Special Time and Location: Martin M104, 3:30-4:45
Power edge ideals
Michael Cowen — Clemson University
Every electrical power system can be modeled by a graph G whose vertices represent buses and whose edges represent power lines. A phasor measurement unit (PMU) is a device that can be placed at a bus to observe the voltage at that bus as well as the current and its phase through all incident power lines. The problem of monitoring the entire electric power system using the fewest number of PMUs is closely related to the well-known vertex covering and dominating set problems in graph theory.
In this talk, we will give an overview of the PMU placement problem and its connections to commutative ring theory. We define the edge ideal I^P_G of a graph G vertices in a polynomial ring R=k[X_1, . . . , X_n] and we describe some algebraic properties of the quotient R/I^P_G. In particular, we will show that, when we restrict to trees, the Cohen- Macaulay property is implied by the unmixed property and implies the complete intersection property. We will also give examples to show that for non-trees, these implications can fail.
Toward DG-Algebra Resolutions for Fiber Products
Hugh Geller — Clemson University
In 2017, Nasseh and Sather-Wagstaff proved that if M and N are finitely generated modules over a non-trivial fiber product R such that Tor^i_R(M,N)=0 for i»0, then M or N has finite projective dimension. Their proof uses Moore’s explicit free resolution of the second syzygy of M. Recently, Avramov, Iyengar, Nasseh, and Sather-Wagstaff have proved that the same conclusion holds for other classes of rings using differential graded algebra techniques. Thus, it is natural to ask whether one can explicitly describe DG algebra resolutions of fiber products. We will present progress on this question.
Gorenstein Injective Modules under Flat Base Change
James Gossell — Clemson University
For A→B a flat homomorphism of commutative noetherian rings and M an injective A-module, we can calculate the injective dimension of M⊗_AB as a B-module by calculating the injective dimensions of the fibers F(p)=B_p/pB_p for each p∈Ass(M) by the formula id_B(M⊗B) = sup_{p∈Ass(M)} id_{F(p)}F(p) (Foxby, 1975). This formula can be used to recover the fact that injective modules remain injective under certain flat base changes, such as a localization. We work towards a generalization of Foxby’s Theorem to calculate the Gorenstein injective dimension of the base change of a Gorenstein injective module.