On the Frobenius Complexity of Stanley-Reisner Rings Irina Ilioaea Georgia State University, Atlanta, GA, USA June, 2020 Irina Ilioaea Early Commutative Algebra Researchers June, 2020 1 / 24
Stanley-Reisner Rings and Simplicial Complexes Non-faces : { x 1 , x 3 } , { x 1 , x 4 } Facets: { x 1 , x 2 } , { x 2 , x 3 , x 4 } Figure: Simplicial complex ∆ Irina Ilioaea Early Commutative Algebra Researchers June, 2020 2 / 24
Stanley-Reisner Rings and Simplicial Complexes Non-faces : { x 1 , x 3 } , { x 1 , x 4 } Facets: { x 1 , x 2 } , { x 2 , x 3 , x 4 } Figure: Simplicial complex ∆ The Stanley-Reisner ring associated to our simplicial complex ∆ is given by k [∆] = k [ x 1 , x 2 , x 3 , x 4 ] ( x 1 x 3 , x 1 x 4 ) . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 2 / 24
Frobenius Operators Let ( R , m , k ) a local ring of characteristic p . Let F : R → R be the Frobenius map, that is F ( r ) = r p . We have that ( a + b ) p = a p + b p , ( a · b ) p = a p · b p , for all a , b ∈ R . Therefore, the Frobenius map is a ring homomorphism. Let F e : R → R be the e -th iteration of the Frobenius map, that is F e ( r ) = r q , where q = p e , e ∈ N . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 3 / 24
Frobenius Operators Let ( R , m , k ) a local ring of characteristic p . Let F : R → R be the Frobenius map, that is F ( r ) = r p . We have that ( a + b ) p = a p + b p , ( a · b ) p = a p · b p , for all a , b ∈ R . Therefore, the Frobenius map is a ring homomorphism. Let F e : R → R be the e -th iteration of the Frobenius map, that is F e ( r ) = r q , where q = p e , e ∈ N . For any e ≥ 0, we let R ( e ) be the R -algebra defined as follows: as a ring R ( e ) equals R while the R -algebra structure is defined by rs = r q s , for all r ∈ R , s ∈ R ( e ) . In the same way, starting with an R -module M, we can define a new R -module M ( e ) . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 3 / 24
Frobenius Operators Let E R := E R ( k ) denote the injective hull of the residue field k . An e -th Frobenius operator(action) on E R is an additive map φ : E R → E R such that φ ( rz ) = r q φ ( z ), for all r ∈ R and z ∈ E R . The collection of e -th Frobenius operators(actions) on E R is an R -module, denoted by F e ( E R ). Irina Ilioaea Early Commutative Algebra Researchers June, 2020 4 / 24
Frobenius Operators Let E R := E R ( k ) denote the injective hull of the residue field k . An e -th Frobenius operator(action) on E R is an additive map φ : E R → E R such that φ ( rz ) = r q φ ( z ), for all r ∈ R and z ∈ E R . The collection of e -th Frobenius operators(actions) on E R is an R -module, denoted by F e ( E R ). Definition (The Frobenius Algebra of Operators) The algebra of the Frobenius operators on E R is defined by � F e ( E R ). F ( E R ) = e ≥ 0 This is a N -graded noncommutative ring under composition of maps and due to Matlis duality, its zero degree component is R . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 4 / 24
The Frobenius Algebra of Operators If ( R , m , k ) is d -dimensional, local and Gorenstein ring, E R ∼ = H d m ( R ) and F ( H d m ( R )) is generated by the canonical action F on H d m ( R ) . In this case, the Frobenius complexity of the ring R is −∞ . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 5 / 24
The Frobenius Algebra of Operators If ( R , m , k ) is d -dimensional, local and Gorenstein ring, E R ∼ = H d m ( R ) and F ( H d m ( R )) is generated by the canonical action F on H d m ( R ) . In this case, the Frobenius complexity of the ring R is −∞ . Question(Lyubeznik, Smith - 1999) Is F ( E R ) always finitely generated as a ring over R ? Irina Ilioaea Early Commutative Algebra Researchers June, 2020 5 / 24
The Frobenius Algebra of Operators If ( R , m , k ) is d -dimensional, local and Gorenstein ring, E R ∼ = H d m ( R ) and F ( H d m ( R )) is generated by the canonical action F on H d m ( R ) . In this case, the Frobenius complexity of the ring R is −∞ . Question(Lyubeznik, Smith - 1999) Is F ( E R ) always finitely generated as a ring over R ? In 2009, Katzman gave an example of a ring R such that F ( E R ) is not finitely generated as a ring over R . The ring is R = k [ x , y , z ] / ( xy , xz ) . The Frobenius complexity of the ring R equals 0. Irina Ilioaea Early Commutative Algebra Researchers June, 2020 5 / 24
The Frobenius Algebra of Operators Katzman raised the finite generation question for the determinantal ring of 2 x 2 minors in a 2 x 3 matrix. Enescu and Yao showed that the Frobenius complexity of determinantal rings can be positive, irrational and depends upon the characteristic. Irina Ilioaea Early Commutative Algebra Researchers June, 2020 6 / 24
The Frobenius Algebra of Operators Katzman raised the finite generation question for the determinantal ring of 2 x 2 minors in a 2 x 3 matrix. Enescu and Yao showed that the Frobenius complexity of determinantal rings can be positive, irrational and depends upon the characteristic. In 2012, ` Alvarez, Boix and Zarzuela completely described what happens in the case of Stanley-Reisner rings: when finite generation occurs, then F ( E R ) is principally generated. Irina Ilioaea Early Commutative Algebra Researchers June, 2020 6 / 24
The Frobenius Algebra of Operators Problem Find ways to measure the generation of F ( E R ) over R . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 7 / 24
The Frobenius Algebra of Operators Problem Find ways to measure the generation of F ( E R ) over R . Enescu and Yao were motivated by the finite generation question to introduce a new invariant of a local ring of prime characteristic, called the Frobenius complexity. Irina Ilioaea Early Commutative Algebra Researchers June, 2020 7 / 24
The Frobenius Algebra of Operators Problem Find ways to measure the generation of F ( E R ) over R . Enescu and Yao were motivated by the finite generation question to introduce a new invariant of a local ring of prime characteristic, called the Frobenius complexity. In the case when this Frobenius algebra is infinitely generated over R , they used the complexity sequence { c e } e ≥ 0 in order to describe how far it is from being finitely generated. Irina Ilioaea Early Commutative Algebra Researchers June, 2020 7 / 24
Frobenius Complexity Let G e := G e ( F ( E R )) be the subring of F ( E R ) generated by elements of degree less or equal to e . Note that G e − 1 ⊆ G e , for all e . Moreover, ( G e ) i = F ( E R ) i , for all 0 ≤ i ≤ e and ( G e − 1 ) e ⊆ F ( E R ) e . We will denote the minimal number of homogeneous generators of G e as a subring of F ( E R ) over F ( E R ) 0 = R by k e . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 8 / 24
Frobenius Complexity Let G e := G e ( F ( E R )) be the subring of F ( E R ) generated by elements of degree less or equal to e . Note that G e − 1 ⊆ G e , for all e . Moreover, ( G e ) i = F ( E R ) i , for all 0 ≤ i ≤ e and ( G e − 1 ) e ⊆ F ( E R ) e . We will denote the minimal number of homogeneous generators of G e as a subring of F ( E R ) over F ( E R ) 0 = R by k e . Proposition [Enescu, Yao] The minimal number of generators of the R -module F ( E R ) e equals ( G e − 1 ) e k e − k e − 1 , for all e . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 8 / 24
Frobenius Complexity Definition [Enescu, Yao] The sequence { k e } e is called the growth sequence for F ( E R ). The complexity sequence is given by { c e = k e − k e − 1 } e . The complexity of F ( E R ) is cx ( F ( E R )) = inf { n > 0 : c e = O ( n e ) } . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 9 / 24
Frobenius Complexity Definition [Enescu, Yao] The Frobenius complexity of the ring R is defined by cx F ( R ) = log p ( cx ( F ( E R ))) . It is easy to note that F ( E R ) is finitely generated as a ring over R if and only if cx ( F ( E R ))) = 0 if and only if { c e } e ≥ 0 is eventually zero. In this case, cx F ( R ) = −∞ . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 10 / 24
Frobenius Complexity Definition [Enescu, Yao] The Frobenius complexity of the ring R is defined by cx F ( R ) = log p ( cx ( F ( E R ))) . It is easy to note that F ( E R ) is finitely generated as a ring over R if and only if cx ( F ( E R ))) = 0 if and only if { c e } e ≥ 0 is eventually zero. In this case, cx F ( R ) = −∞ . If the sequence { c e } e ≥ 0 is bounded by above, but not eventually zero, cx ( F ( E R ))) = 1 . Hence, cx F ( R ) = 0 . Irina Ilioaea Early Commutative Algebra Researchers June, 2020 10 / 24
Frobenius Complexity Let k be a field of characteristic p , S = k [[ x 1 , . . . , x n ]] and q = p e , for e ≥ 0. Let I ≤ S be an ideal in S and R = S / I . We denote I [ q ] = ( i q : i ∈ I ). Proposition(Fedder) There exists an isomorphism of R -modules: = I [ q ] : S I F e ( E R ) ∼ . I [ q ] Irina Ilioaea Early Commutative Algebra Researchers June, 2020 11 / 24
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