0_V^{!}{0}_{V\ast }[H]={\lambda }_{-1}(V^{\vee })[H]={\lambda }_{-1}(B)[H]$. The last equality is the actual zero-differential coordinate Koszul complex on the zero section. It is valid for every coherent class and every nonreduced $F$, with its equivariant structure retained. This proves the self-intersection formula without invoking virtual-pullback functoriality.

Once multiplication by ${\lambda }_{-1}(B)$ is invertible after character localization, the preceding formula is a left inverse to $i_{\ast }$, namely ${\lambda }_{-1}(B{)}^{-1}{i}^{!}$. The moving-complement support argument gives surjectivity, so together they give concentration in degree zero, without invoking higher equivariant K-groups or the problematic augmentation-ideal argument in Kiem–Savvasv1.

Actual conormal calculation for the global deformation normal sheaf. The required mapping-cone normal-SHEAF identity admits the following local module computation. This does not identify raw cones under nonflat parameter restriction.

Let $F=\mathrm{Spec}A\mathrm{/}I\subset X=\mathrm{Spec}A\mathrm{/}J\subset {\mathbb{A}}^n$, with $J\subset I$, and put $D=\mathrm{Spec}A[t,J\mathrm{/}t]$. The zero lift of $F\times {\mathbb{A}}^1$ into $D$ has ideal $K=(I,J\mathrm{/}t)$. As an $(A\mathrm{/}I)[t]$ module, $K\mathrm{/}{K}^2\cong ((J\mathrm{/}IJ)[t]\oplus (I\mathrm{/}{I}^2)[t])\mathrm{/}⟨(tj,-\bar{j}):j\in J\mathrm{/}IJ⟩$. The two summands send $j$ to $j\mathrm{/}t$ and $i$ to $i$. Their displayed relation is $t(j\mathrm{/}t)-j=0$. Products of any two generators vanish modulo $K^2$: these impose $I^2$, $IJ\mathrm{/}t$, and $J^2\mathrm{/}{t}^2$. Every negative-degree Rees monomial with at least two factors $J\mathrm{/}t$ is such a product. Hence every remaining element has an expression linear in $I$ and $J\mathrm{/}t$; the original equality in $A[t,t^{-1}]$ between two linear expressions is generated by the displayed $t(j\mathrm{/}t)-j$ relations and the two module relations $I^2$ and $IJ$. This proves the module presentation with its relations, including the possible $t$ torsion.

Use the scaled translation action on the smooth ambient affine space: $x\mapsto x+ta$. It preserves the Rees algebra, since $f(x+ta)\mathrm{/}t=f(x)\mathrm{/}t+d{f}_x(a)+t(\cdots )$ for $f\in J$. Its infinitesimal map into the preceding normal sheaf is dual to $(j,i)\mapsto dj+tdi\in {\mathrm{\Omega }}_A{\mathrm{\mid }}_F[t]$. Thus the relative normal stack of $F\times {\mathbb{A}}^1$ into the intrinsic deformation stack is the vector-bundle/cone-stack object represented by the dual of this two-term conormal complex; quotienting by the ambient tangent action is part of this description.

On the other hand, put $P=[J\mathrm{/}IJ\to {\mathrm{\Omega }}_A{\mathrm{\mid }}_F]$ and $Q=[I\mathrm{/}{I}^2\to {\mathrm{\Omega }}_A{\mathrm{\mid }}_F]$. These are the truncated derived restrictions of the two cotangent complexes relevant to the normal stack. For $f:P\to Q$, form $c((tid,f):P\to P\oplus Q)$. Its three terms are $J\mathrm{/}IJ⟶(J\mathrm{/}IJ)\oplus (I\mathrm{/}{I}^2)\oplus {\mathrm{\Omega }}_A{\mathrm{\mid }}_F⟶{\mathrm{\Omega }}_A{\mathrm{\mid }}_F\oplus {\mathrm{\Omega }}_A{\mathrm{\mid }}_F$, with maps $j\mapsto (tj,\bar{j},-dj)$ and $(j,i,\omega )\mapsto (dj+t\omega ,di+\omega )$. Eliminate the second differential summand using the identity on $\omega$. The remaining complex is the cokernel module $((J\mathrm{/}IJ)\oplus (I\mathrm{/}{I}^2))\mathrm{/}⟨(tj,\bar{j})⟩$ mapping to ${\mathrm{\Omega }}_A{\mathrm{\mid }}_F$ by $(j,i)\mapsto dj-tdi$. Replacing $i$ by $-i$ gives exactly the conormal complex above. This proves the truncated mapping-cone identity used by $h^1\mathrm{/}{h}^0$.

The two affine parameter charts of ${\mathbb{P}}^1$ give the usual homogeneous form with the domain twisted by $\mathcal{O}(-1)$ and map $(z_{0}id,z_{1}f)$. The construction uses the actual ideal modules, their universal derivations and the scaled tangent action. It is functorial under étale change of the ambient chart and compatible with the comparison of smooth embeddings in their product. Consequently these identifications descend as maps of the intrinsic normal sheaf stacks and their equivariant structures. This is descent of modules and representable morphisms, not an inference that an atlas detects $G_0$ equality.

Global compatible virtual-cone class comparison. This construction uses the global moving/fixed two-term POT and the chosen geometric fixed embedding after torus isogeny as in the preceding constructions. It compares coherent classes through actual deformation spaces. It does not assert equality of raw cones under nonflat base change.

Two elementary operations on global deformations. For a DM morphism $f$, its intrinsic deformation $D_f\to {\mathbb{P}}^1$ is constructed on smooth-embedding groupoid charts by the extended ideal algebra and the scaled ambient tangent action. In the essential chart $X\subset {\mathbb{A}}^n$, it is $[\mathrm{Spec}A[t,J\mathrm{/}t]\mathrm{/}{\mathbb{G}}_a^n]$, with translation $x\mapsto x+ta$ and its induced action on $J\mathrm{/}t$. The generic quotient is the target and the special quotient is the intrinsic normal cone. Embedding comparisons in the product of smooth ambient charts identify these constructions; their groupoid cocycles therefore descend the actual algebraic deformation stack. Equivariance descends functorially from the original action, so an equivariant affine atlas is not required for this global-resolution construction.

Specialization of coherent sheaves means derived restriction of an extension from the ordinary open part; its independence is the principal-divisor self-intersection computation already proved in the relative-pullback note. For a POT $E_f$ with obstruction bundle stack ${\mathcal{E}}_f$, define $f^{!}$ by this specialization, the actual closed cone pushforward, and the zero-section Koszul operation in a vector-bundle presentation. The common-atlas proof in the K postimage makes the operation independent of the presentation. Proper pushforward and flat pullback commute with it: the maps of extended ideal algebras give the corresponding maps of deformations, and the usual finite-Tor projection and flat tensor identities identify the derived special restrictions. This is a global construction of operations; no injectivity of $G_0$ pullback along a chart is used.

We also need commutation with the refined pullback of a parameter divisor, including after a nonflat base change. Put $W=D_f{\times }_Y{D}_{\nu }$ and write $s,t$ for its parameters, $W_s=V(s)=C_f{\times }_Y{D}_{\nu }$, $W_{00}=C_f{\times }_Y{C}_{\nu }$. For a coherent $Q$ define ${\kappa }_s[Q]=[Q\mathrm{/}sQ]-[{\mathrm{Ann}}_Q(s)]$, the Euler class of $K(s)\otimes Q$, $K(s)=[\mathcal{O}\stackrel{s}{\to }\mathcal{O}]$. This is the refined operation for the parameter Cartier divisor on the base. When $s$ is a zero divisor on $W$, it is not being identified with derived pullback along the possibly nonregular immersion $W_s\to W$.

Choose a common coherent extension $\tilde{H}$ from the invertible-parameter open and set $\beta ={\kappa }_s[\tilde{H}]\in G_0(W_s)$. On $t\mathrm{\ne }0$ it restricts to the pullback of ${\sigma }_f[H]$. For the base change $\xi :C_f{\times }_Y{Y}^{\mathrm{\prime }}\to C_f$ of $\nu$, the actual extended-ideal algebra gives a closed map $q:D_{\xi }\to W_s$ which is an isomorphism for $t\mathrm{\ne }0$: on charts the base-changed extended ideal algebra surjects onto the extended algebra of the image ideal. This is a surjection, not a false assertion that the normal cone commutes with arbitrary base change. The intrinsic maps descend with their ambient tangent actions.

Represent ${\sigma }_f[H]$ by a finite difference of coherent sheaves and choose corresponding extensions $L$ on $D_{\xi }$. By coherent localization, $\beta -q_{\ast }[L]=j_{\ast }\gamma$ for a class on $V(t)\subset W_s$. The operation ${\kappa }_t$ kills this boundary-image subgroup: a sheaf annihilated by $t$ has complex $[Q\stackrel{0}{\to }Q]$, whose two homology classes cancel; powers of $t$ give a finite filtration with this same conclusion. Closed pushforward commutes with $K(t)$ and its homology, so ${\kappa }_t\beta =q_{0\ast }{\kappa }_t[L]=q_{0\ast }{\sigma }_{\xi }{\sigma }_f[H]$. This proves the required pushed-specialization identification, including its parameter Tor terms.

The bounded bicomplex $\tilde{H}\otimes K(s)\otimes K(t)$ has coherent homology. Filtering in one Koszul degree gives the two successive parameter homology operations; Euler additivity on its finite spectral sequence therefore gives ${\kappa }_t{\kappa }_s[\tilde{H}]={\sum }_j(-1{)}^j[H_j(\tilde{H}\otimes K(s)\otimes K(t))]$ in $G_0(W_{00})$. Filtering in the reverse order gives ${\kappa }_s{\kappa }_t$; signed interchange of the two Koszul factors is a chain isomorphism. The total class is independent of the chosen extension: a difference is supported on $V(s)\cup V(t)$; localization and finite support filtrations express it as boundary-image classes, killed by the corresponding parameter operation before or after interchange. Thus the two pushed iterated specializations agree globally. None of this requires the parameters to be a regular sequence on their product deformation, or identifies their raw iterated cones.

The parameter functions, Koszul complexes, closed map $q$, coherent localization and pushforward in this argument are global on the deformation stacks; the chart calculation constructs their Rees map, rather than testing equality of $G_0$ classes on an atlas. After these equal pushed classes reach the common product of the two obstruction bundle stacks, apply its finite zero-section Koszul complexes there. Their signed tensor interchange and finite homology projection formula preserve the equality. No extension of these obstruction bundles or their Koszul complexes over $W$ is assumed. This proves the required parameter-Gysin compatibility and its equivariant version.

The normal-sheaf family and its global POT. Write $g:M\to \mathrm{Spec}\mathbb{C}$, $h:F\to \mathrm{Spec}\mathbb{C}$ and $i:F\to M$. The compatible POTs are $E_g=E$, $E_h={\tilde{E}}_F$, $E_i=B[1]$, as constructed in the relative note. On $F\times {\mathbb{P}}^1$ form $\nu :E{\mathrm{\mid }}_F\otimes \mathcal{O}(-1)\to E{\mathrm{\mid }}_F\oplus {\tilde{E}}_F$, $\nu =(z_{0}id,z_{1}u)$, and the corresponding cotangent map $\mu =(z_{0}id,z_{1}v)$ from $i^{\ast }{L}_M\otimes \mathcal{O}(-1)$ to $i^{\ast }{L}_M\oplus L_F$. Let $C(\nu ),C(\mu )$ be their mapping cones.

The explicit conormal calculation in the conormal calculation above identifies $h^1\mathrm{/}{h}^0(C(\mu {)}^{\vee })$ with the intrinsic normal sheaf of $\kappa :F\times {\mathbb{P}}^1\to D_g$. It gives the equations and tangent quotient in each chart, including the $t$-torsion; the homogeneous transitions give the global identification. The morphism $C(\nu )\to C(\mu )$ is an obstruction theory: take its cone, or the two rows of the compatible triangle, and the POT isomorphisms in degree zero and surjections in degree minus one show its cone has no cohomology in degrees at least minus one. Hence it embeds the intrinsic normal cone of $\kappa$ in $h^1\mathrm{/}{h}^0(C(\nu {)}^{\vee })$.

This complex has a global two-term vector-bundle presentation in the present case. The degree-minus-one map $E^{-1}{\mathrm{\mid }}_F\to {\tilde{E}}_F^{-1}$ is the identity under the fixed/moving decomposition. Thus the degree-minus-two bundle in the literal mapping cone injects as a subbundle into its degree-minus-one term through $(z_{0}id,z_{1}id)$; the Euler exact sequence on ${\mathbb{P}}^1$ gives a locally free cokernel. Taking this cokernel produces the required two-term presentation, equivariantly and globally. All zero-section operations in the following calculation are therefore the finite Koszul operations already defined, not operations on an unspecified sheaf stack.

Fibre comparison and the zero-section reduction. At $z_1=0$ the cone of $(id,0)$ cancels the $E{\mathrm{\mid }}_F$ summand and leaves ${\tilde{E}}_F$. At $z_0=0$ it is $E{\mathrm{\mid }}_F\oplus E_i$, by the compatible triangle. Apply ${\kappa }^{!}$ to the structure sheaf of $D_g$ and use the proved parameter-Gysin commutation to restrict to these two fibres. At the ordinary fibre the result is ${\tilde{\mathcal{O}}}_F^{vir}$. At the cone fibre it is the virtual pullback for the composition $F\to M\to C_g$, applied to ${\mathcal{O}}_{C_g}$, with the special POT $E{\mathrm{\mid }}_F\oplus E_i$.

Here the middle map is the zero section of the intrinsic cone, with its embedding in ${\mathcal{E}}_g$. The needed composition formula reduces to the elementary interchange of a zero-section operation with $i^{!}$, as follows. Use the Cartesian square of the zero sections $M\to {\mathcal{E}}_g$ and $F\to i^{\ast }{\mathcal{E}}_g$. Base change of $i$ gives $I:i^{\ast }{\mathcal{E}}_g\to {\mathcal{E}}_g$. The parameter-Gysin commutation just proved interchanges the zero-section pullback and $i^{!}$, leaving the composition $F\to i^{\ast }{\mathcal{E}}_g\stackrel{I}{\to }{\mathcal{E}}_g$. Deform $I$ to its relative cone. Its obstruction bundle stack is the product $i^{\ast }{\mathcal{E}}_g{\times }_F{\mathcal{E}}_i$, since $I$ is the pullback of $i$ and the first factor is already a vector-bundle stack over $F$. The resulting operation is the zero-section operation of that product. In global bundle presentations this is exactly the total complex of the two coordinate Koszul complexes; associativity of derived tensor identifies it with their successive zero-section operations. The affine-bundle denominator terms are treated by the canonical common-atlas construction, so this is an equality of the globally defined operations. This proves that the special-fibre result is $i^{!}(0_{{\mathcal{E}}_g}^{!}{\mathcal{O}}_{C_g})=i^{!}{\mathcal{O}}_M^{vir}$.

Finally the two refined fibre pullbacks of any class in $G_0(F\times {\mathbb{P}}^1)$ have the same value in $G_0(F)$. Tensor the sequence $0\to \mathcal{O}(-1)\to \mathcal{O}\to {\mathcal{O}}_p\to 0$ with the class and push along the projective projection; its alternating result is independent of $p$. Applied to ${\kappa }^{!}{\mathcal{O}}_{D_g}$ this gives $i^{!}{\mathcal{O}}_M^{vir}={\tilde{\mathcal{O}}}_F^{vir}$. The extra zero-obstruction summand $A[1]$ has the direct zero-differential Koszul factor ${\lambda }_{-1}(A)$. Therefore $i^{!}{\mathcal{O}}_M^{vir}={\lambda }_{-1}(A){\mathcal{O}}_F^{vir}$.