|
Using combinatorial structures to obtain resolutions of monomial
ideals is an idea that traces back to Diana Taylor’s thesis, where a simplex associated to the generators of a monomial ideal was used to construct a free resolution
of the ideal. This concept has been expanded over the years, with various authors determining conditions under which simplicial or cellular complexes can be
associated to monomial ideals in ways that produce a free resolution.
In a research project initiated at a BIRS workshop “Women in Commutative
Algebra” in Fall 2019, the authors studied simplicial and cellular structures that
produced resolutions of powers of monomial ideals. The optimal structure to use
depends upon the structure of the monomial ideal. This talk will focus on powers
of square-free monomial ideals of projective dimension one. Faridi and Hersey
proved that a monomial ideal has projective dimension one if and only if there
is an associated tree (one dimensional acyclic simplicial complex) that supports
a free resolution of the ideal. The talk will show how, for each power r > 1, to
use the tree associated to a square-free monomial ideal I of projective dimension
one to produce a cellular complex that supports a free resolution of I
r
. Moreover,
each of these resolutions will be minimal resolutions. These cellular resolutions
can also be viewed as strands of the resolution of the Rees algebra of I. This talk
will contain joint work with Susan Cooper, Sabine El Khoury, Sara Faridi, Sarah
Mayes-Tang, Liana Sega, and Sandra Spiroff.
https://zoom.us/join
Meeting ID: 908 611 6889
Passcode: 362880
|
|