Lifting low-dimensional local systems

Let k be a field of characteristic p>0 . Denote by 𝐖_r(k) the ring of truntacted Witt vectors of length r ≥ 2 , built out of k . In this text, we consider the following question, depending on a given profinite group G . Q ( G ): Does every (continuous) representation G⟶GL_d(k) lift to a representation G⟶GL_d(𝐖_r(k)) ? We work in the class of cyclotomic pairs (Definition 4.3 ), first introduced in De Clercq and Florence ( https://arxiv.org/abs/2009.11130 , 2018) under the name “smooth profinite groups”. Using Grothendieck-Hilbert’ theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over ℤ[1/p] , smooth curves over algebraically closed fields, and affine schemes over 𝔽_p . In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to Q ( G ), for a cyclotomic profinite group G : the answer is positive, when d=2 and r=2 . When d=2 and r=∞ , we show that any 2-dimensional representation of G stably lifts to a representation over 𝐖(k) : see Theorem 6.1 . When p=2 and k=𝔽_2 , we prove the same results, up to dimension d=4 . We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3 ).