Proof. 1. Since C is conical, equals the inverse image of ${\mathbf P}(C)$ .

2. U is the complement of B and $U'$ is the complement of the image of ${\mathbf P}(C)$ by the projection ${\mathbf P}(V)\to X$ .

Proof. Since $\dim C\leqq \dim X$ , the projectivization ${\mathbf P}(C)$ has dimension strictly less than $\dim X$ . Hence it follows from Lemma 1.2.

Proof. 1. h is $T^*X$ -transversal if and only if $W\times _XT^*X\to T^*W$ is an injection. This means that $h^*\Omega ^1_{X/k}\to \Omega ^1_{W/k}$ is a locally split injection.

2. and 3. Clear from the definition and 1.

4. It suffices to apply Lemma 1.2.2 to the closed conical subset $h^*C\cap \mathrm {Ker}(W\times _XT^*X \to T^*W))$ of $W\times _XT^*X$ .

5. Condition (1) means that $T^*_ZX\times _XW\to T^*W$ is an injection. In other words, ${\mathcal N}_{Z/X}\otimes _{{\mathcal O}_Z} {\mathcal O}_V$ is locally a direct summand of $\Omega ^1_{W/k} \otimes _{{\mathcal O}_W}{\mathcal O}_V$ . This is equivalent to condition (2).

Proof. We consider the composition $(g\circ h)^*C \subset T^*X\times _XW \to T^*U\times _UW \to T^*W$ . The assumption implies that the intersection $(g\circ h)^*C \cap \mathrm {Ker}( T^*X\times _XW \to T^*U\times _UW)$ is a subset of the 0-section. Since $h^*(g^\circ C) \subset T^*U\times _UW$ is the image of $(g\circ h)^*C$ , the intersection $(g\circ h)^*C \cap \mathrm {Ker}( T^*X\times _XW \to T^*W)$ is a subset of the 0-section if and only $h^*(g^\circ C) \cap \mathrm {Ker}( T^*U\times _UW \to T^*W)$ is a subset of the 0-section.

Proof. 1. By decomposing h into $W\to X\times W\to X$ , it suffices to consider the case where h is an immersion by Lemma 2.5. Since the immersion $h\colon W\to X$ is $g_\circ C'$ -transversal, the intersection $\mathrm {Ker}(T^*X\times _XX'\to T^*X')\times _{X'}V \cap \mathrm {Ker}(T^*X\times _XW\to T^*W) \times _WV$ is a subset of the 0-section on a neighbourhood $V'$ of $ \mathrm {pr}_1^{-1}(B')\subset V$ . Namely, the restriction of $(T^*X\times _XX'\to T^*X')\times _{X'}V'$ on $\mathrm {Ker}(T^*X\times _XW\to T^*W) \times _WV'$ is a locally split injection and the assertion follows.

2. Similarly as in the proof of 1, we may assume that $h\colon W\to X$ is an immersion. Then, further, we have a commutative diagram

of exact sequences where the left vertical arrow is an isomorphism. Hence the C-transversality of h implies the $C'$ -transversality of $h'\colon V'\to X'$ and the equality ${h^\circ g_\circ C' =g^{\prime }_\circ h^{\prime \circ } C'}$ follows.

Proof. 1. Let ${\mathcal I}\subset {\mathcal O}_Z$ and ${\mathcal J}\subset {\mathcal O}_X$ be the ideal sheaves defining $W\subset Z$ and $W\subset X$ . Then the section $s\colon T^*Z\to T^*X|_Z$ induces a section $T^*_WZ\to T^*_WX$ on sub vector bundles, namely ${\mathcal I}/{\mathcal I}^2 \to {\mathcal J}/{\mathcal J}^2$ . Locally on W, we take a basis of $s({\mathcal I}/{\mathcal I}^2) \subset {\mathcal J}/{\mathcal J}^2$ . By taking its lifting to ${\mathcal J}$ , we define a closed subscheme $V\subset X$ . Then, for the conormal sheaf, we have an isomorphism ${\mathcal N}_{V/X}\otimes _{{\mathcal O}_V} {\mathcal O}_W\to {\mathcal N}_{W/Z}$ . Hence after shrinking V if necessary, V is smooth over k and $\mathrm {codim}_XV= \mathrm {codim}_ZW$ . The isomorphism ${\mathcal N}_{V/X}\otimes _{{\mathcal O}_V} {\mathcal O}_W\to {\mathcal N}_{W/Z}$ means the equality $T^*_VX|_W=s(T^*_WZ)$ .

2. Since $V\cap Z=W$ , the equality $T^*_VX|_W=s(T^*_WZ)$ implies that the section s induces a bijection $s^{-1}(C)\cap \mathrm {Ker}(T^*Z|_W\to T^*W) \to C\cap \mathrm {Ker}(T^*X|_V\to T^*V)$ .

Proof. 1. (1) $\Rightarrow $ (2): Let $(a,b)$ be a point of the intersection $\bigl (h^*C\times _W (T^*Y\times _YW)\bigr ) \cap \mathrm {Ker}((T^*X\times _XW)\times _W (T^*Y\times _YW) \to T^*W)$ . Then, $b\in T^*Y\times _YW$ is in the inverse image of $-dh(a)\in h^\circ C$ and the $h^\circ C$ -acyclicity of f implies that $b=0$ . Further the C-transversality of h implies $a=0$ .

(2) $\Rightarrow $ (1): Since the injection $T^*X\times _XW \to (T^*X\times _XW)\times _W (T^*Y\times _YW)$ to the first factor maps $h^*C \cap \mathrm {Ker}(T^*X\times _XW \to T^*W)$ to $\bigl (h^*C\times _W (T^*Y\times _YW)\bigr ) \cap \mathrm {Ker}((T^*X\times _XW)\times _W (T^*Y\times _YW) \to T^*W)$ , the $C\times T^*Y$ -transversality of $(h,f)$ implies the C-transversality of h. If $b\in T^*Y\times _YW$ is in the inverse image of $dh(a)\in h^\circ C$ for $a\in h^* C$ , then $(-a,b)$ is contained in $\bigl (h^*C\times _W (T^*Y\times _YW)\bigr ) \cap \mathrm {Ker}((T^*X\times _YW)\times _W (T^*Y\times _YW) \to T^*W)$ and is $0$ .

If $f\colon W\to Y$ is $h^\circ C$ -acyclic, then $\mathrm {Ker}(W\times _YT^*Y\to T^*W)|_{h^{-1}(B)}$ is a subset of the $0$ -section.

2. The morphism f is $T^*_XX$ -acyclic if and only if f is $T^*Y$ -transversal. Hence the assertion follows from Lemma 2.3.1.

Proof. 1. Since h is $T^*_XX$ -transversal by Lemma 2.3.2 and $h^\circ T^*_XX=T^*_WW$ , condition (1) means that $f\colon W\to Y$ is $T^*_WW$ -acyclic. This is equivalent to (2) by Lemma 3.2.2.

2. By Lemma 3.2.1, condition (1) means that $(h,f)\colon W\to X\times Y$ is $T^*X\times T^*Y$ -transversal. This is equivalent to (2) by Lemma 2.3.1.

Proof. By Lemma 3.2.1, the conditions are equivalent to the conditions: (1) ${(g\times 1_Y)\circ (h,f)}$ is $C\times T^*Y$ -transversal, and (2) $(h,f)$ is $(g\times 1_Y)^\circ (C\times T^*Y)$ -transversal, respectively. Hence they are equivalent by Lemma 2.5.

Proof. Since $g_\circ C'\subset T^*X$ is defined as the image of the inverse image of $C'$ by $T^*X\times _XX'\to T^*X'$ , either of the conditions means that the inverse image of $C'$ by $T^*Y\times _YX'\to T^*X'$ is a subset of the $0$ -section.

Proof. Since the inverse image of $h^\circ C$ by $T^*Y\times _YW\to T^*W$ is a subset of the $0$ -section and since $T^*Y'\times _{Y'}Y\to T^*Y$ is an injection, the inverse image of $h^\circ C$ by $T^*Y'\times _{Y'}W\to T^*W$ is a subset of the $0$ -section.

Proof. By Lemma 2.7, there exists an open neighbourhood $U' \subset W'$ of $h^{\prime -1}(B')$ such that g and h are transversal on $U'$ and $h^\circ g_\circ C'=g'|_{U'\circ } h^{\prime \circ } C'$ . Since f is $h^\circ g_\circ C'$ -acyclic, $f'$ is $h^{\prime \circ } C'$ -acyclic by Lemma 3.5.

Proof. Let $g\colon Y'\to Y$ be a morphism of smooth schemes over k and consider the commutative diagram (3.1). By Lemma 3.2.1, $f\colon W\to Y$ is smooth on a neighbourhood $U\subset W$ of $h^{-1}(B)$ . Hence the morphism g is transversal to f on the inverse image $U'\subset W'$ of U. To show that $(h',f')$ is C-acyclic on $U'$ , it suffices to show that

$$ \begin{align*} &\mathrm{Ker}((T^*X\times_XW) \times_W (T^*Y\times_YW) \to T^*W)\times_WU'\\ &\to \mathrm{Ker}((T^*X\times_XU') \times_{U'} (T^*Y'\times_{Y'}U') \to T^*U') \end{align*} $$

is an isomorphism further by Lemma 3.2.1. By the factorization $Y'\to Y\times Y'\to Y$ , it suffices to prove the cases where $Y'\to Y$ is smooth and $Y'\to Y$ is an immersion respectively. Since the morphism $f\colon W\to Y$ is transversal to $g\colon Y'\to Y$ in the second case, the assertion follows.

Proof. 1. The implication (1) $\Rightarrow $ (2) is the local acyclicity of smooth morphisms [Reference Artin2, Théorème 2.1] and (2) $\Rightarrow $ (3) is clear. The implication (3) $\Rightarrow $ (1) is [Reference Artin1, Proposition 2.13 (i)].

Proof. Let $s,t$ be geometric points of Y and $t\to Y_{(s)}$ be a specialization and consider the cartesian diagram

where the vertical arrows are induced by $g'$ . Since g is assumed proper on the support $B'$ of ${\mathcal F}'$ , the vertical arrows in the commutative diagram

are isomorphisms. If (2) is satisfied, then the upper horizontal arrow is an isomorphism. Hence the lower horizontal arrow is an isomorphism and (1) holds. Conversely, if g is finite on B and if the lower horizontal arrow is an isomorphism, the upper horizontal arrow is an isomorphism before taking $g^{\prime }_{s*}$ and (1) holds.

Proof. 1. By Lemma 3.3.1, a pair $(h,f)$ of morphisms $h\colon W\to X$ and $f\colon W\to Y$ of smooth schemes over k, is $T^*_XX$ -acyclic if and only if f is smooth.

(1) $\Rightarrow $ (2): If f is smooth and ${\mathcal F}$ is locally constant, then f is $h^*{\mathcal F}$ -acyclic by Theorem 4.2.1 (1) $\Rightarrow $ (2).

(2) $\Rightarrow $ (1): Since the pair $(1_X,1_X)$ is $T^*_XX$ -acyclic, the condition (2) implies that the identity $1_X$ is ${\mathcal F}$ -acyclic. By Theorem 4.2.1 (3) $\Rightarrow $ (1), this implies that ${\mathcal F}$ is locally constant.

2. Suppose that a pair $(h,f)$ of morphisms $h\colon W\to X$ and $f\colon W\to Y$ of smooth schemes over k, is $T^*X$ -acyclic. Then $(h,f)\colon W\to X\times Y$ is smooth by Lemma 3.3.2. Since the question is étale local on W, we may assume that $W={\mathbf A}^n_{X\times Y}$ . Then, we have a cartesian diagram

By Theorem 4.2.2, the morphism $W\to {\mathbf A}^n_Y$ is ${\mathcal F}_W$ -acyclic for the pull-back ${\mathcal F}_W$ of ${\mathcal F}$ . Since ${\mathbf A}^n_Y\to Y$ is smooth, the composition $W\to Y$ is also ${\mathcal F}_W$ -acyclic by [Reference Illusie8, Corollaire 2.7].

3. Since the pair $(1,0)$ of the identity $1\colon U\to U$ and the $0$ -mapping $U\to {\mathbf A}^1$ is C-acyclic, the constant morphism $0\colon U\to {\mathbf A}^1$ is ${\mathcal F}$ -acyclic. Hence we have ${\mathcal F}=0$ .

If ${\mathcal F}=0$ , every pair $(h,f)$ of morphisms $h\colon W\to X$ and $f\colon W\to Y$ of smooth schemes over k is ${\mathcal F}$ -acyclic.

4. Since $(h,f)$ is universally C-acyclic by Lemma 3.8, the assertion follows.

Proof. Let $h\colon W\to X$ and $f\colon W\to Y$ be morphisms of smooth schemes over k such that the pair $(h,f)$ is $C"$ -acyclic. Then, since $C,C'\subset C"$ , the pair $(h,f)$ is C-acyclic and $C'$ -acyclic. Hence f is $h^*{\mathcal F}$ -acyclic and $h^*{\mathcal F}'$ -acyclic. By the distinguished triangle $ {\mathcal F}'\to {\mathcal F}\to {\mathcal F}"\to $ , f is $h^*{\mathcal F}"$ -acyclic.

Proof. 1. Let $h\colon W\to V$ and $f\colon W\to Y$ be morphisms of smooth schemes over k such that the pair $(h,f)$ is $g^\circ C$ -acyclic. Then, $(gh,f)$ is C-acyclic by Lemma 3.4 (2) $\Rightarrow $ (1). Hence f is locally acyclic relatively to $(gh)^*{\mathcal F}=h^*(g^*{\mathcal F})$ .

2. Let $h\colon W\to X$ and $f\colon W\to Y$ be morphisms of smooth schemes over k such that the pair $(h,f)$ is C-acyclic. Then, for every $i\in I$ , the pair of $h_i\colon W_i=W\times _XU_i\to U_i$ and $f_i\colon W_i\to Y$ is $j_i^\circ C$ -acyclic by Lemma 3.4 (2) $\Rightarrow $ (1). Further it is $C_i$ -acyclic since $C_i\subset j_i^\circ C$ by Lemma 2.3.3. Hence $f_i$ is locally acyclic relatively to $h_i^*({\mathcal F}_{U_i})=(h^*{\mathcal F})_{W_i}$ for every $i\in I$ . Since $(W_i\to W)_{i\in I}$ is an étale covering, f is $h^*{\mathcal F}$ -acyclic.

3. Let $h\colon W\to X$ and $f\colon W\to Y$ be morphisms of smooth schemes over k and assume that $(h,f)$ is $\overline {C'}\cup C|_Z$ -acyclic. Then, $(h,f)$ is $\overline {C'}$ -acyclic on U and is C-acyclic on a neighbourhood V of Z. Hence f is $h^*{\mathcal F}$ -acyclic on $X=U\cup V$ .

Proof. 1. Let $h\colon W\to X$ and $f\colon W\to Y$ be morphisms of smooth schemes over k such that the pair $(h,f)$ is $C|_Z$ -acyclic. Then, by Lemma 2.3.4, there exists an open neighbourhood U of Z such that the pair of $h_U\colon W\times _XU\to X$ and $f_U\colon W\times _XU\to Y$ is C-acyclic. Hence $f_U$ is $(h^*{\mathcal F})_{W\times _XU}$ -acyclic. For the complement , the restriction $(h^*{\mathcal F})_{W\times _XV}$ is $0$ since ${\mathcal F} =i_*{\mathcal F}_Z$ . Hence $f_V\colon W\times _XV\to Y$ is $(h^*{\mathcal F})_{W\times _XV}$ -acyclic. Since $U\cup V=X$ , f is ${\mathcal F}$ -acyclic.

2. First, we show that ${\mathcal F}_Z$ is micro supported on $s^{-1}(C)$ . Let $h\colon W\to Z$ and $f\colon W\to Y$ be morphisms of smooth schemes over k such that the pair $(h,f)$ is $s^{-1}(C)$ -acyclic. We show that we may assume h is an immersion. We consider the commutative diagram

For the section $\widetilde s=s\times 1\colon T^*Z\times T^*W=T^*(Z\times W) \to T^*X|_Z\times T^*W=T^*(X\times W)|_{Z\times W}$ extending s, we have $\mathrm {pr}_1^\circ (s^{-1}(C)) =\widetilde s^{-1}(\mathrm {pr}_1^\circ C)$ . Since the projection $Z\times W\to Z$ is smooth, we may replace $h\colon W\to Z$ , $X\supset Z$ and C by $(h,1)\colon W\to Z\times W$ , $X\times W \supset Z\times W$ and $\mathrm {pr}_1^\circ C$ , by Lemma 4.7.1. Thus we may assume that h is an immersion.

Since the assertion is local on X, by Lemma 2.8.1, we may extend the immersion $(h,f)\colon W\to Z\times Y$ to an immersion $(h',f')\colon V\to X\times Y$ transversal to the immersion $Z\times Y\to X\times Y$ such that the restriction $T^*_V(X\times Y)|_W$ of the conormal bundle is the image of $T^*_W(Z\times Y)$ by the section $s|_W\times 1\colon T^*Z|_W\times _W (T^*Y\times _YW) \to T^*X|_W\times _W (T^*Y\times _YW)$ .

We consider the cartesian diagram

Since $T^*_V(X\times Y)|_W$ is the image of $T^*_W(Z\times Y)$ by the section, the $s^{-1}(C)$ -acyclicity of $(h,f)\colon W\to Z\times Y$ implies the C-acyclicity of $(h',f')\colon V\to X\times Y$ by Lemma 3.2.1 and Lemma 2.8.2, after shrinking V if necessary. Since ${\mathcal F}$ is micro supported on C, this implies that $f'\colon V\to Y$ is $h^{\prime *}{\mathcal F}$ -acyclic. This means that $f\colon W\to Y$ is $h^*{\mathcal F}_Z$ -acyclic since $h^{\prime *}{\mathcal F}$ is the direct image of $h^*{\mathcal F}_Z$ by the closed immersion $W\to V$ . Hence ${\mathcal F}_Z$ is micro supported on $s^{-1}(C)$ .

We show that ${\mathcal F}_Z$ is micro supported on $C_Z$ . Since the assertion is local on Z, we may assume that there exists a section $s\colon T^*Z\to T^*X|_Z$ . Then, since $s^{-1}(C)\subset C_Z$ , the assertion follows.

3. Let $h\colon W\to X$ and $f\colon W\to Y$ be morphisms of smooth schemes over k such that the pair $(h,f)$ is $i_\circ C$ -acyclic. By Lemma 4.5.3, the support of ${\mathcal F}$ is a subset of the base B of C. By Lemma 2.7 and Lemma 3.5, there exists an open neighbourhood $V'\subset V=Z\times _XW$ of the inverse image of B such that $V'$ is smooth over k, that the pair of $h'\colon V'\to Z$ and the composition $f'\colon V'\to W\to Y$ is C-acyclic. Hence $f'$ is $h^{\prime *}{\mathcal F}$ -acyclic. Since $V'$ contains the inverse image of the support of ${\mathcal F}$ , the morphism f is $h^*i_*{\mathcal F}$ -acyclic.

Proof. Let $h\colon W\to X$ and $f\colon W\to Y$ be morphisms of smooth schemes over k such that $(h,f)$ is C-acyclic. Let

be the cartesian diagram. Then, by Lemma 3.7, there exists an open regular neighbourhood $U'\subset W'=W\times _XX'$ of the inverse image of $B'$ where h is transversal to g and $(h',fg')$ is $C'$ -acyclic. Hence the assertion follows from Lemma 4.3.

Proof. 1. The restriction ${\mathcal F}|_U$ is micro supported on $C|_U$ by Lemma 4.7.1. We show that $C|_U$ is the smallest. Let $C'\subset T^*U$ be a closed conical subset on which ${\mathcal F}|_U$ is micro supported and be the complement. Then ${\mathcal F}$ is micro supported on $\overline {C'}\cup C|_Z$ by Lemma 4.7.3. Since C is the smallest, we have $C\subset \overline {C'}\cup C|_Z$ and $C|_U\subset (\overline {C'}\cup C|_Z)|_U=C'$ .

2. For every $i,j$ , the restrictions $C_i|_{U_i\cap U_j}$ and $C_j|_{U_i\cap U_j}$ are the singular support of ${\mathcal F}|_{U_i\cap U_j}$ and are the same by 1. Hence the union $C=\bigcup _iC_i$ is a closed conical subset of $T^*X$ . Since $C_i=C|_{U_i}$ , ${\mathcal F}$ is micro supported on C by Lemma 4.7.1.

We show that C is the smallest. Let $C'\subset T^*X$ be a closed conical subset on which ${\mathcal F}$ is micro supported. Then, for each i, we have $C_i\subset C'|_{U_i}$ . Hence we have $C\subset C'$ .

Proof. 1. Let be the complement. Since ${\mathcal F}|_U=0$ is micro supported on $\varnothing $ by Lemma 4.5.3, ${\mathcal F}$ is micro supported on $C_P|_X\subset T^*P$ by Lemma 4.7.3. Since $C_P$ is the smallest, we have $C_P=C_P|_X \subset T^*P|_X$ .

2. By Lemma 4.8.2, ${\mathcal F}$ is micro supported on C. We show that C is the smallest. Assume that ${\mathcal F}$ is micro supported on a closed conical subset $C'\subset T^*X$ . Then, since $i_*{\mathcal F}$ is micro supported on $i_\circ C'$ by Lemma 4.8.3, we have $C_P\subset i_\circ C'$ . By taking the closure of the image by the surjection $T^*P|_X\to T^*X$ , we obtain $C\subset C'$ .

Since $i_*{\mathcal F}$ is micro supported on $i_\circ C$ , we have $C_P\subset i_\circ C$ . We show the other inclusion $i_\circ C\subset C_P$ . Since the assertion is local, we may assume that there exists a section $s\colon T^*X\to T^*P|_X$ of the surjection $T^*P|_X\to T^*X$ . Since ${\mathcal F}$ is micro supported on $s^{-1}(C_P)$ by Lemma 4.8.2, we have $C\subset s^{-1}(C_P)$ . Hence we have $i_\circ C\subset C_P$ .

Proof. 1. The kernel $\mathrm {Ker}((T^*{\mathbf P}\times T^*{\mathbf P}^\vee )|_Q\to T^*Q)$ equals the conormal bundle $L_Q$ and the first projection induces an isomorphism $L_Q\to L\subset T^*{\mathbf P}\times _{\mathbf P}Q$ to the tautological bundle. By this isomorphism, the intersection $(C\times T^*{\mathbf P}^\vee )|_Q \cap L_Q$ is identified with ${q^*C\cap L}$ . Hence $(q,q^\vee )$ is C-acyclic on $U\subset Q$ if and only if the restriction $(q^*C\cap L)|_U$ is a subset of the $0$ -section. Since the projectivization $E={\mathbf P}(C)\subset {\mathbf P}(T^*{\mathbf P})$ equals ${\mathbf P}(q^* C\cap L)\subset {\mathbf P}(L)=Q$ , the assertion follows.

2. The Legendre transform $C^\vee $ is the image of the intersection $q^\circ C\cap q^{\vee \circ }T^*{\mathbf P}^\vee \subset T^*Q$ by $q^{\vee \circ }T^*{\mathbf P}^\vee \to T^*{\mathbf P}^\vee $ . We identify the intersection $q^\circ T^*{\mathbf P} \cap q^{\vee \circ }T^*{\mathbf P}^\vee \subset T^*Q$ with the tautological line bundle $L\subset T^*{\mathbf P}\times _{\mathbf P}Q$ . Then, the intersection $q^\circ C\cap q^{\vee \circ }T^*{\mathbf P}^\vee \subset T^*Q$ is identified with $q^* C\cap L$ . Since the projectivization $E={\mathbf P}(C)\subset {\mathbf P}(T^*{\mathbf P})$ equals ${\mathbf P}(q^* C\cap L) \subset {\mathbf P}(L)=Q$ , the closed conical subset $q^* C\cap L$ equals $L|_E$ outside the $0$ -section.

Since $L|_E$ is identified with $L^\vee |_E$ inside $T^*Q$ , we have ${\mathbf P}(C^\vee )= {\mathbf P}(L^\vee |_E) =E\subset {\mathbf P}(T^*{\mathbf P}^\vee )=Q$ .

3. By 2 and symmetry, we have ${\mathbf P}(C) ={\mathbf P}(C^{\vee }) ={\mathbf P}(C^{\vee \vee })$ . Hence we have $C^{\vee \vee }\subset C^+$ .

Proof. 1. Since $(h,f)$ is $C^{\vee +}$ -acyclic and $C^{\vee +}$ contains the $0$ -section, the morphism $f\colon W\to Y$ is smooth. By Proposition 5.4.1, $(q,q^\vee )$ is C-acyclic outside E. Hence $(qh',q^\vee _W)$ is C-acyclic outside $E_W$ by Lemma 3.8 and $(qh',f')$ is C-acyclic outside $E_W$ by Lemma 3.6.

2. By the description of $C^\vee $ in Proposition 5.4.2 and by the open condition Lemma 2.3.4 and Lemma 3.2.1, the $C^{\vee }$ -acyclicity of $(h,f)$ implies the $T^*{\mathbf P}$ -acyclicity of $(qh',f')$ on a neighbourhood of $E_W\subset Q_W$ .

Proof. 1. By Lemma 4.7.1 and Lemma 4.9, the Radon transform $R{\mathcal F}=q^\vee _*q^*{\mathcal F}$ is micro supported on the Legendre transform $C^\vee =q^\vee _\circ q^\circ C$ .

2. By 1, $R^\vee R{\mathcal F}$ is micro supported on $C^{\vee \vee }\subset C^+$ .

Proof. 1. The immersion i induces a morphism

(5.3) $$ \begin{align} (p\times p)^* p^\vee_*\Lambda =\mathrm{pr}_{13*}\Lambda \to (q\times q)_*\Lambda_{ Q\times_{{\mathbf P}^\vee}Q } \end{align} $$

and we have isomorphisms $R^sp^\vee _*\Lambda \to \Lambda (-t)[-2t]$ for $s=2t, 0\leqslant t\leqslant n$ and $R^sp^\vee _*\Lambda =0$ otherwise. The restriction of the closed immersion $ i\colon Q\times _{{\mathbf P}^\vee }Q \to {\mathbf P}\times {\mathbf P}^\vee \times {\mathbf P}$ on the diagonal ${\mathbf P}\subset {\mathbf P}\times {\mathbf P}$ is the sub ${\mathbf P}^{n-1}$ -bundle $Q \subset {\mathbf P}\times {\mathbf P}^\vee $ . On the complement , $Q\times _{{\mathbf P}^\vee }Q$ is a sub ${\mathbf P}^{n-2}$ -bundle. Hence (5.3) induces an isomorphism of cohomology sheaves except for degree $s=2(n-1)$ and induces an isomorphism $\delta _{{\mathbf P}*} p^*R^{2(n-1)}p^\vee _*\Lambda \to R^{2(n-1)}(q\times q)_*\Lambda _{ Q\times _{{\mathbf P}^\vee }Q}$ .

2. This follows from the isomorphisms (5.1).

Proof. 1. By the cartesian diagram

we have $R^\vee R{\mathcal F}= Rq_*q^{\vee *}Rq^\vee _*q^*{\mathcal F}$ . By the proper base change theorem, we have a canonical isomorphism $Rq_*q^{\vee *}Rq^\vee _*q^*{\mathcal F} \to R(q\circ \mathrm {pr}_2)_* (q\circ \mathrm {pr}_1)^*{\mathcal F}$ . In the notation of (5.4), the latter is identified with $R(\mathrm {pr}_2\circ (q\times q))_* (\mathrm {pr}_1\circ (q\times q))^*{\mathcal F}$ . This is identified with $R\mathrm {pr}_{2*}(\mathrm {pr}_{1*}{\mathcal F} \otimes R(q\times q)_*\Lambda _{Q\times _{{\mathbf P}^\vee }Q})$ by the projection formula.

2. This follows from the isomorphism (5.5) and the distinguished triangle (5.2).

Proof. (1) $\Rightarrow $ (2): The pair $(q,q^\vee )$ of $q\colon Q\to {\mathbf P}$ and $q^\vee \colon Q\to {\mathbf P}^\vee $ is C-acyclic outside $E={\mathbf P}(C)$ by Proposition 5.4.1. Hence (1) implies that $q^\vee $ is universally $q^*{\mathcal F}$ -acyclic on the complement by Lemma 4.5.4.

(2) $\Rightarrow $ (3): Assume that a pair of morphisms $h\colon W\to {\mathbf P}^\vee , f\colon W\to Y$ is $C^{\vee +}$ -acyclic and show that f is $h^*R{\mathcal F}$ -acyclic. We consider the commutative diagram

with cartesian square as in Lemma 5.5. Since $q^\vee $ is proper, it suffices to show that $f'$ is $h^{\prime *}q^*{\mathcal F}$ -acyclic by Lemma 4.3.

By (2), $q^\vee _W$ is $(qh')^*{\mathcal F}$ -acyclic on the complement . By Lemma 5.5.1, the morphism $f\colon W\to Y$ is smooth on W. Hence $f'$ is $(qh')^*{\mathcal F}$ -acyclic on by [Reference Illusie8, Corollaire 2.7].

By Lemma 5.5.2, the pair $(qh',f')$ is $T^*{\mathbf P}$ -acyclic on a neighbourhood of $E_W\subset Q_W$ . Since ${\mathcal F}$ is micro supported on $T^*{\mathbf P}$ by Lemma 4.5.2, $f'$ is $(qh')^*{\mathcal F}$ -acyclic on a neighbourhood of $E_W$ . Thus $f'$ is $(qh')^*{\mathcal F}$ -acyclic as required.

(3) $\Rightarrow $ (4) By (3) and (1) $\Rightarrow $ (3), $R^\vee R{\mathcal F}$ is micro supported on $(C^{\vee +}) ^{\vee +}= C^+$ . Since $p^*p_*{\mathcal F}$ is micro supported on the 0-section $T^*_{\mathbf P}{\mathbf P}$ , by the distinguished triangle in Proposition 5.8.2, ${\mathcal F}$ is also micro supported on $C^+$ .

Proof. By Proposition 5.9 (2) $\Rightarrow $ (4), ${\mathcal F}$ is micro supported on $C^+$ . Hence ${\mathcal F}$ is micro supported on $ C=C^+|_B$ by Lemma 4.8.1.

Assume that ${\mathcal F}$ is micro supported on $C' \subset T^*{\mathbf P}$ . Then, by Proposition 5.9 (1) $\Rightarrow $ (2), we have ${\mathbf P}(C')\supset E={\mathbf P}(C)$ since E is the smallest. Since the base of $C'$ contains $B=\mathrm {supp}\, {\mathcal F}$ as a subset, we have $C'\supset C$ .

Proof. By Lemma 5.2, the assertion is local on X. Hence, we may assume that X is affine and is a closed subscheme of ${\mathbf A}^n$ . By Proposition 5.3, we may assume that $X={\mathbf A}^n$ . Further by Lemma 5.2, we may assume that $X={\mathbf P}^n$ . This case is proved in Corollary 5.10.

Proof. We may assume that $X={\mathbf P}$ is a projective space and it suffices to show the inequality $E'\subset E_{k'}$ or the equality $E'=E_{k'}$ of the projectivizations $E={\mathbf P}(SS {\mathcal F})$ and $E'={\mathbf P}(SS {\mathcal F}')$ . Since the complement is the largest open subset on which $p^\vee \colon Q\to {\mathbf P}^\vee $ is universally $p^*{\mathcal F}$ -acyclic and similarly for , we have . Further if $k'$ is an algebraic extension, we have .

Proof. 1. There exists a dense open scheme $U\subset X$ such that ${\mathcal F}_U$ is micro supported on the $0$ -section by Lemma 2.1 and an easy case of Lemma 4.7.1. By Lemma 4.5.1 (1) $\Rightarrow $ (2), ${\mathcal F}_U$ is locally constant.

2. By Lemma 4.8.1, we may assume that $(U_i\to X)_{i\in I}$ is an étale covering.

(1) $\Rightarrow $ (2): If ${\mathcal F}$ is micro supported on $C\subset T^*X$ and if $\dim C\leqq \dim X$ , then ${\mathcal F}_{U_i}$ is micro supported on $C_{U_i}\subset T^*U_i$ by Lemma 4.7.1 and $\dim C_{U_i}\leqq \dim U_i$ for every $i\in I$ .

(2) $\Rightarrow $ (1): Suppose ${\mathcal F}_{U_i}$ is micro supported on $C_i\subset T^*U_i$ and $\dim C_i\leqq \dim U_i$ for every $i\in I$ . Then, ${\mathcal F}$ is micro supported on $C=\overline {\bigcup _{i\in I}\mathrm {Im}(C_i)} \subset T^*X$ by Lemma 4.7.2. Since I is finite, we have $C = \bigcup _{i\in I}\overline {\mathrm {Im}(C_i)}$ and $\dim C =\max _{i\in I}\dim C_i\leqq \dim X$ .

3. (1) $\Rightarrow $ (2): If ${\mathcal F}$ is micro supported on $C\subset T^*X$ and if $\dim C\leqq \dim X$ , then $i_*{\mathcal F}$ is micro supported on $i_\circ C\subset T^*P$ by Lemma 4.8.3 and $\dim i_\circ C\leqq \dim P$ .

(2) $\Rightarrow $ (1): Suppose that the singular support $C_P=SS i_*{\mathcal F}$ satisfies $\dim C_P\leqq \dim P$ . By Lemma 5.3.2, the singular support $C=SS{\mathcal F}$ exists and $C_P=i_\circ C$ . Hence, we have $\dim C=\dim C_P-\mathrm {codim}_PX \leqq \dim X$ .

4. Suppose that ${\mathcal F}$ and ${\mathcal F}'$ are micro supported on C and $C'$ respectively and that $\dim C,\dim C'\leqq \dim X$ . By Lemma 4.6, ${\mathcal F}"$ is micro supported on $C"=C\cup C'$ and we have $\dim C"\leqq \dim X$ .

Proof. We prove the theorem first assuming that k is perfect by induction on $\dim X$ . If $X=\varnothing $ , the assertion is clear. By Lemma 6.2.1, the largest open subset $U\subset X$ on which ${\mathcal F}$ is constructible is nonempty. To prove the equality $X=U$ by contradiction, we assume that the complement is not empty. Since k is assumed perfect, after shrinking X if necessary, we may assume that Z is smooth over k.

Let $j\colon U\to X$ be the open immersion, $i\colon Z\to X$ be the closed immersion and consider the distinguished triangle $j_!j^*{\mathcal F}\to {\mathcal F} \to i_*i^*{\mathcal F}\to $ . Since $j_!j^*{\mathcal F}$ is constructible, it suffices to show that $i_*i^*{\mathcal F}$ is constructible. By Theorem 5.13, $j_!j^*{\mathcal F}$ is holonomic. Hence by Lemma 6.2.4, $i_*i^*{\mathcal F}$ is also holonomic. Since replacing ${\mathcal F}$ by $i_*i^*{\mathcal F}$ does not change U, we may assume that ${\mathcal F}=i_*{\mathcal F}_Z$ for ${\mathcal F}_Z=i^*{\mathcal F}$ . Since $i_*{\mathcal F}_Z$ is holonomic, ${\mathcal F}_Z$ is also holonomic by Lemma 6.2.3. Since $\dim Z<\dim X$ , ${\mathcal F}_Z$ is constructible. This contradicts the assumption that U is the largest.

We show the general case. Let $k'$ be a perfect closure of k. Then, by Corollary 5.12, the pull-back ${\mathcal F}'$ on the base change $X_{k'}$ is holonomic. Hence by the perfect case, ${\mathcal F}'$ is constructible. Since the canonical morphism $X_{k'}\to X$ is a homeomorphism, ${\mathcal F}$ is also constructible.

Proof. 1. It suffices to show (1) $\Rightarrow $ (2). By replacing Q by $P\times Q$ , we may assume that there exists a smooth morphism $Q\to P$ compatible with i and $i'$ . Since $i'$ induces a section of a smooth morphism $Q\times _PX\to X$ and since the question is local on X by Lemma 6.2.2, we may assume that there exists a closed subscheme $Q'$ of Q containing X and étale over P by [Reference Grothendieck6, Chapitre 0_IV (15.1.16) a)⇔c)] and [Reference Grothendieck6, Théorème (17.6.1) a)⇔c′)]. By Lemma 6.2.3, replacing Q by $Q'$ , we may assume that there exists an étale morphism $h\colon Q\to P$ compatible with i and $i'$ . Since $X\to Q$ induces an open immersion $X\to X\times _PQ$ , after shrinking Q, we may assume that $X\to X\times _PQ$ is an isomorphism. If $i_*{\mathcal F}$ is micro supported on $C\subset T^*P$ and $\dim C\leqq \dim P$ , then $i^{\prime }_*{\mathcal F}$ is micro supported on $h^*C\subset T^*Q$ by Lemma 4.7.1 and $\dim h^*C =\dim C\leqq \dim P=\dim Q$ .

2. The implication (1) $\Rightarrow $ (2) is clear.

(2) $\Rightarrow $ (1): Since X is quasi-compact, we may assume that J is finite. By [Reference Grothendieck6, Corollaire (18.4.7)], for each $j\in J$ , there exist an open covering $(U_{jh}\to U\times _XU_j)_{h\in H_j}$ , closed immersions $i_{jh}\colon U_{jh}\to P_{jh}$ to schemes étale over P and closed immersions $i^{\prime }_{jh}\colon U_{jh}\to Q_{jh}$ to schemes étale over $Q_j$ . Then since $U_{jh}\to U_j\times _{Q_j}Q_{jh}$ is an open and closed immersion, $i^{\prime }_{jh*}({\mathcal F}|_{U_{jh}})$ is holonomic for every $j,h$ by Lemma 6.2.2. Hence $i_{jh*}({\mathcal F}|_{U_{jh}})$ is also holonomic by 1. Since $(P_{jh}\to P)_{j\in J,h\in H_j}$ is an étale covering of the support of an open neighbourhood of the image $i(U)$ , $i_*({\mathcal F}|_U)$ is holonomic by Lemma 6.2.2.

Proof. Since the question is local on X, we may assume that there exists a closed immersion $i\colon X\to P$ to a smooth scheme over k. Then $i_*{\mathcal F}$ is constructible if and only if it is holonomic by Theorem 5.13 and Theorem 6.3.