We construct spectrum level splittings of BPGL<1> ^ BPGL<1> and BPGL<0> ^ BPGL<0> in terms of motivic Adams covers. These calculations are performed at all primes p and over R, C, and all finite fields Fq where char(Fq) ≠ p. This lifts classical results in spite of the current lack of motivic Brown--Gitler spectra, and opens the door to the study of the BPGL<1>-motivic Adams spectral sequence. 

We compute the ring of cooperations for the very effective Hermitian K-theory spectrum kq over finite fields Fq of where char(Fq)≠2. To do this, we use the motivic Adams spectral sequence and show that the differentials are determined by the motivic cohomology of Fq. We extend our results to compute the E1-page of the kq-resolution.

We compute the ring of cooperations for the the very effective Hermitian K-theory spectrum kq over the real numbers. This result involves a delicate series of spectral sequences and is a first step towards the kq-resolution. Additionally, we prove a splitting result for the very effective symplectic K-theory spectrum ksp using a motivic integral Brown-Gitler spectrum.

We examine Li’s double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double determinantal varieties are smooth. We use this framework to give a straighforward formula for their dimension. Finally, we use the smallest nontrivial toric double determinantal variety to provide some empirical evidence concerning an open problem in local algebra.