Framed P² monads and stable ADHM data

Direct multiplication gives $\beta \alpha =(-z_0{B}_2+z_2)(z_0{B}_1-z_1)+(z_0{B}_1-z_1)(z_0{B}_2-z_2)+z_0^{2}IJ=z_0^2([B_1,B_2]+IJ)=0$. The $z_0{z}_1$ and $z_0{z}_2$ cross terms cancel pairwise, and the $z_1{z}_2$ terms cancel. On $L_{\mathrm{\infty }}$, $\alpha =(-z_1,-z_2,0{)}^T$ is injective at every point because $z_1,z_2$ are not simultaneously zero. Thus $\alpha$ has full rank generically; its sheaf kernel is a torsion subsheaf of the locally free source, and hence is zero.

Conversely, let $\mathcal{E}$ be a framed torsion-free sheaf with the stated Hilbert polynomial. The preceding Beilinson lemma constructs its monad, with terms $V_1\mathcal{O}(-1)$, $M\mathcal{O}$, and $V_2\mathcal{O}(1)$ of ranks $k,2k+N,k$. Its restriction to $L_{\mathrm{\infty }}$ remains exact at the ends because $\mathcal{E}$ is locally free there. Put $K=\ker \beta {\mathrm{\mid }}_{L_{\mathrm{\infty }}}$. The sequence $0\to V_1\mathcal{O}(-1)\to K\to W\mathcal{O}\to 0$ splits since ${\mathrm{Ext}}^1(W\mathcal{O},V_1\mathcal{O}(-1))=V_1\otimes W^{\ast }\otimes H^1(\mathcal{O}(-1))=0$.

At a point of $L_{\mathrm{\infty }}$, $\beta =(z_2,-z_1,0)$ is onto. At an affine point $[1:u_1:u_2]$, failure of surjectivity would give a nonzero $\lambda \in V^{\ast }$ with $\lambda B_1=u_1\lambda$, $\lambda B_2=u_2\lambda$, and $\lambda I=0$. Then $\lambda$ annihilates every vector $B_{i_1}\cdots B_{i_m}I(w)$, contradicting stability. Hence $\beta$ is everywhere onto, so $K=\ker \beta$ is a vector bundle of rank $k+N$.

A fiber kernel of $\alpha$ over $[1:u_1:u_2]$ is a nonzero vector $v$ with $B_{1}v=u_{1}v$, $B_{2}v=u_{2}v$, and $Jv=0$. Each $u_i$ must be an eigenvalue of the fixed matrix $B_i$, so the rank-drop locus is contained in the finite product of the two finite spectra. Such a vector spans a common invariant line in $\ker J$, so costability fails. Conversely, if costability fails, choose a nonzero common invariant subspace $S\subseteq \ker J$. On $S$, the complex equation implies $[B_1,B_2]{\mathrm{\mid }}_S=-IJ{\mathrm{\mid }}_S=0$, so the restrictions commute. By [F6] they have a common eigenvector; its eigenvalues give an affine point where $\alpha$ has a fiber kernel. Therefore $\alpha$ has full fiber rank everywhere exactly for costable data.

The remaining line extension $0\to V_1\mathcal{O}(-1)\oplus W\mathcal{O}\to M\mathcal{O}\to V_2\mathcal{O}(1)\to 0$ has class in $\mathrm{Hom}(V_2,V_1)\otimes H^1(\mathcal{O}(-2))$, since $H^1(\mathcal{O}(-1))=0$. Twist by $\mathcal{O}(-1)$ and take cohomology. The connecting map $V_2\to V_1\otimes H^1(\mathcal{O}(-2))$ is an isomorphism: the adjacent groups $H^0(\mathcal{O}(-1))$ and $H^1(\mathcal{O}(-1))$ vanish, while $H^1(\mathcal{O}(-2))\cong \mathbb{C}$. Thus $V_2$ is identified with $V_1$ by this canonical extension class. Choosing a basis of $V_1$ and using the fixed framed basis of $W$ identifies $M=V\oplus V\oplus W$ and the line maps with the two Euler blocks in the Statement. Any other such choice differs by one simultaneous element of $GL(V)$.

The sequence $0\to V\mathcal{O}(-1)\stackrel{\alpha }{\to }K\to \mathcal{E}\to 0$ has locally free first two terms. Away from the finite rank-drop set of step 2.2, $\mathcal{E}$ is locally free; hence any torsion is supported at finitely many points. At such a point, the local ring is regular of dimension two and this sequence gives $\mathrm{pd}\mathcal{E}\le 1$. The Auslander–Buchsbaum formula [F7] gives $\mathrm{depth}\mathcal{E}\ge 1$. If a nonzero finite-length submodule $T$ existed, choose the least $n\ge 1$ with ${\mathfrak{m}}^{n}T=0$. A nonzero element of ${\mathfrak{m}}^{n-1}T$ is killed by all of $\mathfrak{m}$; hence no element of $\mathfrak{m}$ acts injectively on $\mathcal{E}$, contradicting depth at least one. Thus no such torsion exists. Thus $\mathcal{E}$ is torsion-free of rank $(2k+N)-k-k=N$. The rank drops only off $L_{\mathrm{\infty }}$, so $\mathcal{E}$ is locally free near the framing line.

The remaining coefficients of the two linear maps are their $z_0$ coefficients, so write $\alpha =(z_0{B}_1-z_1,z_0{B}_2-z_2,z_{0}J{)}^T$ and $\beta =(-z_0{C}_1+z_2,z_0{C}_2-z_1,z_{0}I)$. The coefficients of $z_0{z}_1$ and $z_0{z}_2$ in $\beta \alpha =0$ force $C_1=B_2$ and $C_2=B_1$; the $z_0^2$ coefficient is then $[B_1,B_2]+IJ=0$. This recovers the stated ADHM form. The surjectivity of $\beta$ implies stability: if $U=\mathrm{span}\{B_{i_1}\cdots B_{i_m}I(W)\}$ were proper, it is invariant and the induced $B_1,B_2$ on $V\mathrm{/}U$ commute because $[B_1,B_2]=-IJ$ has image in $U$. Their dual actions on the nonzero annihilator $U^{\perp }\subset V^{\ast }$ commute, so [F6] gives a common eigen-covector $\lambda \mathrm{\ne }0$ with $\lambda I=0$. At the corresponding affine point, $\lambda \beta =0$, contradicting surjectivity.

On $L_{\mathrm{\infty }}$, the complex is $0\to V{\mathcal{O}}_{L_{\mathrm{\infty }}}(-1)\stackrel{(-z_1,-z_2,0{)}^T}{\to }(V\oplus V\oplus W){\mathcal{O}}_{L_{\mathrm{\infty }}}\stackrel{(z_2,-z_1,0)}{\to }V{\mathcal{O}}_{L_{\mathrm{\infty }}}(1)\to 0$. The first two $V$ summands form the Euler resolution of the zero sheaf on the line, while the $W$ summand passes unchanged to middle cohomology. The framing is therefore the canonical identification $\mathcal{E}{\mathrm{\mid }}_{L_{\mathrm{\infty }}}=W\otimes {\mathcal{O}}_{L_{\mathrm{\infty }}}$. In the Grothendieck group, the monad gives $[\mathcal{E}]=[(V\oplus V\oplus W)\mathcal{O}]-[V\mathcal{O}(-1)]-[V\mathcal{O}(1)]$. Taking Euler characteristics after twisting by $\mathcal{O}(m)$ and using $\chi (\mathcal{O}(d))=\left(\genfrac{}{}{0px}{}{d+2}{2}\right)$ yields $\chi (\mathcal{E}(m))=N\left(\genfrac{}{}{0px}{}{m+2}{2}\right)-k$.

The Beilinson construction is natural under framed sheaf isomorphisms, so an isomorphism of framed sheaves induces maps on the cohomology vector spaces used in steps 1.2, 2.3 and 3.2 and hence a single change of basis $g\in GL(V)$; the normalized Euler blocks force the recovered tuples to differ by $g\cdot (B_1,B_2,I,J)$. Conversely such a $g$ is an isomorphism of monads acting trivially on $W$, and therefore induces a framed sheaf isomorphism. This proves the pointwise inverse equivalence modulo $GL(V)$. For a finite-type data family, step 2.1 makes $\beta$ surjective on the total space, so $K=\ker \beta$ is locally free. To establish total-space injectivity of $\alpha$, trivialize $\mathcal{V}$ over an affine base $\mathrm{Spec}A$. On $z_0\mathrm{\ne }0$ its first square block is $B_1-u_{1}I$, with determinant monic up to sign in $u_1$ over $A[u_2]$. On $z_1\mathrm{\ne }0$ its second block is $a{B}_2-bI$, where $a=z_0\mathrm{/}{z}_1$, $b=z_2\mathrm{/}{z}_1$; its determinant is monic up to sign in $b$ over $A[a]$. On $z_2\mathrm{\ne }0$ use the first block $a{B}_1-bI$, with $a=z_0\mathrm{/}{z}_2$, $b=z_1\mathrm{/}{z}_2$. Multiplication by any of these determinants is injective on the polynomial coordinate module by the highest nonzero coefficient, even when $A$ has nilpotents. The adjugate identity then makes each square block, and hence $\alpha$, sheaf-injective; localization preserves this. The sequence $0\to \mathcal{V}\mathcal{O}(-1)\to K\to \mathcal{E}\to 0$ is therefore exact and its terms before $\mathcal{E}$ are flat over the base. After tensoring with any residue field of the base, the first map remains sheaf-injective by step 1.1; hence ${\mathrm{Tor}}_1(k(s),\mathcal{E})=0$. Since $\mathcal{E}$ is finite over each local ring of ${\mathbb{P}}^2\times S$ and its residue fiber is a vector space over the residue field, [F8] gives flatness over $S$. Under the stated relative cohomology and base-change hypotheses, the same constructions use vector bundles and are inverse for families.

Step 2.2 proves that costability is equivalent to fiberwise injectivity of $\alpha$. Since $\beta$ is surjective and $K$ is locally free, the quotient $K\mathrm{/}\mathrm{im}\alpha$ is locally free exactly when $\alpha$ has constant full rank; failure at any point gives a fiber dimension jump. Hence the locally-free sheaf locus is precisely the regular ADHM locus. Combining this with [F4] gives the positive-FI and zero-FI identifications in the Statement.

Let $D=\mathrm{Spec}A$ for $A=\mathbb{C}[ϵ]\mathrm{/}({ϵ}^2)$, and let $\mathcal{F}$ be any first-order framed-sheaf family in the Statement, with special fiber $\mathcal{E}$. Put $Q_1=\mathcal{O}(-2)$, $Q_2={\mathrm{\Omega }}^1$, and $Q_3=\mathcal{O}(-1)$ on ${\mathbb{P}}^2$, and set ${\mathcal{F}}_i=\mathcal{F}\otimes {\mathrm{pr}}_1^{\ast }{Q}_i$ and ${\mathcal{E}}_i=\mathcal{E}\otimes Q_i$. Each ${\mathcal{F}}_i$ is flat over $A$. If $i_0:{\mathbb{P}}_{\mathbb{C}}^2\hookrightarrow {\mathbb{P}}_D^2$ is the special-fiber inclusion, flatness gives the exact sequence of sheaves on ${\mathbb{P}}_D^2$ $0⟶(i_0{)}_{\ast }{\mathcal{E}}_i⟶{\mathcal{F}}_i⟶(i_0{)}_{\ast }{\mathcal{E}}_i⟶0,$ where the last map is reduction modulo $ϵ$ and the first identifies the special fiber with $ϵ{\mathcal{F}}_i$. The closed-immersion pushforward is exact and $H^q({\mathbb{P}}_D^2,(i_0{)}_{\ast }{\mathcal{E}}_i)=H^q({\mathbb{P}}^2,{\mathcal{E}}_i)$, so this sequence gives the corresponding long exact cohomology sequence. The absolute Beilinson conclusions give $H^0({\mathbb{P}}^2,{\mathcal{E}}_i)=H^2({\mathbb{P}}^2,{\mathcal{E}}_i)=0$ and $h^1({\mathbb{P}}^2,{\mathcal{E}}_i)=d_i$; [F10] also gives $H^q({\mathbb{P}}^2,{\mathcal{E}}_i)=0$ for $q>2$, with $(d_1,d_2,d_3)=(k,2k+N,k)$. The long exact sequence therefore gives $H^0({\mathbb{P}}_D^2,{\mathcal{F}}_i)=H^2({\mathbb{P}}_D^2,{\mathcal{F}}_i)=0$ and $0⟶H^1({\mathbb{P}}^2,{\mathcal{E}}_i)⟶M_i:=H^1({\mathbb{P}}_D^2,{\mathcal{F}}_i)⟶H^1({\mathbb{P}}^2,{\mathcal{E}}_i)⟶0.$ The three standard affine opens of ${\mathbb{P}}_D^2$ and their affine intersections form a finite cover; the separated-scheme ordered-\v{C}ech theorem [F10] computes quasi-coherent cohomology and gives $H^q({\mathbb{P}}_D^2,{\mathcal{F}}_i)=0$ for $q>2$. By the cohomology-presheaf description of $R^q{p}_{\ast }$ and the fact that $D$ has one topological point, $R^q{p}_{\ast }{\mathcal{F}}_i$ corresponds to the $A$-module $H^q({\mathbb{P}}_D^2,{\mathcal{F}}_i)$. Multiplication by $ϵ$ on ${\mathcal{F}}_i$ factors as reduction followed by the first inclusion; since reduction is surjective on $H^1$, the displayed sequence identifies $ϵM_i$ with its left term and $M_i\mathrm{/}ϵM_i$ with its right term. Thus $M_i$ has complex dimension $2d_i$ and $M_i\mathrm{/}ϵM_i\cong H^1({\mathbb{P}}^2,{\mathcal{E}}_i)$. Lifting a basis of this quotient gives a surjection $A^{d_i}\to M_i$ by Nakayama; both sides have complex dimension $2d_i$, so it is an isomorphism. Hence $R^1{p}_{\ast }{\mathcal{F}}_i$ is locally free of rank $d_i$, $R^0{p}_{\ast }{\mathcal{F}}_i=R^2{p}_{\ast }{\mathcal{F}}_i=0$, and all three canonical derived base-change maps in degrees $0,1,2$ are isomorphisms: in degree one the map is the displayed reduction isomorphism, and in degrees zero and two both sides vanish. This proves the preceding Beilinson family hypotheses for every first-order framed-sheaf family. The ADHM family construction has flat middle cohomology by step 4.2 and retains the relative framing; applying this argument to its sheaf family puts every first-order ADHM deformation in the same range. Expanding ${\mu }_{\mathbb{C}}(x+ϵ\dot{x})=0$ gives $d_1(\dot{x})=0$. A change of basis $1+ϵ\xi$, which is the identity on the identified special fiber and on $W$, changes $\dot{x}$ by $d_0(\xi )$. Thus the data-side tangent vector space is $\ker d_1\mathrm{/}\mathrm{im}{d}_0$. Differentiating the relative monad maps gives $\beta \dot{\alpha }+\dot{\beta }\alpha =0$; a change by $d_0(\xi )$ induces the corresponding chain homotopy fixing the relative framing and special-fiber identification. By the family equivalence in step 4.2, now applicable by the preceding base-change verification, the two constructions commute with the dual-number base change, preserve the specified identifications, and induce inverse linear maps on first-order deformation classes and tangent vector spaces.

Let the rotation torus act on homogeneous coordinates with characters $1,t_1,t_2$, and let $h=\mathrm{diag}(e_1^{-1},\dots ,e_N^{-1})$ act on $W$. On the monad vector spaces choose characters $V(t_1{t}_2{)}^{-1}$ in the source, $V{t}_2^{-1}\oplus V{t}_1^{-1}\oplus Wh$ in the middle, and $V$ in the target, with the natural linearizations of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$. The first and second blocks are equivariant because the ratios of middle to source characters are respectively $t_1,t_2$; the $J$ block has ratio $t_1{t}_{2}h$; in $\beta$, the first two ratios are $t_2,t_1$ and the framing block ratio is $h^{-1}$. These are exactly the raw transformations in [F1]. On $L_{\mathrm{\infty }}$ the Euler blocks cancel, leaving the framing factor $Wh$; its character is ${\sum }_a{e}_a^{-1}$ and its dual has character ${\sum }_a{e}_a$. Thus the monad correspondence is torus-equivariant in the specified dual-framing convention.

If $k=0$, $V=0$, all four maps and both equations are the unique zero-dimensional maps, stability and costability are vacuous, and the monad is $0\to W\mathcal{O}\to 0$. Its middle cohomology is the framed bundle $W\mathcal{O}$ with Hilbert polynomial $N\left(\genfrac{}{}{0px}{}{m+2}{2}\right)$. Conversely, [F3] constructs this same zero-term monad from every framed sheaf with that polynomial, so the framing identifies each such sheaf with $W\mathcal{O}$. Thus both moduli spaces are points, and all empty-matrix and Euler-block conventions are satisfied.