Reading a sheaf at the normal scale

The proofs in this unit use the explicit prerequisite contracts below; using a contract does not close its proof obligation. Kashiwara and Schapira’s Microlocal Study of Sheaves, §2.2, provides the classical specialization construction and its functorial comparisons. Its positive-chamber construction is the starting point for the map and support calculations below. The source account at the end identifies its mechanisms and the additional arguments written out here. No constructibility or finite-generation assumption is made.

The route through the proofs starts with a normalization problem: which boundary map fixes the shift? The neighborhood and support formulas then show what normal data can detect. Ordinary restriction and the exceptional costalk are checked separately, with their actual unit and counit, before the punctured recovery is deduced. Agreement of the objects alone would not identify the arrow in that recovery triangle.

Next follow the directions of the direct and inverse comparison maps before asking when they are isomorphisms. The compactness proof separates escape in the original manifold from escape in normal velocity; the orientation calculation keeps the relative shifts visible. Finally, homogeneous sheaves calibrate the construction, and two arcs that have the same tangent ray show why tensor comparison need not be invertible. The four solved problems test the zero-rank, endpoint, properness and orientation mechanisms.

SH02-SP-CONVENTIONS — The object being constructed

Let $k$ be a commutative ring of finite global dimension. All manifolds are finite-dimensional, Hausdorff and countable at infinity, and maps are $C^\infty$; the real analytic setting is also allowed. In the comparison theorems, a map called smooth means a submersion, not merely a $C^\infty$ map. Let $i:M\hookrightarrow X$ be a closed embedded submanifold. A locally closed embedding is treated in an open neighborhood in which it is closed; every construction below is compatible with that restriction. Write

$$E=T_MX,\quad \tau:E\longrightarrow M,\quad e:M\longrightarrow E, \quad D=D_MX,\quad p:D\longrightarrow X,\quad t:D\longrightarrow\mathbb R.$$

Here $D$ is the deformation to the normal bundle. Its central fibre is $E$, embedded by $s:E\hookrightarrow D$. Its positive chamber is $\Omega=\{t>0\}$, embedded by $j$, with $r=p|_\Omega:\Omega\simeq X\times\mathbb R_{>0}\to X$. In coordinates $x=(a,b)$ with $M=\{a=0\}$, the deformation coordinates are $(v,b,t)$ and $p(v,b,t)=(tv,b)$. The positive parameter is oriented by increasing $t$. The scaling action is $(v,b,t)\mapsto(\lambda v,b,t/\lambda)$ for $\lambda>0$; it fixes $p$ and restricts to ordinary dilation on $E$.

All input complexes belong to $D^b(k_X)$ or the corresponding bounded category on another manifold. The notation does not mean perfect or constructible. A shift satisfies $H^a(K[n])=H^{a+n}(K)$. Thus a module with its only cohomology in degree one is written $L[-1]$. Tensor products are derived unless a displayed factor is an invertible orientation complex. The support of a complex means the closed support: the closure of the union of the supports of its cohomology stalks.

The proof dependencies are as follows. The identifiers are contracts, not claims that these inputs are proved.

SH02-IMPORT-TAUTNESS — Tautness (SH-01 import contract)

Derived cohomology of a locally closed subspace of a metrizable manifold is the filtered colimit over its neighborhoods, in every degree; the same holds for restriction to a closed submanifold.

SH02-OPS-SIX — Six-operation prerequisite contract

Exact inverse image, derived direct images, their units/counits, localization, $Rf_!\dashv f^!$, proper-support base change, projection formula, smooth base change, and trace-compatible pasting of these maps.

SH02-OPS-BOUNDS — Boundedness prerequisite contract

The preceding operations on finite-dimensional manifolds have the finite cohomological amplitudes needed to carry bounded inputs to the bounded outputs displayed below.

SH02-OPS-ORIENTATION — Orientation prerequisite contract

$\omega_{Y/X}=\operatorname{or}_Y\otimes f^{-1}\operatorname{or}_X^{-1}[\dim Y-\dim X]$, its trace, and the smooth formula $f^!A\simeq f^{-1}A\otimes\omega_{Y/X}$.

Tautness here includes a noncompact locally closed subset; replacing it by a compact-set version would leave the section theorem below unproved. The interval condition concerns fibres of the projection, not merely the image of a scaling orbit. No nonproper closed-base-change assertion is used.

SH02-SP-BOUNDARY — Fixing the boundary shift

Let $H=r^{-1}F$. The localization triangle for the open inclusion $j$, applied to $j_!H$, is

$$R\Gamma_{D\setminus\Omega}(j_!H)\longrightarrow j_!H \longrightarrow Rj_*H\xrightarrow{+1}.$$

The first term is supported on $E$: it vanishes on both $\Omega$ and $\{t<0\}$. It is consequently $s_*s^!j_!H$. Restriction by $s^{-1}$ kills the middle term and gives a natural isomorphism

$$s^{-1}Rj_*r^{-1}F\simeq s^!j_!r^{-1}F[1].$$

The orientation of the positive parameter identifies $r^!F\simeq r^{-1}F[1]$. We therefore obtain

$$s^{-1}Rj_*r^{-1}F\simeq s^!j_!r^!F. \qquad\text{(SH02-SP-BOUNDARY-EQ)}$$

This isomorphism is the connecting isomorphism in the displayed triangle, followed by the oriented smooth-pullback isomorphism. That specification fixes the map and its sign; an unspecified equivalence of the two objects would not suffice when traces are used later.

Define the specialization by

$$\nu_MF=s^{-1}Rj_*r^{-1}F\in D^b(k_E). \qquad\text{(SH02-SP-DEFINITION-EQ)}$$

The boundedness assertion follows from SH02-OPS-BOUNDS. The second expression in SH02-SP-BOUNDARY-EQ is an alternative formula for this same functor, with its stated natural identification.

SH02-SP-CONIC — Directions and support

The complex $r^{-1}F$ is invariant under the scaling action on $\Omega$: its pullback from $X$ is unchanged because $r$ is unchanged. The inclusions $j$ and $s$ are equivariant. SH02-CON-FUNCTORS therefore gives

$$\nu_MF\in D^b_{\mathbb R_{>0}}(k_E).$$

Put $Z=\operatorname{supp}F$. Outside the closure of $r^{-1}Z$, the complex $Rj_*r^{-1}F$ vanishes: each point has an open neighborhood whose intersection with $\Omega$ misses that closed support. Restriction to $E$ gives

$$\operatorname{supp}(\nu_MF)\subset C_M(Z). \qquad\text{(SH02-SP-SUPPORT-BOUND)}$$

This argument proves containment. It does not assert equality; derived sections in a direction can vanish even when the geometric support approaches that direction.

SH02-SP-SECTIONS — A neighborhood test in the original manifold

For an open conic subset $V\subset E$, let $\mathcal U(V)$ consist of open subsets $U\subset X$ satisfying

$$C_M(X\setminus U)\cap V=\varnothing.$$

The transition map for $U'\subset U$ is restriction from $U$ to $U'$. The family is filtered in that direction: $U\cap U'$ is again admissible, because normal cones commute with finite unions. For every integer $a$,

$$H^a(V;\nu_MF)\simeq \underset{U\in\mathcal U(V)}{\operatorname{colim}}H^a(U;F). \qquad\text{(SH02-SP-SECTIONS-EQ)}$$

We first specify the map. Pull a section complex on $U$ back to $r^{-1}U$. Cone avoidance says that $V\cup r^{-1}U$ is a neighborhood of $V$ in $D_{\ge0}$. Extend that neighborhood to an open subset of $D$ and restrict $Rj_*r^{-1}F$ to $V$. The resulting map $R\Gamma(U;F)\to R\Gamma(V;\nu_MF)$ is independent of the extension: two choices agree after their common refinement. These maps respect restriction in $U$.

For the proof, tautness and the definition of $Rj_*$ give

$$H^a(V;\nu_MF) \simeq\underset{W\supset V}{\operatorname{colim}} H^a(W\cap\Omega;r^{-1}F),$$

where $W$ runs through open neighborhoods in $D$. The interval-neighborhood contract permits a cofinal choice for which $r:W\cap\Omega\to U_W=r(W\cap\Omega)$ has nonempty interval fibres. The positive chamber is the product $X\times\mathbb R_{>0}$, so the interval descent theorem identifies the last group with $H^a(U_W;F)$.

The images $U_W$ are admissible. Indeed, a smaller open neighborhood of $V$ contained in $W$ has its positive part in $r^{-1}U_W$, so cone avoidance applies. Conversely, given $U\in\mathcal U(V)$, apply the same cofinal refinement inside a neighborhood whose positive part is contained in $r^{-1}U$. Its image $U_W$ lies in $U$. Thus these images are cofinal in $\mathcal U(V)$, and their descent identifications give precisely the previously specified map. This proves the formula in every degree.

For a vector $v\in E$, this becomes the stalk formula

$$H^a(\nu_MF)_v\simeq \underset{v\notin C_M(X\setminus U)}{\operatorname{colim}}H^a(U;F). \qquad\text{(SH02-SP-STALK-EQ)}$$

To justify using conic neighborhoods for this stalk, first take a small product chart in the quotient of $E\setminus M$ by positive dilation. Its inverse image is a conic neighborhood with a radial interval factor; conic descent computes the stalk by these charts. At the zero section, intersect with fibrewise balls and use radial contraction before taking the base neighborhood colimit. Finally, if $v$ is outside a closed conic set, the complement itself is an open conic neighborhood of $v$. These observations identify the two filtered systems and prove the displayed formula, including zero vectors.

SH02-SP-SUPPORTS — Keeping track of closed supports

Let $A\subset E$ be closed and conic. Consider pairs $(U,Z)$ with $U$ an open neighborhood of $M$ in $X$, $Z$ closed in $X$, and $C_M(Z)\subset A$. Restriction to a smaller $U$ and enlargement to a larger $Z$ give the transition maps. Intersecting the neighborhoods and taking the union of two supports shows that this system is filtered. There are natural isomorphisms

$$H_A^a(E;\nu_MF)\simeq \underset{(U,Z)}{\operatorname{colim}}H^a_{Z\cap U}(U;F). \qquad\text{(SH02-SP-SUPPORTS-EQ)}$$

Here is the map and the proof. A supported section on $U$ pulls back to a section supported on $r^{-1}(Z\cap U)$. Its closure in $p^{-1}U\cap D_{\ge0}$ meets $E$ only in $C_M(Z)\subset A$. Restriction to the central fibre therefore gives a section with support in $A$. The localization triangles for $Z\cap U\subset U$ and $A\subset E$ give a morphism of long exact cohomology sequences after taking this filtered colimit.

Two of its three terms are already known. For the term without supports, an open set has $C_M(X\setminus U)=\varnothing$ exactly when it contains $M$. Formula SH02-SP-SECTIONS-EQ with $V=E$ applies. For the complement term, the sets $U\setminus Z$ lie in $\mathcal U(E\setminus A)$ because

$$C_M\bigl(X\setminus(U\setminus Z)\bigr) \subset C_M(X\setminus U)\cup C_M(Z)\subset A.$$

They give the full admissible system: any admissible open $W$ is represented by $U=X$ and $Z=X\setminus W$. Common refinements preserve that representation at the level of the colimit. Hence the complement maps are isomorphisms by the section theorem as well. Exactness of filtered colimits of $k$-modules and the five lemma give the supported isomorphism. This proof also establishes compatibility with the maps in the localization triangles.

SH02-SP-ZERO — What survives when the direction is forgotten

There are canonical identifications

$$e^{-1}\nu_MF\simeq R\tau_*\nu_MF\simeq i^{-1}F, \qquad e^!\nu_MF\simeq R\tau_!\nu_MF\simeq i^!F. \qquad\text{(SH02-SP-ZERO-EQ)}$$

We describe the maps rather than choosing isomorphisms after the fact. The unit $p^{-1}F\to Rj_*j^{-1}p^{-1}F$, restricted to $s\circ e$, gives $i^{-1}F\to e^{-1}\nu_MF$. For the second identification, the boundary formula and the counit $j_!j^!p^!F\to p^!F$ give

$$e^!\nu_MF\simeq e^!s^!j_!r^!F \longrightarrow e^!s^!p^!F\simeq i^!F.$$

To prove the first map invertible, work over each open subset of $M$ and use the section formula with the entire normal bundle over that subset. Its admissible sets are just neighborhoods of that part of $M$; tautness identifies the colimit with the cohomology of $i^{-1}F$. The conic ordinary contraction identifies the other side with $e^{-1}\nu_MF$. The described unit induces exactly these restriction maps, so it is the isomorphism just computed.

For the second map, apply the support formula with $A=e(M)$. We need a small geometric fact. If $Z$ is closed and $C_M(Z)$ is contained in the zero section, then there is a neighborhood $U_0$ of $M$ such that $Z\cap U_0\subset M$. Otherwise some $m\in M$ is approached by points of $Z\setminus M$. In a normal chart write them as $(a_n,b_n)$ with $a_n\ne0$, divide the normal coordinate by $\lVert a_n\rVert$, and pass to a subsequence of the unit sphere. The sequence criterion puts a nonzero normal vector in $C_M(Z)$, a contradiction. Taking the union of the resulting neighborhoods over $m$ gives $U_0$. Thus the supported colimit reduces to $H^a_M(U;F)$ over neighborhoods of $M$, which is the cohomology of $i^!F$. Localizing on $M$ gives the corresponding isomorphism of cohomology sheaves. It remains to check the particular counit map displayed above, rather than an unspecified object-level isomorphism.

Set $F_M=i_*i^!F$ and use the support counit $F_M\to F$. The support formula shows that $e^!\nu_MF_M\to e^!\nu_MF$ is an isomorphism: in its colimit, $R\Gamma_M(U;F_M)\to R\Gamma_M(U;F)$ is an isomorphism by idempotence of sections with support. Naturality therefore reduces the required counit calculation to $F=i_*L$, for $L\in D^b(k_M)$.

Let $h:M\times\mathbb R\hookrightarrow D$ be the zero-normal-coordinate axis in the deformation, and let $z:M\hookrightarrow M\times\mathbb R$ be time zero. The positive pullback of $i_*L$ is supported on this axis. The oriented product formula and proper base change for its closed embedding identify

$$j_!r^!i_*L\simeq h_*(L\boxtimes k_{(0,\infty)}[1]), \qquad \nu_Mi_*L\simeq e_*z^!(L\boxtimes k_{(0,\infty)}[1])\simeq e_*L.$$

To identify the map, put $\pi:M\times\mathbb R\to M$. Composition of exceptional inverse images gives

$$h^!p^!i_*L\simeq(p\circ h)^!i_*L \simeq\pi^!i^!i_*L\simeq\pi^!L.$$

Under $h_*\dashv h^!$, the counit from $h_*(L\boxtimes k_{(0,\infty)}[1])$ to $p^!i_*L$ therefore becomes the open-extension counit on the axis with target $\pi^!L$. There is no claim that $p^!i_*L$ itself is supported on that axis. Applying $z^!$ shows that the counit $e^!\nu_Mi_*L\to i^!i_*L=L$ is the oriented endpoint boundary map tensored with $L$. The endpoint costalk of $k_{(0,\infty)}$ is $k[-1]$, its projection orientation contributes $[1]$, and the connecting map used in SH02-SP-BOUNDARY-EQ identifies their composite with the identity of $k$. This follows also by writing the localization triangle for the half-line: the restriction map from the constant sheaf on the closed half-line to its endpoint is the identity on sections. Hence the counit is the identity of $L$. The naturality square for $F_M\to F$, whose other vertical arrows are isomorphisms, proves that the original counit is an isomorphism. Proper-support contraction supplies $R\tau_!\nu_MF\simeq e^!\nu_MF$.

Let $a:X\setminus M\hookrightarrow X$, $b:E\setminus M\hookrightarrow E$, and $\tau^\circ=\tau|_{E\setminus M}$. The two zero-section identifications fit into the localization triangles and imply

$$R\tau^\circ_*b^{-1}\nu_MF\simeq i^{-1}Ra_*a^{-1}F. \qquad\text{(SH02-SP-PUNCTURE-EQ)}$$

Indeed, compare the triangles

$$i^!F\longrightarrow i^{-1}F\longrightarrow i^{-1}Ra_*a^{-1}F\xrightarrow{+1}$$

and

$$e^!\nu_MF\longrightarrow R\tau_*\nu_MF \longrightarrow R\tau^\circ_*b^{-1}\nu_MF\xrightarrow{+1}.$$

Here is the first-arrow compatibility with the specified maps. Specialize the support counit $i_*i^!F\to F$ and use $\nu_Mi_*i^!F\simeq e_*i^!F$. Its adjoint is a map $\sigma:i^!F\to e^!\nu_MF$. Naturality of the exceptional counit and its endpoint identity give $c_F\sigma=\operatorname{id}$, where $c_F:e^!\nu_MF\to i^!F$ is the isomorphism just proved. Thus $\sigma=c_F^{-1}$. Naturality of the ordinary restriction unit, applied to the same support counit, identifies the maps from supported to ordinary restriction. The first two vertical maps are therefore compatible isomorphisms, so the induced map of the third terms is an isomorphism. This statement retains the whole punctured normal bundle, rather than choosing a sphere or a metric.

SH02-SP-DIRECT — Sending normal data through a map

Consider a map of pairs $f:(Y,N)\to(X,M)$, where $N$ and $M$ are closed embedded submanifolds and $f(N)\subset M$. Write $E_Y=T_NY$, $E_X=T_MX$, $q=T_Nf:E_Y\to E_X$, and $g:D_NY\to D_MX$ for the induced maps. Decorate $s,j,r,p,t$ by $X$ or $Y$ when necessary, and put $g_+=g|_{\Omega_Y}=f\times\operatorname{id}_{\mathbb R_{>0}}$. The central and positive squares are Cartesian. No assertion is made that the square involving $p_Y,p_X,f,g$ is Cartesian at time zero.

For $G\in D^b(k_Y)$ there is a commuting square, with both vertical arrows forgetting proper supports:

$$\begin{array}{ccc} Rq_!\nu_NG&\xrightarrow{c_!}&\nu_M Rf_!G\\ \downarrow&&\downarrow\\ Rq_*\nu_NG&\xleftarrow{c_*}&\nu_M Rf_*G. \end{array} \qquad\text{(SH02-SP-DIRECT-SQUARE)}$$

Here are definitions that determine all four maps. Set $K=Rj_{Y*}r_Y^{-1}G$. Proper-support base change at the central fibre gives $Rq_!s_Y^{-1}K\simeq s_X^{-1}Rg_!K$. The comparison

$$Rg_!Rj_{Y*}A\longrightarrow Rj_{X*}Rg_{+!}A$$

is the adjoint, under $j_X^{-1}\dashv Rj_{X*}$, of proper-support base change followed by the restriction counit $j_Y^{-1}Rj_{Y*}A\to A$. Apply it with $A=r_Y^{-1}G$. Base change in the positive product square gives $Rg_{+!}r_Y^{-1}G\simeq r_X^{-1}Rf_!G$. Restriction by $s_X^{-1}$ defines $c_!$.

For the other arrow, smooth base change along the projection $r_X$ gives

$$\nu_M Rf_*G\simeq s_X^{-1}Rg_*Rj_{Y*}r_Y^{-1}G.$$

Compose this with the ordinary base-change map $s_X^{-1}Rg_*K\to Rq_*s_Y^{-1}K$ to define $c_*$. This last map is a comparison, not an assumed isomorphism.

For completeness, the square’s commutativity is a statement about these maps. In the first construction, replace $Rg_!$ and $Rg_{+!}$ by their maps to ordinary direct image; naturality of restriction identifies the result with the composition map for $Rg_*Rj_{Y*}$. Then restrict to the central fibre and apply the ordinary base-change map. The result is the map $Rq_!s_Y^{-1}K\to Rq_*s_Y^{-1}K$: the intermediate unit and counit cancel by their triangle identity. This verifies that going across the top, down, and back along the bottom equals the left vertical arrow. It also shows why the arrow on the bottom points in the opposite direction.

SH02-SP-PROPER-GEOMETRY — The extra compactness condition

Let $Z\subset Y$ be closed. Suppose that

1. $f|_Z:Z\to X$ is proper;
2. $q|_{C_NZ}:C_NZ\to E_X$ is proper;
3. $Z\cap f^{-1}(M)\subset N$.

The conditions rule out three different failures. Properness on the original support keeps the underlying points from escaping. The support-intersection condition places a limit over the target submanifold on the chosen source submanifold. Properness on the normal cone then keeps rescaled normal vectors from escaping while their images converge. Each condition is used separately in the proof, so ordinary properness alone cannot replace this list.

Then the closed subset

$$D_Z=\overline{r_Y^{-1}Z}\subset D_NY$$

is proper over $D_MX$ under $g$. Its central fibre is $C_NZ$ and it has no negative-time points.

We prove compactness over a compact set $K\subset D_MX$. These spaces are metrizable, so it suffices to take a sequence $u_n\in D_Z\cap g^{-1}K$ and find a convergent subsequence. Pass first to a subsequence for which $g(u_n)$ converges to $v$.

If infinitely many $u_n$ have time zero, properness in condition 2 gives the required subsequence. We may otherwise assume $t_n=t(u_n)>0$. The points $y_n=p_Y(u_n)$ belong to $Z$, and $f(y_n)=p_Xg(u_n)$ stays in a compact subset of $X$. Condition 1 permits passing to $y_n\to y\in Z$. If $t(v)>0$, the positive product chart gives $u_n\to(y,t(v))$, as required.

It remains to consider $t_n\to0$. Now $f(y)\in M$, so condition 3 is needed to conclude $y\in N$. Choose coordinates $y_n=(a_n,b_n)$ with $N=\{a=0\}$ and $y=(0,b)$. Then

$$u_n=(w_n,b_n,t_n),\qquad w_n=a_n/t_n.$$

Write the normal component of $f$ in compatible target coordinates as $F(a,b)$, with $F(0,b)=0$. Convergence of $g(u_n)$ says that $F(a_n,b_n)/t_n$ is bounded. Suppose that $w_n$ is unbounded; pass to a subsequence with $\lVert w_n\rVert\to\infty$ and $w_n/\lVert w_n\rVert\to w$, where $\lVert w\rVert=1$. Define

$$\sigma_n=t_n\lVert w_n\rVert=\lVert a_n\rVert\longrightarrow0.$$

The points $(w_n/\lVert w_n\rVert,b_n,\sigma_n)$ are positive lifts of the same $y_n\in Z$. Their limit $(w,b,0)$ belongs to $C_NZ$. Their target normal coordinates are

$$\frac{F(a_n,b_n)}{\sigma_n} =\frac{F(a_n,b_n)/t_n}{\lVert w_n\rVert}\longrightarrow0.$$

Thus $q(w,b)$ is the zero vector over $f(y)$. Since $C_NZ$ is closed and conic, its entire closed ray through $(w,b)$ lies in that single fibre of $q$. A nonzero closed ray is not compact, contradicting condition 2. Therefore $(w_n)$ is bounded. A final subsequence converges in the deformation chart to a point of the closed set $D_Z$. This proves the properness assertion in all cases.

SH02-SP-PROPER — When both direct comparisons are isomorphisms

Apply the preceding lemma to $Z=\operatorname{supp}G$. Under its three hypotheses, every arrow in SH02-SP-DIRECT-SQUARE is an isomorphism.

Indeed, $K=Rj_{Y*}r_Y^{-1}G$ is supported on $D_Z$, so $Rg_!K\to Rg_*K$ is an isomorphism. Proper base change on this closed support makes the central ordinary base-change map an isomorphism. On the positive chamber, $g_+$ is proper on the support of $r_Y^{-1}G$ by condition 1, so its proper and ordinary direct images agree as well. The comparison used in $c_!$ is now the usual composition isomorphism of ordinary direct images. Finally, $\operatorname{supp}\nu_NG\subset C_NZ$, so condition 2 makes $Rq_!\nu_NG\to Rq_*\nu_NG$ an isomorphism. These observations prove the claim for the specified maps, not merely for the resulting objects.

A useful sufficient condition is that $N=f^{-1}M$, that $f$ is clean with respect to $M$, and that $f$ is proper on $\operatorname{supp}G$. Clean means that the induced linear map on each normal fibre is injective. Condition 3 is then automatic. To verify condition 2, let $K\subset E_X$ be compact. The base points of $q^{-1}K\cap C_NZ$ lie in the compact set

$$L=Z\cap N\cap f^{-1}(\tau_XK).$$

Choose bundle metrics. Fibrewise injectivity of $q$, over the compact set $L$, gives a uniform lower bound $\lVert q_yv\rVert\ge c\lVert v\rVert$ with $c>0$: take the positive minimum over the unit sphere bundle above $L$. Thus $q^{-1}K\cap C_NZ$ is a closed subset of a bounded disk bundle over a compact base and is compact. The three hypotheses now apply.

The use of metrics in this last proof is only a compactness test. The comparison maps and their isomorphism statement contain no metric choice.

SH02-SP-ORIENTATIONS — The determinant calculation

For a map $h:P\to Q$ of manifolds, write

$$\omega_h=\operatorname{or}_P\otimes h^{-1}\operatorname{or}_Q^{-1} [\dim P-\dim Q].$$

It is an invertible complex, whether or not $h$ is smooth. We write its tensor inverse as $\omega_h^{\otimes-1}$; this is not Verdier duality of an arbitrary sheaf. The comparison

$$\theta_h(A):\omega_h\otimes h^{-1}A\longrightarrow h^!A \qquad\text{(SH02-SP-THETA-EQ)}$$

is adjoint to the projection formula followed by the relative trace $Rh_!\omega_h\to k_Q$. This fixes its direction and normalization.

For the normal bundle projection followed by the inclusion $p_E=i\tau:E\to X$, the exact normal sequence gives a canonical orientation identification

$$p_E^!k_X\simeq k_E=p_E^{-1}k_X. \qquad\text{(SH02-SP-NORMAL-ORIENTATION)}$$

Indeed, along the zero section the tangent orientation of $E$ is the product of the orientation of $M$ and that of its normal bundle. The same product is the orientation of $X|_M$, by the exact sequence $0\to TM\to TX|_M\to E\to0$. Also $\dim E=\dim X$, so the degree shift cancels. A choice of splitting proves the identification; the space of splittings is affine, so the identification does not depend on it. Fibre contraction extends the orientation identification over $E$.

Applying this calculation on $Y$ and $X$ gives

$$\omega_q\simeq\tau_Y^{-1}(\omega_f|_N). \qquad\text{(SH02-SP-ORIENTATION-Q)}$$

The restriction symbol includes the map $N\to Y$. To record both factors, let $h=f|_N:N\to M$ and factor the normal map as

$$E_Y\xrightarrow{u}N\times_M E_X\xrightarrow{v}E_X.$$

With pullbacks to the displayed source understood explicitly, one has

$$\omega_u\simeq\tau_Y^{-1} \bigl((\omega_f|_N)\otimes\omega_h^{\otimes-1}\bigr). \qquad\text{(SH02-SP-ORIENTATION-LINEAR)}$$

For the fibrewise transpose $u^t:N\times_M E_X^*\to E_Y^*$,

$$\omega_{u^t}\simeq\pi^{-1} \bigl(\omega_h\otimes(\omega_f|_N)^{\otimes-1}\bigr), \qquad\text{(SH02-SP-ORIENTATION-TRANSPOSE)}$$

where $\pi:N\times_M E_X^*\to N$. To check the shifts, set $r_Y=\dim Y-\dim N$ and $r_X=\dim X-\dim M$. The first linear map has relative shift $r_Y-r_X$ and the transpose has $r_X-r_Y$. Their orientation lines are inverse to one another because a real vector space and its dual have canonically identified orientation lines. This proves both formulas with their stated inverses. All reordering of shifted factors uses the Koszul symmetry; the normal exact sequence is ordered with tangent-to-base before normal directions.

SH02-SP-TWISTS — Moving a locally constant factor through the limit

If $L$ is a locally constant invertible complex on $Y$, there is a natural isomorphism

$$\nu_N(L\otimes G)\simeq \tau_Y^{-1}(L|_N)\otimes\nu_NG. \qquad\text{(SH02-SP-TWIST-EQ)}$$

Pull $L$ to the deformation by $p_Y$. Tensoring by this invertible locally constant complex commutes with $Rj_{Y*}$: tensoring with it and with its inverse are mutually inverse equivalences. Move the inverse tensor factor across a derived Hom, use $j_Y^{-1}\dashv Rj_{Y*}$, and move the factor back on $\Omega_Y$. Yoneda then gives the required projection isomorphism. This argument does not assert that an arbitrary invertible $k$-module is free. Restriction to $E_Y$ gives $s_Y^{-1}p_Y^{-1}L=\tau_Y^{-1}(L|_N)$ and proves the formula. The same argument allows a locally constant finite projective coefficient factor. Mere flatness does not assert commutation with an arbitrary ordinary direct image.

SH02-SP-INVERSE — Pullback and exceptional pullback

An invertible relative orientation complex does not by itself identify exceptional inverse image with twisted ordinary inverse image. The comparison map is available for a general map of manifolds; smoothness makes the general isomorphism theorem used here applicable. Keep this distinction between a determinant identification and an isomorphism theorem when reading the next square.

For $F\in D^b(k_X)$ there are natural comparison maps

$$\alpha:q^{-1}\nu_MF\longrightarrow\nu_N f^{-1}F, \qquad \beta:\nu_N f^!F\longrightarrow q^!\nu_MF. \qquad\text{(SH02-SP-INVERSE-MAPS)}$$

They form the commuting orientation square

$$\begin{array}{ccc} \omega_q\otimes q^{-1}\nu_MF &\xrightarrow{\ \omega_q\otimes\alpha\ }& \nu_N(\omega_f\otimes f^{-1}F)\\ \theta_q\downarrow&&\downarrow\nu_N(\theta_f)\\ q^!\nu_MF&\xleftarrow{\ \beta\ }&\nu_N f^!F. \end{array} \qquad\text{(SH02-SP-INVERSE-SQUARE)}$$

The top arrow incorporates SH02-SP-ORIENTATION-Q and SH02-SP-TWIST-EQ. Thus its right-hand orientation factor lives on $Y$, and its left-hand factor lives on $E_Y$; the notation does not confuse these spaces.

We first construct $\alpha$. Write $A=r_X^{-1}F$. The ordinary base-change map for the positive square gives

$$q^{-1}s_X^{-1}Rj_{X*}A \simeq s_Y^{-1}g^{-1}Rj_{X*}A \longrightarrow s_Y^{-1}Rj_{Y*}g_+^{-1}A \simeq\nu_N f^{-1}F.$$

To construct $\beta$, orient both positive projections by the same increasing parameter. Smooth pullback and composition give

$$r_Y^{-1}f^!F\simeq g_+^!r_X^{-1}F.$$

Explicitly, $r_Y^{-1}f^!F\simeq r_Y^!f^!F[-1] \simeq g_+^!r_X^!F[-1]\simeq g_+^!r_X^{-1}F$; the two parameter shifts cancel in the displayed order. Proper-support base change for the positive square, by taking right adjoints, gives

$$Rj_{Y*}g_+^!A\simeq g^!Rj_{X*}A.$$

Finally, the central square has the exceptional base-change comparison

$$s_Y^{-1}g^!B\longrightarrow q^!s_X^{-1}B.$$

Its adjoint is $Rq_!s_Y^{-1}g^!B\simeq s_X^{-1}Rg_!g^!B\to s_X^{-1}B$, using the counit of $Rg_!\dashv g^!$. Composing these three maps with $B=Rj_{X*}A$ defines $\beta$.

Here is a check of the square at the level of maps. On the deformation, the orientation line $\omega_g$ restricts on the positive chamber to $r_Y^{-1}\omega_f$ and on the central fibre to $\omega_q$. The determinant identifications agree with SH02-SP-ORIENTATION-Q: both cancel the same positively oriented parameter factor. Tensor the ordinary positive base-change map by $\omega_g$, then apply the trace comparisons for $g_+$ and $g$. These two routes agree. To verify this equality, take the adjoint under $Rg_!\dashv g^!$, and then under $j_X^{-1}\dashv Rj_{X*}$. Both routes become the projection formula followed by the trace of $g_+$ on $\Omega_X$; the inserted restriction unit and counit cancel.

Now restrict that equality to $E_Y$. Composing with exceptional central base change sends $s_Y^{-1}\theta_g$ to $\theta_q$. Taking its adjoint under $Rq_!\dashv q^!$ checks this assertion: proper-support base change identifies the adjoint with the restriction of the trace of $g$, which is the trace of $q$. This is exactly the trace-compatible base-change identity in SH02-OPS-SIX. Together with the twist identification, the equality is SH02-SP-INVERSE-SQUARE. The proof uses one fixed trace and the units/counits, so it introduces neither an arbitrary scalar nor an unrecorded orientation sign.

All four maps in SH02-SP-INVERSE-SQUARE are isomorphisms on the open subset of $E_Y$ where $q$ is smooth, meaning a submersion. To see the relevant geometry, at a central point the deformation derivative is block triangular: on the tangent space of $E_Y$ it is $dq$, and on the final time coordinate it is the identity. Surjectivity of $dq$ makes $g$ a submersion near that point. Smooth base change therefore makes $\alpha$ and the exceptional central comparison defining $\beta$ isomorphisms there; all other arrows used to construct $\beta$ are already isomorphisms. The comparison $\theta_q$ is an isomorphism on this locus. The commuting square now also makes its right vertical arrow an isomorphism on the same locus, regardless of its behavior elsewhere.

In particular, if both $f:Y\to X$ and $h:N\to M$ are smooth, then $q$ is smooth everywhere. Indeed, surjectivity of $df$ implies surjectivity on the normal quotients, and the base map $h$ is a submersion. In bundle coordinates these two statements give surjectivity of the derivative of $q$. Thus all comparisons in the square are isomorphisms globally under these two hypotheses.

SH02-SP-ADJUNCTION — The direct and inverse maps are mates

The previous constructions obey two identities useful when another functor is applied to the normal bundle. The map $\beta_F$ is exactly

$$\nu_N f^!F\longrightarrow q^!Rq_!\nu_Nf^!F \xrightarrow{q^!c_!}q^!\nu_M Rf_!f^!F \longrightarrow q^!\nu_MF, \qquad\text{(SH02-SP-SHRIEK-MATE)}$$

where the first arrow is the unit for $Rq_!\dashv q^!$ and the last uses the counit for $Rf_!\dashv f^!$. Likewise, $\alpha_F$ is exactly

$$q^{-1}\nu_MF\longrightarrow q^{-1}\nu_M Rf_*f^{-1}F \xrightarrow{q^{-1}c_*}q^{-1}Rq_*\nu_Nf^{-1}F \longrightarrow\nu_Nf^{-1}F, \qquad\text{(SH02-SP-STAR-MATE)}$$

using the unit for $f^{-1}\dashv Rf_*$ and the counit for $q^{-1}\dashv Rq_*$. To prove these identities, substitute the definitions of $c_!$ and $c_*$ from SH02-SP-DIRECT. Move their positive-square base-change maps across the displayed adjunctions. In the proper-support case this gives $Rj_{Y*}g_+^!\simeq g^!Rj_{X*}$; in the ordinary case it gives $g^{-1}Rj_{X*}\to Rj_{Y*}g_+^{-1}$. Moving the central-square map gives respectively $s_Y^{-1}g^!\to q^!s_X^{-1}$ and $q^{-1}s_X^{-1}=s_Y^{-1}g^{-1}$. The product-projection comparison moves to the positive-parameter identification used in defining $\beta$, and to the evident inverse-image identification used in defining $\alpha$. All remaining inserted units and counits cancel by their triangle identities. The resulting composites are precisely the definitions in SH02-SP-INVERSE, which proves the two formulas with their fixed maps.

SH02-SP-EXTERNAL — Synchronizing the deformation parameter

Let $M\subset X$ and $N\subset Y$ be closed embedded submanifolds, and let $F\in D^b(k_X)$ and $G\in D^b(k_Y)$. There is a natural morphism

$$\nu_MF\boxtimes^L\nu_NG\longrightarrow \nu_{M\times N}(F\boxtimes^L G) \quad\text{on }T_MX\times T_NY. \qquad\text{(SH02-SP-EXTERNAL-EQ)}$$

The reason for a comparison rather than an equality in its definition is that the product of two deformations has two time coordinates, while the deformation of the product has one. In charts there is a closed embedding

$$d:D_{M\times N}(X\times Y)\hookrightarrow D_MX\times D_NY, \qquad (v,w,t)\longmapsto((v,t),(w,t)).$$

It is the equal-time locus. Its central map is the identity under the canonical normal-bundle identification. Its positive restriction is the equal-time embedding $d_+$ in $\Omega_X\times\Omega_Y$, and the square with the positive inclusions is Cartesian.

Write $A=r_X^{-1}F$ and $B=r_Y^{-1}G$. The external-product comparison

$$Rj_{X*}A\boxtimes^L Rj_{Y*}B \longrightarrow R(j_X\times j_Y)_*(A\boxtimes^L B)$$

is adjoint to its restriction on $\Omega_X\times\Omega_Y$, where the counits give the identity on $A\boxtimes^L B$. Pull it back along $d$ and use ordinary base change

$$d^{-1}R(j_X\times j_Y)_*(A\boxtimes^L B) \longrightarrow Rj_*d_+^{-1}(A\boxtimes^L B).$$

The last inverse image is $r^{-1}(F\boxtimes^L G)$. Restricting the composite to the central fibre gives SH02-SP-EXTERNAL-EQ. Finite global dimension of $k$ ensures the tensor products of bounded inputs remain in the bounded categories being used. No Künneth isomorphism for a nonproper map, and no general isomorphism assertion for this comparison, is assumed.

SH02-SP-TENSOR — Multiplication on one normal bundle

For $F,G\in D^b(k_X)$, pull the external comparison back by the normal diagonal $\delta_E:E\to E\times E$. The map of pairs $\delta_X:(X,M)\to(X\times X,M\times M)$ has normal map $\delta_E$. Its comparison $\alpha$ gives the composite

$$\begin{aligned} \nu_MF\otimes^L\nu_MG &\simeq\delta_E^{-1}(\nu_MF\boxtimes^L\nu_MG)\\ &\longrightarrow\delta_E^{-1}\nu_{M\times M}(F\boxtimes^L G)\\ &\longrightarrow\nu_M\delta_X^{-1}(F\boxtimes^L G) \simeq\nu_M(F\otimes^L G). \end{aligned} \qquad\text{(SH02-SP-TENSOR-EQ)}$$

Naturality follows from that of the two comparisons. Their associativity and symmetry follow by using a product of three deformation spaces and restricting to the locus where all time coordinates agree: both composites are adjoint to the same tensor product of restriction counits. The tensor symmetry is the usual Koszul symmetry. The constant sheaf gives the unit, since locally the positive half of a deformation chart is a product half-neighborhood and $\nu_Mk_X\simeq k_E$. The displayed morphism need not be an isomorphism; the example below exhibits a failure with elementary closed supports.

SH02-SP-HOMOGENEOUS — A calibration on a vector space

Let $V$ be a finite-dimensional real vector space, specialized along its origin. Identify $T_{\{0\}}V$ with $V$. If $F\in D^b(k_V)$ is conic, then there is a canonical identification

$$\nu_{\{0\}}F\simeq F. \qquad\text{(SH02-SP-HOMOGENEOUS-EQ)}$$

In the deformation chart $p(v,t)=tv$, the positive pullback is canonically identified with the pullback of $F$ under $(v,t)\mapsto v$, by normalized conic transport. The stalk of its direct image at $(v,0)$ is computed on products $U\times(0,\epsilon)$; interval descent gives $R\Gamma(U;F)$. Passing to the stalk colimit gives $F_v$. These maps are the restrictions of the conic transport map and hence glue and are natural. This proves the claim without a finiteness hypothesis on the stalk modules.

One consequence is that specialization is not a tangent approximation of supports alone. It retains the sheaf maps between angular pieces and the zero section, as well as every cohomological shift.

SH02-SP-EXAMPLE-SEPARATING-ARCS — Tensor products can lose a direction

In $X=\mathbb R^2$, specialize at the origin. Let

$$A_+=\{(x,x^3):x\ge0\},\qquad A_-=\{(x,-x^3):x\ge0\}, \qquad L=\{(u,0):u\ge0\}.$$

Write $k_A$ for the constant sheaf on a closed subset $A$, extended by zero. The maps $f_\pm:\mathbb R\to\mathbb R^2$ defined by $f_\pm(x)=(x,\pm x^3)$ are proper closed embeddings, are clean with respect to the origin, and induce the same normal map $x\mapsto(x,0)$. The homogeneous calibration gives $\nu_{\{0\}}k_{[0,\infty)}\simeq k_{[0,\infty)}$ on the line. The proper direct-image theorem therefore gives

$$\nu_{\{0\}}k_{A_+}\simeq k_L, \qquad \nu_{\{0\}}k_{A_-}\simeq k_L.$$

These sheaves are flat over $k$ stalkwise. Their derived tensor product before specialization is $k_{A_+\cap A_-}=k_{\{0\}}$. Consequently the tensor comparison has the form

$$k_L\longrightarrow k_{\{0\}}.$$

At a nonzero point of $L$, its source stalk is $k$ and its target stalk is zero. For any nonzero coefficient ring, it is not an isomorphism. At the origin the map is the identity under the zero-section identifications, because the comparisons are made from the restriction counits. Thus the map is the ordinary restriction from the closed ray to its endpoint. The failure has a geometric explanation: the two curved supports meet only at the origin, while their normal directions agree along the whole ray.

SH02-SP-EXERCISE-ZERO-RANK — Specializing along the whole space

Problem. Compute specialization when $M=X$. Verify both formulas in SH02-SP-ZERO-EQ and the boundary shift directly, for an arbitrary bounded complex $F$.

Solution. The normal bundle has rank zero and equals $X$; the deformation is $X\times\mathbb R$, and $p$ is projection. Restriction of $Rj_*p^{-1}F|_{t>0}$ to $t=0$ is $F$ by interval descent. Thus $\nu_XF=F$, and $e,\tau,i$ are all identities. For the exceptional expression, the costalk at the endpoint of extension by zero from $(0,\infty)$ has the parameter contribution $k[-1]$. The smooth projection contributes $[1]$. Their composite is $F$, agreeing with the defining expression. This test would detect a missing parameter shift in the boundary formula.

SH02-SP-EXERCISE-OPEN-RAY — A nonzero costalk with zero stalk

Problem. Let $X=\mathbb R$, $M=\{0\}$ and $F=k_{(0,\infty)}$, using extension by zero for the open inclusion. Compute the specialization, its zero stalk, its zero costalk, and the punctured direct image. Check the localization triangle and its shift.

Solution. The sheaf is conic, so the calibration gives $\nu_MF=F$ on the normal line. Its stalk at zero is zero. On a small interval around zero, the ordinary derived sections of $F$ vanish: in the triangle $k_{(0,\infty)}\to k_{[0,\infty)}\to k_{\{0\}}\xrightarrow{+1}$, the latter two section complexes are $k$ and the map is the identity. On the punctured interval the section complex is $k$. The local-support triangle is therefore

$$k[-1]\longrightarrow0\longrightarrow k\xrightarrow{+1}.$$

It proves $i^!F=k[-1]$. Equivalently, the normal proper direct image is $R\Gamma_c((0,\infty);k)=k[-1]$, while the ordinary direct image from the punctured normal line is $k$. All identifications agree with SH02-SP-ZERO-EQ and SH02-SP-PUNCTURE-EQ. Zero ordinary stalk does not imply zero costalk.

SH02-SP-EXERCISE-PROPERNESS — Proper on the original support is insufficient

Problem. Let $f:\mathbb R\to\mathbb R$ be $f(y)=y^2$, and let $N=M=\{0\}$ and $G=k_{\mathbb R}$, with $k\ne0$. Determine why the direct comparison theorem cannot use only properness of $f$ on $\operatorname{supp}G$.

Solution. The map $f$ is proper, and $\operatorname{supp}G\cap f^{-1}M=N$. Its normal map is the zero linear map $q:\mathbb R\to\mathbb R$, since $df_0=0$. The normal cone of the support is the whole line, on which $q$ is not proper.

The sheaf $Rf_*G=f_*G$ has stalk $k\oplus k$ at every positive point, stalk $k$ at zero, and zero at negative points. There is no higher direct image: inverse images of sufficiently small intervals are unions of intervals, each acyclic for the constant sheaf. Positive scaling of the target lifts to scaling by the positive square root on the source, so this sheaf is conic. Its specialization is itself by the homogeneous calibration. On the other hand,

$$Rq_*\nu_NG=k_{\{0\}},\qquad Rq_!\nu_NG=k_{\{0\}}[-1].$$

The first formula uses $R\Gamma(\mathbb R;k)=k$, the second uses $R\Gamma_c(\mathbb R;k)=k[-1]$. At a positive normal vector, either direct image under $q$ has zero stalk, whereas $\nu_MRf_*G$ has stalk $k\oplus k$. Thus $c_*$ is not an isomorphism; the analogous positive-stalk test shows the failure of $c_!$ as well. The missing hypothesis is precisely properness on the normal cone. In the deformation, points can run to infinity in the rescaled normal coordinate while their target deformation points remain bounded.

SH02-SP-EXERCISE-ORIENTATION — Test a smooth projection

Problem. Let $Y=X\times\mathbb R^d$, let $N=M\times\mathbb R^d$, and let $f$ be projection. Compute the normal map and the relative complexes in SH02-SP-INVERSE-SQUARE. Does the calculation require orientability of $M$ or $X$?

Solution. There are canonical identifications $E_Y=E_X\times\mathbb R^d$ and $q$ is projection. Both $f$ and $f|_N$ are smooth, so $\alpha$ and $\beta$ are isomorphisms. The standard orientation of $\mathbb R^d$ gives

$$\omega_f=k_Y[d],\qquad\omega_q=k_{E_Y}[d].$$

Thus $f^!F=f^{-1}F[d]$, and the square identifies $\nu_Nf^{-1}F[d]$ with $q^{-1}\nu_MF[d]$ by the same normalized pullback comparison. The orientation of $X$ cancels against its inverse, as does the orientation of $M$ in the normal determinant calculation. Neither manifold is required to be orientable. With a nontrivial vector bundle in place of $\mathbb R^d$, keep its orientation local system instead of replacing it by $k$; the shift is unchanged.

SH02-SP-DEPENDENCY-BOUNDARY — What the arguments establish

The construction, neighborhood and support formulas, zero and punctured recoveries, direct and inverse comparison maps, the three-hypothesis properness result, the orientation square, and both tensor comparisons have proofs in this unit relative to the contracts in SH02-SP-CONVENTIONS. The properness argument and the solved comparisons also test the exceptional cases that a support-only or unshifted account would miss.

This lesson uses deformation geometry, noncompact tautness, the bounded six-operation package with trace-compatible base change, the orientation trace and conic contraction as prerequisites; their full proofs are not given here. A next mathematical use is to apply Fourier–Sato transform in the normal fibres; that step belongs to the microlocalization unit and requires its sign and orientation conventions.

SH02-SP-SOURCE-ACCOUNT — Sources and proof mechanisms

Kashiwara and Schapira, Microlocal Study of Sheaves, Astérisque 128 (1985), §2.2, supplies the deformation construction used here. Definition 2.2.1 takes ordinary derived direct image from the positive chamber and restricts it to the normal bundle. Proposition 2.2.1 then records conicity, ordinary and supported recovery on the zero section, and the neighborhood and supported-section formulas. The present proof makes the passage from a deformation neighborhood to sections explicit: cone avoidance chooses the neighborhood system, interval-fibre descent removes the positive parameter, and tautness passes to the central fibre. Localization supplies the supported formula. The endpoint calculation fixes the particular exceptional counit and its compatibility with the ordinary restriction map; these checks cannot be replaced by a comparison of supports.

Propositions 2.2.2 and 2.2.3 of that work construct the direct and inverse comparison maps by the positive and central Cartesian squares. Their mechanism is retained here: use base change where its hypotheses hold, and otherwise retain the natural comparison arrow. The source’s short proper-direct-image statement uses transversality and properness on the support. For the arbitrary map of pairs used in this lesson, the explicit support-intersection and normal-cone properness conditions in SH02-SP-PROPER-GEOMETRY must still be checked. The rescaling argument there is the proof of compactness needed by SH02-SP-PROPER, rather than an appeal to a shorter criterion. For inverse image, the source also gives the orientation square and the isomorphism result when the map and its restriction to the submanifold are submersions. Here the determinant calculation, the common oriented parameter, and the adjunction-mate checks specify the maps and their shifts.

The coefficient and range conventions must be compared separately. Microlocal Study of Sheaves, §1.3.1, starts with a unital coefficient ring and assumes finite weak global dimension when forming the derived tensor product. Its specialization definition and the three propositions above use bounded-below complexes. This lesson deliberately fixes a commutative ring of finite global dimension and bounded inputs, and lists the finite cohomological amplitudes it needs as a prerequisite. It neither imports a constructibility restriction nor establishes an unbounded version.

Schapira’s An Introduction to Sheaves on Grothendieck Topologies (1 August 2026) provides a second view of the six-operation mechanisms. Theorem 4.5.3 proves proper-support base change by fibrewise acyclicity. Theorem 4.6.1 constructs exceptional inverse image as a right adjoint under a finite cohomological-dimension hypothesis; Proposition 4.6.4 constructs its tensor comparison from projection formula and adjunction. Definitions 5.1.4 and 5.1.6 record the orientation and relative orientation complexes, and Proposition 5.1.9 proves the submersion formula by reduction to a product, compactly supported cohomology of a convex fibre, and adjunction. These are the operations used in the boundary, twist and inverse-comparison arguments here, rather than a reason to assume all comparison arrows are invertible.

Those notes do not discharge every prerequisite in this lesson: the existence argument for the right adjoint invokes representability, the identification with the classical differentiable orientation in Proposition 5.1.5(d) refers elsewhere, and the final product step in the submersion proof invokes Exercise 3.18. In particular, a source reference does not close the noncompact tautness, interval-neighborhood, conic contraction or trace-compatible orientation obligations listed above. The common-time external comparison, homogeneous calibration, separating arcs and solved problems remain part of this lesson’s argument; no general tensor isomorphism is inferred from the cited specialization statements.

The independently written explanations in this lesson are offered under CC0. This dedication does not change the terms of the cited human-authored works or of any separately licensed reader components.