Abstract

Let $\mathrm{kq}<!--FUNCTION\; APPLICATION-->$ denote the very effective cover of the motivic hermitian K-theory spectrum. We analyze the ring of cooperations ${\pi }_{\ast \ast }^{ℝ}(\mathrm{kq}<!--FUNCTION\; APPLICATION-->\otimes \mathrm{kq}<!--FUNCTION\; APPLICATION-->)$ in the stable motivic homotopy category $\mathrm{SH}<!--FUNCTION\; APPLICATION-->(ℝ)$, giving a full description in terms of Brown–Gitler comodules. To do this, we decompose the ${E<!--FUNCTION\; APPLICATION-->}_2$-page of the motivic Adams spectral sequence and show that it must collapse. The description of the ${E<!--FUNCTION\; APPLICATION-->}_2$-page is accomplished by a series of algebraic Atiyah–Hirzebruch spectral sequences which converge to the summands of the ${E<!--FUNCTION\; APPLICATION-->}_2$-page. Along the way, we prove a splitting result for the very effective symplectic K-theory ksp over any base field of characteristic not  two.

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