Add support for spectral sequences in Cech cohomology

[HechtiDerLachs]

This PR contains my recent work on speeding up computations of spectral sequences in Cech cohomology; see the accompanying preprint on the ArXiv.

I make this a draft for the beginning and clean everything up, including documentation.

TODOs:

• Link the article here once it appears on the ArXiv.
• Add documentation and examples.

[codecov]

Codecov Report

❌ Patch coverage is 84.81625% with 157 lines in your changes missing coverage. Please review.
✅ Project coverage is 84.88%. Comparing base (c09a2e1) to head (48897a8).
⚠️ Report is 193 commits behind head on master.

Files with missing lines	Patch %	Lines
experimental/Schemes/src/DerivedPushforward.jl	75.61%	59 Missing ⚠️
experimental/Schemes/src/SpectralSequences.jl	87.80%	55 Missing ⚠️
...bleAndHyperComplexes/src/Morphisms/induced_homs.jl	50.00%	34 Missing ⚠️
src/Modules/Iterators.jl	90.90%	4 Missing ⚠️
...dHyperComplexes/src/Morphisms/SummandProjection.jl	93.75%	2 Missing ⚠️
...perComplexes/src/Morphisms/new_koszul_complexes.jl	96.55%	1 Missing ⚠️
...ndHyperComplexes/src/Morphisms/strand_morphisms.jl	97.36%	1 Missing ⚠️
...l/DoubleAndHyperComplexes/src/Morphisms/strands.jl	95.23%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #4951      +/-   ##
==========================================
+ Coverage   84.81%   84.88%   +0.07%     
==========================================
  Files         698      707       +9     
  Lines       94290    95347    +1057     
==========================================
+ Hits        79972    80937     +965     
- Misses      14318    14410      +92     

Files with missing lines	Coverage Δ
...tal/DoubleAndHyperComplexes/src/Morphisms/Types.jl	81.42% <100.00%> (+4.23%)	⬆️
...dHyperComplexes/src/Morphisms/shifted_complexes.jl	100.00% <100.00%> (ø)
...perComplexes/src/Morphisms/strand_functionality.jl	100.00% <100.00%> (ø)
...eAndHyperComplexes/src/Morphisms/tensor_product.jl	100.00% <100.00%> (ø)
...AndHyperComplexes/src/Morphisms/total_complexes.jl	100.00% <100.00%> (ø)
...tal/DoubleAndHyperComplexes/src/Morphisms/views.jl	100.00% <100.00%> (ø)
...oubleAndHyperComplexes/src/Objects/Constructors.jl	95.23% <100.00%> (ø)
...DoubleAndHyperComplexes/src/Objects/induced_ENC.jl	96.42% <ø> (ø)
...HyperComplexes/src/Objects/new_koszul_complexes.jl	100.00% <100.00%> (+6.81%)	⬆️
...perComplexes/src/Morphisms/new_koszul_complexes.jl	96.55% <96.55%> (ø)
... and 7 more

... and 25 files with indirect coverage changes

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[jankoboehm]

I am ok with it. Tests failing?

[HechtiDerLachs]

Thanks for having a look. This is still awaiting merge of #4955 and rebase on that.

[HechtiDerLachs]

All failing tests are now because of the non-supported syntax findfirst(bla(x) for x in ...) in julia 1.6. Since there is an ongoing discussion about dropping support for the latter, I will not adjust the code here for the moment.

[HechtiDerLachs]

@benlorenz : Could we have a look together at some point today to see what's actually causing the failure on 1.6? I can not reproduce the error here.

[benlorenz]

1.6 seems to be working after the latest change, 1.10 and 1.12 are failing with some index out of bounds errors in the spectral sequences tests? (https://github.com/oscar-system/Oscar.jl/actions/runs/15959406222/job/45009887648?pr=4951#step:11:3241)

[HechtiDerLachs]

Yes, but I suppose these "index out of bounds" errors are secondary to some findfirst(...) not returning the correct result. Or do you think differently? The same tests pass just fine on my local machine with julia 1.11.1.

Edit: I guess, what I'm asking is that if you, as one of the people who probably see the most failing tests, have any list of buzzwords, like e.g. the findfirst with the unsupported syntax, that we could search for in the contents of this PR, then adjusting these could probably already solve the problem. Of course, I can not ask you to dive into the internals and start debugging with or even for me.

[HechtiDerLachs]

Just a heads up: Yesterday we found that the generic code for computing spectral sequences can not (yet) be applied in the toric setup in its full generality. We have to overhaul that. However, since we are actively working on this together with Friedemann Groh (external), @HereAround, and @emikelsons, I found it in the interest of the Oscar project to get a preliminary version in, so that, for instance, we can continue working on #5032 to eventually resolve the issues. The caveats for the toric code have a temporary safeguard built in by throwing an error, unless the user manually sets a "global exponent vector" to compute with in the toric context. This should leave us ample room for further experiments with Friedemann's code while, at the same time, not giving wrong functionality to the user.

[fingolfin]

This should work also in 1.6? (I didn't test the code)

Suggested change

	while !(ind in rngs[cur_rng_ind])
	cur_rng_ind+=1
	end
	k = cur_rng_ind # findfirst(ind in r for r in rngs)
	# manual way to project (saves allocations)
	vv = FreeModElem(coordinates(v)[rngs[cur_rng_ind]], summands[cur_rng_ind]) #canonical_projection(dom, k)(v)
	k = findfirst(r -> ind in r, rngs)
	# manual way to project (saves allocations)
	vv = FreeModElem(coordinates(v)[rngs[k]], summands[k]) #canonical_projection(dom, k)(v)

[HechtiDerLachs]

Thanks for having a look! However, the point here was not to start looking for k from scratch every time, since the ind come in in an ordered way. Note that cur_rng_ind is another running variable in this loop. This was actually crucial for speed at some point (O(n) vs. O(n^2)).

[HechtiDerLachs]

Just some cleanup of the code.

[HechtiDerLachs]

Thanks for having a look! However, the point here was not to start looking for k from scratch every time, since the ind come in in an ordered way. Note that cur_rng_ind is another running variable in this loop. This was actually crucial for speed at some point (O(n) vs. O(n^2)).

[jankoboehm]

@simonbrandhorst will take another look

[lgoettgens]

This is currently mentioned in the Other changes part of the release notes. Can someone (@HechtiDerLachs @simonbrandhorst @jankoboehm) please add one of the following labels to choose in which section it should appear instead? topic: commutative algebra, topic: algebraic geometry, experimental

[lgoettgens]

This is currently mentioned in the Other changes part of the release notes. Can someone (@HechtiDerLachs @simonbrandhorst @jankoboehm) please add one of the following labels to choose in which section it should appear instead? topic: commutative algebra, topic: algebraic geometry, experimental

I now picked topic: algebraic geometry. If someone disagrees with this, please just change the labels.