Abstract
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.
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Notes
Here \({\mathbb {P}}^n\) is the usual n-dimensional projective space over k. Sometimes we denote this by \({\mathbb {P}}^n_k\) in order to make the base field explicit.
Note that we define \(P(X)(T) = \sum \nolimits _{i \ge 0} \dim _{\mathbb {Q}_\ell } {{\,\mathrm{H}\,}}^i(X,\mathbb {Q}_\ell )T^i\) (2.31).
We refer the reader who is unfamiliar with model theory terminology to the book [4] for all the necessary background that will be required in this article.
Here, \({\mathbb {A}}^n\) denotes the usual n-dimensional affine space over k.
Note that in loc. cit. it is shown that \(R\Gamma (X,R) \otimes ^{{\mathbb {L}}} R\Gamma (Y,R) \cong R\Gamma (X \times Y,R)\). This gives rise to the standard \(\mathrm{Tor}\) spectral sequence. If \(R ={\mathbb {Z}}/\ell ^n{\mathbb {Z}}\), then all \(\mathrm{Tor}^i\)’s vanish for \(i > 1\), and the spectral sequence gives the Kunneth short exact sequence.
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Saugata Basu would like to acknowledge support from the National Science Foundation award CCF-1618981, DMS-1620271, and CCF-1910441. Deepam Patel would like to acknowledge support from the National Science Foundation award DMS-1502296.
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Basu, S., Patel, D. Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda’s theorem. Sel. Math. New Ser. 26, 71 (2020). https://doi.org/10.1007/s00029-020-00596-0
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DOI: https://doi.org/10.1007/s00029-020-00596-0