Lemma 42.59.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $i : X \to Y$ and $j : Y \to Z$ be regular immersions of schemes locally of finite type over $S$. Then $j \circ i$ is a regular immersion and $(j \circ i)^! = i^! \circ j^!$.
42.59 Gysin maps for local complete intersection morphisms
Before reading this section, we suggest the reader read up on regular immersions (Divisors, Section 31.21) and local complete intersection morphisms (More on Morphisms, Section 37.62).
Let $(S, \delta )$ be as in Situation 42.7.1. Let $i : X \to Y$ be a regular immersion1 of schemes locally of finite type over $S$. In particular, the conormal sheaf $\mathcal{C}_{X/Y}$ is finite locally free (see Divisors, Lemma 31.21.5). Hence the normal sheaf
\[ \mathcal{N}_{X/Y} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{C}_{X/Y}, \mathcal{O}_ X) \]
is finite locally free as well and we have a surjection $\mathcal{N}_{X/Y}^\vee \to \mathcal{C}_{X/Y}$ (because an isomorphism is also a surjection). The construction in Section 42.54 gives us a canonical bivariant class
\[ i^! = c(X \to Y, \mathcal{N}_{X/Y}) \in A^*(X \to Y)^\wedge \]
We need a couple of lemmas about this notion.
Proof. The first statement is Divisors, Lemma 31.21.7. By Divisors, Lemma 31.21.6 there is a short exact sequence
\[ 0 \to i^*(\mathcal{C}_{Y/Z}) \to \mathcal{C}_{X/Z} \to \mathcal{C}_{X/Y} \to 0 \]
Thus the result by the more general Lemma 42.54.10. $\square$
Lemma 42.59.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $p : P \to X$ be a smooth morphism of schemes locally of finite type over $S$ and let $s : X \to P$ be a section. Then $s$ is a regular immersion and $1 = s^! \circ p^*$ in $A^*(X)^\wedge $ where $p^* \in A^*(P \to X)^\wedge $ is the bivariant class of Lemma 42.33.2.
Proof. The first statement is Divisors, Lemma 31.22.8. It suffices to show that $s^! \cap p^*[Z] = [Z]$ in $\mathop{\mathrm{CH}}\nolimits _*(X)$ for any integral closed subscheme $Z \subset X$ as the assumptions are preserved by base change by $X' \to X$ locally of finite type. After replacing $P$ by an open neighbourhood of $s(Z)$ we may assume $P \to X$ is smooth of fixed relative dimension $r$. Say $\dim _\delta (Z) = n$. Then every irreducible component of $p^{-1}(Z)$ has dimension $r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$. Observe that $s(X) \cap p^{-1}(Z) = s(Z)$ scheme theoretically. Hence by the same reference as used above $s(X) \cap p^{-1}(Z)$ is a closed subscheme regularly embedded in $\overline{p}^{-1}(Z)$ of codimension $r$. We conclude by Lemma 42.54.5. $\square$
Let $(S, \delta )$ be as in Situation 42.7.1. Consider a commutative diagram
\[ \xymatrix{ X \ar[rd]_ f \ar[rr]_ i & & P \ar[ld]^ g \\ & Y } \]
of schemes locally of finite type over $S$ such that $g$ is smooth and $i$ is a regular immersion. Combining the bivariant class $i^!$ discussed above with the bivariant class $g^* \in A^*(P \to Y)^\wedge $ of Lemma 42.33.2 we obtain
\[ f^! = i^! \circ g^* \in A^*(X \to Y) \]
Observe that the morphism $f$ is a local complete intersection morphism, see More on Morphisms, Definition 37.62.2. Conversely, if $f : X \to Y$ is a local complete intersection morphism of locally Noetherian schemes and $f = g \circ i$ with $g$ smooth, then $i$ is a regular immersion. We claim that our construction of $f^!$ only depends on the morphism $f$ and not on the choice of factorization $f = g \circ i$.
Lemma 42.59.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. The bivariant class $f^!$ is independent of the choice of the factorization $f = g \circ i$ with $g$ smooth (provided one exists).
Proof. Given a second such factorization $f = g' \circ i'$ we can consider the smooth morphism $g'' : P \times _ Y P' \to Y$, the immersion $i'' : X \to P \times _ Y P'$ and the factorization $f = g'' \circ i''$. Thus we may assume that we have a diagram
\[ \xymatrix{ & P' \ar[d]^ p \ar[rd]^{g'} \\ X \ar[r]^ i \ar[ru]^{i'} & P \ar[r]^ g & Y } \]
where $p$ is a smooth morphism. Then $(g')^* = p^* \circ g^*$ (Lemma 42.14.3) and hence it suffices to show that $i^! = (i')^! \circ p^*$ in $A^*(X \to P)$. Consider the commutative diagram
\[ \xymatrix{ & X \times _ P P' \ar[d]^{\overline{p}} \ar[r]_ j & P' \ar[d]^ p \\ X \ar[ru]^ s \ar[r]^1 & X \ar[r]^ i & P } \]
where $s =(1, i')$. Then $s$ and $j$ are regular immersions (by Divisors, Lemma 31.22.8 and Divisors, Lemma 31.21.4) and $i' = j \circ s$. By Lemma 42.59.1 we have $(i')^! = s^! \circ j^!$. Since the square is cartesian, the bivariant class $j^!$ is the restriction (Remark 42.33.5) of $i^!$ to $P'$, see Lemma 42.54.2. Since bivariant classes commute with flat pullbacks we find $j^! \circ p^* = \overline{p}^* \circ i^!$. Thus it suffices to show that $s^! \circ \overline{p}^* = \text{id}$ which is done in Lemma 42.59.2. $\square$
Definition 42.59.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. We say the gysin map for $f$ exists if we can write $f = g \circ i$ with $g$ smooth and $i$ an immersion. In this case we define the gysin map $f^! = i^! \circ g^* \in A^*(X \to Y)$ as above.
It follows from the definition that for a regular immersion this agrees with the construction earlier and for a smooth morphism this agrees with flat pullback. In fact, this agreement holds for all syntomic morphisms.
Lemma 42.59.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. If the gysin map exists for $f$ and $f$ is flat, then $f^!$ is equal to the bivariant class of Lemma 42.33.2.
Proof. Choose a factorization $f = g \circ i$ with $i : X \to P$ an immersion and $g : P \to Y$ smooth. Observe that for any morphism $Y' \to Y$ which is locally of finite type, the base changes of $f'$, $g'$, $i'$ satisfy the same assumptions (see Morphisms, Lemmas 29.34.5 and 29.30.4 and More on Morphisms, Lemma 37.62.8). Thus we reduce to proving that $f^*[Y] = i^!(g^*[Y])$ in case $Y$ is integral, see Lemma 42.35.3. Set $n = \dim _\delta (Y)$. After decomposing $X$ and $P$ into connected components we may assume $f$ is flat of relative dimension $r$ and $g$ is smooth of relative dimension $t$. Then $f^*[Y] = [X]_{n + s}$ and $g^*[Y] = [P]_{n + t}$. On the other hand $i$ is a regular immersion of codimension $t - s$. Thus $i^![P]_{n + t} = [X]_{n + s}$ (Lemma 42.54.5) and the proof is complete. $\square$
Lemma 42.59.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ and $g : Y \to Z$ be local complete intersection morphisms of schemes locally of finite type over $S$. Assume the gysin map exists for $g \circ f$ and $g$. Then the gysin map exists for $f$ and $(g \circ f)^! = f^! \circ g^!$.
Proof. Observe that $g \circ f$ is a local complete intersection morphism by More on Morphisms, Lemma 37.62.7 and hence the statement of the lemma makes sense. If $X \to P$ is an immersion of $X$ into a scheme $P$ smooth over $Z$ then $X \to P \times _ Z Y$ is an immersion of $X$ into a scheme smooth over $Y$. This prove the first assertion of the lemma. Let $Y \to P'$ be an immersion of $Y$ into a scheme $P'$ smooth over $Z$. Consider the commutative diagram
\[ \xymatrix{ X \ar[r] \ar[d] & P \times _ Z Y \ar[r]_ a \ar[ld]^ p & P \times _ Z P' \ar[ld]^ q \\ Y \ar[r]_ b \ar[d] & P' \ar[ld] \\ Z } \]
Here the horizontal arrows are regular immersions, the south-west arrows are smooth, and the square is cartesian. Whence $a^! \circ q^* = p^* \circ b^!$ as bivariant classes commute with flat pullback. Combining this fact with Lemmas 42.59.1 and 42.14.3 the reader finds the statement of the lemma holds true. Small detail omitted. $\square$
Lemma 42.59.7. Let $(S, \delta )$ be as in Situation 42.7.1. Consider a commutative diagram of schemes locally of finite type over $S$ with both square cartesian. Assume $f : X \to Y$ is a local complete intersection morphism such that the gysin map exists for $f$. Let $c \in A^*(Y'' \to Y')$. Denote $res(f^!) \in A^*(X' \to Y')$ the restriction of $f^!$ to $Y'$ (Remark 42.33.5). Then $c$ and $res(f^!)$ commute (Remark 42.33.6).
\[ \xymatrix{ X'' \ar[d] \ar[r] & X' \ar[d] \ar[r] & X \ar[d]^ f \\ Y'' \ar[r] & Y' \ar[r] & Y } \]
Proof. Choose a factorization $f = g \circ i$ with $g$ smooth and $i$ an immersion. Since $f^! = i^! \circ g^!$ it suffices to prove the lemma for $g^!$ (which is given by flat pullback) and for $i^!$. The result for flat pullback is part of the definition of a bivariant class. The case of $i^!$ follows immediately from Lemma 42.54.8. $\square$
Lemma 42.59.8. Let $(S, \delta )$ be as in Situation 42.7.1. Consider a cartesian diagram of schemes locally of finite type over $S$. Assume
$f$ is a local complete intersection morphism and the gysin map exists for $f$,
$X$, $X'$, $Y$, $Y'$ satisfy the equivalent conditions of Lemma 42.42.1,
for $x' \in X'$ with images $x$, $y'$, and $y$ in $X$, $Y'$, and $Y$ we have $n_{x'} - n_{y'} = n_ x - n_ y$ where $n_{x'}$, $n_ x$, $n_{y'}$, and $n_ y$ are as in the lemma, and
for every generic point $\xi \in X'$ the local ring $\mathcal{O}_{Y', f'(\xi )}$ is Cohen-Macaulay.
\[ \xymatrix{ X' \ar[d]_{f'} \ar[r] & X \ar[d]^ f \\ Y' \ar[r] & Y } \]
Then $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are as in Remark 42.42.2.
Proof. Recall that $n_{x'}$ is the common value of $\delta (\xi )$ where $\xi $ is the generic point of an irreducible component passing through $x'$. Moreover, the functions $x' \mapsto n_{x'}$, $x \mapsto n_ x$, $y' \mapsto n_{y'}$, and $y \mapsto n_ y$ are locally constant. Let $X'_ n$, $X_ n$, $Y'_ n$, and $Y_ n$ be the open and closed subscheme of $X'$, $X$, $Y'$, and $Y$ where the function has value $n$. Recall that $[X'] = \sum [X'_ n]_ n$ and $[Y'] = \sum [Y'_ n]_ n$. Having said this, it is clear that to prove the lemma we may replace $X'$ by one of its connected components and $X$, $Y'$, $Y$ by the connected component that it maps into. Then we know that $X'$, $X$, $Y'$, and $Y$ are $\delta $-equidimensional in the sense that each irreducible component has the same $\delta $-dimension. Say $n'$, $n$, $m'$, and $m$ is this common value for $X'$, $X$, $Y'$, and $Y$. The last assumption means that $n' - m' = n - m$.
Choose a factorization $f = g \circ i$ where $i : X \to P$ is an immersion and $g : P \to Y$ is smooth. As $X$ is connected, we see that the relative dimension of $P \to Y$ at points of $i(X)$ is constant. Hence after replacing $P$ by an open neighbourhood of $i(X)$, we may assume that $P \to Y$ has constant relative dimension and $i : X \to P$ is a closed immersion. Denote $g' : Y' \times _ Y P \to Y'$ the base change of $g$ and denote $i' : X' \to Y' \times _ Y P$ the base change of $i$. It is clear that $g^*[Y] = [P]$ and $(g')^*[Y'] = [Y' \times _ Y P]$. Finally, if $\xi ' \in X'$ is a generic point, then $\mathcal{O}_{Y' \times _ Y P, i'(\xi )}$ is Cohen-Macaulay. Namely, the local ring map $\mathcal{O}_{Y', f'(\xi )} \to \mathcal{O}_{Y' \times _ Y P, i'(\xi )}$ is flat with regular fibre (see Algebra, Section 10.142), a regular local ring is Cohen-Macaulay (Algebra, Lemma 10.106.3), $\mathcal{O}_{Y', f'(\xi )}$ is Cohen-Macaulay by assumption (4) and we get what we want from Algebra, Lemma 10.163.3. Thus we reduce to the case discussed in the next paragraph.
Assume $f$ is a regular closed immersion and $X'$, $X$, $Y'$, and $Y$ are $\delta $-equidimensional of $\delta $-dimensions $n'$, $n$, $m'$, and $m$ and $m' - n' = m - n$. In this case we obtain the result immediately from Lemma 42.54.6. $\square$
Remark 42.59.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. Assume the gysin map exists for $f$. Then $f^! \circ c_ i(\mathcal{E}) = c_ i(f^*\mathcal{E}) \circ f^!$ and similarly for the Chern character, see Lemma 42.59.7. If $X$ and $Y$ satisfy the equivalent conditions of Lemma 42.42.1 and $Y$ is Cohen-Macaulay (for example), then $f^![Y] = [X]$ by Lemma 42.59.8. In this case we also get $f^!(c_ i(\mathcal{E}) \cap [Y]) = c_ i(f^*\mathcal{E}) \cap [X]$ and similarly for the Chern character.
Lemma 42.59.10. Let $(S, \delta )$ be as in Situation 42.7.1. Consider a cartesian square of schemes locally of finite type over $S$. Assume
both $f$ and $f'$ are local complete intersection morphisms, and
the gysin map exists for $f$
\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]
Then $\mathcal{C} = \mathop{\mathrm{Ker}}(H^{-1}((g')^*\mathop{N\! L}\nolimits _{X/Y}) \to H^{-1}(\mathop{N\! L}\nolimits _{X'/Y'}))$ is a finite locally free $\mathcal{O}_{X'}$-module, the gysin map exists for $f'$, and we have
\[ res(f^!) = c_{top}(\mathcal{C}^\vee ) \circ (f')^! \]
in $A^*(X' \to Y')$.
Proof. The fact that $\mathcal{C}$ is finite locally free follows immediately from More on Algebra, Lemma 15.85.5. Choose a factorization $f = h \circ i$ with $h : P \to Y$ smooth and $i$ an immersion. Then we can factor $f' = h' \circ i'$ where $h' : P' \to Y'$ and $i' : X' \to P'$ the base changes. Picture
\[ \xymatrix{ X' \ar[r] \ar[d] & P' \ar[r] \ar[d] & Y' \ar[d] \\ X \ar[r]^ i & P \ar[r]^ h & Y } \]
In particular, we see that the gysin map exists for $f'$. By More on Morphisms, Lemmas 37.13.13 we have
\[ \mathop{N\! L}\nolimits _{X/Y} = \left( \mathcal{C}_{X/P} \to i^*\Omega _{P/Y} \right) \]
where $\mathcal{C}_{X/P}$ is the conormal sheaf of the embedding $i$. Similarly for the primed version. We have $(g')^*i^*\Omega _{P/Y} = (i')^*\Omega _{P'/Y'}$ because $\Omega _{P/Y}$ pulls back to $\Omega _{P'/Y'}$ by Morphisms, Lemma 29.32.10. Also, recall that $(g')^*\mathcal{C}_{X/P} \to \mathcal{C}_{X'/P'}$ is surjective, see Morphisms, Lemma 29.31.4. We deduce that the sheaf $\mathcal{C}$ is canonicallly isomorphic to the kernel of the map $(g')^*\mathcal{C}_{X/P} \to \mathcal{C}_{X'/P'}$ of finite locally free modules. Recall that $i^!$ is defined using $\mathcal{N} = \mathcal{C}_{Z/X}^\vee $ and similarly for $(i')^!$. Thus we have
\[ res(i^!) = c_{top}(\mathcal{C}^\vee ) \circ (i')^! \]
in $A^*(X' \to P')$ by an application of Lemma 42.54.4. Since finally we have $f^! = i^! \circ h^*$, $(f')^! = (i')^! \circ (h')^*$, and $(h')^* = res(h^*)$ we conclude. $\square$
Lemma 42.59.11 (Blow up formula). Let $(S, \delta )$ be as in Situation 42.7.1. Let $i : Z \to X$ be a regular closed immersion of schemes locally of finite type over $S$. Let $b : X' \to X$ be the blowing up with center $Z$. Picture Assume that the gysin map exists for $b$. Then we have in $A^*(E \to Z)$ where $\mathcal{F}$ is the kernel of the canonical map $\pi ^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'}$.
\[ \xymatrix{ E \ar[r]_ j \ar[d]_\pi & X' \ar[d]^ b \\ Z \ar[r]^ i & X } \]
\[ res(b^!) = c_{top}(\mathcal{F}^\vee ) \circ \pi ^* \]
Proof. Observe that the morphism $b$ is a local complete intersection morphism by More on Algebra, Lemma 15.31.2 and hence the statement makes sense. Since $Z \to X$ is a regular immersion (and hence a fortiori quasi-regular) we see that $\mathcal{C}_{Z/X}$ is finite locally free and the map $\text{Sym}^*(\mathcal{C}_{Z/X}) \to \mathcal{C}_{Z/X, *}$ is an isomorphism, see Divisors, Lemma 31.21.5. Since $E = \text{Proj}(\mathcal{C}_{Z/X, *})$ we conclude that $E = \mathbf{P}(\mathcal{C}_{Z/X})$ is a projective space bundle over $Z$. Thus $E \to Z$ is smooth and certainly a local complete intersection morphism. Thus Lemma 42.59.10 applies and we see that
\[ res(b^!) = c_{top}(\mathcal{C}^\vee ) \circ \pi ^! \]
with $\mathcal{C}$ as in the statement there. Of course $\pi ^* = \pi ^!$ by Lemma 42.59.5. It remains to show that $\mathcal{F}$ is equal to the kernel $\mathcal{C}$ of the map $H^{-1}(j^*\mathop{N\! L}\nolimits _{X'/X}) \to H^{-1}(\mathop{N\! L}\nolimits _{E/Z})$.
Since $E \to Z$ is smooth we have $H^{-1}(\mathop{N\! L}\nolimits _{E/Z}) = 0$, see More on Morphisms, Lemma 37.13.7. Hence it suffices to show that $\mathcal{F}$ can be identified with $H^{-1}(j^*\mathop{N\! L}\nolimits _{X'/X})$. By More on Morphisms, Lemmas 37.13.11 and 37.13.9 we have an exact sequence
\[ 0 \to H^{-1}(j^*\mathop{N\! L}\nolimits _{X'/X}) \to H^{-1}(\mathop{N\! L}\nolimits _{E/X}) \to \mathcal{C}_{E/X'} \to \ldots \]
By the same lemmas applied to $E \to Z \to X$ we obtain an isomorphism $\pi ^*\mathcal{C}_{Z/X} = H^{-1}(\pi ^*\mathop{N\! L}\nolimits _{Z/X}) \to H^{-1}(\mathop{N\! L}\nolimits _{E/X})$. Thus we conclude. $\square$
Lemma 42.59.12. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$ such that both $X$ and $Y$ are quasi-compact, regular, have affine diagonal, and finite dimension. Then $f$ is a local complete intersection morphism. Assume moreover the gysin map exists for $f$. Then in $\mathop{\mathrm{CH}}\nolimits ^*(X) \otimes \mathbf{Q}$ where the intersection product is as in Section 42.58.
\[ f^!(\alpha \cdot \beta ) = f^!\alpha \cdot f^!\beta \]
Proof. The first statement follows from More on Morphisms, Lemma 37.62.11. Observe that $f^![Y] = [X]$, see Lemma 42.59.8. Write $\alpha = ch(\alpha ') \cap [Y]$ and $\beta = ch(\beta ') \cap [Y]$ where $\alpha ', \beta ' \in K_0(\textit{Vect}(X)) \otimes \mathbf{Q}$ as in Section 42.58. Setting $c = ch(\alpha ')$ and $c' = ch(\beta ')$ we find $\alpha \cdot \beta = c \cap c' \cap [Y]$ by construction. By Lemma 42.59.7 we know that $f^!$ commutes with both $c$ and $c'$. Hence
\begin{align*} f^!(\alpha \cdot \beta ) & = f^!(c \cap c' \cap [Y]) \\ & = c \cap c' \cap f^![Y] \\ & = c \cap c' \cap [X] \\ & = (c \cap [X]) \cdot (c' \cap [X]) \\ & = (c \cap f^![Y]) \cdot (c' \cap f^![Y]) \\ & = f^!(\alpha ) \cdot f^!(\beta ) \end{align*}
as desired. $\square$
Lemma 42.59.13. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$ such that both $X$ and $Y$ are quasi-compact, regular, have affine diagonal, and finite dimension. Then $f$ is a local complete intersection morphism. Assume moreover the gysin map exists for $f$ and that $f$ is proper. Then in $\mathop{\mathrm{CH}}\nolimits ^*(Y) \otimes \mathbf{Q}$ where the intersection product is as in Section 42.58.
\[ f_*(\alpha \cdot f^!\beta ) = f_*\alpha \cdot \beta \]
Proof. The first statement follows from More on Morphisms, Lemma 37.62.11. Observe that $f^![Y] = [X]$, see Lemma 42.59.8. Write $\alpha = ch(\alpha ') \cap [X]$ and $\beta = ch(\beta ') \cap [Y]$ $\alpha ' \in K_0(\textit{Vect}(X)) \otimes \mathbf{Q}$ and $\beta ' \in K_0(\textit{Vect}(Y)) \otimes \mathbf{Q}$ as in Section 42.58. Set $c = ch(\alpha ')$ and $c' = ch(\beta ')$. We have
\begin{align*} f_*(\alpha \cdot f^!\beta ) & = f_*(c \cap f^!(c' \cap [Y]_ e)) \\ & = f_*(c \cap c' \cap f^![Y]_ e) \\ & = f_*(c \cap c' \cap [X]_ d) \\ & = f_*(c' \cap c \cap [X]_ d) \\ & = c' \cap f_*(c \cap [X]_ d) \\ & = \beta \cdot f_*(\alpha ) \end{align*}
The first equality by the construction of the intersection product. By Lemma 42.59.7 we know that $f^!$ commutes with $c'$. The fact that Chern classes are in the center of the bivariant ring justifies switching the order of capping $[X]$ with $c$ and $c'$. Commuting $c'$ with $f_*$ is allowed as $c'$ is a bivariant class. The final equality is again the construction of the intersection product. $\square$
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