Figure 1 - uploaded by Nikola Milicevic
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Sheaf convolution of interval modules : k[a, b) * k[c, d) = k[max(a+ d, b + c),b + d)
Source publication
We conduct a study of real-valued multi-parameter persistence modules as sheaves and cosheaves. Using the recent work on the homological algebra for persistence modules, we define two different convolution operations between derived complexes of persistence modules. We show that one of these operations is canonically isomorphic to the derived tenso...
Contexts in source publication
Context 1
... follows by Proposition 3.7 that (M * N) x is the k-vector subspace of s+t=r M a ⊗ k N b that is the limit of the digram of vector spaces M p ⊗ k N q with p + q ≥ x and the maps in the diagram are given by 1 Mp ⊗ k N q≤q ′ and M p≤p ′ ⊗ k 1 Nq and their compositions. Suppose that b + c ≤ a + d as in Figure 1. Note that M a ⊗ k N c ∼ = k. ...
Context 2
... for any other a ′ = a, c ′ = c such that a ′ + c ′ = a + c we have M a ′ ⊗ k N b ′ = 0. Therefore, we have (M * N) a+c = lim s+t≥a+c M s ⊗ k N t = 0. Same reasoning shows that along any antidiagonal l < a + d we have (M * N) l = lim s+t≥l M s ⊗ k N t = 0. The antidiagonal a + d is the first one where we have a nontrivial limit as all the relevant maps that contribute to the limit computation outside the rectangle in Figure 1 are mapping into trivial vector spaces. Furthermore lim s+t≥a+d (M s ⊗ k N t ) ∼ = (M * N) a+d ∼ = k. ...
