The kernel of the Gysin homomorphism for positive characteristic
Let $k$ be an uncountable algebraically closed field of positive characteristic and let $S$ be a smooth projective connected surface over $k$. We extend the theorem on the Gysin kernel from [20, Theorem 5.1] to also be true over $k$, where it was proved over $\mathbb{C}$. This is done by showing that almost all results still hold true over $k$ via the same argument or by using \'{e}tale base arguments and then using a lift with the Comparison theorems [16, Theorems 21.1 & 20.5] and Tate's Conjecture for finitely generated fields [27] and [31] as needed.
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