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We consider a theory of noncommutative Gr\"obner bases on decreasinglyfiltered algebras whose associated graded algebras are commutative. We transfermany algorithms that use commutative Gr\"obner bases to this context. As animportant application, we implement very efficient algorithms to compute theExt 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 algebraictopology, is calculated up to total degree of 261, including the ring structurein this range.

Noncommutative Gr\"obner Bases and Ext groups; Application to the Steenrod Algebra