Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety.

Let $Y \subset X$ be a (possibly) singular closed subvariety of codimension $c$.

I am interested in hypothesis on the singularities of $Y$ that could allow to have a Thom-Gysin isomorphism in étale cohomology $$H^{*-2c}_{\text{ét}}(Y; \mathbf Q_\ell(-c)) \simeq H^*_{\text{ét}}(X,X-Y; \mathbf Q_\ell) $$

As mentionned in the answer to the question The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base, if $Y$ is the zero section in the total space of a line bundle (and more generally of a vector bundle) over a singular base, there is such an isomorphism.

Moreover, in general, when $Y$ is a complete intersection, there is a well defined morphism:

$$H^{*-2c}_{\text{ét}}(Y; \mathbf Q_\ell(-c)) \to H^*_{\text{ét}}(X,X-Y; \mathbf Q_\ell).$$ (I got this from the article of Joël Riou in the book "Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents", Exposé XVI, section 2. It is available at http://arxiv.org/abs/1207.3648)

In such generality, it is certainly not an isomorphism, but are there known conditions on $Y$ that would ensure the previous map to be an isomorphism?



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