Decomposition of Exact pfd Persistence Bimodules

We characterize the class of persistence modules indexed over R² 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 R, yet it departs from it at key stages due to the product order on R² 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.