Decomposition of Exact pfd Persistence Bimodules

We characterize the class of persistence modules indexed over \(\mathbb {R}^2\) that are decomposable into summands whose supports have the shape of a block—i.e. a horizontal band, a vertical band, an upper-right quadrant, or a lower-left quadrant. Assuming the modules are pointwise finite dimensional (pfd), we show that they are decomposable into block summands if and only if they satisfy a certain local property called exactness. Our proof follows the same scheme as the proof of decomposition for pfd persistence modules indexed over \(\mathbb {R}\), yet it departs from it at key stages due to the product order on \(\mathbb {R}^2\) not being a total order, which leaves some important gaps open. These gaps are filled in using more direct arguments. Our work is motivated primarily by the stability theory for zigzags and interlevel-sets persistence modules, in which block-decomposable bimodules play a key part. Our results allow us to drop some of the conditions under which that theory holds, in particular the Morse-type conditions.