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Abstract
We show that the triangulated category of bounded constructible complexes on an algebraic variety $X$ over an algebraically closed field is equivalent to the bounded derived category of the abelian category of constructible sheaves on $X$, extending a theorem of Nori to the case of positive characteristic. The coefficients are allowed to be finite of torsion, a finite extension of $\mathbf{Q}_\ell$, or the ring of integers of such. A series of reductions due to Nori, independent of the characteristic of the ground field, show that the aforementioned equivalence of categories is implied by the property that the (étale or $\ell$-adic) cohomology in degrees strictly greater than zero of a constructible sheaf on an affine space $\mathbf{A}^n$ is effaceable by a monomorphism into another constructible sheaf, which is a theorem of Nori in characteristic zero.
To extend Nori's theorem on the effaceability of the cohohomology of a constructible sheaf on affine space to positive characteristic, we use a variation of a theorem of Achinger to show that the hypotheses needed to proceed with Nori's construction can be secured after twisting our given constructible sheaf $\mathcal F$ by an automorphism of $\mathbf{A}^n$. We show that we can find an automorphism $g$ of $\mathbf{A}^n$ so that if we extend our twisted sheaf $g^*\mathcal F$ by zero to $\mathbf{A}^{n-1}\times\mathbf{P}^1$, the restriction of its singular support to a neighborhood of $\mathbf{A}^{n-1}\times\{\infty\}$ is either empty or is the union of the zero section and the conormal to this divisor. This condition implies that the formation of the direct image of $g^*\mathcal F$ along the open immersion $\mathbf{A}^n\hookrightarrow\mathbf{A}^{n-1}\times\mathbf{P}^1$ commutes with restriction to the fibers of the projection to $\mathbf{A}^{n-1}$. Following Nori, we show that given any constructible sheaf $\mathcal G$ on $\mathbf{A}^n$ that has this property and which is the extension by zero of a lisse sheaf on the complement of a closed subset of $\mathbf{A}^n$ finite and surjective over $\mathbf{A}^{n-1}$, $\mathcal G$ can be embedded in a constructible sheaf $\mathcal H$ so that $R\pi_*\mathcal H=0$, where $\pi:\mathbf{A}^n\to\mathbf{A}^{n-1}$ is the projection. This is enough to make the rest of Nori's argument work in positive characteristic.