Model short and long sequences with optional zeros. See exactness status at every object instantly. Export clean CSV and PDF outputs for quick submission.
| Sequence | Object dimensions | Map ranks | Expected |
|---|---|---|---|
| 0 → A → B → C → 0 | (0, 2, 3, 1, 0) | (0, 2, 1, 0) | Exact by dimension tests; short exact at A,B,C. |
| A → B → C | (2, 4, 2) | (2, 2) | Exact at B if rank(f1)=dim(B)−rank(f2)=2. |
Use the “Load example” button to prefill the first example.
This tool helps you design and validate finite exact sequences of vector spaces using dimension and rank information. It supports chains from three to five objects, optionally fixing the first and last object as 0 to model standard constructions. The checker focuses on whether images and kernels can match in dimension at every position, giving a fast feasibility test before you invest time building explicit maps in a proof or computation. It enables quick scenario comparisons across runs.
For each object you provide a label and a non‑negative dimension. For each map you provide a rank that must satisfy rank(f) ≤ min(dim(domain), dim(codomain)). When “compute missing ranks” is enabled, the builder tries to fill blank ranks using the interior exactness rule rank(f_i) = dim(V_i) − rank(f_{i+1}) together with endpoint requirements for injectivity and surjectivity in common 0‑anchored sequences.
After submission, the results panel summarizes the whole chain and reports exactness at every object. For interior objects it compares dim(im left)=rank(f_i) with dim(ker right)=dim(V_i)−rank(f_{i+1}). For the first object it tests nullity(f1)=0, and for the last object it tests rank(f_last)=dim(V_last). A map table additionally reports nullity, and flags injective and surjective behavior directly from ranks.
Matrix mode adds an optional computational layer. If you paste matrices, the calculator computes ranks using Gaussian elimination, and it can test whether consecutive compositions are zero by multiplying matrices and checking for the zero matrix. This composition check is important because exact sequences are complexes, so the feasibility of “image equals kernel” is meaningful only when f_{i+1}∘f_i=0. Shape warnings highlight when matrix dimensions do not match the chosen object sizes.
Professional reporting is built in through export controls. The CSV export bundles both result tables so you can archive runs, share them with collaborators, or import them into analysis notes. The PDF export produces a clean, printable report that includes generation time, the exactness table, and the map summary in one document. These exports make it easy to document alternative rank choices when exploring short exact sequences or longer resolutions.
It checks dimension‑level exactness conditions using ranks and dimensions, reporting whether dim(im)=dim(ker) at each object and whether endpoint maps are injective or surjective.
No. It shows the rank and dimension data are compatible with exactness. You still need maps with zero consecutive compositions and the required image–kernel equality, which matrix mode can partially test.
Start with object dimensions, then keep each rank within min(domain, codomain). For interior exactness, enforce rank(f_i)=dim(V_i)−rank(f_{i+1}) and adjust until every object shows “Exact”.
If some rank boxes are blank, the tool tries to infer them from neighboring ranks and dimensions using the interior exactness equation, plus endpoint injective/surjective requirements in common 0‑anchored sequences.
Use it when you have explicit linear maps. Paste matrices to compute ranks automatically and, when consecutive matrices are provided, to test whether compositions are zero for the chain.
CSV includes both result tables for spreadsheets and archives. PDF produces a printable report with generation time, the exactness‑by‑object table, and the map summary for documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.