Affine morphisms have one-sided perverse exactness

Let $U$ be smooth irreducible affine of dimension $i\ge 1$ and let $L$ be a nonzero finite-rank local system. By [F3] choose $i$ functions with finite nonempty common zero set $S$. Local complex orientation and local constancy identify $H_S^{2i}(U,L)$ with ${⨁}_{s\in S}{L}_s$, which is nonzero, while [F1] gives $H^{2i}(U,L)=0$. The support long exact sequence therefore gives $H^{2i-1}(U\setminus S,L)\mathrm{\ne }0$. Cover $U\setminus S$ by the $i$ principal opens $D(f_j)$. Every finite intersection is affine, so [F1] bounds its ordinary cohomology by $i$; the Čech index is at most $i-1$. In total degree $2i-1$ only the term $H^i(D(f_1\cdots f_i),L)$ can contribute, and nonvanishing of the total group forces it to be nonzero. For $i=0$ the detector is simply the nonzero stalk on an isolated point.

The forward direction of the affine-open characterization is [F1], applied to each restriction. Conversely suppose all positive $H^q(V,K)$ vanish. Write $U_i=Y\setminus F_{i-1}$ and argue downwards in $i$ that the stalk degrees on $U_i$ obey the upper perverse support bound. Suppose the assertion is established on all larger-dimensional layers. On $U_i$ let $A={\tau }_{\le -i}K$ and $Q={\tau }_{>-i}K$. On higher-dimensional layers the induction hypothesis already gives the upper perverse stalk bound, preserved by ordinary truncation; on $W_i$ truncation makes the degree at most $-i$. These are exactly the upper perverse bounds for $A$ on $U_i$, without any claim that $U_i$ has dimension $i$. Thus on each affine $V\subset U_i$, [F1] and the triangle $A\to K\to Q$ give $H^q(V,Q)=0$ for $q>0$. All ordinary cohomology sheaves of $Q$ are supported on the closed smooth layer $W_i\subset U_i$ and are local systems there.

If $Q$ is nonzero, let $b>-i$ be its largest nonzero cohomological degree and let $L={\mathcal{H}}^b(Q){\mathrm{\mid }}_{W_i}$ be nonzero on a connected component. The finitely many smooth irreducible components of $W_i$ are disjoint and closed in $U_i$; remove the components where $L$ is zero and all but one nonzero component, and choose an affine open $V_0\subset U_i$ in the resulting open, meeting the retained component. Then $V_0\cap W_i$ is smooth affine and $L$ is a nonzero local system on it. Step 1.1 supplies a principal open of it with nonzero $H^i(L)$. Since $W_i\cap V_0$ is closed in the affine $V_0$, the defining function lifts from its coordinate-ring quotient to a function on $V_0$; localizing the latter gives an affine $V\subset U_i$ with $H^i(V\cap W_i,L)\mathrm{\ne }0$. The closed embedding $e:V\cap W_i\hookrightarrow V$ has exact pushforward, and $Q{\mathrm{\mid }}_V=e_{\ast }{e}^{\ast }Q{\mathrm{\mid }}_V$ because all its cohomology is supported there. Thus $H^a(V,{\mathcal{H}}^t(Q))=H^a(V\cap W_i,{\mathcal{H}}^t(Q){\mathrm{\mid }}_{W_i})$. In the bounded spectral sequence $H^a(V\cap W_i,{\mathcal{H}}^t(Q){\mathrm{\mid }}_{W_i})\Rightarrow H^{a+t}(V,Q)$, [F1] gives $a\le i$ and maximality of $b$ gives $t\le b$. The nonzero term $(i,b)$ has no incoming differential, since those would start at a larger $t$, and no outgoing differential, since those would end at a larger $a$. It is the sole term in its total degree $i+b>0$, contradicting step 1.2. Hence $Q=0$, completing the descending dimension induction and the converse.

Let $P\in {}^p{D}_c^{\le 0}(X)$. For every affine Zariski open $V\subset Y$ its inverse image under the affine morphism $u$ is affine, and derived composition with [F1] gives $H^q(V,R{u}_{\ast }P)=H^q(u^{-1}V,P)=0$ for $q>0$; the characterization proved in step 2.1 therefore puts $R{u}_{\ast }P$ in the upper perverse aisle. For $P$ in the lower aisle, apply this to $DP$ and dualize by [F2] to obtain the claimed lower-aisle preservation by $R{u}_{!}$. All zero and empty cases follow from the same zero triangles.