How to construct cup-product in a general site?Sheaves of Principal partsWhen is the K-theory presheaf a sheaf?Cup products and hypercohomologyInterpreting $f^*f_*$Etale site is useful - examples of using the small fppf site?Rational singularities under flat morphismsParabolic cohomology of modular groups and cup-productsUniversal homeomorphism of stacks and etale sites$p$-adic Hodge Theory for rigid spaces, after P. ScholzeIn the definition of big/small étale/fppf/… site, is their covering set really a set?
How to construct cup-product in a general site?
Sheaves of Principal partsWhen is the K-theory presheaf a sheaf?Cup products and hypercohomologyInterpreting $f^*f_*$Etale site is useful - examples of using the small fppf site?Rational singularities under flat morphismsParabolic cohomology of modular groups and cup-productsUniversal homeomorphism of stacks and etale sites$p$-adic Hodge Theory for rigid spaces, after P. ScholzeIn the definition of big/small étale/fppf/… site, is their covering set really a set?
$begingroup$
Let $mathcalC$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $mathcalC$. Do we have cup-product as follows?
$$H^i(X,F)otimes H^j(X,G)rightarrow H^i+j(X,Fotimes_mathbbZG)$$
or even the relative version
$$R^if_*Fotimes R^jf_*Grightarrow R^i+jf_*(Fotimes_mathbbZG)$$
along a map $f:Xrightarrow Y$ of schemes.
For etale cohomology, it seems that Godement resolution is needed in the construction, see [Etale cohomology theory, the paragraph before Prop. 7,4,10] by Fu Lei, see also Appendix B of http://math.stanford.edu/~conrad/Weil2seminar/Notes/L12-13.pdf.
But in a general site (e.g. fppf site), we may not have Godement resolution. I checked the chapter "cohomology of sites" of Stack-Project, unfortunately, cup -product is in the list of topics that "should be discussed in this chapter, and have not yet been written"
It would be very helpful, if someone knows a reference.
ag.algebraic-geometry etale-cohomology grothendieck-topology sites cup-product
asked 8 hours ago
HeerHeer
477314
$endgroup$
add a comment |
$begingroup$
Let $mathcalC$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $mathcalC$. Do we have cup-product as follows?
$$H^i(X,F)otimes H^j(X,G)rightarrow H^i+j(X,Fotimes_mathbbZG)$$
or even the relative version
$$R^if_*Fotimes R^jf_*Grightarrow R^i+jf_*(Fotimes_mathbbZG)$$
along a map $f:Xrightarrow Y$ of schemes.
For etale cohomology, it seems that Godement resolution is needed in the construction, see [Etale cohomology theory, the paragraph before Prop. 7,4,10] by Fu Lei, see also Appendix B of http://math.stanford.edu/~conrad/Weil2seminar/Notes/L12-13.pdf.
But in a general site (e.g. fppf site), we may not have Godement resolution. I checked the chapter "cohomology of sites" of Stack-Project, unfortunately, cup -product is in the list of topics that "should be discussed in this chapter, and have not yet been written"
It would be very helpful, if someone knows a reference.
ag.algebraic-geometry etale-cohomology grothendieck-topology sites cup-product
asked 8 hours ago
HeerHeer
477314
$endgroup$
add a comment |
$begingroup$
Let $mathcalC$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $mathcalC$. Do we have cup-product as follows?
$$H^i(X,F)otimes H^j(X,G)rightarrow H^i+j(X,Fotimes_mathbbZG)$$
or even the relative version
$$R^if_*Fotimes R^jf_*Grightarrow R^i+jf_*(Fotimes_mathbbZG)$$
along a map $f:Xrightarrow Y$ of schemes.
For etale cohomology, it seems that Godement resolution is needed in the construction, see [Etale cohomology theory, the paragraph before Prop. 7,4,10] by Fu Lei, see also Appendix B of http://math.stanford.edu/~conrad/Weil2seminar/Notes/L12-13.pdf.
But in a general site (e.g. fppf site), we may not have Godement resolution. I checked the chapter "cohomology of sites" of Stack-Project, unfortunately, cup -product is in the list of topics that "should be discussed in this chapter, and have not yet been written"
It would be very helpful, if someone knows a reference.
ag.algebraic-geometry etale-cohomology grothendieck-topology sites cup-product
asked 8 hours ago
HeerHeer
477314
$endgroup$
Let $mathcalC$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $mathcalC$. Do we have cup-product as follows?
$$H^i(X,F)otimes H^j(X,G)rightarrow H^i+j(X,Fotimes_mathbbZG)$$
or even the relative version
$$R^if_*Fotimes R^jf_*Grightarrow R^i+jf_*(Fotimes_mathbbZG)$$
along a map $f:Xrightarrow Y$ of schemes.
For etale cohomology, it seems that Godement resolution is needed in the construction, see [Etale cohomology theory, the paragraph before Prop. 7,4,10] by Fu Lei, see also Appendix B of http://math.stanford.edu/~conrad/Weil2seminar/Notes/L12-13.pdf.
But in a general site (e.g. fppf site), we may not have Godement resolution. I checked the chapter "cohomology of sites" of Stack-Project, unfortunately, cup -product is in the list of topics that "should be discussed in this chapter, and have not yet been written"
It would be very helpful, if someone knows a reference.
ag.algebraic-geometry etale-cohomology grothendieck-topology sites cup-product
ag.algebraic-geometry etale-cohomology grothendieck-topology sites cup-product
asked 8 hours ago
HeerHeer
477314
asked 8 hours ago
HeerHeer
477314
edited 8 hours ago
Heer
asked 8 hours ago
HeerHeer
477314
asked 8 hours ago
HeerHeer
477314
asked 8 hours ago
HeerHeer
477314
477314
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes. This is treated in detail in Section 8.4
of Jardine's book “Local homotopy theory”.
See also the introduction to Chapter 8 there
for a historical comment on cup products and Godement resolutions.
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
$endgroup$
$begingroup$
The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right?
$endgroup$
– Heer
5 hours ago
$begingroup$
Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear.
$endgroup$
– Heer
5 hours ago
$begingroup$
@Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions.
$endgroup$
– Dmitri Pavlov
4 hours ago
$begingroup$
As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX).
$endgroup$
– Jesse Silliman
3 hours ago
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
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votes
$begingroup$
Yes. This is treated in detail in Section 8.4
of Jardine's book “Local homotopy theory”.
See also the introduction to Chapter 8 there
for a historical comment on cup products and Godement resolutions.
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
$endgroup$
$begingroup$
The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right?
$endgroup$
– Heer
5 hours ago
$begingroup$
Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear.
$endgroup$
– Heer
5 hours ago
$begingroup$
@Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions.
$endgroup$
– Dmitri Pavlov
4 hours ago
$begingroup$
As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX).
$endgroup$
– Jesse Silliman
3 hours ago
add a comment |
$begingroup$
Yes. This is treated in detail in Section 8.4
of Jardine's book “Local homotopy theory”.
See also the introduction to Chapter 8 there
for a historical comment on cup products and Godement resolutions.
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
$endgroup$
$begingroup$
The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right?
$endgroup$
– Heer
5 hours ago
$begingroup$
Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear.
$endgroup$
– Heer
5 hours ago
$begingroup$
@Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions.
$endgroup$
– Dmitri Pavlov
4 hours ago
$begingroup$
As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX).
$endgroup$
– Jesse Silliman
3 hours ago
add a comment |
$begingroup$
Yes. This is treated in detail in Section 8.4
of Jardine's book “Local homotopy theory”.
See also the introduction to Chapter 8 there
for a historical comment on cup products and Godement resolutions.
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
$endgroup$
Yes. This is treated in detail in Section 8.4
of Jardine's book “Local homotopy theory”.
See also the introduction to Chapter 8 there
for a historical comment on cup products and Godement resolutions.
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
answered 6 hours ago
Dmitri PavlovDmitri Pavlov
13.9k43486
13.9k43486
$begingroup$
The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right?
$endgroup$
– Heer
5 hours ago
$begingroup$
Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear.
$endgroup$
– Heer
5 hours ago
$begingroup$
@Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions.
$endgroup$
– Dmitri Pavlov
4 hours ago
$begingroup$
As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX).
$endgroup$
– Jesse Silliman
3 hours ago
add a comment |
$begingroup$
The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right?
$endgroup$
– Heer
5 hours ago
$begingroup$
Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear.
$endgroup$
– Heer
5 hours ago
$begingroup$
@Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions.
$endgroup$
– Dmitri Pavlov
4 hours ago
$begingroup$
As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX).
$endgroup$
– Jesse Silliman
3 hours ago
$begingroup$
The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right?
$endgroup$
– Heer
5 hours ago
$begingroup$
The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right?
$endgroup$
– Heer
5 hours ago
$begingroup$
Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear.
$endgroup$
– Heer
5 hours ago
$begingroup$
Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear.
$endgroup$
– Heer
5 hours ago
$begingroup$
@Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions.
$endgroup$
– Dmitri Pavlov
4 hours ago
$begingroup$
@Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions.
$endgroup$
– Dmitri Pavlov
4 hours ago
$begingroup$
As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX).
$endgroup$
– Jesse Silliman
3 hours ago
$begingroup$
As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX).
$endgroup$
– Jesse Silliman
3 hours ago
add a comment |
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