Algebra Seminar | Dharm Veer (Dalhousie University)
Mar 9, 2026
10:30AM to 11:30AM
Date/Time
Date(s) - 09/03/2026
10:30 am - 11:30 am
Speaker: Dharm Veer (Dalhousie University)
Location: Hamilton Hall, Room 312
Title: Complements and complementary homologies
Abstract: In a finite lattice L, two elements are said to be complements if their meet is the minimal element \hat{0} of L and their join is the maximal element \hat{1} of L. For m in L, let C(m) denote the set of all complements of m in L. It is known that if L non-complemented, i.e., there exists an element in L that does not have a complement, then the poset \Bar{L} = (\hat{0}, \hat{1}) is contractible.
One of our main results shows that if the k-th reduced homology of \Bar{L} with coefficients in a field K is nonzero for some k>0, then for every m in \Bar{L}, there exists an m’ in C(m), an n in the interval (\hat{0},m] that is also a complement of m’ such that m’ and n have “complementary homologies”.
We apply above result to derive consequences for the Betti numbers of monomial ideals. This is an ongoing joint project with Sara Faridi, and Volkmar Welker.