Videos

Recent Developments in Commutative Algebra: "Polarization and Gorenstein liaison"

Presenter
April 19, 2024
Keywords:
  • Commutative rings
  • modules
  • ideals
  • mixed characteristic
  • Frobenius powers
  • test ideals
  • tight closure
  • perfectoid methods
  • singularities
  • birational algebraic geometry
  • multiplier ideals
  • symbolic powers
  • syzygies
  • free resolutions
  • homological methods
  • derived categories
  • polynomials
  • monomial ideals
  • toric varieties
  • Schubert varieties
  • combinatorial commutative algebra
  • equivariant ideals
  • maximal Cohen-Macaulay modules
  • applications of representation theory
  • twisted commutative algebras
  • D-modules
  • local cohomology
  • computational commutative algebra
MSC:
  • 05Exx - Algebraic combinatorics
  • 11Sxx - Algebraic number theory: local fields
  • 11Txx - Finite fields and commutative rings (number-theoretic aspects)
  • 13-XX - Commutative algebra
  • 14-XX - Algebraic geometry
  • 16Exx - Homological methods in associative algebras {For commutative rings
  • see \newline 13Dxx
  • for general categories
  • see 18Gxx}
  • 18Gxx - Homological algebra in category theory
  • derived categories and functors [See also 13Dxx
  • 16Exx
  • 20Jxx
  • 55Nxx
  • 55Uxx
  • 57Txx]
  • 19Axx - Grothendieck groups and $K_0$K_0 [See also 13D15
  • 18F30]
  • 19Lxx - Topological $K$K-theory [See also 55N15
  • 55R50
  • 55S25]
  • 20Jxx - Connections of group theory with homological algebra and category theory
Abstract
The major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then indeed it can be after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely-related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. After reviewing the key aspects of liaison and polarization, we will describe conditions that allow for the lifting of a basic double G-link induced from a vertex decomposition of the Stanley--Reisner complex of the polarization of an ideal $I$ to a basic double G-link of $I$ itself. We will introduce a notion of polarization of a Gr\"obner basis of an arbitrary homogeneous ideal and give a result for geometric vertex decomposition and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage. Time permitting, we will give an application to vertex decompositions of Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals and of artinian monomial ideals, extending a recent result of Fl{\o}ystad and Mafi. The work described in this talk is joint with Sara Faridi, Jenna Rajchgot and Alexandra Seceleanu.