SH02-MO-UNIT — Transporting directional obstructions

Original AI programme expression: CC0 1.0 Universal. The proofs use the exact local tests, propagation results, six-operation identities and Fourier equivalence identified below. Those dependencies require their own verification; a conditional proof here does not close them. This lesson retains bounded complexes of arbitrary sheaves over a commutative ring $k$ of finite global dimension. Only the Morse inequalities impose field coefficients.

All manifolds are finite dimensional, Hausdorff and countable at infinity. A map is $C^\infty$ unless stated otherwise. We use submersion for a map with surjective differential; mere differentiability is not sufficient for any submersion assertion. Test functions and exhaustion functions may be $C^1$. Tensor products and internal Hom are derived. Put $A^a=\{(x,-\xi):(x,\xi)\in A\}$ for a subset of a cotangent bundle.

The organizing question is whether a sheaf operation can create a failed local support test. Products separate the variables of such tests. Proper images collect tests along compact fibres. Noncharacteristic inverse images preserve the direction of a test. An exhaustion handles some nonproper maps by excluding the directions in which information could escape. Fourier transformation then exchanges a vector and a covector.

SH02-MO-DEPENDENCIES — The contracts used in this lesson

The following are dependencies, not assertions that a heading elsewhere establishes their complete proof.

We also use proper base change, proper-support base change, localization, compact-neighborhood continuity, and the projection formula at their manifold bounded-derived generality. Compact-neighborhood continuity is only applied to compact sets. It is not used as base change across an arbitrary closed set for a nonproper map. The noncharacteristic deformation input uses the intersection of the closures of shrinking increments, as in the corrected foundation contract. Local constancy along interval fibres implies local derived descent; that contract includes higher extension data, not just descent of each cohomology sheaf.

SH02-MO-COTANGENT — The correspondence attached to a map

For $f:Y\to X$, write

$$T^*Y\xleftarrow{f_d}Y\times_XT^*X\xrightarrow{f_\pi}T^*X, \qquad f_d(y,\xi)=(y,d f_y^t\xi),\quad f_\pi(y,\xi)=(f(y),\xi).$$

The relative conormal is $\ker f_d$. For a submanifold $M\subset X$ it is the usual $T_M^*X$. For $A,B\subset T^*X$ their ordinary fibrewise sum is

$$A+B=\{(x,\xi+\eta):(x,\xi)\in A,\ (x,\eta)\in B\}.$$

This operation is not automatically closed. It must not replace the enlarged limiting operations used later for characteristic maps.

SH02-MO-CONE-PROPER — No cancellation and closed images

If $A,B$ are closed conic sets and $A\cap B^a$ is contained in the zero section, addition $A\times_XB\to T^*X$ is proper. In particular $A+B$ is closed. Here properness is local over the base and hence global over compact subsets of the target.

Proof. Choose a relatively compact coordinate neighborhood and a Euclidean norm. If there were no uniform $c>0$ after shrinking the neighborhood, there would be $x_n\to x$, $\xi_n\in A_{x_n}$, $\eta_n\in B_{x_n}$ with

$$|\xi_n|+|\eta_n|=1,\qquad |\xi_n+\eta_n|\longrightarrow0.$$

Compactness of the unit sphere and closedness give a limit $\xi=-\eta\neq0$, contrary to the hypothesis. Thus

$$|\xi+\eta|\geq c(|\xi|+|\eta|). \qquad\text{(MO1)}$$

The inverse image of a compact set under addition is closed, has base in a compact set and has bounded fibre coordinates by MO1; it is compact. Finite coordinate covers handle an arbitrary compact target set. A proper map between locally compact Hausdorff spaces is closed, proving the assertion. The same normalization argument proves a second useful fact: if $f_d$ has no nonzero kernel vector in $f_\pi^{-1}A$, then its restriction to this closed conic set is proper. $\square$

SH02-MO-CONE-AC — A support calculation in opposite cone topologies

Let $V$ be a finite-dimensional real vector space, $\gamma$ a closed pointed convex cone containing $0$, and $\phi_\gamma:V\to V_\gamma$ the identity into the topology of ordinary open sets $U$ satisfying $U+\gamma=U$. Suppose

$$R\phi_{\gamma*}H=0,\qquad H\in D^b(k_V).$$

Let $\Omega$ be open and invariant under addition by $-\gamma$, and assume that $\Omega\cap(K+\gamma)$ is relatively compact for every compact $K\subset V$. Then

$$R\phi_{\gamma*}(H_\Omega)=0. \qquad\text{(MO2)}$$

If $\Omega'\subset\Omega$ is also $(-\gamma)$-open and $\Omega\setminus\Omega'$ is relatively compact, extension of compactly supported sections induces an isomorphism

$$R\Gamma_c(\Omega';H)\xrightarrow{\sim}R\Gamma_c(\Omega;H). \qquad\text{(MO3)}$$

Here $H_\Omega=j_!j^{-1}H$; it is not $Rj_*j^{-1}H$.

Proof. It suffices to prove that $R\Gamma(U;H_\Omega)=0$ on a basis of $\gamma$-open sets with $U\cap\Omega$ relatively compact. Sets $U=B_\epsilon(v)+\gamma$ form such a basis by the support hypothesis. The sheaf extension-by-zero definition gives

$$H^r(U;H_\Omega)=\underset{\rightarrow}{\lim}_K H^r_K(U;H), \qquad\text{(MO4)}$$

where $K$ is closed in $U$ and contained in $\Omega$. The identity follows by resolving with sheaves acyclic for the relevant supports; their support families form a filtered union. Every such $K$ has compact closure in $V$. Put $C=\overline K-\gamma$. This set is closed and is $\gamma$-closed. If $z=w-v\in C\cap U$ with $v\in\gamma$, then $w=z+v\in U$; hence $w\in\overline K\cap U=K$. It follows that $z\in K-\gamma\subset\Omega$. Thus

$$K\subset C\cap U\subset\Omega.$$

These enlarged supports are cofinal in MO4. Directional support compatibility identifies their cohomology with cohomology with support $C\cap U$ of $R\phi_{\gamma*}H$, which is zero. This proves MO2 without any closed-fibre base-change assertion.

For MO3, the cone of $H_{\Omega'}\to H_\Omega$ is $H_{\Omega\setminus\Omega'}$. Its support is contained in the compact closure of $\Omega\setminus\Omega'$. Compactly supported and ordinary global cohomology therefore agree on that cone. MO2 gives $R\Gamma(V;H_\Omega)=R\Gamma(V;H_{\Omega'})=0$, since global sections factor through $R\phi_{\gamma*}$. Its global cohomology vanishes, so its compactly supported cohomology vanishes too. The compact-support localization triangle proves MO3. $\square$

SH02-MO-EXTERNAL-TENSOR — Product tests

For $F\in D^b(k_X)$ and $G\in D^b(k_Y)$,

$$\operatorname{SS}(F\boxtimes G)\subset \operatorname{SS}(F)\times\operatorname{SS}(G). \qquad\text{(MO5)}$$

Proof. Work in coordinate neighborhoods. Suppose $(x,\xi)\notin\operatorname{SS}(F)$. The compact-cap criterion gives a closed cone $\gamma$, a cap $C_z$ and its base $B_z$, depending on nearby $z$, such that restriction $R\Gamma(C_z;F)\to R\Gamma(B_z;F)$ is an isomorphism. Use $\gamma\times\{0\}$ in the product. Its cap and base at $(z,w)$ are $C_z\times\{w\}$ and $B_z\times\{w\}$. Compact base change and the projection formula give

$$R\Gamma(C_z\times\{w\};F\boxtimes G) \simeq R\Gamma(C_z;F)\otimes G_w,$$

and the same formula for $B_z$. The restriction is therefore an isomorphism. It excludes every covector whose first component is $(x,\xi)$; the test is valid in a neighborhood, as required by the criterion. If instead $(y,\eta)\notin\operatorname{SS}(G)$, interchange the factors. This proves MO5. No finite-rank Künneth assertion is needed: the tensor factor is a fixed stalk complex and the cap is compact. $\square$

SH02-MO-EXTERNAL-HOM — Reversing the second direction

For the projections $q_X,q_Y$ from $X\times Y$,

$$\operatorname{SS}\bigl(R\mathcal Hom(q_Y^{-1}G,q_X^{-1}F)\bigr) \subset\operatorname{SS}(F)\times\operatorname{SS}(G)^a. \qquad\text{(MO6)}$$

The same estimate holds with $q_X^!F$ in place of $q_X^{-1}F$: locally the two differ by an invertible constant orientation complex and a shift, which do not affect support-test vanishing.

Proof. Put $K=R\mathcal Hom(q_Y^{-1}G,q_X^!F)$. First compute on an open rectangle $U\times V$. Exceptional adjunction for $p:U\times V\to U$, proper-support base change from $V\to\{*\}$, and the constant-sheaf adjunction give

$$\begin{aligned} R\Gamma(U\times V;K) &\simeq R\operatorname{Hom}_{k_U} \bigl(Rp_!q_V^{-1}(G|_V),F|_U\bigr)\\ &\simeq R\operatorname{Hom}_k \bigl(R\Gamma_c(V;G),R\Gamma(U;F)\bigr). \end{aligned} \qquad\text{(MO7)}$$

Here $q_V:U\times V\to V$ is the second projection. The proper-support base-change isomorphism identifies $Rp_!q_V^{-1}(G|_V)$ with the constant complex on $U$ having value $R\Gamma_c(V;G)$. All maps in MO7 commute with restriction to smaller open rectangles. The finite-dimensional manifold and finite-global-dimension hypotheses supply the boundedness required by these adjunctions. In particular, this calculation uses open restriction of internal Hom; it does not replace either open neighborhood by a compact subset inside a Hom argument.

Suppose $(x,\xi)\notin\operatorname{SS}(F)$. For $\xi=0$, the constant-function support tests imply that $F$ vanishes near $x$, so $K$ vanishes on the corresponding product neighborhood. For $\xi\ne0$, SH02-MST-EQUIVALENCE supplies a pointed closed convex cone $\gamma$ and a bounded representative $H$ on the coordinate vector space $E$, agreeing with $F$ near $x$, such that

$$R\phi_{\gamma*}H=0, \qquad \langle v,\xi\rangle<0 \quad(v\in\gamma\setminus\{0\}).$$

Replace $F$ by $H$ in $K$ and denote the resulting complex by $K_H$. It is bounded by the bounded internal-Hom estimate, and agrees with $K$ near $\{x\}\times Y$. If $U$ is $\gamma$-open, derived direct-image composition gives $R\Gamma(U;H)=0$. Formula MO7 therefore gives $R\Gamma(U\times V;K_H)=0$ for every ordinary open $V\subset Y$.

These rectangles form a basis for the directional topology with cone $\gamma\times\{0\}$. Indeed, an ordinary open set invariant under this cone contains, around any one of its points, an ordinary rectangle $B\times V$ and hence $(B+\gamma)\times V$. Derived sections of $R\phi_{\gamma\times\{0\}*}K_H$ on each such basis member are the just-computed zero complexes. Their filtered stalk colimits are zero, so

$$R\phi_{\gamma\times\{0\}*}K_H=0.$$

For every second covector $\eta$, pairing $(\xi,\eta)$ with a nonzero vector $(v,0)$ in this pointed cone gives the strictly negative number $\langle\xi,v\rangle$. The cone-to-test implication SH02-MST-CONE-TO-TEST therefore excludes $(x,y;\xi,\eta)$ from $\operatorname{SS}(K_H)$ and hence from $\operatorname{SS}(K)$. This implication allows cones of empty interior. Compactness of their unit directions makes strict negativity persist in a covector neighborhood, as the definition of microsupport requires.

Now suppose $(y,\eta)\notin\operatorname{SS}(G)$. The directional test equivalence permits replacement of $G$ near $y$ by a bounded $H$ with $R\phi_{\gamma*}H=0$, where $\eta$ lies strictly in the polar direction detected by $\gamma$. To test the opposite direction in $K$, choose nested $(-\gamma)$-open sets $\Omega'\subset\Omega$ whose difference is relatively compact, with $\Omega\cap(L+\gamma)$ relatively compact for compact $L$. Such sets are supplied by bounded transverse caps of the pointed cone. MO3 makes

$$R\operatorname{Hom}_k(R\Gamma_c(\Omega;H),R\Gamma(U;F)) \longrightarrow R\operatorname{Hom}_k(R\Gamma_c(\Omega';H),R\Gamma(U;F))$$

an isomorphism for every open $U\subset X$. By external adjunction these are exactly the restriction maps for $K$ on $U\times\Omega$ and $U\times\Omega'$. The cone form of the local test excludes $(x,y;\xi,-\eta)$, uniformly near the prescribed point. This gives MO6. $\square$

SH02-MO-PROPER-PUSH — Collecting tests along a fibre

Let $f:Y\to X$ be any manifold map and assume that $f$ is proper on $\operatorname{supp}(G)$. Then

$$\operatorname{SS}(Rf_*G)\subset f_\pi f_d^{-1}\operatorname{SS}(G). \qquad\text{(MO8)}$$

For a closed embedding this is an equality.

Proof. The set on the right is closed: $f_\pi$ is proper on the closed subset in question, because its base lies in $\operatorname{supp}(G)$ and the covector in its source is the target covector. Take a covector outside it and a sufficiently small cotangent neighborhood disjoint from it. Every test function $h$ whose differential lies there satisfies $d(hf)_y\notin\operatorname{SS}(G)$ at every relevant point of the fibre. Localization commutes with ordinary direct image, and proper base change on the support gives

$$\bigl(R\Gamma_{\{h\geq h(x)\}}Rf_*G\bigr)_x \simeq R\Gamma\bigl(f^{-1}(x); R\Gamma_{\{hf\geq h(x)\}}G\bigr)=0. \qquad\text{(MO9)}$$

The last complex restricts to zero on the fibre because each of its stalks is a vanishing local test. This proves MO8.

For a closed embedding, straighten $Y$ to a vector subspace in a chart. A compact cap test for $i_*G$ is exactly its intersection with $Y$ tested against $G$, since $i$ is proper and $i_*G$ is supported on $Y$. Intersecting the cap and its base with $Y$ replaces the testing covector by its restriction to $TY$; their proper-cone conditions are preserved. Thus a covector absent from $\operatorname{SS}(i_*G)$ has restricted covector absent from $\operatorname{SS}(G)$. Together with MO8 this gives equality, including covectors conormal to $Y$. $\square$

SH02-MO-SUBMERSION — Exact pullback and local descent

If $f:Y\to X$ is a submersion, then

$$\operatorname{SS}(f^{-1}F)=f_df_\pi^{-1}\operatorname{SS}(F). \qquad\text{(MO10)}$$

For $G\in D^b(k_Y)$ the following conditions are equivalent:

1. every $H^j(G)$ restricts to a locally constant sheaf on each fibre;
2. locally on $Y$, the complex $G$ is isomorphic to $f^{-1}F$ for a bounded complex on an open subset of $X$;
3. $\operatorname{SS}(G)\subset f_d(Y\times_XT^*X)$.

Proof. Locally $f$ is a projection $U\times V\to U$, and $f^{-1}F=F\boxtimes k_V$. MO5 proves the inclusion in MO10, since the constant sheaf has only zero covectors. Conversely, pull a test function $h$ on $U$ back to the product. Its local support complex is the pullback of that on $U$: this follows from open-product base change in the localization triangle. The stalks at $(x,y)$ and $x$ are equal. If a covector downstairs fails a support test arbitrarily near $(x,\xi)$, its horizontal pullback fails the corresponding tests arbitrarily near $(x,y;\xi,0)$. This proves equality.

The interval-fibre descent contract gives (1)$\Leftrightarrow$(2), locally in a contractible coordinate box in $V$; it applies to the whole bounded complex and preserves its extension data. MO10 gives (2)$\Rightarrow$(3). For (3)$\Rightarrow$(1), use the directional cutoff characterization with the entire vector space $V$ as cone. Its polar is $\{0\}$, so its directional topology forgets the $V$ coordinate. It identifies $G$ locally with a pullback, giving (2) and hence (1). $\square$

SH02-MO-FIBER-REMARK — A special cohomology-sheaf criterion

The horizontal bundle $V_f=f_d(Y\times_XT^*X)$ is a smooth coisotropic, hence involutive, submanifold. In projection coordinates it is defined by $\eta_1=\cdots=\eta_r=0$; its symplectic orthogonal is spanned by the vertical vectors $\partial/\partial y_j$, which are tangent to it. The equivalence above gives

$$\operatorname{SS}(G)\subset V_f \quad\Longleftrightarrow\quad \operatorname{SS}(H^jG)\subset V_f\text{ for every }j.$$

This statement concerns this particular horizontal set. It does not assert $\operatorname{SS}(H^jG)\subset\operatorname{SS}(G)$ for an arbitrary complex.

SH02-MO-BOUNDARY — Four ways to impose a boundary

For a subset $S\subset X$, let $N_x(S)$ be its strict normal cone and let $N_x^*(S)=N_x(S)^\circ$ use the nonnegative polar. Thus the convention from the subset lesson is

$$N_x(S)=T_xX\setminus C_x(X\setminus S,S),\qquad N^*(X\setminus S)=-N^*(S).$$

These strict normal polars need not be conormal bundles, even when $S$ is a submanifold. The following table describes the four operations on $F\in D^b(k_X)$. Every intersection condition is required over the full base, and “zero” means contained in $T_X^*X$.

Here $F_Z=i_*i^{-1}F$ and $R\Gamma_ZF=i_*i^!F$. The two closed-set constructions have different signs. The assumptions make the indicated sums closed by MO1.

Proof of the ordinary open image. Work near $x$ in a vector space and put $A=\operatorname{SS}(F)_x$, $N=N_x^*(\Omega)$. Outside the closed support of $F$ the assertion is immediate, so assume $0\in A$. Take $\xi\notin A+N$. The cone $N+\mathbb R_{\geq0}(-\xi)$ has no nonzero intersection with $-A$: an equality $n-t\xi=-a$ with $t>0$ would give $\xi=t^{-1}(a+n)\in A+N$, and $t=0$ is excluded by the transversality assumption. Separate their compact angular parts and enlarge the first cone slightly to a closed pointed convex cone $K$ such that

$$N+\mathbb R_{\geq0}(-\xi)\subset\operatorname{Int}K\cup\{0\}, \qquad K\cap(-A)\subset\{0\}. \qquad\text{(MO11)}$$

If the right-hand bound is the entire fibre there is nothing to test; otherwise this pointed enlargement is possible. One direct construction is to choose a strictly positive linear functional on the first pointed cone, take its compact affine section, enlarge that section inside the complement of $-A$, and take its cone. Compactness gives a strict angular margin. With $\gamma=K^\circ$, every nonzero $v\in\gamma$ satisfies $\langle v,\xi\rangle<0$. The strict-normal criterion makes $\Omega$ locally $\gamma$-open. Closedness of microsupport permits a neighborhood $U$ of $x$ on which

$$\operatorname{SS}(F)\cap(U\times(-\gamma^\circ)) \subset U\times\{0\}.$$

Choose nested $\gamma$-open caps $\Omega_0\subset\Omega_1$ with $\Omega_1\setminus\Omega_0\subset U$ and compact forward slices, so that their difference tests a neighborhood of $x$. Propagation gives

$$\bigl(R\phi_{\gamma*}R\Gamma_{X\setminus\Omega_0}F\bigr)|_{\Omega_1}=0. \qquad\text{(MO12)}$$

Both $\Omega$ and the caps are open in the directional topology. Consequently direct-image composition, open restriction and the directional support identity carry MO12 to the same statement with $F$ replaced by $Rj_*j^{-1}F$. The cone test therefore excludes $(x,\xi)$ from its microsupport. This argument only uses identities for open subsets in the two topologies; it does not commute a nonproper direct image with arbitrary closed restriction.

Proof of extension by zero. Now take $\xi\notin A-N$ and assume $A\cap N$ is zero. The same separation, with the opposite normal, provides a cone $\gamma$ detecting $\xi$ for which $\Omega$ is locally $(-\gamma)$-open and $\operatorname{SS}(F)$ avoids the required negative polar cone. The supported version of propagation produces a bounded representative

$$F'=R\Gamma_{\Omega_1\setminus\Omega_0}F, \qquad R\phi_{\gamma*}F'=0,$$

which agrees with $F$ near $x$. Shrink the local model for $\Omega$ by an opposite directional cap, retaining its germ at $x$, so that $\Omega\cap(K+\gamma)$ is relatively compact for every compact $K$. This is possible because $\gamma$ is pointed: a linear functional strictly positive on its nonzero directions cuts off every forward slice. MO2 then gives $R\phi_{\gamma*}(F'_\Omega)=0$. Near $x$ this is the extension by zero of $F|_\Omega$, so the local cone test proves the second row of the table.

For closed $Z$, apply the two open results to $\Omega=X\setminus Z$ and use

$$R\Gamma_ZF\longrightarrow F\longrightarrow Rj_*j^{-1}F\xrightarrow{+1}, \qquad j_!j^{-1}F\longrightarrow F\longrightarrow F_Z\xrightarrow{+1}.$$

The triangle inequality for microsupport and $N^*(\Omega)=-N^*(Z)$ give the last two rows. $\square$

The four rows align with Kashiwara–Schapira, Microlocal Study of Sheaves, Astérisque 128 (1985), Propositions 4.3.1–4.3.2, pp. 67–69, and the two localization triangles. The source uses the nonnegative polar convention, as here. Its Corollary 4.3.3 already places the point on the boundary of the closed subset. The boundary qualification in the next statement therefore agrees with that source; it corrects an overbroad formulation, not that corollary.

SH02-MO-VANISH — An outward direction forces a costalk to vanish

Let $Z$ be closed, let $x\in\partial Z$ be a boundary point, and suppose

$$N_x^*(Z)\neq T_x^*X, \qquad \operatorname{SS}(F)_x\cap N_x^*(Z)\subset\{0\}. \qquad\text{(MO13)}$$

Then $(R\Gamma_ZF)_x=0$. The boundary qualification is necessary: if $Z=X$ in positive dimension, then $N_x^*(Z)=\{0\}\neq T_x^*X$, but $R\Gamma_Zk_X=k_X$. A hypothesis merely saying $x\in Z$ would therefore give a false statement. At a boundary point $N_x^*(Z)$ is nonzero; the first condition of MO13 also makes it proper.

Proof. Put $H=R\Gamma_ZF$ and $N=N_x^*(Z)$. Under the first condition in MO13 the strict normal $N_x(Z)$ is a nonempty open convex cone, so its polar $N$ contains no nonzero line. The boundary estimate gives $\operatorname{SS}(H)_x\subset\operatorname{SS}(F)_x-N$. This set has only the zero vector in common with $N$: if $a-n\in N$, then $a\in N+N=N$, so $a=0$ by MO13; pointedness then forces $n=0$. Separate this closed conic bound from a sufficiently small pointed convex enlargement of $N$.

Choose a closed pointed $\gamma$ with

$$N\subset\operatorname{Int}(-\gamma^\circ)\cup\{0\}, \qquad \operatorname{SS}(H)_x\cap(-\gamma^\circ)\subset\{0\}. \qquad\text{(MO14)}$$

The strict-normal criterion makes $Z$ locally $\gamma$-closed. Propagation now supplies $\gamma$-open $\Omega_0\subset\Omega_1$ such that $\Omega_1\setminus\Omega_0$ contains a relatively compact neighborhood of $x$ and

$$\bigl(R\phi_{\gamma*}R\Gamma_{X\setminus\Omega_0}H\bigr)|_{\Omega_1}=0.$$

Here is the needed neighborhood comparison in detail. MO14 gives $-\gamma\setminus\{0\}\subset N_x(Z)$. Enlarge $\gamma$ slightly to a closed cone $\Gamma$ with $\gamma\setminus\{0\}\subset\operatorname{Int}\Gamma$ and $-\Gamma\setminus\{0\}\subset N_x(Z)$. Compactness of its angular section gives a coordinate ball $U$ in which displacements in $-\Gamma$ preserve $Z$ whenever both endpoints remain in $U$. No point $z\in Z\cap U$ can satisfy $z-x\in\operatorname{Int}\Gamma$: the allowed displacements near $x-z$ would otherwise put a whole ball around $x$ inside $Z$, contradicting $x\in\partial Z$.

There is $c>0$ such that $\operatorname{dist}(v,E\setminus\operatorname{Int}\Gamma)\geq c|v|$ for $v\in\gamma$. If $z=x+b+v\in Z\cap U$, $|b|<\epsilon$ and $v\in\gamma$, the preceding exclusion gives $c|v|\leq\epsilon$. Therefore

$$\bigl(B_\epsilon(x)+\gamma\bigr)\cap Z\cap U \subset B_{(1+c^{-1})\epsilon}(x).$$

This proves cofinality in the ordinary neighborhoods of $x$ in $Z\cap U$. All propagation here is on the local domain $U$, as permitted by its exact contract; no claim is made about distant components of the original $Z$. Since $H|_U$ is supported on $Z\cap U$, its stalk is the colimit of $H^r(\Omega\cap U;H|_U)$ over these directional neighborhoods $\Omega\subset\Omega_1$. Propagation identifies each group with $H^r(\Omega\cap\Omega_0\cap U;H|_U)$. Choose $\Omega$ so that $\Omega\cap Z\cap U\subset\Omega_1\setminus\Omega_0$. Then $\Omega\cap\Omega_0\cap Z\cap U$ is empty. The colimit is zero in every degree. $\square$

SH02-MO-MICROLOCAL-SUPPORT — Where directional sheaves can live

For a closed submanifold $M\subset X$ and bounded complexes $F,G$,

$$\operatorname{supp}(\mu_MF)\subset T_M^*X\cap\operatorname{SS}(F), \qquad \operatorname{supp}(\mu hom(G,F)) \subset\operatorname{SS}(G)\cap\operatorname{SS}(F). \qquad\text{(MO15)}$$

Proof. A stalk of $\mu_MF$ at $p\in T_M^*X$ is the filtered colimit of $(R\Gamma_ZF)_x$ over the strict closed-support tests in SH02-MIC-STALKS. If $p\neq0$ and $p\notin\operatorname{SS}(F)$, its angular neighborhood may be chosen disjoint from microsupport. The strict support tests can be refined so that their nonzero normal polars lie in that neighborhood and their strict normal cone is nonempty. The strict positivity condition excludes an ordinary neighborhood of $x$ from such a support, so $x$ is either outside the support or on its boundary. In the first case its costalk is zero directly; in the second case the corrected MO13 applies. This is a cofinal refinement, so it annihilates the stalk, proving the first inclusion. At a zero covector outside microsupport the sheaf itself vanishes near the base point, and the conclusion is immediate.

To spell out the support refinement, use normal coordinates $(x',x\text{”})$ with $M=\{x\text{”}=0\}$. The strict condition on $C_M(Z)_x$ gives, after shrinking, a constant $c>0$ with $\langle p,x\text{”}\rangle\geq c|x\text{”}|$ for points of $Z$ near $x$. Otherwise a sequence of violating normalized normal coordinates would give a nonzero vector in $C_M(Z)_x$ pairing nonpositively with $p$. Choose a small $0<\epsilon<c$ and enlarge the support locally to $\{\langle p,x\text{”}\rangle\geq\epsilon|x\text{”}|\}$. This remains an allowed strict support, contains the original germ, and has at $x$ a nonzero strict normal polar concentrated in an arbitrarily small angular neighborhood of $p$ when $\epsilon$ is small. Tangential components of that polar vanish. The point $x$ lies on its boundary. These enlargements therefore give the asserted cofinal family and make the application of MO13 explicit.

For the second inclusion apply the first one on $X\times X$ to $R\mathcal Hom(q_2^{-1}G,q_1^!F)$. Its microsupport lies in $\operatorname{SS}(F)\times\operatorname{SS}(G)^a$ by MO6. The diagonal conormal is $(x,x;\xi,-\xi)$. Intersecting the product bound with it gives exactly the second set in MO15. $\square$

SH02-MO-EMBEDDING — Restricting across a transverse submanifold

Let $i:M\hookrightarrow X$ be a closed submanifold and assume

$$\operatorname{SS}(F)\cap T_M^*X\subset T_X^*X. \qquad\text{(MO16)}$$

Then

$$\operatorname{SS}(F_M)\subset\operatorname{SS}(F)+T_M^*X, \qquad i^{-1}F\otimes\omega_i\xrightarrow{\sim}i^!F, \qquad\text{(MO17)}$$

where the second arrow is the canonical tensor–exceptional comparison and

$$\omega_i=\operatorname{or}_{M/X}[\dim M-\dim X].$$

Proof of the bound. For a hypersurface, work in a chart where it separates two open sides $\Omega_+$ and $\Omega_-$. Their strict normal polars are the two opposite conormal rays. Apply the extension-by-zero boundary estimate to both sides and the triangle

$$F_{\Omega_+}\oplus F_{\Omega_-}\longrightarrow F \longrightarrow F_M\xrightarrow{+1}.$$

Their two bounds lie in $\operatorname{SS}(F)+T_M^*X$, proving the claim. The equality for closed direct images in MO8 simultaneously gives

$$\operatorname{SS}(i^{-1}F)\subset i_di_\pi^{-1}\operatorname{SS}(F) \qquad\text{(MO18)}$$

for a hypersurface: applying $i_d$ kills the added conormal covector. For higher codimension use a local flag of hypersurfaces ending at $M$ and induct on this pair of assertions. At each stage the next inclusion is noncharacteristic: a restricted covector annihilating its tangent space lifts to a covector of $\operatorname{SS}(F)$ annihilating $TM$, which must vanish by MO16. The conormal additions along the flag sum to $T_M^*X$. This proves MO17 and MO18 without identifying $N^*(M)$ with the conormal bundle.

Proof of the comparison. MO15 says that $\mu_MF$ is supported on its zero section $s:M\hookrightarrow T_M^*X$. The localization triangle for the complement of that section makes the natural arrow

$$s^!\mu_MF\longrightarrow R\pi_*\mu_MF$$

an isomorphism. The zero-section and Sato identifications identify its source with $i^{-1}F\otimes\omega_i$ and its target with $i^!F$. Their trace compatibility identifies this particular arrow with the canonical tensor–exceptional comparison. Thus MO17 is an isomorphism of the specified maps, conditional on that map-identification contract. $\square$

SH02-MO-NONCHAR — A map transverse to a conic set

A map $f:Y\to X$ is noncharacteristic for $A\subset T^*X$ if

$$f_\pi^{-1}A\cap\ker f_d \subset Y\times_XT_X^*X. \qquad\text{(MO19)}$$

For a closed conic $A$, MO1 shows that $f_d:f_\pi^{-1}A\to T^*Y$ is proper. Thus its image is a closed conic set. A submersion is noncharacteristic for every $A$, since its transposed differential is injective. A closed embedding need not be noncharacteristic.

SH02-MO-PULLBACK — The general noncharacteristic inverse image

If $f$ is noncharacteristic for $\operatorname{SS}(F)$, then

$$\operatorname{SS}(f^{-1}F)\subset f_df_\pi^{-1}\operatorname{SS}(F), \qquad f^{-1}F\otimes\omega_f\xrightarrow{\sim}f^!F. \qquad\text{(MO20)}$$

Proof. Factor $f$ as its graph $g:Y\hookrightarrow Y\times X$, followed by the projection $q:Y\times X\to X$. MO10 describes the microsupport of $q^{-1}F$ as the covectors $(y,f(y);0,\xi)$ with $(f(y),\xi)\in\operatorname{SS}(F)$. Such a covector is conormal to the graph precisely when $d f_y^t\xi=0$. Thus MO19 is exactly the hypothesis needed to apply MO17 to the graph. MO18 followed by the cotangent map of the projection proves the estimate. The canonical comparisons for $q$ and $g$ compose to that for $qg=f$, while

$$\omega_g\otimes g^{-1}\omega_q\simeq\omega_f.$$

The submersion formula and MO17 therefore prove the second statement with its canonical map. $\square$

SH02-MO-DIAGONAL — Tensor and Hom on one manifold

If $\operatorname{SS}(F)\cap\operatorname{SS}(G)^a$ is zero, then

$$\operatorname{SS}(F\otimes G)\subset\operatorname{SS}(F)+\operatorname{SS}(G). \qquad\text{(MO21)}$$

If $\operatorname{SS}(F)\cap\operatorname{SS}(G)$ is zero, then

$$\operatorname{SS}(R\mathcal Hom(G,F)) \subset\operatorname{SS}(F)+\operatorname{SS}(G)^a. \qquad\text{(MO22)}$$

Under the second assumption, if $G$ also has cohomologically constructible local data in the precise biduality sense, evaluation gives

$$R\mathcal Hom(G,k_X)\otimes F \xrightarrow{\sim}R\mathcal Hom(G,F). \qquad\text{(MO23)}$$

Proof. Let $\delta:X\hookrightarrow X\times X$ be the diagonal. The identities

$$F\otimes G=\delta^{-1}(F\boxtimes G), \qquad R\mathcal Hom(G,F) \simeq\delta^!R\mathcal Hom(q_2^{-1}G,q_1^!F)$$

are respectively tensor restriction and exceptional internal adjunction. The cotangent map of $\delta$ adds the two covectors; its kernel consists of $(\xi,-\xi)$. MO5 and MO6 show that the two different intersection assumptions make this embedding noncharacteristic for the corresponding external complex. Apply MO20. Its orientation factor does not change microsupport, so it applies equally to $\delta^!$ and $\delta^{-1}$; this proves MO21–MO22.

For MO23, the external constructibility comparison identifies $R\mathcal Hom(q_2^{-1}G,q_1^!F)$ with the external dual–tensor expression. Pull its evaluation map back to the diagonal and apply the canonical noncharacteristic comparison. The relative diagonal orientation cancels the orientation in $q_1^!$, since $q_1\delta=\mathrm{id}$. The resulting map is precisely ordinary internal evaluation in MO23. No constructibility assumption is used in MO21 or MO22. $\square$

SH02-MO-CHARACTERISTIC-BOUNDARY — The range of these estimates

The ordinary sum in MO21 and the ordinary cotangent image in MO20 depend on the no-cancellation hypotheses. Their characteristic analogues involve limiting cone operations. This lesson does not infer such an analogue by dropping the hypotheses from either formula.

SH02-MO-RELATIVE-CUTOFF — Replacing a nonproper map by a bounded part

Let $f:Y\to X$, $G\in D^b(k_Y)$, and $\varphi:Y\to\mathbb R$ be $C^1$. Write

$$Y_t=\{\varphi<t\},\quad \overline Y_t=\{\varphi\leq t\}, \quad j_t:Y_t\hookrightarrow Y,\quad i_t:\overline Y_t\hookrightarrow Y.$$

The bar here denotes a closed sublevel, not an assertion that it is the closure of the open sublevel. Set $f_t=fj_t$ and $\overline f_t=fi_t$. Assume

$$f:\operatorname{supp}(G)\cap\overline Y_t\longrightarrow X \text{ is proper for every real }t. \qquad\text{(MO24)}$$

Put $V_f=f_d(Y\times_XT^*X)$, and fix $t_0\in\mathbb R$.

If

$$d\varphi_y\notin\operatorname{SS}(G)_y+(V_f)_y \quad\text{for }\varphi(y)>t_0, \qquad\text{(MO25+)}$$

restriction gives the isomorphisms

$$Rf_*G\xrightarrow{\sim}R(f_t)_*j_t^{-1}G\quad(t>t_0), \qquad Rf_*G\xrightarrow{\sim}R(\overline f_t)_*i_t^{-1}G\quad(t\geq t_0). \qquad\text{(MO26+)}$$

If instead

$$-d\varphi_y\notin\operatorname{SS}(G)_y+(V_f)_y \quad\text{for }\varphi(y)>t_0, \qquad\text{(MO25-)}$$

extension and the exceptional counit give

$$R(f_t)_!j_t^{-1}G\xrightarrow{\sim}Rf_!G\quad(t>t_0), \qquad R(\overline f_t)_*i_t^!G\xrightarrow{\sim}Rf_!G\quad(t\geq t_0). \qquad\text{(MO26-)}$$

In the closed formula of MO26−, the ordinary direct image is legitimate because its support is proper by MO24. Under MO25+ for $Rf_*$, or under MO25− for $Rf_!$, respectively, one has

$$\operatorname{SS}(Rf_{\diamond}G) \subset f_\pi f_d^{-1}\bigl(\operatorname{SS}(G)\cap T^*Y|_{\overline Y_{t_0}}\bigr) \subset f_\pi f_d^{-1}\operatorname{SS}(G), \qquad \diamond\in\{*,!\}. \qquad\text{(MO27)}$$

The second relation is an inclusion. Equality of those two geometric images does not follow from the hypotheses; an example below shows why.

The antecedent for nonproper control is Kashiwara–Schapira, Microlocal Study of Sheaves, Theorem 4.4.1, pp. 73–75 (PDF pp. 76–78): an exhausting family with proper support and one-sided strict-normal exclusion gives stable open cutoffs and the direct-image estimate. That theorem explains why horizontal covectors must be included in the exclusion. Its printed statement is not the four-map endpoint statement (MO26) or the localized bound (MO27). Those statements are proved below by the proper map formed from the original map and the exhaustion, the two one-sided mechanisms (MO29)–(MO30), and the separate compact-neighborhood and support-colimit passages. In particular, the relative cutoff proof is part of this unit; neither its negative-sign case nor its closed endpoint is being supplied by a change of citation.

SH02-MO-ONE-SIDED — The one-parameter mechanism

We first justify the section argument used in the proof. Suppose $H\in D^b(k_{U\times\mathbb R})$ satisfies

$$\operatorname{SS}(H)|_{s>t_0}\subset \{(x,s;\xi,\sigma):\sigma\leq0\}. \qquad\text{(MO28)}$$

For any convex coordinate open $W\subset U$ and $t>t_0$,

$$R\Gamma(W\times\mathbb R;H) \xrightarrow{\sim}R\Gamma(W\times(-\infty,t);H). \qquad\text{(MO29)}$$

Indeed, on the upper tail the cone characterization with the positive vertical ray makes $H$ a pullback from that directional topology. The ordinary convex set $W\times(t_0,t)$ has directional saturation $W\times(t_0,\infty)$. Derived convex continuation identifies their section complexes by restriction. Apply Mayer–Vietoris to the cover of $W\times\mathbb R$ by $W\times(-\infty,t)$ and $W\times(t_0,\infty)$: since the latter term maps isomorphically to the intersection, the former computes the global sections. The same argument is valid after an increasing coordinate change identifying a tail with a full line; microsupport transforms with the positive derivative, so the sign is unchanged.

With $\sigma\geq0$ in MO28, the opposite cone gives, for $t_0<t<t'$,

$$R\Gamma_{W\times(-\infty,t]}(W\times\mathbb R;H) \xrightarrow{\sim} R\Gamma_{W\times(-\infty,t']}(W\times\mathbb R;H). \qquad\text{(MO30)}$$

To check this without dualizing an arbitrary sheaf, use the support triangle for the difference $(t,t']$. On $W\times(t,\infty)$, restriction to $W\times(t',\infty)$ is an isomorphism: both have the same saturation for the negative vertical cone. Their localization fibre, the derived sections with support in $(t,t']$, is zero. This is precisely the cone of MO30. The argument proves the supported statement directly and does not require reflexive Verdier duality.

SH02-MO-CUTOFF-PROOF — Proof of the four cutoff maps

The map $h=(f,\varphi):Y\to X\times\mathbb R$ is proper on $\operatorname{supp}(G)$. The inverse image of a compact subset is closed in the inverse image of a compact base set intersected with some closed sublevel, which is compact by MO24. Put $H=Rh_*G$ and let $q:X\times\mathbb R\to X$ be projection. Proper base change identifies every cutoff in MO26 with the corresponding cutoff for $H$, and $Rf_*G=Rq_*H$, $Rf_!G=Rq_!H$.

Under MO25+, MO8 implies MO28. In fact a covector $(x,s;\xi,\sigma)$ in the estimated microsupport comes from

$$d f_y^t\xi+\sigma\,d\varphi_y\in\operatorname{SS}(G)_y.$$

If $s>t_0$ and $\sigma>0$, division by $\sigma$ contradicts MO25+, since $V_f$ is a vector space in each fibre. MO29 on a basis of $X$ proves the open ordinary formula in MO26+.

To obtain the closed formula, fix $x\in X$ and $b>t$. The set $C=\operatorname{supp}(H)\cap(\{x\}\times(-\infty,t])$ is compact. Neighborhoods $W\times(-\infty,u)$, with $W\ni x$ shrinking and $t<u<b$ decreasing to $t$, are cofinal neighborhoods of $C$ relative to the support. Otherwise points escaping a prescribed neighborhood would have base tending to $x$ and height at most $u$; properness below $b$, after fixing a compact base neighborhood, gives a convergent subsequence with limit in $C$, a contradiction. Compact-neighborhood continuity therefore identifies the colimit of their cohomology with $H^r(\{x\}\times(-\infty,t];H)$. Proper base change identifies the latter with the closed-cutoff image stalk. The maps from the full image into all the open-cutoff stalk systems are already isomorphisms. Their colimit proves the closed comparison at $x$. This includes $t=t_0$, since only $u>t_0$ was used. The argument always keeps both the compact support sublevel and the shrinking base neighborhood; it never treats an unbounded half-strip as a compact set.

Under MO25− the same argument gives $\sigma\geq0$ on the upper tail. MO30 makes the supported closed cutoffs constant for $t>t_0$. Their filtered colimit is $Rq_!H$: locally over a compact base set, every support proper over that base is compact and therefore lies below some finite level; conversely supports in a closed sublevel are proper by MO24. This is a cofinality statement about proper supports, not a claim that ordinary sections commute with an arbitrary closed exhaustion. The finite cohomological dimension of proper direct image permits the support-family colimit on bounded resolutions. It proves the closed formula for $t>t_0$. For $t=t_0$, use localization and the increasing union $\{s>t_0\}=\bigcup_{t>t_0}\{s>t\}$. On a flabby resolution, sections on that increasing union are computed by the standard inverse-limit complex, with surjective termwise restrictions. The already constant cohomology systems give zero derived-limit obstruction. Taking fibres in the localization triangle proves the endpoint formula. Open cutoffs are the filtered union of smaller closed support cutoffs; their extension maps are the maps in MO26−. This proves the open exceptional formula for $t>t_0$.

It remains to prove the localized bound. For the ordinary case apply the open boundary estimate to $H$ on $\{s<t\}$, and then proper image to its ordinary extension $Rj_*j^{-1}H$, whose support lies in $\{s\leq t\}$. At the new boundary, both its added normal covector and a preexisting vertical covector of $H$ are nonpositive. A sum with vertical component zero must have both components zero. Away from the boundary no normal is added. The result is

$$\operatorname{SS}(Rq_*H) \subset\{(x,\xi): (x,s;\xi,0)\in\operatorname{SS}(H) \text{ for some }s\leq t\}.$$

Use MO8 for $h$ to lift such a covector to $\operatorname{SS}(G)$ over $\varphi\leq t$. For the exceptional case use the closed supported cutoff and its boundary estimate; both vertical signs are now nonnegative and the same no-cancellation conclusion holds. Finally let $t\downarrow t_0$. The lifting points lie over a fixed base point in the compact support sublevel for any fixed $t_1>t_0$, and their covectors equal $d f_y^t\xi$. A convergent subsequence and closedness give a lift with $\varphi\leq t_0$. This proves MO27. $\square$

SH02-MO-RELATIVE-COVECTOR — A quotient formulation

If $f$ is a submersion, the quotient bundle

$$T^*(Y/X)=T^*Y/V_f$$

is the relative cotangent bundle. If $p$ is its quotient map, MO25± says exactly

$$\pm p(d\varphi_y)\notin p(\operatorname{SS}(G)_y) \quad(\varphi(y)>t_0).$$

Indeed two covectors have the same quotient precisely when their difference is horizontal. For a general map the same fibrewise quotient statement makes sense as a quotient of vector spaces; it need not form a vector bundle when the differential changes rank.

SH02-MO-MORSE — Cohomology between two levels

Let $\varphi:X\to\mathbb R$ be $C^1$, proper on $\operatorname{supp}(F)$, and let $a<b$.

If $d\varphi_x\notin\operatorname{SS}(F)$ whenever $a\leq\varphi(x)<b$, then the natural restrictions

$$R\Gamma(\{\varphi<b\};F) \xrightarrow{\sim}R\Gamma(\{\varphi\leq a\};F) \xrightarrow{\sim}R\Gamma(\{\varphi<a\};F) \qquad\text{(MO31)}$$

are isomorphisms. If $-d\varphi_x\notin\operatorname{SS}(F)$ for $a<\varphi(x)\leq b$, then

$$R\Gamma_{\{\varphi\leq a\}}(X;F) \xrightarrow{\sim}R\Gamma_{\{\varphi\leq b\}}(X;F). \qquad\text{(MO32)}$$

If the negative exclusion instead holds on $a\leq\varphi(x)<b$, then

$$R\Gamma_c(\{\varphi<a\};F) \xrightarrow{\sim}R\Gamma_c(\{\varphi<b\};F). \qquad\text{(MO33)}$$

Proof. Properness permits replacement of $F$ by $H=R\varphi_*F$ on the line, using MO8 and proper base change for both ordinary and supported sections. Its microsupport has no positive covectors on $[a,b)$ in the first case and no negative covectors on the corresponding interval in the other cases. The one-sided argument proves invariance between levels strictly inside these intervals. Endpoint passage is local near $a$ or $b$: split off a fixed lower open tail by Mayer–Vietoris, so the varying part lies in a compact interval. Compact-neighborhood continuity gives a closed endpoint as a colimit of larger open neighborhoods. An open endpoint is the increasing union of smaller opens; their already constant cohomology system has no derived-limit obstruction. This gives the two arrows of MO31 with the stated inclusion of $a$ and exclusion of $b$.

For MO32 use the supported argument MO30; the varying support is the half-open band $(a,b]$, so precisely that band requires the negative exclusion. For MO33 write compactly supported cohomology of an open sublevel as the filtered colimit over closed supports strictly inside it and use the same supported localization argument. The changing band is now $[a,b)$. The properness of $\varphi$ on the support makes every closed bounded band compact; the unchanged lower tail may be noncompact and is retained throughout. No bounded-below exhaustion of the whole support is being assumed here. $\square$

SH02-MO-FIELD-BOUNDARY — Local jumps and finite Morse data

Only in this section and the following proof, assume that $k$ is a field. Let $F\in D^b(k_X)$ and let $\varphi$ be $C^\infty$. Suppose:

1. $\operatorname{supp}(F)\cap\{\varphi\leq t\}$ is compact for every $t$;
2. the graph $\{(x,d\varphi_x):x\in X\}$ meets $\operatorname{SS}(F)$ at finitely many points $p_1,\ldots,p_N$;
3. for $x_i=\pi(p_i)$, the jump complex

$$J_i=\bigl(R\Gamma_{\{\varphi\geq\varphi(x_i)\}}F\bigr)_{x_i}$$

has finite-dimensional cohomology in a bounded range.

The graph of $d\varphi$ is Lagrangian: pullback of the tautological one-form is $d\varphi$, hence pullback of its exterior derivative vanishes, and the graph has half the cotangent dimension. It need not be conic. No nondegenerate critical-point hypothesis is imposed on these finitely many intersections.

SH02-MO-MORSE-INEQUALITIES — A finite filtration by local tests

Under those three hypotheses, $R\Gamma(X;F)$ has bounded finite-dimensional cohomology. Set

$$b_j=\dim_k H^j(X;F),\qquad m_j=\sum_{i=1}^N\dim_k H^j(J_i).$$

For every integer $\ell$,

$$\sum_{j\leq\ell}(-1)^{\ell-j}b_j \leq\sum_{j\leq\ell}(-1)^{\ell-j}m_j, \qquad \sum_j(-1)^j b_j=\sum_j(-1)^j m_j. \qquad\text{(MO34)}$$

Proof. First suppose $N=0$. The closed support is bounded below under $\varphi$: otherwise the nonempty nested compact sets $\operatorname{supp}(F)\cap\{\varphi\leq-n\}$ would have nonempty intersection inside one compact sublevel, which is impossible for a real-valued function. Choose $a$ below this support. There are no failed positive tests at any level, so MO31 identifies $R\Gamma(\{\varphi<b\};F)$ with $R\Gamma(\{\varphi<a\};F)=0$ for every $b>a$. An increasing sequence $b\to+\infty$ exhausts $X$. The open-union Milnor comparison gives $R\Gamma(X;F)=0$, since both the inverse limit and its first derived limit of these zero groups vanish in every degree. If the support is empty, the conclusion holds immediately. Thus $b_j=m_j=0$ for every $j$, proving both assertions in this case.

Now suppose $N>0$. Put $H=R\varphi_*F$. The function is proper on the support because every compact interval lies in a compact sublevel. Let $c_1<\cdots<c_r$ be the distinct values of the $\varphi(x_i)$. Proper base change and localization identify the jump of $H$ at $c_\nu$ with

$$L_\nu=\bigl(R\Gamma_{[c_\nu,\infty)}H\bigr)_{c_\nu} \simeq\bigoplus_{\varphi(x_i)=c_\nu}J_i. \qquad\text{(MO35)}$$

Indeed the supported sheaf restricted to that compact fibre has zero stalk away from the finitely many $x_i$, by the definition of microsupport. A sheaf complex supported at finitely many isolated points has global sections equal to the direct sum of its stalk complexes. This proves MO35 without assuming constructibility of $F$ elsewhere.

There is no contribution below $c_1$. One way to see this is to use compact sublevels: their nonempty intersection as the level tends to $-\infty$ would contradict compactness of any one sublevel, so the support is bounded below. MO31 then propagates the zero section complex up to the first exceptional level. Between consecutive $c_\nu$, and above $c_r$, MO31 identifies the section complexes. At a level $c_\nu$, localization supplies the triangle