# Microlocal composition at prescribed covectors

Ordinary convolution integrates across an entire intermediate manifold. A kernel germ at one pair of covectors does not retain all that information. To compose two germs, we must isolate the intermediate covector that contributes to the chosen output. This lesson constructs that composition, explains why a formal system of convolutions is represented by a bounded sheaf germ, and proves its basic compatibilities.

Coefficients are arbitrary modules over a commutative ring \(k\) of finite global dimension; all sheaf complexes are bounded. Manifolds are finite dimensional, real smooth, Hausdorff and second countable. No constructibility, finite-rank or global support-properness hypothesis is assumed. Tensor products are derived over \(k\). A subscript \(F_V\) for an open set means extension by zero of the restriction to \(V\), with its natural arrow \(F_V\to F\).

The proof has three stages: use a directional cutoff to remove remote middle covectors, use a boundary cutoff to make the relevant image proper, and then compare incoming denominators. The following prerequisites supply those distinct operations.

| Prerequisite and exact statement | Role here |
| --- | --- |
| [Directional localization and incoming fractions](../../sheaf-proof-readings/src/SH02/microlocal-categories.md#sh02-mc-local--a-category-that-records-specified-directions) and [formal operations](../../sheaf-proof-readings/src/SH02/microlocal-categories.md#sh02-mc-four--four-microlocal-operations-and-their-variance) | A denominator has a cone whose microsupport misses the selected point. Morphisms from an incoming formal system are filtered colimits of morphism groups, with equality tested after a common refinement. |
| [Refined incoming replacement, (MC.7)](../../sheaf-proof-readings/src/SH02/microlocal-categories.md#sh02-mc-cutoff--representatives-with-a-prescribed-directional-bound) | For a proper closed convex cone \(C\), an open cone \(U\subset C\), and a conic neighborhood \(W\) of the nonzero part of \(C\cap\operatorname{SS}(F)\) in one cotangent fibre, obtain \(F_-\to F\) invertible on \(U\), with fibre microsupport in \(W\cup\{0\}\). Only the incoming construction is used. |
| [Boundary control, (MC.20)–(MC.24)](../../sheaf-proof-readings/src/SH02/microlocal-categories.md#sh02-mc-tilted--removing-boundary-contributions-from-a-direct-image) | Isolated incidence gives arbitrarily small open restrictions with proper support over the target, no new boundary incidence, and the same exclusion for the comparison cones. The zero target covector is included. |
| [Direct-image germ formula, (MC.25)](../../sheaf-proof-readings/src/SH02/microlocal-categories.md#sh02-mc-direct-germs--direct-images-use-the-germ-at-the-chosen-basepoint) and [isolated-incidence representation, (MC.26)–(MC.28)](../../sheaf-proof-readings/src/SH02/microlocal-categories.md#sh02-mc-direct-rep--isolated-incidence-and-proper-support) | The formal \(!\)-image is computed by shrinking base neighborhoods *after applying the image functor*. Isolated incidence represents it by a bounded germ and confines its witnesses to any prescribed incidence neighborhood. A globally proper representative with entire-fibre isolation computes it by its ordinary image. |
| [Noncancellation and proper addition](../../sheaf-proof-readings/src/SH02/microsupport-operations.md#sh02-mo-cone-proper--no-cancellation-and-closed-images), [noncharacteristic tensor](../../sheaf-proof-readings/src/SH02/microsupport-operations.md#sh02-mo-diagonal--tensor-and-hom-on-one-manifold), [submersion pullback](../../sheaf-proof-readings/src/SH02/microsupport-operations.md#sh02-mo-submersion--exact-pullback-and-local-descent) and [proper direct image](../../sheaf-proof-readings/src/SH02/microsupport-operations.md#sh02-mo-proper-push--collecting-tests-along-a-fibre) | These give the closed ordinary-sum estimate (9), its horizontal matching equation, and the closed-embedding calculation in (10). None of these estimates uses constructible biduality. |
| [Bounded kernel calculus](../../sheaf-proof-readings/src/SH02/kernel-calculus.md#sh02-ker-003-bounds-that-make-the-operators-well-defined) and [elimination of the middle variable](../../sheaf-proof-readings/src/SH02/kernel-calculus.md#sh02-ker-006-eliminating-the-middle-variable) | Finite global dimension bounds derived tensor products; finite-dimensional manifold fibres bound \(!\)-images. Proper-support base change, projection formula and \(!\)-Fubini give the ordinary comparison maps. |

The [external-product bound](../../sheaf-proof-readings/src/SH02/microsupport-operations.md#sh02-mo-external-tensor--product-tests) will also be used. The suppliers identify their own lower inputs, including noncharacteristic deformation, directional projectors, finite cohomological bounds and the proper-support base-change and projection-formula contracts.

*Original lesson text and solutions: CC0 1.0 Universal. Human mathematical sources are credited below.*

## Which middle covector is being composed

Fix \(p_X=(x_0;\xi_0)\), \(p_Y=(y_0;\eta_0)\), and \(p_Z=(z_0;\zeta_0)\). Zero covectors are allowed. Let \(K_1\) be a germ on \(X\times Y\) at \((p_X,p_Y^a)\), and \(K_2\) a germ on \(Y\times Z\) at \((p_Y,p_Z^a)\). A superscript \(a\) reverses the covector and leaves its base point fixed.

For chosen representatives form the twisted matching set

\[
\begin{split}
\mathcal T(K_1,K_2)=\{(u,v,w):
 &(u,v^a)\in\operatorname{SS}(K_1),\\
 &(v,w^a)\in\operatorname{SS}(K_2)\},
\qquad \rho(u,v,w)=(u,w^a).
\end{split}
\tag{1}
\]

The opposite physical middle covectors are \(-\eta\) in \(K_1\) and \(+\eta\) in \(K_2\). Their sum is zero in the ordinary tensor kernel. We call the pair **microlocally composable** if

\[
\mathcal T(K_1,K_2)\cap
(\{p_X\}\times T^*Y\times\{p_Z\})
\subset\{(p_X,p_Y,p_Z)\}
\quad\text{near }(p_X,p_Y,p_Z).
\tag{2}
\]

The allowed singleton need not occur. Condition (2) is local around the specified middle covector. It does not exclude a different witness far away in that fibre. The germ of microsupport is invariant under an isomorphism in each point-localized category, so (2) depends only on the two kernel germs.

Ordinary convolution is
\(K_1\circ K_2=Rq_{13!}(q_{12}^{-1}K_1\otimes^Lq_{23}^{-1}K_2)\).
For example, if \(K_1\in D^{[a,b]}\), \(K_2\in D^{[c,d]}\), \(g\) bounds the global dimension of \(k\), and \(d_Y\) bounds the proper-support cohomological dimension of the projection with fibre \(Y\), then

\[
 K_1\circ K_2\in D^{[a+c-g,\,b+d+d_Y]}.
\]

Thus each ordinary convolution below is bounded. This estimate does not yet make its value depend only on the selected microlocal germs.

For germs we first form the formal system

\[
P=\text{“}\!\lim_{K_1',K_2'}\!\text{”}\;K_1'\circ K_2',
\tag{3}
\]

where \(K_i'\to K_i\) runs over incoming denominators at the appropriate physical pair. A denominator means its cone misses that pair. The notation in (3) means a pro-object in the output localized category \(\mathcal D=D^b(k_{X\times Z};(p_X,p_Z^a))\). More explicitly, its value on a test germ \(A\) is

\[
 \operatorname{Hom}(P,A)=
 \underset{K_1'\to K_1,\ K_2'\to K_2}{\operatorname{colim}}
 \operatorname{Hom}_{\mathcal D}(K_1'\circ K_2',A).
\]

A further incoming denominator induces the transition by precomposition. All indexing categories are small in a fixed universe. This is not an inverse limit of sheaves, and it makes no assertion that one fixed ordinary convolution computes the answer.

## Replacing a kernel to control the whole middle fibre

At the fixed base points, two additional conditions are useful:

\[
\mathcal T\cap
(\{p_X\}\times T_{y_0}^*Y\times\{p_Z\})
\subset\{(p_X,p_Y,p_Z)\},
\tag{4}
\]

\[
\mathcal T\cap
(\{(x_0;0)\}\times \dot T_{y_0}^*Y\times\{(z_0;0)\})
=\varnothing.
\tag{5}
\]

The dot removes the zero covector. Condition (4) controls every covector in the middle fibre at \(y_0\), not just those near \(\eta_0\). Condition (5) rules out nonzero middle cancellation when both endpoint covectors vanish.

Here is the cutoff consequence we need. If \(\xi_0\ne0\), a microlocally composable pair admits an incoming replacement \(K_1'\to K_1\), invertible at \((p_X,p_Y^a)\), for which (4) holds and

\[
\operatorname{SS}(K_1')\cap
(\{(x_0;0)\}\times\dot T_{y_0}^*Y)=\varnothing.
\tag{6}
\]

The same construction works with any specified closed conic set in place of \(\operatorname{SS}(K_2)\). It also works simultaneously with a finite union of such sets. This last clause will allow common refinements of two denominators.

**Proof of the cutoff consequence.** Work in the cotangent fibre at \((x_0,y_0)\), and write \(q=(\xi_0,-\eta_0)\). Local isolation gives a compact neighborhood \(L\) of \(\eta_0\) such that matching the two original microsupports at the fixed endpoints \(p_X,p_Z\), for \(\eta\in L\), occurs only at \(\eta_0\). Choose two proper closed convex cones \(C_-\) and \(C_+\) around \(q\), with \(q\in\operatorname{Int}C_-\), \(C_-\setminus0\subset\operatorname{Int}C_+\), and with the fixed-first-component slice of \(C_+\) contained in \(\{\xi_0\}\times(-L)\). They can also be chosen to contain no nonzero vector with first component zero. Indeed, because \(\xi_0\ne0\), choose a linear functional \(\ell\) of the first component with \(\ell(\xi_0)=1\), take two sufficiently small nested closed convex balls around \(q\) in the hyperplane \(\ell(\xi)=1\), and take their positive cones. Their normalized sections and their fixed-first-component slices are compact.

Choose an open conic neighborhood \(U\) of \(q\) contained in \(\operatorname{Int}C_-\). The refined incoming replacement will be an isomorphism throughout \(U\). On a compact normalized section, \(C_-\cap\operatorname{SS}(K_1)\) is disjoint from the set of second-kernel matching directions with fixed first component \(\xi_0\) that lie in \(C_+\) but outside \(U\): an intersection would be an original matching covector in \(L\), hence would equal \(q\), which is in \(U\). Use the section \(\ell(\xi)=1\) to make this precise. The first set is \(C_-\cap\operatorname{SS}(K_1)\cap\{\ell=1\}\). The second consists of points of \(C_+\cap\{\ell=1\}\) with first component exactly \(\xi_0\), outside \(U\), whose middle component matches \(\operatorname{SS}(K_2)\) at \(p_Z\). It is closed, although fixing \(p_Z\) need not leave a conic set in the middle covector alone. Both sets are compact and disjoint. Separation on this section, followed by positive dilation, gives a conic neighborhood \(W\) of \((C_-\cap\operatorname{SS}(K_1))\setminus0\), contained in \(\operatorname{Int}C_+\), whose fixed-first-component slice contains no second-kernel matching covector outside \(U\).

Apply the refined incoming replacement theorem with the closed cone \(C_-\), the inner isomorphism cone \(U\), and the directional neighborhood \(W\). It gives \(K_1'\to K_1\) invertible throughout \(U\), with microsupport at the base point contained in \(W\cup\{0\}\). A matching covector inside \(U\) belongs to the original microsupport, because the replacement is an isomorphism there, and so it is the chosen covector. A matching covector outside \(U\) is excluded by the choice of \(W\). This proves (4). The containment in \(C_+\cup\{0\}\) excludes every nonzero covector with first component zero and proves (6). For any prescribed closed conic matching set the same argument applies under the stated local isolation; for finitely many such sets choose \(L\) for all of them and separate their finite closed union outside \(U\). Inside \(U\), agreement with the original microsupport preserves each isolation condition. The construction uses the refined replacement theorem at its stated arbitrary bounded coefficient scope. \(\square\)

If \(\zeta_0\ne0\), apply the argument to \(K_2\) after interchanging the endpoint roles. In either case (6), or its endpoint analogue, supplies (5).

There are two remaining cases. If both endpoint covectors are zero but \(\eta_0\ne0\), positive scaling preserves their zeros. If both kernel microsupports contained the specified pairs, scaling \(\eta_0\) by numbers tending to one would contradict (2). Thus one germ is zero and can be replaced by the zero kernel. If all three covectors are zero, (2) isolates the intersection of the two supports in the middle base variable. At the fixed bases, any nonzero matching middle covector could be scaled down towards zero, again contradicting (2). This proves (4) and (5) in the last case as well.

Consequently every composable pair admits incoming representatives satisfying (4) and (5), including the cases involving zero covectors.

## Why the formal composition is a bounded germ

**Theorem.** For a microlocally composable pair, the pro-object (3) is represented by an object of \(D^b(k_{X\times Z};(p_X,p_Z^a))\). Denote it by \(K_1\circ_\mu K_2\). For every neighborhood \(W\) of the specified triple,

\[
\operatorname{SS}(K_1\circ_\mu K_2)
\subset\rho\bigl(W\cap\mathcal T(K_1,K_2)\bigr)
\quad\text{near }(p_X,p_Z^a).
\tag{7}
\]

If the representatives already satisfy (4) and (5), there is a canonical formal identification

\[
K_1\circ_\mu K_2
\simeq\text{“}\!\lim_{V\ni y_0}\!\text{”}\;(K_1)_{X\times V}\circ K_2.
\tag{8}
\]

If \(\xi_0\ne0\), one may instead hold \(K_2\) fixed and vary only \(K_1'\to K_1\) in (3).

**Proof.** First (2), (4) and (5) imply isolated matching over the fixed endpoint covectors on a neighborhood of the middle *base point*, with no bound imposed on the middle covector. If this were false, choose offending \((y_j;\eta_j)\), with \(y_j\to y_0\), matching those endpoints. If \(\eta_j\) is bounded, closedness and (4) force a subsequence to converge to \(p_Y\); then (2) excludes it. If \(|\eta_j|\to\infty\), divide both physical kernel covectors by \(|\eta_j|\). A subsequence of unit middle covectors converges to a nonzero vector at \(y_0\), while the endpoint covectors tend to zero. Conicity and closedness give the forbidden pair in (5). This proves the asserted base-neighborhood isolation.

The same argument gives the uniform control needed later in (7). Choose a middle-base neighborhood whose compact closure is contained in the isolation neighborhood. Choose endpoint base neighborhoods with compact closures small enough that (5) holds throughout the product of all three closures. On this compact base set, matching covectors with endpoint covectors in a sufficiently small compact neighborhood of \((p_X,p_Z)\) have bounded middle covector: otherwise normalization gives a nonzero cancellation at a limiting base point. If such witnesses stayed outside a prescribed neighborhood of \((p_X,p_Y,p_Z)\) while their endpoints approached \((p_X,p_Z)\), a convergent subsequence would give a forbidden fixed-endpoint match. Therefore, after shrinking the output cotangent neighborhood, every retained matching witness lies in the prescribed neighborhood. This controls the intermediate **covector**, as well as its base point.

Put
\(H=q_{12}^{-1}K_1\otimes^Lq_{23}^{-1}K_2\)
on \(X\times Y\times Z\). Condition (5) is exactly the noncharacteristic tensor condition at the fixed base point: possible cancellation has components \((0,-\eta,0)+(0,\eta,0)\). It persists on a smaller base neighborhood. Indeed a sequence of nonzero cancellations at convergent base points, normalized to unit length, would yield a prohibited limiting cancellation at the fixed bases. The ordinary tensor estimate therefore gives

\[
\operatorname{SS}(H)\subset
(\operatorname{SS}(K_1)\times T_Z^*Z)
+(T_X^*X\times\operatorname{SS}(K_2))
\tag{9}
\]

there. A covector on the right of (9) whose middle component vanishes comes from a matching triple (1). By the preceding isolation, the incidence for \(q_{13}\) over \((p_X,p_Z^a)\) is confined to
\((x_0,y_0,z_0;\xi_0,0,-\zeta_0)\).

Apply the isolated-incidence direct-image prerequisite to \(H\). In its exact form it represents the formal proper-support image by a bounded germ, identifies it with the formal ordinary image, and allows its microsupport witnesses to be confined to any prescribed neighborhood of that incidence point. Its direct-germ formula and boundary-control construction give (8). Here is why only middle neighborhoods remain in that formula. Product neighborhoods \(U_X\times V\times U_Z\) form a neighborhood basis at \((x_0,y_0,z_0)\), and

\[
 H_{U_X\times V\times U_Z}
 =H_{X\times V\times Z}\otimes
 q_{13}^{-1}k_{U_X\times U_Z}.
\]

Projection formula moves the last factor outside \(Rq_{13!}\). It is the identity on the output base germ, so shrinking \(U_X,U_Z\) has no effect there. Also \(H_{X\times V\times Z}=q_{12}^{-1}(K_1)_{X\times V}\otimes q_{23}^{-1}K_2\). Thus the neighborhood system is exactly (8). Restricting open middle neighborhoods is legitimate *after this direct-image argument*; they have not been declared cofinal among all microlocal denominators beforehand.

Finally choose the direct-image incidence neighborhood small enough for the uniform matching control above. Each horizontal covector from (9) then has its full matching triple inside the prescribed \(W\). This gives (7). Properness of the retained support and the uniform bound on the middle covector make these retained witness sets closed over a compact output neighborhood; no extra limiting witness is introduced.

It remains to identify this represented image with the system (3), rather than just one choice of representatives. Consider denominators \(K_1'\to K_1\) and \(K_2'\to K_2\). When \(\xi_0\ne0\), make a further incoming replacement of \(K_1'\) using the union of the matching sets of \(K_2'\) and \(K_2\). It satisfies (4) and (5) for both. Write \(C_2=\operatorname{Cone}(K_2'\to K_2)\). Its microsupport is contained in \(\operatorname{SS}(K_2')\cup\operatorname{SS}(K_2)\), and it misses \((p_Y,p_Z^a)\). Let \(B\to K_1'\) be the further replacement. Condition (6) makes \(q_{12}^{-1}B\) noncharacteristic for both second kernels and for \(q_{23}^{-1}C_2\) near the fixed bases. The cone of the tensor comparison is

\[
 q_{12}^{-1}B\otimes q_{23}^{-1}C_2.
\]

If this cone had the horizontal incidence covector \((\xi_0,0,-\zeta_0)\), the tensor estimate would give a matching middle covector for \(B\) and \(C_2\). Whole-fibre isolation for the union forces that covector to be \(\eta_0\), which is excluded by the denominator condition on \(C_2\). Hence the cone has no selected incidence. The empty-incidence boundary-control construction and proper-image estimate represent its formal \(!\)-image by zero at the output point. The preceding product-neighborhood calculation identifies this formal image with its middle-neighborhood system. The induced comparison of the formal systems (8) is therefore invertible; this does not assert that each ordinary neighborhood term is zero.

For a further arrow between two good first replacements, its cone has microsupport in their union and misses \((p_X,p_Y^a)\). Each replacement satisfies (6), so does their union. Both have whole-fibre isolation against the fixed second kernel; the same cone argument applies with the roles of the two factors in the tensor estimate exchanged. Arbitrary first denominators admit these good common refinements by the incoming fraction calculus followed by the cutoff consequence. Thus all comparisons become isomorphisms in the formal neighborhood systems after suitable refinements.

For precision, call a first denominator good relative to a finite list of second kernels when its microsupport satisfies (6) and (4) against each member. Every first denominator admits such a good refinement. Given finitely many denominators and comparison arrows, the incoming fraction calculus first supplies a common denominator and equalizes any required parallel arrows after one more refinement. Applying the cutoff consequence to that common denominator and the finite union of the second microsupports makes it good while preserving all those commutative diagrams. Restricting it to a sufficiently small middle neighborhood is still an incoming denominator and is still good, because that restriction agrees with it near the selected bases.

Now test against an arbitrary output germ \(A\). A morphism class in the system (3) is represented at one ordinary term. Pass to a good first refinement and then to its middle-neighborhood system. The cone argument identifies this formal neighborhood system with the one in which \(K_2\) is fixed; after a further neighborhood refinement, the class therefore has a representative in that controlled system. Conversely every such term is an allowed term of (3). If two classes become equal in either description, the filtered-colimit zero criterion realizes that equality after finitely many further denominator choices. Choose a common good refinement for those choices as above; the same comparisons of formal systems preserve the equality in the other description. This proves both surjectivity and injectivity on the Hom colimits. Finally comparison to any one fixed good first representative identifies this controlled system with its middle-neighborhood system (8). The colimits are consequently identical. This proves (3) equals (8) as a pro-object, without asserting eventual equality of ordinary sheaves or a uniform stabilization of all denominators. It also proves that fixing \(K_2\) gives the same pro-object when \(\xi_0\ne0\).

For \(\zeta_0\ne0\) reverse the construction. When both endpoints are zero and the middle is nonzero the zero representative described above makes all systems zero. When all are zero, a denominator is an isomorphism on some ordinary base neighborhood: exclusion of the zero covector means that the cone vanishes near that base point. Product base neighborhoods are then a common refinement of the kernel denominators, and the preceding boundary-controlled construction applies to the tensor germ. Thus the same formal comparison holds in this case. These exhaust the cases and complete the proof. \(\square\)

The construction respects morphisms of germs. Represent a morphism by a fraction, make the common incoming replacements just used, apply tensor and proper-support image to its ordinary numerator, and invert the resulting denominator. A further common refinement gives the same arrow. Composition of fractions and identity arrows are preserved because ordinary convolution preserves them before localization. This supplies the functoriality used below.

## Three compatibility calculations

First, composition can be expressed as a transformation with one enlarged output manifold. Let

\[
i:X\times Y\times Z\hookrightarrow
(X\times Z)\times(Y\times Z),
\qquad i(x,y,z)=(x,z;y,z),
\]

and put \(\widetilde K_1=i_*(K_1\boxtimes k_Z)\). Then

\[
K_1\circ_\mu K_2\simeq\widetilde K_1\circ_\mu K_2,
\tag{10}
\]

where the right output point is \((p_X,p_Z^a)\), its middle point is \((p_Y,p_Z^a)\), and its last manifold is a point. The physical point of \(\widetilde K_1\) is
\((p_X,p_Z^a,p_Y^a,p_Z)\).
To check composability, the closed-embedding microsupport formula forces the two physical \(Z\)-covectors to sum to zero; the constant \(k_Z\) has no tangential \(Z\)-covector. Matching with \(K_2\) therefore reduces exactly to (2), with no extra middle choice. For ordinary representatives, tensor with \(K_2\) restricts to the closed diagonal in the two \(Z\) copies. Integration over the second \(Z\) removes that diagonal and leaves the original integration over \(Y\). Projection formula gives (10). The construction of \(\widetilde K_1\) also descends to the specified kernel germ: the submersion and closed-embedding estimates send a cone invisible at \((p_X,p_Y^a)\) to a cone invisible at \((p_X,p_Z^a,p_Y^a,p_Z)\). Start with representatives satisfying (4) and (5). Their enlarged pair also satisfies these conditions: the diagonal forces its extra middle \(Z\)-covector, and a nonzero cancellation with zero enlarged output covector would be a nonzero cancellation in the original middle \(Y\)-covector. Apply (8) to both pairs. Product neighborhoods \(V_Y\times V_Z\) form a basis for the enlarged middle base. On the diagonal the \(V_Z\) factor becomes an output-base restriction, which disappears in the output germ; the remaining \(V_Y\) restriction is exactly the original system (8). This identifies the formal systems without assuming that every enlarged-kernel denominator is itself an enlarged original denominator. Naturality of the ordinary diagonal and projection-formula maps makes the resulting isomorphism independent of choices. Closed direct image introduces no exceptional orientation shift here.

Second, we give an associativity statement under a strong whole-matching isolation hypothesis. Joint conicity will force all selected covectors in this hypothesis to be zero; the diagonal exercise below explains why the ordinary nonzero graph case needs a different associativity theorem. Let \(K_3\) be a germ from \(W\) to \(Z\), with fixed \(p_W\). Suppose the triple matching set, after both antipodal matches, has

\[
\mathcal T(K_1,K_2,K_3)
=\{(p_X,p_Y,p_Z,p_W)\}
\tag{11}
\]

on a neighborhood of that point. This condition isolates the whole matching germ; unlike (2), it does not fix the endpoints before taking the intersection. Then the two adjacent pairs and the two pairs involving their composites are composable, and

\[
(K_1\circ_\mu K_2)\circ_\mu K_3
\simeq K_1\circ_\mu(K_2\circ_\mu K_3).
\tag{12}
\]

Here is the isolation check. The specified triple actually occurs by (11). Any other nearby middle witness for the first pair can be appended to the fixed \(K_3\) witness, contradicting (11); the other adjacent pair is treated using the fixed \(K_1\) witness. For a pair involving a composite, choose product cotangent neighborhoods \(U_X\times U_Y\times U_Z\times U_W\) contained in the neighborhood where (11) holds. Use (7) for \(K_1\circ_\mu K_2\), confining its witnesses to \(U_X\times U_Y\times U_Z\), and shrink the composite's output neighborhood as required by that estimate. A nearby match of this composite with \(K_3\), with outer endpoints fixed at \(p_X,p_W\), has an actual lift to a witness of \(K_1,K_2\) in the chosen product. Appending its \(K_3\) witness gives a point of the fourfold matching set in that product. By (11) it is the designated tuple. The other parenthesization is identical with the roles reversed. This proves composability of both outer pairs, even when a composite has no microsupport at the selected point.

To prove (12) as an actual natural isomorphism, take the formal systems over all three denominators and over the two intermediate base neighborhoods. Common refined cutoffs and the just-proved isolation identify either iterated construction with that system. For every ordinary term, associativity of tensor and \(!\)-Fubini identify both parenthesizations with integration of
\(q_{12}^{-1}K_1'\otimes q_{23}^{-1}K_2'\otimes q_{34}^{-1}K_3'\)
over \(Y\times Z\). The ordinary associativity maps commute with refinements and neighborhood maps. Testing against every output germ, iterated filtered colimits agree with the colimit over the joint indexing category: each finite set of choices has a common refinement. Thus the two formal systems are naturally isomorphic, and representability gives (12). The ordinary coherence equalities survive this construction, since they hold for each common ordinary term.

The literal hypothesis (11) is particularly strong. A nonzero matching tuple has nearby distinct tuples obtained by scaling all its covectors by the same positive number. Thus it cannot be an isolated singleton in the entire matching set. We retain (11) for this associativity statement; an associativity theorem with only endpoint-fibre isolation would require its own hypotheses and proof. Ordinary global convolution and the earlier admissible regional composition have their separately established associativity statements.

Third, take another composable pair \((K_1',K_2')\) on \(X',Y',Z'\), with its own selected covectors. The external pair is composable, and

\[
(K_1\boxtimes^L K_1')\circ_\mu
(K_2\boxtimes^L K_2')
\simeq (K_1\circ_\mu K_2)\boxtimes^L
(K_1'\circ_\mu K_2').
\tag{13}
\]

Indeed external-product microsupport is contained in the product of the two microsupports, so every nearby matching witness has a witness for each original pair. Conditions (2) for those pairs force the prescribed two middle covectors. Choose good incoming representatives for both pairs. Their external products are denominators for the external kernels, by the external-product bound applied to their comparison cones. The two factorwise versions of (4) imply (4) for the product pair; a nonzero middle cancellation with zero product endpoints would violate (5) in at least one factor. Thus these are good representatives for the product pair too.

Use the proper restricted tensor representatives supplied by the two factorwise direct-image proofs. Their external product has proper support over the product output and isolated incidence there. The product direct-image theorem therefore computes the external composition by its ordinary \(!\)-image. Proper-support base change, projection formula and \(!\)-Fubini identify this image with the external product of the two factorwise images, giving (13). These maps commute with refinements, so the isomorphism descends to the germs. This argument does not assume that an arbitrary denominator of an external product factors into two external denominators. This proves the compatibility without requiring the microsupport of an external product to equal the full product for arbitrary coefficients.

## Kernels that act on every incoming germ

Define \(\mathcal N(X,Y;p_X,p_Y)\) as the full subcategory of kernel germs satisfying

\[
\operatorname{SS}(K)\cap
(\{p_X\}\times T^*Y)
\subset\{(p_X,p_Y^a)\}
\quad\text{near }(p_X,p_Y^a).
\tag{14}
\]

It is triangulated: shifts preserve microsupport, and the microsupport of a term of a triangle is contained in the union of the other two. For any input kernel germ \(L\) from \(Z\) to \(Y\), (14) forces the sole possible middle witness to be \(p_Y\). Thus the pair is composable and there is a bifunctor

\[
\mathcal N(X,Y;p_X,p_Y)\times
D^b(k_{Y\times Z};(p_Y,p_Z^a))
\longrightarrow D^b(k_{X\times Z};(p_X,p_Z^a)).
\tag{15}
\]

If \(L\) satisfies the analogous condition at \(p_Y,p_Z\), the composite satisfies (14) at \(p_X,p_Z\). To check it, take the confined microsupport estimate (7) inside neighborhoods where both versions of (14) hold. A witness with first component \(p_X\) first has middle component \(p_Y\), then last component \(p_Z\). The proper confined representatives exclude additional limiting witnesses. Hence composition also induces

\[
\mathcal N(X,Y;p_X,p_Y)\times
\mathcal N(Y,Z;p_Y,p_Z)
\longrightarrow\mathcal N(X,Z;p_X,p_Z).
\tag{16}
\]

In particular take \(Z\) to be a point. By a graph kernel here we mean a kernel whose **twisted microsupport** near the selected pair lies in the graph of a map from the output cotangent neighborhood to the input cotangent neighborhood, taking \(p_X\) to \(p_Y\). A kernel of a local homogeneous contact transformation has this property. Fixing its output at \(p_X\) forces its input to be \(p_Y\), proving (14); hence it acts on every input sheaf germ.

To compare this action with an ordinary regional convolution, require a representative with [selected-region admissibility](../../sheaf-proof-readings/src/SH03/kernels-that-preserve-chosen-cotangent-directions.md#the-localized-composition-theorem) and require its **entire selected twisted relation over an output neighborhood** to lie in that single graph. This second condition excludes a remote input branch with the same output; local condition (14) and admissibility alone do not exclude it. Under these hypotheses the germ action agrees with the regional localized operator. One checks the comparison by its proper-support kernel map: selected cotangent properness confines all middle covectors and base points over a compact output neighborhood, and the graph selects a unique input germ. The boundary-controlled direct-image comparison in the proof of (8) then removes the part outside a neighborhood of that input point. The actual comparison cones have no output incidence. Consequently a kernel isomorphism at the selected physical pair gives an isomorphism of these point-localized actions. For a diagonal kernel the selected input equals the output. This is the kernel-germ descent used in the diagonal coefficient model of the identity quantization.

**A remote branch changes a regional action.** Let \(k\ne0\), \(X=Y=\mathbb R\), and

\[
 K=k_{\{x=y\}}\oplus k_{\{x=y+1\}},\qquad
 p_X=p_Y=(0;1),\qquad F=k_{[-1,\infty)}.
\]

Near the selected physical pair, the second summand is absent and the twisted relation is the diagonal graph. On the positive cotangent regions the full relation is admissible: it is the union of the two graphs \((x;\xi)\mapsto(x;\xi)\) and \((x;\xi)\mapsto(x-1;\xi)\), each proper over the output. But \(F\) is constant near zero, so its germ at \((0;1)\) is zero, whereas ordinary convolution gives

\[
 K\circ F\simeq k_{[-1,\infty)}\oplus k_{[0,\infty)}.
\]

The second summand has the positive boundary covector \((0;1)\) in its microsupport. Its contribution comes from the remote input \((-1;1)\). Thus the regional action at the output is nonzero while the action on the specified input germ is zero. This is exactly the extra branch excluded by the entire-relation hypothesis.

## Exercises with complete solutions

### Check the middle sign

*Difficulty: Introductory.*

Let a witness of (1) have middle covector \(\eta\). Write the covectors of both pulled-back kernels on \(X\times Y\times Z\) and their sum. What goes wrong if the first physical middle covector is taken to be \(+\eta\)?

**Solution.** The two covectors are \((\xi,-\eta,0)\) and \((0,\eta,-\zeta)\). Their sum is \((\xi,0,-\zeta)\), which is the pullback of the output covector under \(q_{13}\). With \(+\eta\) in the first kernel the sum has middle component \(2\eta\). It is not such a pullback unless \(\eta=0\), so that convention fails to describe the pushforward incidence for nonzero middle covectors.

### Local isolation is weaker than whole-fibre isolation

*Difficulty: Intermediate.*

In one-dimensional cotangent fibres, let the two twisted relations be the unions of graphs \(\xi=\eta\), \(\xi=2\eta\) and \(\eta=\zeta\), \(\eta=\zeta/2\), respectively. Fix endpoint covectors \(\xi=\zeta=1\) and chosen middle covector \(\eta=1\). Compare (2) and (4) for these relations.

**Solution.** There are two matched middle values: \(1\) uses the first branch of each relation, and \(1/2\) uses the second branch of each. A neighborhood of \(\eta=1\) excludes \(1/2\), so the local isolation condition (2) holds. Condition (4) fails because it inspects the whole middle fibre at the fixed bases and sees both values. This is a calculation of closed conic relation models; it does not presume that every such relation is the microsupport of a particular kernel. The cutoff step is precisely what removes the unwanted remote middle branch from an appropriate representative.

### Cancellation with zero endpoints is an obstruction at infinity

*Difficulty: Intermediate.*

Take both \(X\) and \(Z\) to be points, \(Y=\mathbb R\), and both kernels to be the skyscraper \(k_{\{0\}}\), with \(k\ne0\). At any chosen middle covector in \(T_0^*\mathbb R\), show why the pair fails (2), and exhibit the obstruction (5). Does ordinary convolution nevertheless exist?

**Solution.** Each skyscraper has the entire conormal fibre at zero as microsupport. Thus every \(\eta\in T_0^*\mathbb R\) matches zero endpoint covectors. No middle covector is isolated, including \(\eta=0\), so (2) fails. Every nonzero middle covector also violates (5). Ordinary convolution does exist and is \(R\Gamma_c(\mathbb R;k_{\{0\}}\otimes^L k_{\{0\}})=k\). The obstruction concerns whether a result is determined by the specified two microlocal germs under the theorem's hypotheses; it is not a failure of ordinary \(!\)-convolution to be defined.

### Why the graph realization carries opposite Z covectors

*Difficulty: Advanced.*

For the embedding \(i\) in (10), compute its conormal condition on the two \(Z\) covectors. Explain the selected physical point of \(\widetilde K_1\) and the absence of a codimension shift.

**Solution.** The tangent vector in the two \(Z\) positions is \((v_Z,v_Z)\), so a conormal covector has \(\zeta_1+\zeta_2=0\). The original external factor \(k_Z\) has zero tangential covector, leaving the same sum-zero condition for the lifted microsupport. For output \((p_X,p_Z^a)\) and middle \((p_Y,p_Z^a)\), the physical kernel point is \((p_X,p_Z^a,p_Y^a,p_Z)\). Its \(Z\) components are opposite as required. Formula (10) uses closed direct image and a diagonal support constraint; it does not apply exceptional restriction along \(i\). No relative dualizing complex appears, and hence no extra shift arises.

### Isolating endpoints is different from isolating the whole graph

*Difficulty: Advanced.*

Take three diagonal kernels on copies of the same manifold and a nonzero covector \(p\). At the triple with all selected covectors \(p\), compare the pair condition (2), endpoint-fibre isolation for the triple, and the whole-matching condition (11). Does failure of the last condition negate ordinary associativity?

**Solution.** A diagonal kernel has twisted relation \(u=v\). Thus either adjacent pair has matching triples \(u=v=w\); after fixing its endpoints at \(p\), its middle is forced to be \(p\), proving (2). The three-kernel matching set is \(u=v=w=t\), so fixing the outer endpoints also forces both middle points to be \(p\). But the full set contains distinct nearby tuples with all covectors scaled from \(p\) by a common positive factor. Therefore (11) fails. The ordinary diagonal kernels have proper identity projections, and their convolutions are again the diagonal kernel; ordinary associativity follows from the kernel calculus. Its proof uses those global support conditions, not (11). A sufficient condition in a particular theorem need not be necessary for the conclusion by another theorem.

### A formal pro-object is not a chosen term

*Difficulty: Advanced.*

In abelian groups consider the inverse system \(\mathbb Z\xleftarrow{2}\mathbb Z\xleftarrow{2}\cdots\). Compute morphisms from its formal pro-object into the constant object \(\mathbb Z\). Compare them with morphisms from one term and from the ordinary inverse limit.

**Solution.** By the pro-object convention the morphism group is the direct limit of \(\operatorname{Hom}(\mathbb Z,\mathbb Z)=\mathbb Z\) under precomposition by multiplication by two. It is \(\mathbb Z[1/2]\). One term instead gives \(\mathbb Z\). The ordinary inverse limit is zero: its first component would have to be divisible by every power of two, and the same holds for every component. Its morphism group into \(\mathbb Z\) is zero. Thus neither one term nor an ordinary inverse limit computes the formal pro-object. In the theorem, representability is proved using denominator refinements and isolated direct images; it is not obtained by erasing the quotation marks in (3).

## References

Masaki Kashiwara and Pierre Schapira's [Microlocal study of sheaves](https://webusers.imj-prg.fr/~pierre.schapira/BooksMono/Ast128.pdf), Astérisque 128 (1985), §6.3, pp. 108–114, supplies the microlocal kernel and contact-transformation setting. Its Proposition 6.3.1 imposes properness of a selected microsupport projection; Proposition 6.3.3 proves composition under the corresponding regional hypotheses. Those statements do not by themselves prove the isolated-covector formal representability theorem used here. In particular, this lesson has not acquired a constructibility or global properness assumption by comparison with that source.

The linked prerequisite proofs at the beginning supply the incoming directional cutoff, boundary control, formal direct-image calculation and isolated-incidence representation used here. The proof separates these mechanisms: cone separation removes remote middle covectors; normalization rules out escape through unbounded cancellation; proper boundary restrictions permit direct-image estimates; and common denominator refinements make the result a construction on localized categories. The ordinary tensor and proper-support comparison maps then yield the displayed compatibilities.