Rees algebras of square-free monomial ideals Louiza Fouli New - PowerPoint PPT Presentation

Rees algebras of square-free monomial ideals Louiza Fouli New Mexico State University University of Nebraska AMS Central Section Meeting October 15, 2011

Rees algebras This is joint work with Kuei-Nuan Lin.

Rees algebras This is joint work with Kuei-Nuan Lin. Let R be a Noetherian ring and I an ideal.

Rees algebras This is joint work with Kuei-Nuan Lin. Let R be a Noetherian ring and I an ideal. i ≥ 0 I i t i ⊂ R [ t ] . The Rees algebra of I is R ( I ) = R [ It ] = ⊕

Rees algebras This is joint work with Kuei-Nuan Lin. Let R be a Noetherian ring and I an ideal. i ≥ 0 I i t i ⊂ R [ t ] . The Rees algebra of I is R ( I ) = R [ It ] = ⊕ The Rees algebra is the algebraic realization of the blowup of a variety along a subvariety.

Rees algebras This is joint work with Kuei-Nuan Lin. Let R be a Noetherian ring and I an ideal. i ≥ 0 I i t i ⊂ R [ t ] . The Rees algebra of I is R ( I ) = R [ It ] = ⊕ The Rees algebra is the algebraic realization of the blowup of a variety along a subvariety. It is an important tool in the birational study of algebraic varieties, particularly in the study of desingularization.

Rees algebras This is joint work with Kuei-Nuan Lin. Let R be a Noetherian ring and I an ideal. i ≥ 0 I i t i ⊂ R [ t ] . The Rees algebra of I is R ( I ) = R [ It ] = ⊕ The Rees algebra is the algebraic realization of the blowup of a variety along a subvariety. It is an important tool in the birational study of algebraic varieties, particularly in the study of desingularization. The Rees algebra facilitates the study of the asymptotic behavior of I.

Rees algebras This is joint work with Kuei-Nuan Lin. Let R be a Noetherian ring and I an ideal. i ≥ 0 I i t i ⊂ R [ t ] . The Rees algebra of I is R ( I ) = R [ It ] = ⊕ The Rees algebra is the algebraic realization of the blowup of a variety along a subvariety. It is an important tool in the birational study of algebraic varieties, particularly in the study of desingularization. The Rees algebra facilitates the study of the asymptotic behavior of I. i ≥ 0 I i t i = R ( I ) . Integral Closure: R ( I ) = ⊕

Rees algebras This is joint work with Kuei-Nuan Lin. Let R be a Noetherian ring and I an ideal. i ≥ 0 I i t i ⊂ R [ t ] . The Rees algebra of I is R ( I ) = R [ It ] = ⊕ The Rees algebra is the algebraic realization of the blowup of a variety along a subvariety. It is an important tool in the birational study of algebraic varieties, particularly in the study of desingularization. The Rees algebra facilitates the study of the asymptotic behavior of I. i ≥ 0 I i t i = R ( I ) . So I = [ R ( I )] 1 . Integral Closure: R ( I ) = ⊕

An alternate description We will consider the following construction for the Rees algebra

An alternate description We will consider the following construction for the Rees algebra Let I = ( f 1 , . . . , f n ) , S = R [ T 1 , . . ., T n ] , where T i are indeterminates.

An alternate description We will consider the following construction for the Rees algebra Let I = ( f 1 , . . . , f n ) , S = R [ T 1 , . . ., T n ] , where T i are indeterminates. There is a natural map φ : S − → R ( I ) that sends T i to f i t.

An alternate description We will consider the following construction for the Rees algebra Let I = ( f 1 , . . . , f n ) , S = R [ T 1 , . . ., T n ] , where T i are indeterminates. There is a natural map φ : S − → R ( I ) that sends T i to f i t. Then R ( I ) ≃ S / ker φ .

An alternate description We will consider the following construction for the Rees algebra Let I = ( f 1 , . . . , f n ) , S = R [ T 1 , . . ., T n ] , where T i are indeterminates. There is a natural map φ : S − → R ( I ) that sends T i to f i t. Then R ( I ) ≃ S / ker φ . Let J = ker φ .

An alternate description We will consider the following construction for the Rees algebra Let I = ( f 1 , . . . , f n ) , S = R [ T 1 , . . ., T n ] , where T i are indeterminates. There is a natural map φ : S − → R ( I ) that sends T i to f i t. Then R ( I ) ≃ S / ker φ . Let J = ker φ . � ∞ Then J = J i is a graded ideal. i = 1

An alternate description We will consider the following construction for the Rees algebra Let I = ( f 1 , . . . , f n ) , S = R [ T 1 , . . ., T n ] , where T i are indeterminates. There is a natural map φ : S − → R ( I ) that sends T i to f i t. Then R ( I ) ≃ S / ker φ . Let J = ker φ . � ∞ Then J = J i is a graded ideal. A minimal i = 1 generating set of J is often referred to as the defining equations of the Rees algebra.

Example in degree 2 Example Let R = k [ x 1 , x 2 , x 3 , x 4 ] and I = ( x 1 x 2 , x 2 x 3 , x 3 x 4 , x 1 x 4 ) .

Example in degree 2 Example Let R = k [ x 1 , x 2 , x 3 , x 4 ] and I = ( x 1 x 2 , x 2 x 3 , x 3 x 4 , x 1 x 4 ) . Then R ( I ) ≃ R [ T 1 , T 2 , T 3 , T 4 ] / J and J is minimally generated by

Example in degree 2 Example Let R = k [ x 1 , x 2 , x 3 , x 4 ] and I = ( x 1 x 2 , x 2 x 3 , x 3 x 4 , x 1 x 4 ) . Then R ( I ) ≃ R [ T 1 , T 2 , T 3 , T 4 ] / J and J is minimally generated by x 3 T 1 − x 1 T 2 , x 4 T 4 − x 2 T 3 , x 1 T 3 − x 3 T 4 , x 4 T 1 − x 2 T 4 , T 1 T 3 − T 2 T 4 .

Example in degree 2 Example Let R = k [ x 1 , x 2 , x 3 , x 4 ] and I = ( x 1 x 2 , x 2 x 3 , x 3 x 4 , x 1 x 4 ) . Then R ( I ) ≃ R [ T 1 , T 2 , T 3 , T 4 ] / J and J is minimally generated by x 3 T 1 − x 1 T 2 , x 4 T 4 − x 2 T 3 , x 1 T 3 − x 3 T 4 , x 4 T 1 − x 2 T 4 , T 1 T 3 − T 2 T 4 . I is the edge ideal of a graph:

b b b b Example in degree 2 Example Let R = k [ x 1 , x 2 , x 3 , x 4 ] and I = ( x 1 x 2 , x 2 x 3 , x 3 x 4 , x 1 x 4 ) . Then R ( I ) ≃ R [ T 1 , T 2 , T 3 , T 4 ] / J and J is minimally generated by x 3 T 1 − x 1 T 2 , x 4 T 4 − x 2 T 3 , x 1 T 3 − x 3 T 4 , x 4 T 1 − x 2 T 4 , T 1 T 3 − T 2 T 4 . I is the edge ideal of a graph: x 1 x 2 x 4 x 3

b b b b Example in degree 2 Example Let R = k [ x 1 , x 2 , x 3 , x 4 ] and I = ( x 1 x 2 , x 2 x 3 , x 3 x 4 , x 1 x 4 ) . Then R ( I ) ≃ R [ T 1 , T 2 , T 3 , T 4 ] / J and J is minimally generated by x 3 T 1 − x 1 T 2 , x 4 T 4 − x 2 T 3 , x 1 T 3 − x 3 T 4 , x 4 T 1 − x 2 T 4 , T 1 T 3 − T 2 T 4 . I is the edge ideal of a graph: x 1 x 2 x 4 x 3 Notice that f 1 f 3 = f 2 f 4 = x 1 x 2 x 3 x 4 and that the degree 2 relation “comes” from the “even closed walk”, in this case the square.

The degree 2 case Theorem (Villarreal) Let k be a field and let I = ( f 1 , . . . , f n ) ⊂ R = k [ x 1 , . . . , x d ] be a square-free monomial ideal generated in degree 2 .

The degree 2 case Theorem (Villarreal) Let k be a field and let I = ( f 1 , . . . , f n ) ⊂ R = k [ x 1 , . . . , x d ] be a square-free monomial ideal generated in degree 2 . Let S = R [ T 1 , . . . , T n ] and R ( I ) ≃ S / J.

The degree 2 case Theorem (Villarreal) Let k be a field and let I = ( f 1 , . . . , f n ) ⊂ R = k [ x 1 , . . . , x d ] be a square-free monomial ideal generated in degree 2 . Let S = R [ T 1 , . . . , T n ] and R ( I ) ≃ S / J. Let I s denote the set of all non-decreasing sequences of integers α = ( i 1 , . . ., i s ) of length s.

The degree 2 case Theorem (Villarreal) Let k be a field and let I = ( f 1 , . . . , f n ) ⊂ R = k [ x 1 , . . . , x d ] be a square-free monomial ideal generated in degree 2 . Let S = R [ T 1 , . . . , T n ] and R ( I ) ≃ S / J. Let I s denote the set of all non-decreasing sequences of integers α = ( i 1 , . . ., i s ) of length s. Let f α = f i 1 · · · f i s ∈ I and T α = T i 1 · · · T i s ∈ S.

The degree 2 case Theorem (Villarreal) Let k be a field and let I = ( f 1 , . . . , f n ) ⊂ R = k [ x 1 , . . . , x d ] be a square-free monomial ideal generated in degree 2 . Let S = R [ T 1 , . . . , T n ] and R ( I ) ≃ S / J. Let I s denote the set of all non-decreasing sequences of integers α = ( i 1 , . . ., i s ) of length s. Let f α = f i 1 · · · f i s ∈ I and T α = T i 1 · · · T i s ∈ S. � ∞ Then J = SJ 1 + S ( P s ) , where i = 2 = { T α − T β | f α = f β , for some α, β ∈ I s } P s

The degree 2 case Theorem (Villarreal) Let k be a field and let I = ( f 1 , . . . , f n ) ⊂ R = k [ x 1 , . . . , x d ] be a square-free monomial ideal generated in degree 2 . Let S = R [ T 1 , . . . , T n ] and R ( I ) ≃ S / J. Let I s denote the set of all non-decreasing sequences of integers α = ( i 1 , . . ., i s ) of length s. Let f α = f i 1 · · · f i s ∈ I and T α = T i 1 · · · T i s ∈ S. � ∞ Then J = SJ 1 + S ( P s ) , where i = 2 = { T α − T β | f α = f β , for some α, β ∈ I s } P s = { T α − T β | α, β form an even closed walk }

A generating set for the defining ideal Theorem (D. Taylor) Let R be a polynomial ring over a field and I be a monomial ideal.

A generating set for the defining ideal Theorem (D. Taylor) Let R be a polynomial ring over a field and I be a monomial ideal. Let f 1 , . . . , f n be a minimal monomial generating set of I and let S = R [ T 1 , . . . , T n ] .

A generating set for the defining ideal Theorem (D. Taylor) Let R be a polynomial ring over a field and I be a monomial ideal. Let f 1 , . . . , f n be a minimal monomial generating set of I and let S = R [ T 1 , . . . , T n ] . Let R ( I ) ≃ S / J.