# Pure and simple sheaves from directional tests A constant sheaf on a submanifold can have a nonzero microlocal shift even when its ordinary stalk lies in degree zero. The normalization compares a local half-space test with three tangent Lagrangian planes. Once the test's Morse index is removed, the remaining coefficient complex is independent of the test function. Purity means that this normalized complex has one cohomological degree; simplicity also specifies its coefficient module. Use Composing hypersurface kernels with their shifts and When a kernel quantizes a contact transformation. Throughout, \(X\) is a finite-dimensional smooth real manifold, \(n=\dim X\), and \(k\) is a commutative ring of finite global dimension. Inputs are in \(D^b(k_X)\); coefficient modules need not be finitely generated. We use the exact conormal coefficient-object model, bounded support triangles, and the microlocal half-space test identification stated below. Normal forms and the shift of a submanifold transform binds the exact written parameter Morse proof, and Local existence of contact kernel equivalences proves simultaneous normalization of finitely many smooth conic Lagrangians and transverse auxiliary planes by one hypersurface contact chart. The zero-covector conormal geometry and the conormal index calculation are proved below. The ordered index proof proves alternation, the radical formula, parity, the four-plane cocycle and constant-intersection continuity. The common-pair reduction proof proves the isotropic reduction used below. *Original lesson text and solutions: CC0 1.0 Universal. Human mathematical sources are credited below.* For simplicity and assertions that its coefficient or sheaf is nonzero, assume \(1_k\ne0\). The coefficient-complex, index and transport identities still apply to the zero ring, where all coefficient complexes vanish; we make no simplicity or nonvanishing assertion in that degenerate case. This convention adds no field, finite-generation or perfectness hypothesis. ## The test function includes a covector and a transversality condition Let \(\Lambda\subset T^*X\) be a smooth conic Lagrangian near a point \(p=(x_0;\xi_0)\). A smooth real function \(\varphi\) is a transverse test at \(p\) when \[ \varphi(x_0)=0,\qquad d\varphi(x_0)=\xi_0, \qquad \Lambda\pitchfork\Lambda_\varphi\text{ at }p, \quad \Lambda_\varphi=\{(x;d\varphi(x))\}. \tag{1} \] The graph \(\Lambda_\varphi\) is Lagrangian but usually is not conic. If \(\Lambda=T_M^*X\), the last condition says that \(x_0\) is a nondegenerate critical point of \(\varphi|_M\). Indeed, in coordinates \(x=(a,b)\) with \(M=\{b=0\}\), a tangent vector common to the graph and the conormal has \(\delta b=0\) and \(\delta\xi_a=0\). The graph condition then reads \(\operatorname{Hess}(\varphi|_M)\delta a=0\). The common tangent vanishes exactly when this Hessian is invertible. Assume \(\operatorname{SS}(F)\subset\Lambda\) on a cotangent neighborhood of \(p\). Put \[ C_\varphi(F)=\bigl(R\Gamma_{\{\varphi\geq0\}}F\bigr)_{x_0}, \qquad V=T_p\pi_X^{-1}(x_0), \quad A=T_p\Lambda, \quad B_\varphi=T_p\Lambda_\varphi, \quad \tau_\varphi=\tau(V,A,B_\varphi). \tag{2} \] The support is the **closed** side \(\varphi\geq0\). The ordered index uses \(\omega=d\theta\), as in the preceding lesson. The opposite side would change the local calculation. We need a precise support-test prerequisite: for any bounded \(F\), when \(\varphi(x_0)=0\) and \(d\varphi(x_0)=p\ne0\), \[ C_\varphi(F)\simeq \mu hom(k_{\{\varphi=0\}},F)_p, \tag{3} \] with the positive covector \(d\varphi(x_0)\), no additional shift, and compatibility with representatives and morphisms. Here is the exact derivation from the hypersurface microlocalization and stalk theorems `SH02-MH-SUBMANIFOLD` and `SH02-MIC-STALKS`. Write \(H=\{\varphi=0\}\) and use \(h=\varphi\) as a normal coordinate near \(x_0\). The stalk theorem computes \((\mu_HF)_p\) by local closed supports \(Z\) whose normal cone at \(x_0\) lies in the positive normal half-line, together with zero. Every such support germ is contained in \(\{h\geq0\}\). Indeed, otherwise points \((a_j,h_j)\in Z\) would approach \(x_0\) with \(h_j<0\); in the normal deformation take positive parameters \(t_j=-h_j\). Their normal coordinates \(h_j/t_j=-1\) exhibit a negative vector in \(C_H(Z)_{x_0}\), contrary to the strict pairing condition. Conversely the normal cone of \(\{h\geq0\}\) is the nonnegative normal half-line. Thus this closed half-space is terminal among the allowed support germs. The natural support maps identify the stalk colimit with \((R\Gamma_{\{h\geq0\}}F)_{x_0}\) in every cohomological degree, hence as a bounded derived object. The submanifold-Hom theorem identifies \(\mu hom(k_H,F)_p\) with \((\mu_HF)_p\), without an antipodal map or residual codimension shift. This proves (3) naturally for arbitrary bounded \(F\). Smooth-Lagrangian containment and graph transversality are additionally needed for test independence, rather than for this comparison alone. There is also a direct check that replacing \(F\) by an isomorphic point-localized representative does not change (2). The cone of such a replacement has microsupport avoiding \(p\). The defining microsupport vanishing test for the function \(\varphi\) makes its local support complex zero. Applying the support functor therefore makes the replacement invertible. Thus the conormal coefficient-object model can be used in the calculation below. ## Conormal sheaves give the normalization explicitly Suppose \(\Lambda=T_M^*X\), let \(\ell=\dim M\) and \(c=n-\ell\). The local conormal model gives \(F\simeq Q_M\) at \(p\), for some \(Q\in D^b(k)\). Choose coordinates and use the Morse lemma on \(M\) to write \[ \varphi|_M=|u|^2-|v|^2, \qquad \dim v=m,\quad \dim u=\ell-m. \tag{4} \] Local cohomology depends on this restriction, so \[ C_\varphi(F)\simeq Q[-m]. \tag{5} \] Here is the local calculation, including its degree. On a sufficiently small ball in \(M\), the support triangle is the relative-cohomology triangle for that ball and its open part \(\{|u|^2<|v|^2\}\). When \(m=0\), the open part is empty and the relative complex is \(Q\). When \(m>0\), decrease \(u\) to zero and then radially normalize \(v\); the open part has the homotopy type of \(S^{m-1}\). A finite good cover, with the usual constant-coefficient acyclicity on its contractible intersections, computes its cohomology by the sphere's augmented cochain complex. The relative complex is its reduced cochain complex shifted by \([-1]\), namely \(Q[-m]\). This construction retains the map from constants on the ball to constants on the open part. It works for a complex of arbitrary coefficient modules because the finite sphere complex consists of finite free modules; tensoring it with \(Q\) retains the calculation. For \(m=1\), the open part has two components: the diagonal map \(Q\to Q\oplus Q\) has a cokernel \(Q\), and the relative complex puts it in degree one. Coordinate orientation identifies the resulting rank-one negative-direction line locally; no global trivialization is assumed. Now compute the index directly, including mixed tangential and normal derivatives of the test. Write its Hessian at the point in blocks \(H_{aa},H_{ab},H_{ba},H_{bb}\), with \(H_{ba}=H_{ab}^T\). In the ordered three planes take vectors \[ (0,0;u,v)\in V,\qquad (x,0;0,y)\in A,\qquad (s,t;H_{aa}s+H_{ab}t,H_{ba}s+H_{bb}t)\in B_\varphi. \] For \(\omega=d\xi\wedge dx\), the defining cyclic quadratic form of this triple is \[ u^T(x-s)+(y-v)^Tt-x^TH_{aa}s-x^TH_{ab}t. \] Set \(z=x-s\), \(\widetilde u=u-H_{aa}s\), and \(\delta=y-v-H_{ba}(s+z)\). This is an invertible change of variables when \((s,z,t,v)\) are retained. The form becomes \[ \widetilde u^Tz+\delta^Tt-s^TH_{aa}s. \] The first two terms are hyperbolic pairings of equal positive and negative dimensions. The free \(v\) variables are radical. The signature is therefore \(-\operatorname{sgn}H_{aa}\), where \(H_{aa}=\operatorname{Hess}(\varphi|_M)\). Its \(m\) negative and \(\ell-m\) positive eigenvalues give \[ \tau_\varphi=m-(\ell-m)=2m-\ell. \tag{6} \] Consequently, whenever the displayed shift is integral, \[ C_\varphi(F)[j+\tau_\varphi/2] \simeq Q[j-\ell/2]. \tag{7} \] The right side contains no test function. This is the cancellation that the general definition must preserve: the local Morse group changes with the negative Hessian directions, while the inertia correction changes by exactly the compensating degree. ## A hypersurface contact kernel transports the corrected test Let \(\chi:T^*X\to T^*X'\) be a local contact transformation at the nonzero covector \(p\), with \(\dim X'=n\). Suppose its graph, using the antipodal input convention, is the conormal of a smooth hypersurface \(S\subset X'\times X\). Set \(K=k_S\), let \(p'=\chi(p)\), and let \(T(F)=\Phi_K(F)\) denote the corresponding localized transform. In \(E=T_pT^*X\), set \[ V'=\chi_*^{-1}\bigl(T_{p'}\pi_{X'}^{-1}(\pi_{X'}p')\bigr). \tag{8} \] Choose a function \(\varphi\) with \(\varphi(x_0)=0\), \(d\varphi(x_0)=p\), whose conormal hypersurface germ is carried to \(T^*_{\{\psi=0\}}X'\), with \(\psi(\pi_{X'}p')=0\) and \(d\psi(\pi_{X'}p')=p'\). The comparison in this section holds for every bounded \(F\); no smooth-Lagrangian microsupport assumption is required for it. For the later independence application, a simultaneous choice for the conic Lagrangian and both test-level conormals is proved in the linked local-existence lesson. Let \(A\) be any tangent Lagrangian plane under consideration, including \(T_p\Lambda\). Use \(\tau_\varphi=\tau(V,A,B_\varphi)\) and \(\tau_\psi=\tau(V',A,B_\varphi)\), identifying tangent spaces by \(\chi_*\). Here the latter index uses the transported graph-test plane. When \(A=T_p\Lambda\) for a conic \(\Lambda\), it equals the index of the actual target test \(\psi\), as verified below. Choose \(j\) with \(j+\tau_\varphi/2\in\mathbb Z\). Then \[ C_\varphi(F)[j+\tau_\varphi/2] \simeq C_\psi(TF)[j+\tau_\psi/2+e], \qquad e=\frac12(n-1)+\frac12\tau(V,A,V'). \tag{9} \] **Proof.** Write \(H=\{\varphi=0\}\). Its conormal tangent \(B_H\) contains the radial line \(\rho\) through \(p\). Homogeneity of \(\chi\) gives \(\rho\subset V\cap V'\), independently of \(A\). The proved common-pair reduction, applied to this radial line contained in both vertical planes, shows \(\tau(V,B_\varphi,V')=\tau(V,B_H,V')\). Indeed the reduction of \(B_\varphi\) is the image of \(B_\varphi\cap\rho^\perp\), exactly the reduction of the hypersurface conormal tangent. A tangent vector to the graph has base component tangent to \(H\) precisely when it belongs to \(\rho^\perp\). Conormalizing this base tangent and adjoining the radial direction gives the other plane. Both three-plane reductions meet the required pairwise-intersection condition through \(V\cap V'\). Hypersurface-kernel composition gives \[ T(k_H)\simeq k_{\{\psi=0\}}[a], \qquad a=1-\frac12\bigl[n+1+\tau(V,B_H,V')\bigr]. \tag{10} \] To obtain this particular composition, we need an endpoint that is a point. Here is a direct application of the submanifold transform with its hypotheses checked; the three-nonzero-endpoint hypersurface theorem alone would not supply it. Put \(W=S\subset X'\times X\), \(N=S\cap(X'\times H)\), and \(f=q_{X'}|_S:W\to X'\). Choose a defining function \(h\) for \(S\) with selected physical normal \((p',-p)\). Both \(d_{X'}h\) and \(d_Xh\) are nonzero. The first makes \(dh\) and \(d\varphi\), lifted from \(X\), independent: a linear relation first has zero \(X'\)-component and therefore zero coefficient of \(dh\), then zero coefficient of \(d\varphi\). Hence \(N\) is a smooth closed hypersurface in \(W\). The second makes \(f\) a submersion, since its nonzero \(X\)-derivative can solve the tangent equation for any prescribed \(X'\)-tangent vector. The covector \(\nu=f^*p'\) on \(W\) is nonzero because \(f\) is a submersion. Modulo the normal line of \(W\), \((p',0)\) equals \((0,p)\), so \(\nu\) is the selected conormal of \(N\). Its conormal image is \(T^*_{\{\psi=0\}}X'\) by the assumed contact image of \(T_H^*X\). For transversality, compose the contact graph with \(T_H^*X\). The graph projection to \(T^*X\) is a local diffeomorphism, so its derivative and the derivative of that conormal inclusion are transverse. The general tangent-composition proof applies with the last endpoint symplectic space equal to zero; it does not require a nonzero covector there. Its matching kernel is zero: a vector with zero output must have zero input by the graph isomorphism. The diagonal restriction and subsequent zero-middle-covector restriction therefore have no remaining middle tangent direction. Quotienting the normal line of \(W\) gives precisely the transverse intersection of \(T_N^*W\) with the target cotangent pullback required by the submanifold transform. Thus there are no null or excess directions. We spell out the ordered index, too. Before quotienting the normal of \(W\), lift the triple from \(T^*W\) to the ambient cotangent tangent space. That normal line lies in the vertical plane and the lifted conormal plane, so the common-pair reduction preserves its index. Lift next through the middle diagonal to \(E_{X'}\oplus E_X^a\oplus E_X\), where the planes in order are \[ V_{X'}\oplus V^a\oplus V,\qquad \lambda_S\oplus B_H,\qquad V_{X'}\oplus\Delta_X. \tag{E1} \] These are two comparisons of the same lifted triple (E1). Reducing its middle diagonal normal, which is common to the first and third planes, recovers the triple before that lift; the preceding reduction of the normal of \(W\) identifies this index with \(\tau_W\). To compute the same index in another way, return to (E1) and instead reduce only the endpoint vertical \(V_{X'}\), again common to its first and third planes. The graph carries this endpoint vertical to \(V'\). This alternative reduction leaves the triple \[ (\,V^a\oplus V,\ (V')^a\oplus B_H,\ \Delta_X\,). \tag{E2} \] The same diagonal identity as in the ordered hypersurface-composition proof gives \(\tau_W=\tau(V,V,B_H,V')=\tau(V,B_H,V')\); the triangle index with repeated \(V\) is zero. The point endpoint contributes the zero symplectic space and no index term. Thus the order is exactly the one in (10). Here \(\dim W=2n-1\) and the target hypersurface has dimension \(n-1\). The transverse submanifold-transform formula consequently gives \[ a=\frac{1+(n-1)-(2n-1)-\tau_W}{2} =\frac{1-n-\tau(V,B_H,V')}{2}. \tag{E3} \] This is the exponent in (10). Local coordinate orientations give the same normalized constant coefficient; no global orientation trivialization is asserted. Finally identify the functor, not just this exponent. On a small ambient product neighborhood, stalkwise flatness of the constant closed-support sheaf gives \(k_S\otimes^L q_X^{-1}k_H=k_N\). The closed-embedding projection formula therefore identifies its proper-support image with \(Rf_!k_N\), with the neighborhood restriction retained. Product neighborhoods and their intersections with \(W\) give bases for these local images. The contact graph isolates the chosen middle covector. The refined cutoff and formal-comparison theorem for composition at prescribed covectors identifies the represented formal system of these images with the localized transform \(T(k_H)\), after tensor and direct image. This uses the selected local graph's admissibility; it neither assumes global properness of \(f\) nor substitutes ordinary base restrictions for all microlocal denominators. The submanifold-transform comparison now proves (10) as an isomorphism in the stated output germ category. Contact transport of microlocal Hom, with both its arguments transported, and (3) therefore give \(C_\varphi(F)\simeq C_\psi(TF)[-a]\). The negative sign is forced by shifting the **first** Hom argument by \([a]\). To compare with (9), the remaining degree is \[ 2e=\tau(V,A,B_\varphi)-2a-\tau(V',A,B_\varphi). \tag{11} \] Substitute (10) and apply the four-plane cocycle identity in the order \((V,A,B_\varphi,V')\). Its index terms give \(\tau(V,A,V')+\tau(V,B_H,V')-\tau(V,B_\varphi,V')\). The last two terms cancel by the common-line reduction just proved. The dimension term is \(n-1\), proving (9) for an arbitrary \(A\). For the application \(A=T_p\Lambda\), conicity supplies the additional containment \(\rho\subset A\). Thus reduction through \(V\cap A\) also gives \(\tau(V,A,B_\varphi)=\tau(V,A,B_H)\). After transport, reduction through the target radial line gives equality of the indices for the transported \(B_\varphi\), the target level-hypersurface conormal tangent, and the actual graph-test tangent \(B_\psi\). This verifies the promised interpretation of \(\tau_\psi\) for the type calculation. We did not require the arbitrary plane in the lemma to contain the radial line. Every index order and the Hom-argument shift have now been fixed. \(\square\) For reference, graph-test transversality can also be read on the level hypersurface. At a nonzero covector the intersection of its conormal tangent with \(A\) is just \(\rho\) exactly when \(B_\varphi\cap A=0\). This follows by reducing both statements by the same radial line. It is what permits the hypersurface calculation in (10) for transverse tests. ## A smooth conic Lagrangian at the zero covector is conormal Suppose a smooth conic Lagrangian germ \(\Lambda\subset T^*X\) contains \((x_0,0)\). Then \(\Lambda\) is the conormal of an embedded base submanifold germ near that point, including its zero covectors. **Proof.** Identify the tangent space at a zero covector with \(T_{x_0}X\oplus T_{x_0}^*X\). Its Lagrangian plane \(A\) is invariant under every positive fibre dilation \((v,\eta)\mapsto(v,t\eta)\). If \((v,\eta)\in A\), subtract its image under any fixed \(t\ne1\): both \((0,\eta)\) and \((v,0)\) belong to \(A\). Thus \(A=P\oplus Q\), with \(P\subset T_{x_0}X\) and \(Q\subset T_{x_0}^*X\). Isotropy gives \(Q\subset P^\perp\); dimension \(n\) gives equality. Choose base coordinates \(x=(a,b)\) with \(P=\{b=0\}\), and dual coordinates \((\alpha,\beta)\). The tangent plane is \(\{\delta b=\delta\alpha=0\}\). Projection to \((a,\beta)\) has invertible derivative on \(\Lambda\). After shrinking, the inverse function theorem therefore writes the actual germ as \[ b=g(a,\beta),\qquad \alpha=h(a,\beta). \] Fibre dilation and uniqueness of this graph imply, for \(0