Cohomological dimension of a noetherian separated scheme

Every quasi-coherent sheaf \(F\) on the noetherian scheme is the filtered union of its coherent subsheaves, by P-AGH2515EXTCOH, part (d). These subsheaves form a directed system because the sum of two is again coherent. Cohomology of abelian sheaves on a noetherian space commutes with filtered direct limits [@Har10a, Proposition III.2.9]. Consequently \[ H^i(X,F)\cong\varinjlim_\lambda H^i(X,F_\lambda). \] If the groups on the right vanish for every coherent \(F_\lambda\) in every degree \(i>n\), so do the groups on the left. The converse is immediate because coherent sheaves are quasi-coherent. Thus the same candidate bounds, and hence the same least bound, result from either class of sheaves.

Let \(L\) be the very ample invertible sheaf of a quasi-projective immersion into \(\PP_k^N\). It is ample by [@Har10a, Theorem II.7.6]. For any coherent \(F\), some twist \(F\otimes L^m\) is generated by finitely many global sections: ampleness gives global generation, and quasi-compactness supplies finitely many generators. Untwisting gives a surjection \((L^{-m})^{\oplus a}\twoheadrightarrow F\) from a locally free coherent sheaf. Its kernel is coherent, so the construction may be repeated as many times as needed. This is the quotient-by-twists construction of [@Har10a, II.5.18].

Suppose that \(H^j(X,E)=0\) for all locally free coherent \(E\) and all \(j>n\). For a coherent \(F\), form exact sequences \[ 0\longrightarrow K_{a+1}\longrightarrow E_a\longrightarrow K_a\longrightarrow0, \qquad K_0=F, \] with all \(E_a\) locally free and all \(K_a\) coherent. Fix \(i>n\). Their long exact sequences give successive isomorphisms \[ H^i(X,F)\cong H^{i+1}(X,K_1)\cong\cdots\cong H^{i+\ell}(X,K_\ell), \] because both adjacent cohomology groups of each \(E_a\) vanish in these degrees. Choose \(\ell\) with \(i+\ell>\dim X\), which is finite for a quasi-projective scheme. Grothendieck vanishing [@Har10a, Theorem III.2.7] makes the last group zero. Thus every coherent \(F\) satisfies the same bound, and step 1 extends it to all quasi-coherent sheaves. Only a finite initial segment of a locally free resolution was used; termination of that resolution is not assumed.

Finite intersections of affine opens of a separated scheme are affine. The affine-cover Čech comparison therefore computes \(H^i(X,F)\) for every quasi-coherent \(F\) by a complex with terms only in degrees \(0,\ldots,r\) [@Har10a, Theorem III.4.5]. Its higher cohomology vanishes, yielding the stated bound, as in D-PTIW0. This argument does not use Grothendieck vanishing.

Let \(J\subseteq k[z_0,\ldots,z_N]\) be the homogeneous ideal of \(Z\), and let \(\mathfrak p_1,\ldots,\mathfrak p_s\) be the homogeneous primes of the closures of the irreducible components of \(T\). These primes are incomparable: inclusion between their closures would give inclusion between the corresponding components after intersection with the open set \(\overline X\setminus Z\). Each component meets that open set, so its closure is not contained in \(Z\). Thus \(J\) is not contained in any \(\mathfrak p_a\).

Choose homogeneous \(u_a\in J\setminus\mathfrak p_a\) of positive degree. This is possible by choosing a homogeneous component outside the homogeneous prime and, if necessary, multiplying by a coordinate not in that prime. For every \(b\ne a\), choose a homogeneous \(v_{ab}\in\mathfrak p_b\setminus\mathfrak p_a\). The polynomial \[ w_a=u_a\prod_{b\ne a}v_{ab} \] belongs to \(J\), avoids \(\mathfrak p_a\) by primality, and belongs to every other \(\mathfrak p_b\). Its degree is positive. Let \(D\) be a common positive multiple of the degrees of all the \(w_a\), and define \[ g=\sum_{a=1}^s w_a^{D/\deg w_a}. \] This is homogeneous and belongs to \(J\). Modulo \(\mathfrak p_a\), only its \(a\)th summand survives and is nonzero, so \(g\notin\mathfrak p_a\) for every \(a\). This proves the assertion over any field; no choice of a scalar outside a finite union of proper vector subspaces was used.

Realize \(X\) as an open subscheme of a closed projective subscheme \(\overline X\subseteq\PP_k^N\) and let \(Z=\overline X\setminus X\) with its reduced closed structure. Starting from \(T_0=|X|\), apply step 4 successively to choose positive-degree homogeneous polynomials \(g_j\) vanishing on \(Z\) and not vanishing identically on any component of the nonempty closed subset \(T_j\subseteq X\). Set \[ T_{j+1}=T_j\cap V_+(g_j). \] This intersects each component of \(T_j\) in a proper closed subset, so \(\dim T_{j+1}\le\dim T_j-1\) whenever \(T_{j+1}\) is nonempty. The dimension inequality follows by appending the component to every chain of irreducible closed subsets in its proper closed subset. After at most \(r+1\) choices, the residual set is therefore empty.

For each chosen polynomial, put \[ W_j=\overline X\cap D_+(g_j). \] Because \(g_j\) vanishes on \(Z\), this is an open subset of \(X\). It is affine: \(D_+(g_j)\) is an affine distinguished open of projective space, and \(\overline X\cap D_+(g_j)\) is closed in it. The residual-set construction says that these \(W_j\) cover \(X\). There are at most \(r+1\) of them; repeating a member gives a cover with exactly \(r+1\) indexed members when necessary. Step 3 now gives \(\operatorname{cd}(X)\le r=\dim X\) without using (2.7). No reducedness or irreducibility hypothesis was imposed on \(X\); only the underlying closed subsets were used in constructing the cover.

For \(r\ge1\), choose homogeneous equations \(f_1,\ldots,f_r\) whose projective zero sets intersect set-theoretically in \(Y\), as in the definition of a set-theoretic complete intersection. Then \[ \PP_k^n\setminus Y=\bigcup_{a=1}^r D_+(f_a). \] Each member is affine, and this open subscheme is noetherian and separated. Step 3 with \(r\) affine opens gives the required bound \(r-1\). If \(r=0\), the empty family of defining hypersurfaces has intersection all of \(\PP_k^n\), so the complement is empty and its cohomological dimension is \(-\infty\) by convention. The same inequality holds in this case.

Steps <1>1–<1>3 prove (a)–(c), steps <1>4–<1>5 construct the affine cover and prove (d), and step 6 proves (e).