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ali
ali
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if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they actually take a solidification of this left adjoint). I have no idea what this means and why cancan't we do the same in the étale site? Perhaps we can do this but the result would not be constructible?

So my question is what is a slice in a site, why does it give a left adjoint and why can't we do this in the usual setting?

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they actually take a solidification of this left adjoint). I have no idea what this means and why can we do the same in the étale site? Perhaps we can do this but the result would not be constructible?

So my question is what is a slice in a site, why does it give a left adjoint and why can't we do this in the usual setting?

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they actually take a solidification of this left adjoint). I have no idea what this means and why can't we do the same in the étale site? Perhaps we can do this but the result would not be constructible?

So my question is what is a slice in a site, why does it give a left adjoint and why can't we do this in the usual setting?

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Matthieu Romagny
Matthieu Romagny
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Relative homology in FargauesFargues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, scholzeScholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. theyThey say the left adjoint exist because $f$ is a slice in $v$-site  (they actually take a solidification of this left adjoint). I have no idea what this means and why can we do the same in etalethe étale site? perhapsPerhaps we can do this but the result whouldwould not be constructible?

So my question is what is a slice in a site, why does it give a left adjoint and why we can't we do this in the usual setting?

Relative homology in Fargaues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. they say the left adjoint exist because $f$ is a slice in $v$-site(they actually take a solidification of this left adjoint). I have no idea what this means and why can do the same in etale site? perhaps we can do this but the result whould not be constructible?

So my question is what is a slice in a site, why it give a left adjoint and why we can't do this in the usual setting?

Relative homology in Fargues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site  (they actually take a solidification of this left adjoint). I have no idea what this means and why can we do the same in the étale site? Perhaps we can do this but the result would not be constructible?

So my question is what is a slice in a site, why does it give a left adjoint and why can't we do this in the usual setting?

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ali
ali
  • 1.1k
  • 7
  • 16

Relative homology in Fargaues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. they say the left adjoint exist because $f$ is a slice in $v$-site(they actually take a solidification of this left adjoint). I have no idea what this means and why can do the same in etale site? perhaps we can do this but the result whould not be constructible?

So my question is what is a slice in a site, why it give a left adjoint and why we can't do this in the usual setting?