Constructible sheaves for Proposition II
Only the sheaf theory that Proposition II uses, each item with its smallest example. Throughout, \(k=\overline{\mathbb F}_q\) with \(q\) a power of the prime \(p\), \(\ell\ne p\), and a variety is a separated \(k\)-scheme of finite type.
$$\begin{array}{ll} D^b_c(Y,\overline{\mathbb Q}_\ell) & \text{§1–2: one variety, its functors}\\[2pt] \quad\big\downarrow & \\[2pt] D^b_c([Y/H]) & \text{§3: equivariance under a connected }H\\[2pt] \quad\big\downarrow & \\[2pt] \mathrm{Perv}_H(Y)\ni\mathrm{IC}(\bar o) & \text{§4: perverse heart, simple objects}\\[2pt] \quad\big\downarrow{\ \pi^*[e_M],\ \iota_*} & \\[2pt] \mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)} & \text{§5: levels}\\[2pt] \quad\big\downarrow{\ \mathrm{IC}_w\ast_I-} & \\[2pt] G_w\simeq\bigoplus_{S,i}S[-i]\otimes E_{S,i}(w) & \text{§6–7: convolution, decomposition}\\[2pt] \quad\big\downarrow{\ \operatorname{tr}} & \\[2pt] C_I & \text{§8: Weil structures, traces} \end{array}$$1 · Constructible complexes · 2 · Functors · 3 · Equivariance · 4 · Perverse sheaves · 5 · Levels · 6 · Convolution · 7 · Decomposition · 8 · Weil structures · Where used
1 · Constructible complexes
Definition
A sheaf \(F\) of \(\overline{\mathbb Q}_\ell\)-vector spaces on \(Y\) is lisse if it is étale-locally constant with finite-dimensional stalks, and constructible if it is lisse on the strata of a finite stratification
$$Y=\bigsqcup_{\alpha\in A}Y_\alpha,\qquad Y_\alpha\ \text{locally closed, smooth, connected},\qquad\overline{Y_\alpha}\ \text{a union of strata}.$$ $$D^b_c(Y,\overline{\mathbb Q}_\ell)=\big\{K:\ \mathcal H^i(K)\ \text{constructible for all }i,\ \ \mathcal H^i(K)=0\ \text{for }|i|\gg0\big\}.$$For a fixed stratification \(\mathfrak S\), \(D^b_{\mathfrak S}(Y)\subset D^b_c(Y)\) is the full subcategory with every \(\mathcal H^i(K)|_{Y_\alpha}\) lisse; \(D^b_c(Y)=\bigcup_{\mathfrak S}D^b_{\mathfrak S}(Y)\).
Example · \(Y=\mathbb A^1\). \(\mathfrak S=\{\{0\},\ U\}\), \(U=\mathbb G_m\), with \(j:U\hookrightarrow\mathbb A^1\hookleftarrow\{0\}:i\) and \(\delta_0=i_*\overline{\mathbb Q}_\ell\).
$$\overline{\mathbb Q}_\ell,\ \ \delta_0,\ \ j_!\overline{\mathbb Q}_\ell,\ \ j_*\overline{\mathbb Q}_\ell\ \in D^b_{\mathfrak S}(\mathbb A^1),\qquad\bigoplus_{x\in\mathbb A^1(k)}i_{x,*}\overline{\mathbb Q}_\ell\notin D^b_c(\mathbb A^1).$$The last sheaf is lisse on no dense open set.
\(\overline{\mathbb Q}_\ell\)-coefficients are built from \(\mathbb Z/\ell^n\)-sheaves (SGA 4, SGA 5; Deligne, Weil II); nothing below depends on the construction.
2 · Functors
For a morphism \(f:X\to Y\) of varieties, all functors are derived:
$$f^*,\ f^!:D^b_c(Y)\to D^b_c(X),\qquad f_*,\ f_!:D^b_c(X)\to D^b_c(Y),\qquad f^*\dashv f_*,\quad f_!\dashv f^!,\quad f_!\xrightarrow{\sim}f_*\ \text{if }f\text{ is proper}.$$ $$K\otimes L,\qquad K\boxtimes L=\mathrm{pr}_1^*K\otimes\mathrm{pr}_2^*L,\qquad(K[n])^i=K^{i+n},\ \ \mathcal H^i(K[n])=\mathcal H^{i+n}(K).$$For \(j\) an open and \(i\) a closed immersion, \(j^!=j^*\), \(i_!=i_*\), and
$$j_!j^*K\to K\to i_*i^*K\xrightarrow{+1},\qquad i_*i^!K\to K\to j_*j^*K\xrightarrow{+1}.$$Base change
For a cartesian square \(X'=X\times_YY'\) with \(f':X'\to Y'\), \(g':X'\to X\), \(g:Y'\to Y\):
$$g^*f_!\simeq f'_!g'^*\ \ \text{(proper base change, any }g),\qquad g^*f_*\simeq f'_*g'^*\ \ \text{(smooth base change, }g\text{ smooth)},$$ $$(f_!K)_{\bar y}=R\Gamma_c\big(f^{-1}(\bar y),K\big).$$SGA 4, Exp. XII, XVI, XVII.
Smooth maps and affine-space fibres
Let \(f:X\to Y\) be smooth of relative dimension \(d\). Then
$$f^!\simeq f^*[2d](d),\qquad f^*[d]\ \text{is perverse t-exact (§4)},$$and if the fibres are geometrically connected, \(f^*[d]:\mathrm{Perv}(Y)\to\mathrm{Perv}(X)\) is fully faithful with image closed under subquotients, and sends \(\mathrm{IC}(\bar Z,\xi)\) to \(\mathrm{IC}(f^{-1}\bar Z,f^*\xi)\) (BBD 4.2.5–4.2.6.2).
If moreover \(\pi:X\to Y\) is Zariski-locally \(U\times\mathbb A^d\to U\), then \(R\Gamma(\mathbb A^d,\overline{\mathbb Q}_\ell)=\overline{\mathbb Q}_\ell\) and smooth base change give \(\pi_*\pi^*\simeq\mathrm{id}\), so \(\pi^*\) is fully faithful on all of \(D^b_c\), not only on perverse sheaves:
$$\operatorname{Hom}(\pi^*K,\pi^*L)=\operatorname{Hom}(K,\pi_*\pi^*L)=\operatorname{Hom}(K,L).$$This is why \(\pi_{N,M}^*[e_M]\) in (Lev) is fully faithful and t-exact (§5).
Example · \(\mathbb A^1\). Stalks of the objects of §1:
| object | at \(u\in U\) | at \(0\) |
|---|---|---|
| \(\overline{\mathbb Q}_\ell\) | \(\overline{\mathbb Q}_\ell\) | \(\overline{\mathbb Q}_\ell\) |
| \(\delta_0=i_*\overline{\mathbb Q}_\ell\) | \(0\) | \(\overline{\mathbb Q}_\ell\) |
| \(j_!\overline{\mathbb Q}_\ell\) | \(\overline{\mathbb Q}_\ell\) | \(0\) |
| \(j_*\overline{\mathbb Q}_\ell\) | \(\overline{\mathbb Q}_\ell\) | \(\overline{\mathbb Q}_\ell\oplus\overline{\mathbb Q}_\ell(-1)[-1]\) |
3 · Equivariant complexes
\(H\) is a connected linear algebraic group over \(k\) acting on \(Y\); \(q:Y\to[Y/H]\) is the quotient map, smooth with fibres \(H\).
Definition
$$D^b_{c,H}(Y):=D^b_c\big([Y/H],\overline{\mathbb Q}_\ell\big),\qquad\operatorname{For}=q^*:D^b_c([Y/H])\to D^b_c(Y).$$Laszlo–Olsson (Publ. IHÉS 107, 2008; perverse sheaves: Math. Z. 261, 2009) for Artin stacks; Bernstein–Lunts (LNM 1578, 1994) for the equivariant derived category of a variety.
| fact | statement | source |
|---|---|---|
| perverse equivariance is a property | \(\operatorname{For}:\mathrm{Perv}_H(Y)\to\mathrm{Perv}(Y)\) is fully faithful (\(H\) connected) | BBD 4.2.5 for \(q\); Chen–Yi Lemma 107(ii) for pro-groups |
| on \(D^b_c\) it is a structure | \(\operatorname{Ext}^\bullet_{[\mathrm{pt}/H]}(\overline{\mathbb Q}_\ell,\overline{\mathbb Q}_\ell)=H^\bullet(BH)\); \(H^\bullet(B\mathbb G_m)=\overline{\mathbb Q}_\ell[c]\), \(\deg c=2\) | Zhu A.1.3 |
| unipotent kernels change nothing | \(U\lhd H\) connected unipotent, trivial on \(Y\): pullback along \([Y/H]\to[Y/(H/U)]\) is fully faithful (\(H^\bullet(BU)=\overline{\mathbb Q}_\ell\)), an equivalence on perverse hearts | Zhu A.1.3, Lemma A.1.4(2) |
| local systems on an orbit | \(O=H\cdot y\simeq H/S_y\), \([O/H]\simeq BS_y\): \(\ \operatorname{Loc}_H(O)\simeq\operatorname{Rep}(\pi_0S_y)\), \(\pi_0S_y=S_y/S_y^\circ\) | Chen–Yi Example 108, Lemma 71 |
So under I2 (connected stabilizers) every \(H\)-equivariant local system on an orbit is constant, and \(\mathcal C_I^{(N,M)}\) does not depend on which quotient \(I/I_{N,M}\) with pro-unipotent kernel is used.
Examples.
- \(\mathbb G_m\) on \(\mathbb A^1\), \(t\cdot x=tx\): orbits \(\{0\}\), \(\mathbb G_m\); stabilizers \(\mathbb G_m\), \(1\), connected. Only constant equivariant local systems.
- \(\mathbb G_m\) on \(\mathbb G_m\), \(t\cdot x=t^2x\) (\(p\) odd): \(S=\mu_2\), \(\pi_0S=\mathbb Z/2\), so \(\operatorname{Loc}_H=\{\overline{\mathbb Q}_\ell,\ \mathcal L_{\mathrm{sgn}}\}\), with \(\mathcal L_{\mathrm{sgn}}\) the sign part of \(\mathrm{sq}_*\overline{\mathbb Q}_\ell\), \(\mathrm{sq}(y)=y^2\). Its trace is the Legendre symbol on \(\mathbb F_q^\times\), and \(\mathbb F_q^\times\) has two orbits (squares, non-squares): \(H^1(\mathbb F_q,\mu_2)=\mathbb Z/2\).
- Disconnected \(H=\mathbb Z/2\) on a point: \(\mathrm{Perv}_H(\mathrm{pt})=\operatorname{Rep}(\mathbb Z/2)\) has two simple objects with the same underlying sheaf, so \(\operatorname{For}\) is not full.
4 · Perverse sheaves and IC complexes
Definition · middle perversity
$${}^pD^{\le0}=\big\{K:\dim\operatorname{supp}\mathcal H^{-i}(K)\le i\ \ \forall i\big\},\qquad{}^pD^{\ge0}=\big\{K:\mathbb DK\in{}^pD^{\le0}\big\},$$ $$\mathrm{Perv}(Y)={}^pD^{\le0}\cap{}^pD^{\ge0},\qquad{}^p\mathcal H^i:D^b_c(Y)\to\mathrm{Perv}(Y),\qquad{}^p\tau_{\le i-1}K\to{}^p\tau_{\le i}K\to{}^p\mathcal H^i(K)[-i]\xrightarrow{+1}.$$\(\mathbb D\) is Verdier duality. For \(K\in D^b_{\mathfrak S}(Y)\), with \(i_\alpha:Y_\alpha\hookrightarrow Y\) and \(d_\alpha=\dim Y_\alpha\):
$$K\in\mathrm{Perv}(Y)\iff\mathcal H^j(i_\alpha^*K)=0\ (j>-d_\alpha)\ \text{ and }\ \mathcal H^j(i_\alpha^!K)=0\ (j\lt -d_\alpha)\quad\forall\alpha.$$Equivariantly, \(\mathrm{Perv}_H(Y)=\{K\in D^b_c([Y/H]):\operatorname{For}K\in\mathrm{Perv}(Y)\}\) (Chen–Yi A.7).
Theorem · simple objects (BBD 4.3.1; equivariant: Chen–Yi Example 108)
- Every object of \(\mathrm{Perv}(Y)\) has finite length.
- The simple objects are \(\mathrm{IC}(\bar Z,\xi)=j_{!*}\big(\xi[\dim Z]\big)\), extended by zero from \(\bar Z\), for \(j:Z\hookrightarrow\bar Z\), \(Z\) irreducible, locally closed and smooth, \(\xi\) irreducible lisse on \(Z\).
- If \(H\) has finitely many orbits, the simple objects of \(\mathrm{Perv}_H(Y)\) are the \(\mathrm{IC}(\bar O,\xi)\), with \(O\) an orbit and \(\xi\in\operatorname{Irr}\operatorname{Rep}(\pi_0S_y)\); under I2, only \(\mathrm{IC}(\bar O)=\mathrm{IC}(\bar O,\overline{\mathbb Q}_\ell)\).
On the strata \(Y_\alpha\subseteq\bar Z\smallsetminus Z\) (BBD 1.4.24):
$$\mathrm{IC}(\bar Z,\xi)|_Z=\xi[\dim Z],\qquad\mathcal H^j\big(i_\alpha^*\mathrm{IC}\big)=0\ (j\ge-d_\alpha),\qquad\mathcal H^j\big(i_\alpha^!\mathrm{IC}\big)=0\ (j\le-d_\alpha).$$“Perverse-normalized” is the shift \([\dim Z]\): on a smooth \(Y\) of dimension \(d\), \(\overline{\mathbb Q}_\ell[d]\) is perverse and \(\overline{\mathbb Q}_\ell\) is not if \(d>0\).
Examples. \(Y=\mathrm{pt}\): \(\mathrm{Perv}(\mathrm{pt})\) is finite-dimensional vector spaces, \({}^p\mathcal H^i=\mathcal H^i\), one simple object \(\overline{\mathbb Q}_\ell\). \(Y=\mathbb A^1\): simple objects \(\mathrm{IC}(\mathbb A^1)=\overline{\mathbb Q}_\ell[1]\) and \(\mathrm{IC}(\{0\})=\delta_0\), and two non-split extensions
$$0\to\delta_0\to j_!\overline{\mathbb Q}_\ell[1]\to\overline{\mathbb Q}_\ell[1]\to0,\qquad0\to\overline{\mathbb Q}_\ell[1]\to j_*\overline{\mathbb Q}_\ell[1]\to\delta_0(-1)\to0,\qquad{}^p\mathcal H^1(\overline{\mathbb Q}_\ell)=\overline{\mathbb Q}_\ell[1].$$For \(\mathbb G_m\) acting by scaling, \(\mathrm{Perv}_{\mathbb G_m}(\mathbb A^1)\) has exactly the two simple objects \(\delta_0\), \(\overline{\mathbb Q}_\ell[1]\).
5 · The levels of \(\mathcal C_I\)
\(LX=\varinjlim_NY_N\), \(Y_N=\varprojlim_MY_{N,M}\), \(H_{N,M}=I/I_{N,M}\) connected with \(I_{N,M}\) pro-unipotent, as in (Lev). Put \(D_{N,M}=D^b_c([Y_{N,M}/H_{N,M}])\).
$$\mathcal C_I=\operatorname*{colim}_{(N,M)}D_{N,M}=\bigcup_{N,M}\Phi_{N,M}\big(D_{N,M}\big),\qquad\Phi_{N,M}=i_{N,!}\pi_{N,M}^*[e_M]\ \text{followed by extension by zero along }Y_N\subseteq LX.$$| transition | functor | why fully faithful and t-exact |
|---|---|---|
| \(M\to M'\), same \(N\) | \(p^*[d(M,M')]\), \(p:Y_{N,M'}\to Y_{N,M}\) | affine-space fibres (§2) and unipotent kernel \(I_{N,M}/I_{N,M'}\) (§3) |
| \(N\to N+1\), \(M\gg0\) | \(\iota_*=\iota_!\), \(\iota:Y_{N,M}\hookrightarrow Y_{N+1,M}\) | closed immersion: \(\iota_*\) is t-exact and fully faithful and keeps simple objects (BBD 1.4.16); inflate along the unipotent kernel first |
One integer \(e_M\) per \(M\), independent of \(N\) by (Dict)(a): the shift \(e_M\).
Each object and each Hom of \(\mathcal C_I\) lives at one level, and the transitions are t-exact, so the level t-structures glue (Chen–Yi A.7, (47)):
$${}^p\mathcal C_I^{\le0}=\bigcup\Phi_{N,M}\big({}^pD_{N,M}^{\le0}\big),\qquad{}^p\mathcal C_I^{\ge0}=\bigcup\Phi_{N,M}\big({}^pD_{N,M}^{\ge0}\big),\qquad\mathcal P_I=\bigcup\Phi_{N,M}\,\mathrm{Perv}_{H_{N,M}}(Y_{N,M}).$$Every object of \(\mathcal P_I\) has finite length; its simple objects are \(\mathrm{IC}(P)=\Phi_{N,M}\mathrm{IC}(\bar o)\) for the pieces \(P=\pi_{N,M}^{-1}(o)\), with \(\mathrm{IC}(P)|_P=\overline{\mathbb Q}_\ell[\delta(P)]\), \(\delta(P)=\dim o+e_M\) independent of the level.
Example · \(X=\mathbb A^1\), \(G=\mathbb G_m\). \(Y_{N,M}=t^{-N}k[\![t]\!]/t^Mk[\![t]\!]\simeq\mathbb A^{N+M}\), \(H_{N,M}=(k[\![t]\!]/t^{N+M})^\times\); the orbits are the valuation strata \(\{v(x)=m\}\), \(-N\le m\lt M\), and \(\{0\}\), with connected stabilizers \(1+t^{M-m}k[\![t]\!]\). With \(e_M=-M\), \(\delta(\{v=m\})=-m\) and \(\mathrm{IC}_0\) is the unshifted constant sheaf on \(k[\![t]\!]\). Details: IIa.
6 · Convolution
$$LG\times^ILX=(LG\times LX)/I,\quad h\cdot(g,x)=(gh^{-1},hx),\qquad m:LG\times^ILX\to LX,\ \ [g,x]\mapsto gx.$$For \(\mathcal G\in\mathcal H_{c,I}\) and \(M\in\mathcal C_I\), with \(p:LG\to LG/I\) and \(q:LG\times LX\to LG\times^ILX\) (Chen–Yi §8.1.1, (20)–(21)):
$$q^*\big(\mathcal G\mathbin{\widetilde\boxtimes}M\big)\simeq(p\times\mathrm{id})^*\big(\mathcal G\boxtimes M\big),\qquad\mathcal G\ast_IM=m_!\big(\mathcal G\mathbin{\widetilde\boxtimes}M\big).$$Reduction to finite type
For \(\mathcal G\) supported on \(\mathrm{Fl}_{\le w}\) and \(M\) at level \((N,M)\), (Dict)(d) and the shearing \([g,y]\mapsto(gI,gy)\) turn \(m\) into
$$\mathrm{pr}_2:\ \mathrm{Fl}_{\le w}\times Y_{N',M''}\longrightarrow Y_{N',M''},\qquad\text{proper, so }m_!=m_*,$$ $$\big(\mathcal G\ast_IM\big)_{\bar x}=R\Gamma_c\Big(\{gI\in\mathrm{Fl}_{\le w}:g^{-1}x\in Y_N\},\ \mathcal G_g\otimes M_{g^{-1}x}\Big)\quad\text{(proper base change)}.$$Example · \(G=\mathbb G_m\), \(X=\mathbb A^1\). \(\widetilde W=\mathbb Z\), every \(\mathrm{Fl}_{\le t^n}\) is the point \(t^n\), \(\mathrm{IC}_{t^n}=\delta_{t^n}\):
$$\delta_{t^n}\ast_IM=(t^n\cdot)_*M,\qquad G_{t^n}=\overline{\mathbb Q}_\ell\ \text{on }t^nk[\![t]\!],\qquad\operatorname{tr}G_{t^n}=\mathbf 1_{t^n\mathcal O}=T_{t^n}\ast\varphi_0.$$7 · The decomposition theorem
Varieties (BBD 5.3.8, 5.4.5, 5.4.6, 5.1.14)
Let \(X_0\) be a variety over \(\mathbb F_q\), \(K_0\in D^b_c(X_0,\overline{\mathbb Q}_\ell)\) \(\iota\)-pure, and \(K\) its pullback to \(X=X_0\otimes k\). Then
$$K\simeq\bigoplus_{i\in\mathbb Z}{}^p\mathcal H^i(K)[-i],\qquad{}^p\mathcal H^i(K)\ \text{semisimple, a sum of }\mathrm{IC}(\bar Z,\xi).$$Proper \(f_*\) preserves purity, so \(f_*\mathrm{IC}\) is a sum of shifted IC complexes. Over \(\mathbb C\): BBD 6.2.5, for proper \(f\) and \(K\) semisimple of geometric origin.
Stacks (Sun 2012, Theorem 1.2)
The same splitting of \(Rf_*K\), for \(f_0\) proper with finite diagonal between finite-type Artin stacks over \(\mathbb F_q\) with affine stabilizers and \(K_0\) \(\iota\)-pure; restated on II · Statement. The equivariant case is Bernstein–Lunts 5.3 (as cited in Sun §1).
Summand in the equivariant category. \(S[-i]\) is a direct summand of \(G\) in \(\mathcal C_I\) if there are morphisms in \(D^b_c([Y/H])\)
$$S[-i]\xrightarrow{\ s\ }G\xrightarrow{\ r\ }S[-i],\qquad r\circ s=\mathrm{id}.$$A splitting of \(\operatorname{For}G\) does not give one: \(\operatorname{For}\) is not full on \(D^b_c\) (§3).
Warning · \(\mathbb G_m\) on a point (step 6). With \(c\in H^2(B\mathbb G_m)\), \(c\ne0\):
$$K=\operatorname{Cone}\big(c:\overline{\mathbb Q}_\ell\to\overline{\mathbb Q}_\ell[2]\big),\qquad\operatorname{For}K\simeq\overline{\mathbb Q}_\ell[2]\oplus\overline{\mathbb Q}_\ell[1],\qquad\operatorname{Hom}(\overline{\mathbb Q}_\ell[1],K)=0=\operatorname{Hom}(K,\overline{\mathbb Q}_\ell[2]).$$\(c\) becomes \(0\) on the point, but \(H^1(B\mathbb G_m)=0\) and \(c\cdot:H^0(B\mathbb G_m)\to H^2(B\mathbb G_m)\) is an isomorphism, so no shift of \(\overline{\mathbb Q}_\ell\) is a retract of \(K\). Affine stabilizers are needed too: for \(f:\mathrm{pt}\to BE\), \(E\) an elliptic curve, \(f_*\overline{\mathbb Q}_\ell\) does not split (Drinfeld, in Sun §1).
Example · \(f:\mathbb P^1\to\mathrm{pt}\). \(f_*\overline{\mathbb Q}_\ell\simeq\overline{\mathbb Q}_\ell\oplus\overline{\mathbb Q}_\ell(-1)[-2]\): multiplicity spaces in degrees \(0,2\), with Frobenius \(1\) and \(q\). The splitting is geometric (over \(k\)); Frobenius acts on the multiplicity spaces \(E_{S,i}=\operatorname{Hom}(S,{}^p\mathcal H^i)\), not on a chosen splitting.
8 · Weil structures and traces
Definition
\(X_0\) over \(\mathbb F_q\), \(X=X_0\otimes k\), \(\mathrm{Fr}:X\to X\) the \(q\)-Frobenius. A Weil complex is \((K,\Phi)\), \(\Phi:\mathrm{Fr}^*K\xrightarrow{\sim}K\); every \(K_0\) on \(X_0\) gives one. For \(x\in X_0(\mathbb F_q)=X(k)^{\mathrm{Fr}}\):
$$t_K(x)=\sum_i(-1)^i\operatorname{tr}\big(\Phi_x,\ \mathcal H^i(K)_{\bar x}\big)\in\overline{\mathbb Q}_\ell .$$Sheaf–function dictionary (Laumon 1987, (1.1.1.1)–(1.1.1.4))
$$t_K=t_{K'}+t_{K''}\ \text{for a triangle }K'\to K\to K''\xrightarrow{+1},\qquad t_{K[n]}=(-1)^nt_K,\qquad t_{K_1\otimes K_2}=t_{K_1}t_{K_2},$$ $$t_{f^*L}=t_L\circ f,\qquad t_{f_!K}(y)=\sum_{x\in f^{-1}(y)(\mathbb F_q)}t_K(x)\quad\text{(Grothendieck trace formula)}.$$\(\overline{\mathbb Q}_\ell(-1)\) has \(\Phi=q\). One \(n\) does not determine \(K\); all \(t_{K}\) over \(\mathbb F_{q^n}\) determine its class in \(K_0\) (Laumon, Theorem 1.1.2). Proposition II uses only \(n=1\), through the IC-trace basis.
Example · traces on \(\mathbb A^1(\mathbb F_q)\) (canonical Weil structures; identity on \(\overline{\mathbb Q}_\ell[1]\) for the IC):
| object | on \(\mathbb F_q^\times\) | at \(0\) |
|---|---|---|
| \(\mathrm{IC}(\mathbb A^1)=\overline{\mathbb Q}_\ell[1]\) | \(-1\) | \(-1\) |
| \(\mathrm{IC}(\{0\})=\delta_0\) | \(0\) | \(1\) |
| \(j_*\overline{\mathbb Q}_\ell\) | \(1\) | \(1-q\) |
The last identity is the triangularity of step 5 in its smallest case.
Over \(\mathbb C\). For a complex variety \(Y\), \(D^b_c(Y^{\mathrm{an}},\mathbb C)\) consists of complexes of sheaves on \(Y(\mathbb C)\) whose cohomology is locally constant along an algebraic stratification; §1–7 hold verbatim, with BBD 6.2.5 for decomposition (Sun, Theorem 3.15, states the stack version, proof deferred). At each finite level the Riemann–Hilbert correspondence (Kashiwara 1984; Mebkhout 1984)
$$\mathrm{DR}:D^b_{rh}(\mathcal D_Y)\xrightarrow{\ \sim\ }D^b_c(Y^{\mathrm{an}},\mathbb C),\qquad\mathrm{Mod}_{rh}(\mathcal D_Y)\xrightarrow{\ \sim\ }\mathrm{Perv}(Y^{\mathrm{an}}),$$commutes with \(f_*,f_!,f^*,f^!\) and applies to \(H\)-equivariant objects on \(Y_{N,M}\). There is no Frobenius over \(\mathbb C\), hence no trace; IIc transfers the \(\ell\)-adic conclusion.
Where each is used in Proposition II
| concept | here | used in |
|---|---|---|
| \(D^b_c\), stratifications | §1 | statement, (Lev) |
| smooth pullback, affine fibres | §2 | step 2 (change of \(M\)) |
| proper base change, \(f_!\) | §2 | trace-of-convolution lemma |
| equivariance, connected stabilizers | §3 | step 2, step 5, I2 |
| perverse heart, IC, finite length | §4 | step 2, step 7 |
| gluing the levels | §5 | step 2, step 7 |
| convolution \(\ast_I\) | §6 | trace-of-convolution lemma, step 4 |
| decomposition, equivariant summands | §7 | step 6, step 8, (DT) |
| Weil structures, trace formula | §8 | steps 1, 5, 6, trace-of-convolution lemma |