In this article, we use cohomological techniques to obtain an algebraic version of Toda’s theorem in complexity theory valid over algebraically closed fields of arbitrary characteristic. This result follows from a general ‘connectivity’ result in cohomology. More precisely, given a closed subvariety \(X \subset {\mathbb {P}}^{n}\) over an algebraically closed field k, and denoting by \(\mathrm{J}^{[p]}(X) = \mathrm{J}(X,\mathrm{J}(X,\ldots ,\mathrm{J}(X,X)\cdots )\) the p-fold iterated join of X with itself, we prove that the restriction homomorphism on (singular or \(\ell \)-adic etale) cohomology \(\mathrm{H}^{i}({\mathbb {P}}^{N}) \rightarrow \mathrm{H}^{i}(\mathrm{J}^{[p]}(X))\), with \(N = (p+1)(n+1) - 1\), is an isomorphism for \(0 \le i < p\), and injective for \(i=p\). We also prove this result in the more general setting of relative joins for X over a base scheme S, where S is of finite type over k. We give several other applications of this connectivity result including a cohomological version of classical quantifier elimination in the first order theory of algebraically closed fields of arbitrary characteristic, and to obtain effective bounds on the Betti numbers of images of projective varieties under projection maps.

在本文中，我们使用同调技术获得复杂性理论中Toda定理的代数形式，该形式对任意特征的代数闭合域均有效。该结果来自同调学中一般的“连接性”结果。更精确地说，给定一个封闭的子变量\（X \ subset {\ mathbb {P}} ^ {n} \）在代数封闭字段k上，并用\（\ mathrm {J} ^ {[p]}（X ）= \ mathrm {J}（X，\ mathrm {J}（X，\ ldots，\ mathrm {J}（X，X）\ cdots）\）X与其自身的p倍迭代连接，我们证明（奇异或\（\ ell \）- adic etale）同调上的限制同态\（\ mathrm {H} ^ {i}（{\ mathbb {P}} ^ {N}）\ rightarrow \ mathrm {H} ^ {i}（\ mathrm {J} ^ {[p]}（X） ）\）（\（N =（p + 1）（n + 1）-1 \））是\（0 \ le i <p \）的同构，并且是\（i = p \）的内射形。我们还证明了在基本方案S上X的相对联接的更一般设置中的结果，其中S是k上的有限类型。我们给出了这种连通性结果的其他一些应用，包括在任意特征的代数闭合域的一阶理论中经典量词消除的同调形式，以及在投影映射下获得投影变型图像的贝蒂数的有效边界。