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A cover or hypercover replaces one global object by a simplicial family of local pieces and overlaps. Applying cohomology level by level creates one grading from the simplicial direction and another from internal cohomological degree. Totalisation combines them, while a descent spectral sequence reveals how local classes, compatibility maps and higher obstructions assemble into global cohomology.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Name the resolution and the coefficient object
Let U•→X be an augmented simplicial resolution, such as a Cech nerve or hypercover, and let F be a sheaf or complex for which the relevant descent theorem applies. Each level Up carries local cohomology groups Hq(Up,F).
The site, topology, augmentation and coefficient category are part of the statement. A spectral sequence written without its convergence hypotheses is a page of symbols, not yet a global computation.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Build the two directions before totalising
The simplicial index p records objects, overlaps and higher relations. The cohomological index q records internal resolution degree or local cohomology. Alternating pullbacks along face maps provide the horizontal differential.
An injective, flabby or other suitable resolution supplies the vertical differential. Sign conventions are chosen so the total differential squares to zero; changing a convention changes formulas but not the intended derived object when done consistently.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a bidegree-to-total-degree ledger
Suppose a double complex has nonzero entries in p=0,1,2 and q=0,1. Total degree n=p+q collects entries along a diagonal. Thus total degree two receives contributions from (0,2), (1,1) and (2,0), where present.
| Bidegree (p,q) | Local meaning | Total degree p+q | Possible role |
|---|---|---|---|
| (0,1) | Degree-one classes on local pieces | 1 | Local classes |
| (1,0) | Functions or sections on overlaps | 1 | Compatibility correction |
| (1,1) | Degree-one classes on overlaps | 2 | Higher obstruction input |
| (2,0) | Sections on triple relations | 2 | Coherence correction |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Read the first differential as compatibility
On an E1 page of the form E1^{p,q}=Hq(Up,F), the d1 differential is the alternating sum induced by face maps. It compares restrictions of local classes across overlaps and higher relations.
A class surviving d1 is compatible at the first visible level. Later differentials can still detect obstructions that require more than pairwise agreement. Survival to one page is not yet a global class.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change one row and watch the global degree move
If every positive-q local group vanishes, only the q=0 row remains and the spectral sequence can collapse to a Cech-type complex. If overlap cohomology is nonzero, higher rows can feed or kill classes in later pages.
| Local hypothesis | E1 page shape | Likely simplification | Caution |
|---|---|---|---|
| All Up acyclic for F | Only q=0 row | Cech complex may compute global groups | Check cover and convergence |
| Positive-q overlap terms | Several rows | Higher differentials possible | Pairwise gluing is insufficient |
| Bounded first quadrant | Finite diagonals | Direct-sum totalisation is controlled | Extensions can remain |
| Unbounded directions | Infinite diagonals | Products or completion may matter | Convergence needs care |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Locate higher coherence in simplicial degree
Degree zero stores local objects or sections. Degree one stores identifications on overlaps. Degree two records compatibility among those identifications, and higher degrees continue the coherence tower.
For sets or ordinary sheaves, familiar cocycle conditions can stop the visible story early. For complexes, stacks and higher objects, homotopies between compatibilities and homotopies between those homotopies can be essential.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect the established descent routes
The hypercover guide develops matching objects and degree-by-degree local filling. The Cech and Leray guide gives the ordinary cover comparison.
Use the Mayer–Vietoris spectral-sequence guide for page and differential bookkeeping, and the good-cover guide when geometric intersections support a nerve-level homotopy conclusion.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Choose direct sums, products and filtrations deliberately
For a bounded first-quadrant double complex, each total-degree diagonal has finitely many terms, so direct sums and products agree. In unbounded settings they can differ, and completed or product totalisations may encode different limits.
State the filtration used to create the spectral sequence. Filtering by simplicial degree and filtering by internal cohomological degree expose different intermediate pages even when they converge to the same total cohomology.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
A collapsed page does not always give a canonical direct-sum decomposition of the abutment; extension problems can remain. A visible zero differential can reflect degree reasons rather than a deep vanishing theorem.
Pairwise agreement is not full descent when higher coherence is required. Conversely, adding more simplicial levels without a valid cover, matching or descent condition does not create a correct global object.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use a proof and source routine
Record the site, topology, simplicial resolution, augmentation, coefficient object, local resolution, bicomplex, horizontal and vertical signs, totalisation, filtration, E1 or E2 identification, differential bidegree, boundedness, convergence, edge maps, extension issues and comparison theorem.
The Stacks Project Simplicial Spaces chapter gives descent constructions and spectral sequences for simplicial objects and hypercovers. Its Cohomology of Sheaves chapter constructs Cech double complexes and their total complexes.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why are there two indices?
One records simplicial overlap degree and the other records internal cohomological degree.
Does E2=E∞ give the final groups immediately?
It gives the associated graded pieces under the filtration; extensions may still need to be resolved.
Where is higher coherence visible?
In higher simplicial degrees and in the differentials that test whether lower-degree compatibility extends coherently.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a total-degree transfer check
Draw a bicomplex with p=0,1,2 and q=0,1, place four sample groups in the cells and write both differentials. Totalise it, list degrees one to three and then filter first by p and then by q. Compare what each first page makes visible.
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