Calculator Inputs
Enter the largest shift data from a minimal graded free resolution. Each field accepts comma-separated integers, such as 2, 3, 5. The calculator applies reg(M) = max(ti - i).
Example Data Table
This sample uses quotient data. The resolution gives reg(S/I) = 3, so the matching ideal regularity is reg(I) = 4.
| Object | F0 | F1 | F2 | F3 | ti - i Values | Regularity |
|---|---|---|---|---|---|---|
| S/I | 0 | 2, 3 | 4, 5 | 6 | 0, 2, 3, 3 | reg(S/I) = 3 |
| I | 1 | 3, 4 | 5, 6 | 7 | 1, 3, 4, 4 | reg(I) = 4 |
Formula Used
Let S = k[x0, ..., xn] be a standard graded polynomial ring, and let the minimal graded free resolution of a finitely generated graded object be
0 ← M ← ⨁S(-a0,j) ← ⨁S(-a1,j) ← ... ← ⨁S(-ai,j).
The Castelnuovo-Mumford regularity is reg(M) = maxi,j(ai,j - i).
If you only enter the largest shift at each degree, define ti = maxj(ai,j). Then the same invariant becomes reg(M) = maxi(ti - i).
For quotient data, many users also want the ideal relation reg(I) = reg(S/I) + 1 for nonzero homogeneous ideals.
How to Use This Calculator
- Choose whether your data represents a module, ideal, quotient, or sheaf.
- Enter a label for the object and the number of ring variables.
- Add comma-separated graded shifts in each homological degree Fi.
- Submit the form to compute max shifts, ti - i values, and the final regularity.
- Read the result summary above the form and inspect the contribution graph.
- Use the CSV or PDF buttons to export the computed table and summary.
FAQs
1) What does this calculator compute?
It computes Castelnuovo-Mumford regularity from entered graded shifts in a free resolution. The tool evaluates each homological step, forms ti - i, and returns the largest value.
2) Which input format should I use?
Enter integers separated by commas in each Fi field. Example: 2, 3, 5. Blank fields are ignored, and invalid tokens are skipped with a warning.
3) Why does the calculator use the largest shift in each degree?
Regularity depends on the maximum value of ai,j - i. So only the largest shift in each homological degree is needed to determine the final invariant.
4) Can I use quotient data for S/I?
Yes. Choose the quotient option. The calculator reports reg(S/I) directly and also displays reg(I) = reg(S/I) + 1 as a helpful derived value.
5) Does this replace computer algebra software?
No. It evaluates regularity from shifts you already know. Systems like Macaulay2 or Singular are still needed to compute minimal resolutions from generators or ideals.
6) What happens if my resolution is not minimal?
The result may overstate the invariant. Castelnuovo-Mumford regularity is defined from a minimal graded free resolution, so nonminimal cancellations can distort the entered shifts.
7) Why can the critical degree appear late?
Syzygies can acquire larger degree jumps as the resolution progresses. When that late growth outpaces the homological index, the maximum ti - i occurs in a higher degree.
8) What does the graph show?
The graph plots ti - i for each entered homological degree. The tallest bar marks the contribution that determines regularity and highlights where late-degree growth becomes dominant.