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Noncommutative Gr\"obner Bases and Ext groups; Application to the Steenrod Algebra
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Noncommutative Gr\"obner Bases and Ext groups; Application to the Steenrod Algebra
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We consider a theory of noncommutative Gr\"obner bases on decreasingly filtered algebras whose associated graded algebras are commutative. We transfer many algorithms that use commutative Gr\"obner bases to this context. As an important application, we implement very efficient algorithms to compute the Ext groups over the Steenrod algebra $\mathscr{A}$ at the prime $2$. Especially, the cohomology of the Steenrod algebra $Ext_{\mathscr{A}}^{*, *}(\mathbb{F}_2, \mathbb{F}_2)$, which plays an important role in algebraic topology, is calculated up to total degree of 261, including the ring structure in this range.
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Cited by 1 Pith paper
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Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer
At rank 6 and degree 36 the source of Singer's algebraic transfer is 2-dimensional while the target is 1-dimensional, so the transfer cannot be injective and Singer's conjecture is false.
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