On Kernels of Differentiation on Truncated Polynomial Rings V B Mnukhin School of Computing, Information and Mathematical Sciences, University of the South Pacific, Suva, Fiji Islands, e-mail: mnukhin
[email protected]
In 1942 W. Mayer [2] defined homology groups Hk,i based on a boundary operator ∂ such that ∂ p = 0, (p a prime), instead of the usual ∂ p = 0. For coefficients in a field Fp of characteristic p the Mayers construction is just the ordinary operator of differentiation ∂ := ∂/∂x1 +· · ·+∂/∂xn on the Stanley-Reisner ring Fp [∆] = M0 ⊕M1 ⊕. . . of a simplicial complex ∆, and Hk,i := (Ker ∂ i ∩ Mk )/∂ p−i (Mk+p−i ), (0 < i < p). When ∆ is an (n − 1)simplex, Fp [∆] is the polynomial ring R = Fp [x1 , . . . , xn ]. In 1949 E. Spanier [4] proved that most of the Mayers homology groups are zero and the reminder are isomorphic to ordinary homology groups. In particular, Mayers homology is homotopy invariant. The Mayers construction can be naturally generalized to truncated polynomial rings R(q−1) = Fp [x1 , . . . , xn ]/(xq1 , . . . , xqn ) (and so also truncated Stanley-Reisner rings), pro(q−1) (∞) ducing homology groups Hk,i . Abusing notation, we say that Hk,i is the Mayers homology and so is equivalent to the simplicial homology. (1) Homology Hk,i is known in topological combinatorics as the modular homology [3]. It is not homotopy invariant, but since ∂ commutes with group actions, could be use to study group actions on non-admissable complexes. Generalized homologies with nilpotent differential operators are used in quantum groups theory and theoretical physics, see [1]. (∞) Since the Mayers homology Hk,i (and so the ordinary homology) can be considered (q−1)
as the limit case of Hk,i
(q−1)
when q → ∞, we may expect a “good behaviour” of Hk,i (q−1)
for large q. In the talk we discuss properties of Hk,i and consider branching rules for i (q−1) Ker ∂ on the truncated rings R to explain the results.
References [1] M Dubois-Violette, dN = 0: Generalized homology, K-Theory, 14(1998), 104–118. [2] W Mayer, A new homology theory I, II, Ann. Math. 43(1942), 370–380, and 594–605. [3] V B Mnukhin and I J Siemons, The modular homology of inclusion maps and group actions, J. Comb. Theory, A74(1996), 287–300. [4] E H Spanier, The Mayer homology theory, Bull. Amer. Math. Soc. 55(1949), 103–112.
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