Spectral Sequence Calculator

Calculator Input

Page r

Bidegree p

Bidegree q

Dimension of E_r^p,q

Incoming image rank

Outgoing differential rank

Pages to plot

The calculator uses the standard cohomological differential shape d_r: E_r^p,q → E_r^p+r,q-r+1.

Plotly Graph

Example Data Table

Page r	p	q	Dimension	Incoming Rank	Outgoing Rank	Estimated Next Page
2	1	3	6	1	2	3
3	2	2	5	1	1	3
4	0	5	7	2	3	2
2	4	1	4	0	1	3

Formula Used

This page models a cohomological spectral sequence. It tracks where a class lives, where its differential lands, and how much survives to the next page.

d_r : E_r^{p,q} → E_r^{p+r, q-r+1}

Incoming source into (p,q): E_r^{p-r, q+r-1}

Kernel estimate = dim(E_r^{p,q}) − rank(outgoing d_r)

Next-page estimate = dim(E_r^{p,q}) − incoming rank − outgoing rank

Total degree n = p + q

The next-page estimate assumes independent incoming and outgoing contributions inside a finite-dimensional model. It is useful for teaching, checking examples, and planning computations.

How to Use This Calculator

Enter the page number r.
Enter the bidegree coordinates p and q.
Set the dimension of the selected group.
Enter the incoming image rank.
Enter the outgoing differential rank.
Choose how many pages the graph should show.
Click the calculate button.
Review the result panel above the form.
Use CSV or PDF export if needed.

Frequently Asked Questions

1) What does this calculator actually compute?

It computes bidegree movement, incoming and outgoing differential locations, kernel size, total degree, and an estimated surviving dimension on the next page.

2) Is this a full symbolic spectral sequence solver?

No. It is an educational rank-based calculator. It helps you inspect coordinates and page estimates quickly.

3) Why do incoming and outgoing ranks matter?

Outgoing rank measures classes killed by the differential. Incoming rank measures boundaries entering the current group. Together they affect what survives on the next page.

4) Which differential convention is used?

The page uses the cohomological convention. The differential goes from (p,q) to (p+r, q−r+1).

5) Can I use negative p or q values?

Yes. Many examples allow negative indices. The calculator accepts them directly and still tracks the mapping.

6) Why is the next-page value only an estimate?

It assumes the incoming image and outgoing killed part behave independently. Real spectral sequence structure can be subtler.

7) What does the graph show?

The graph shows the source class, incoming source, outgoing target, and a page-by-page dimension profile for quick visual interpretation.

8) Can I export the computed results?

Yes. Use the CSV button for spreadsheet work. Use the PDF button for printable documentation.