Background Motivation for the problem History of the problem Our main result... ...follows from Finiteness of associated primes of local cohomology modules over Stanley-Reisner rings joint w/ Roberto Barrera and Jeffrey Madsen Ashley K. Wheeler University of Arkansas, Fayetteville comp.uark.edu/~ashleykw 6 January, 2017 Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem History of the problem Our main result... ...follows from Thank-you for the invitation to speak! Local cohomology has applications to Cosmology and String Theory, and it is one of the most active research areas in Commutative Algebra. Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem History of the problem Our main result... ...follows from Thank-you for the invitation to speak! Local cohomology has applications to Cosmology and String Theory, and it is one of the most active research areas in Commutative Algebra. Little is known about local cohomology modules. Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from Local cohomology modules • R = commutative Noetherian ring with 1 • I = ideal in R • M = R -module (may or may not be Noetherian or finitely generated) • j = non-negative integer The j th local cohomology module of M with support in I is defined as the following direct limit of Ext modules: H j Ext j R ( R/I t , M ) . I ( M ) = lim − → t Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from It is the right derived functor of H 0 I (?) : � H 0 Ann M I t I ( M ) := t = { u ∈ M | uI t = 0 for some t } Hom R ( R/I t , M )(= lim Ext 0 R ( R/I t , M )) = lim − → − → t t the global sections of the sheaf ˜ M with support on the closed subscheme Spec R/I ⊂ Spec R . Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from • H 1 I ( M ) measures the obstruction to extending a section of a sheaf to a global section; put X = Spec R and U = X \ Spec( R/I ) I ( M ) → H 0 ( X , ˜ M ) → H 0 ( U , ˜ 0 → H 0 M | U ) → H 1 I ( M ) → 0 Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from • H 1 I ( M ) measures the obstruction to extending a section of a sheaf to a global section; put X = Spec R and U = X \ Spec( R/I ) I ( M ) → H 0 ( X , ˜ M ) → H 0 ( U , ˜ 0 → H 0 M | U ) → H 1 I ( M ) → 0 • If ( R, m ) is a local ring and M is finite generated, then H j m ( M ) can detect regular sequences, compute depth, and reveal the Cohen-Macaulay and Gorenstein properties. Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from In practice, H j I ( M ) is the j th cohomology module of the ˘ Cech complex � � 0 → M → M f i → M f i f j → · · · → M f 1 ··· f s → 0 i i<j where: • f 1 , . . . , f s ∈ R generate I up to radical • given any f ∈ R and any R -module N , N f = N ⊗ R R f , and R f = R [ 1 f ] is the localization of R at f • the maps in the complex are the natural localization maps u �→ u 1 Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from Localization Localizing at a prime ideal gives the stalk at a point in the Zariski topolgy : Spec R = { prime ideals in R } ← topological space V ( J ) = { primes containing the ideal J } ← its closed sets = Spec R/J � � 1 R localized at P is given by R P = R f | f ∈ R \ P and N P = N ⊗ R R P Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from Localization is flat ; as a result, many questions can be reduced to the local case ( local-global principle ). (statement about an R -module N is true) ⇔ (same statement about N P is true for all P ∈ Ass R N ) • Ass R N = assassinator of N , set of all primes associated to N • P is associated to N means P = Ann R ( u ) , the set of ring elements that annihilate some element u ∈ N ; equivalently, P ∈ Ass R N if and only if R/P is isomorphic to a submodule of N . Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Local cohomology modules History of the problem Localization Our main result... ...follows from Localization is flat ; as a result, many questions can be reduced to the local case ( local-global principle ). (statement about an R -module N is true) ⇔ (same statement about N P is true for all P ∈ Ass R N ) • Ass R N = assassinator of N , set of all primes associated to N • P is associated to N means P = Ann R ( u ) , the set of ring elements that annihilate some element u ∈ N ; equivalently, P ∈ Ass R N if and only if R/P is isomorphic to a submodule of N . Local cohomology is the local-global analogue to sheaf cohomology . Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Finiteness of associated primes History of the problem Counterexamples Our main result... Affirmatives ...follows from Finiteness of associated primes Our project is motivated by the following: Question (C. Huneke 1990) Do the local cohomology modules over a Noetherian ring R have finitely many associated primes? Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Finiteness of associated primes History of the problem Counterexamples Our main result... Affirmatives ...follows from Finiteness of associated primes Our project is motivated by the following: Question (C. Huneke 1990) Do the local cohomology modules over a Noetherian ring R have finitely many associated primes? (Answer: No.) Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Finiteness of associated primes History of the problem Counterexamples Our main result... Affirmatives ...follows from Counterexamples • A. Singh 2000: R = Z [ u, v, w, x, y, z ] | Ass R ( H 3 = ⇒ ( x,y,z ) R ) | = ∞ ( ux + vy + wz ) Reason: This local cohomology module has p -torsion for all primes p ∈ Z . Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
Background Motivation for the problem Finiteness of associated primes History of the problem Counterexamples Our main result... Affirmatives ...follows from Counterexamples • A. Singh 2000: R = Z [ u, v, w, x, y, z ] | Ass R ( H 3 = ⇒ ( x,y,z ) R ) | = ∞ ( ux + vy + wz ) Reason: This local cohomology module has p -torsion for all primes p ∈ Z . • M. Katzman 2002: K [ s, t, u, v, x, y ] R = ( K = any field ) ( su 2 x 2 − ( s + t ) uvxy + tv 2 y 2 ) | Ass R ( H 2 = ⇒ ( x,y ) R ) | = ∞ Also shows torsion for infinitely many ring elements. Unlike in Singh’s example, this ring is local. Joint Math Meetings, Atlanta, GA: NAM Granville-Brown-Haynes Session of Presentations by Recent Doctoral Recipients Ashley K. Wheeler Local cohomology over Stanley-Reisner rings
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