[Paper Review] Rationality problem for algebraic tori

Akinari Hoshi, Aiichi Yamasaki|arXiv (Cornell University)|Oct 16, 2012

Algebraic Geometry and Number TheoryMathematics70 references19 citations

TL;DR

This paper provides a complete stably rational classification of algebraic tori of dimensions 4 and 5 over a field $k$, using effective computation of flabby resolutions of $G$-lattices via the GAP computer algebra system. It establishes that there are exactly 487 stably rational, 7 retract rational but not stably rational, and 216 not retract rational tori of dimension 4, and 3051, 25, and 3003 respectively for dimension 5, while also proving that the Krull-Schmidt theorem fails for $G$-lattices of rank 5 and 6.

ABSTRACT

We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\leq { m GL}(n,\mathbb{Z})$ where $n\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.

Motivation & Objective

• To provide a complete classification of stably rational, retract rational, and non-retract rational algebraic tori of dimension 4 and 5 over a field $k$.
• To develop effective algorithms for computing flabby resolutions of $G$-lattices using the GAP computer algebra system.
• To determine the validity of the Krull-Schmidt theorem for $G$-lattices of rank up to 6, and to identify cases where it fails.
• To verify that $H^1(G, [M_G]^{fl}) = 0$ for all Bravais groups $G$ of dimension $n \leq 6$, and for all maximal finite subgroups of $\mathrm{GL}(n,\mathbb{Z})$.
• To apply the developed methods to construct explicit examples of fields that are not retract (stably) rational over $k$.

Proposed method

• The authors use the duality between $G$-lattices and algebraic $k$-tori to translate the rationality problem into a lattice-theoretic problem involving flabby resolutions.
• They implement algorithms in the GAP system to compute the flabby class $[M_G]^{fl}$ of a $G$-lattice $M_G$, and to determine whether it is invertible or zero.
• The method includes three distinct verification techniques (Methods I–III) to confirm whether the flabby class $[M_G]^{fl}$ is trivial, using cohomological and lattice decomposition criteria.
• The classification relies on the computation of $G$-lattices up to rank 6, with a focus on identifying flabby and coflabby lattices and verifying their stable permutability.
• The authors analyze maximal finite subgroups $G \leq \mathrm{GL}(n,\mathbb{Z})$ for $n \leq 6$, and compute their corresponding Bravais groups and associated quadratic forms.
• Tate cohomology computations are used to verify that $H^1(G, [M_G]^{fl}) = 0$ for all such groups, which is a key condition for rationality.

Experimental results

Research questions

• RQ1 How many stably rational, retract rational but not stably rational, and non-retract rational algebraic tori of dimension 4 exist over a field $k$?
• RQ2 How many stably rational, retract rational but not stably rational, and non-retract rational algebraic tori of dimension 5 exist over a field $k$?
• RQ3 Does the Krull-Schmidt theorem hold for $G$-lattices of rank 5 and 6, and if not, how many such lattices are decomposable in multiple ways?
• RQ4 Is $H^1(G, [M_G]^{fl}) = 0$ for all Bravais groups $G$ of dimension $n \leq 6$, and for all maximal finite subgroups of $\mathrm{GL}(n,\mathbb{Z})$?
• RQ5 Can the developed algorithms generate explicit examples of fields that are not retract (stably) rational over $k$?

Key findings

• There are exactly 487 stably rational, 7 not stably but retract rational, and 216 not retract rational algebraic tori of dimension 4 over a field $k$.
• There are exactly 3051 stably rational, 25 not stably but retract rational, and 3003 not retract rational algebraic tori of dimension 5 over a field $k$.
• The Krull-Schmidt theorem fails for $G$-lattices of rank 5 and 6: there are 11 and 131 $G$-lattices of rank 5 and 6, respectively, that decompose into direct sums of different ranks.
• For rank 6, there are exactly 18 $G$-lattices that decompose into the same ranks but with non-isomorphic direct summands, showing a stronger failure of uniqueness.
• The flabby class $[M_G]^{fl}$ is trivial (i.e., $[M_G]^{fl} = 0$) for all $G$-lattices of rank $n \leq 6$ corresponding to Bravais groups, implying $H^1(G, [M_G]^{fl}) = 0$.
• The developed GAP algorithms successfully identify all flabby and coflabby $G$-lattices of rank up to 6, and confirm that all such lattices are stably permutation.