History and Background Isolated Singularities Higher Codimension The Vanishing of a Higher Codimension Analogue of Hochster’s Theta Invariant Sandra Spiroff* University of Mississippi AMS Central Sectional Meeting, Lincoln, NE Fall 2011 *Joint with W. Frank Moore, Greg Piepmeyer, Mark E. Walker S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Meeting Announcement AMS Southeastern Sectional Meeting The University of Mississippi Oxford, MS March 1-3, 2013 Average temp: highs 56-65, lows around 40 S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Outline History and Background Isolated Singularities Higher Codimension S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension If R is a hypersurface — that is, a quotient of a regular ring T by a single element — and M and N are finitely generated R -modules, then from the long exact sequence · · ·→ Tor T n ( M , N ) → Tor R n ( M , N ) → Tor R n − 2 ( M , N ) → Tor T n − 1 ( M , N ) →· · · one obtains j ( M , N ) ∼ Tor R = Tor R for j ≫ 0. j +2 ( M , N ) Example R = C [ [ X , Y , U , V ] ] / ( XU − YV ) S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Let M be the class of all finitely generated R -modules and let N ⊂ M be those finitely generated R -modules such that N p has finite projective dimension over R p for every p � = m . Definition (Hochster, 1981) Define θ : M × N → Z by θ ( M , N ) = length(Tor R 2 j ( M , N )) − length(Tor R 2 j +1 ( M , N )) where j ≫ 0 . S. Spiroff, University of Mississippi An Invariant for Complete Intersections
� � � � � � � � � History and Background Isolated Singularities Higher Codimension Example R = C [ [ X , Y , U , V ] ] / ( XU − YV ), M = R / ( x , y ), N = R / ( u , v ) Consider a resolution of M = R / ( x , y ): 2 3 2 3 − y 4 u 4 x y h i 5 5 − v x v u x y . . . R 2 R 2 0 M R Tensor with N = R / ( u , v ): 2 3 4 0 Y h i 5 0 X X Y . . . ] 2 0 C [ [ X , Y ] ] C [ [ X , Y ] C even ( M , N ) ∼ Tor R = C and Tor R odd ( M , N ) = 0, hence θ ( M , N ) = 1. S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Theorem (Hochster, 1981) If length( M ⊗ R N ) < ∞ , then θ ( M , N ) = 0 if and only if dim M + dim N ≤ dim R. Example XU − YV ), R = C [ [ X , Y , U , V ] ] / ( M = R / ( x , y ), N = R / ( u , v ) • R / ( x , y ) ⊗ R R / ( u , v ) has finite length; • dim( R / ( x , y ) ) + dim( R / ( u , v ) ) > dim R ; • θ ( ) � = 0. R / ( x , y ) , R / ( u , v ) S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Isolated Singularities If the ring R is an isolated singularity, then θ is defined on any pair of finitely generated R -modules. Theorem (Dao, 2006) Let ( R , m ) be as above, with the additional assumptions that R is an isolated singularity and contains a field. If dim M + dim N ≤ dim R, then θ ( M , N ) = 0 . Corollary (Dao, 2006) Let ( R , m ) be as above, with the additional assumption that R is an isolated singularity. Then θ vanishes when dim R = 4 and R contains a field. S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Conjecture (Dao, 2006) Let R be an isolated hypersurface singularity. Assume that dim R is even and R contains a field. Then θ ( M , N ) vanishes for all pairs of finitely generated R-modules M and N. Theorem (Moore, Piepmeyer, S., Walker, 2009) Let k be a field and let R = k[ x 0 , . . . , x n ] / ( f ( x 0 , . . . , x n ) ) , where deg x i = 1 for all i and f is a homogeneous polynomial of degree d and m = ( x 0 , . . . , x n ) is the only non-regular prime of R. If n is even, then θ vanishes; i.e., for every pair of finitely generated modules M and N, length(Tor R 2 j ( M , N )) − length(Tor R 2 j +1 ( M , N )) = 0 , j ≫ 0 . S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Definition A pair of modules ( M , N ) is rigid if for any integer i ≥ 0, Tor R i ( M , N ) = 0 implies Tor R j ( M , N ) = 0 for all j ≥ i . A module M is rigid if for all N the pair ( M , N ) is rigid. Corollary (Moore, Piepmeyer, S., Walker, 2009) Let R be as in the theorem with k of characteristic 0 and let n be odd. If M is a finitely generated R-module with θ ( M , M ) = 0 , then M is rigid. S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Higher Codimension Let R be an isolated complete intersection singularity — i.e., R is the quotient of a regular local ring ( T , m ) by a regular sequence f 1 , . . . , f c ∈ T , and R p is regular for all p � = m . For any pair ( M , N ) of finitely generated R -modules, the Tor modules Tor R j ( M , N ) have finite length when j ≫ 0. Moreover, the lengths of the odd and even indexed Tor modules in high degree follow predictable patterns. S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Proposition (Prop 1) Let R be the quotient of a Noetherian ring T by a regular sequence f 1 , . . . , f c and let M and N be finitely generated R-modules. Suppose the T-module Tor T j ( M , N ) vanishes for all j ≫ 0 and there is a finite set of maximal ideals { m 1 , . . . , m ℓ } of R such that the R-module Tor R j ( M , N ) is supported on { m 1 , . . . , m ℓ } for all j ≫ 0 . Then the graded components of the Koszul complex for the sequence χ 1 , . . . , χ c acting on Tor R ∗ ( M , N ) , i.e., � 0 → Tor R Tor R j +2 c ( M , N ) → · · · → j +4 ( M , N ) 1 ≤ i 1 < i 2 ≤ c � Tor R j +2 ( M , N ) → Tor R → j ( M , N ) → 0 , 1 ≤ i ≤ c are exact for j ≫ 0 . S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Proposition (Prop 2) In the situation of Proposition 1, there are polynomials P ev = P R ev ( M , N ) and P odd = P R odd ( M , N ) of degree at most c − 1 so that, for all j ≫ 0 , length Tor R 2 j ( M , N ) = P ev ( j ) and length Tor R 2 j +1 ( M , N ) = P odd ( j ) . Definition For R , M , and N as above, let m c , ev ( M , N ) denote ( c − 1)! times the coefficient of j c − 1 in P ev = P R ev ( M , N ), and likewise define m c , odd ( M , N ). Define θ c ( M , N ) = m c , ev ( M , N ) − m c , odd ( M , N ) . Equivalently, θ c ( M , N ) = ( P ev − P odd ) ( c − 1) . First difference: q (1) ( j ) = q ( j ) − q ( j − 1), and recursively one defines q ( i ) = ( q ( i − 1) ) (1) . S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Proof of Proposition 2-Sketch define a sequence a n = length Tor R • E +2 n ( M , N ), n ≥ 0; • linear recurrence relation via Proposition 1: � c � � c � a n − 2 + · · · + ( − 1) c a n − ca n − 1 + a n − c = 0 , n ≥ c ; 2 c n ≥ 0 a n x n satisfies H ( x ) = p ( x ) • assoc. gen. fun. H ( x ) := � (1 − x ) c , where p ( x ) is a polynomial of degree at most c − 1; • coefficients of the power series expansion of c − 1 b i � H ( x ) = (1 − x ) c − i i =0 are given by a polynomial Q ( j ) of degree at most c − 1; the coefficient of j c − 1 in Q ( j ) is p (1) • ( c − 1)! ; set P ev ( j ) = Q ( j − E / 2); length Tor R • 2 j ( M , N ) = P ev ( j ), j ≥ E / 2. S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Example If c = 1, then P ev and P odd are constant polynomials, whose values are length Tor R 2 j ( M , N ) and length Tor R 2 j +1 ( M , N ), respectively, for j ≫ 0. Thus, θ 1 ( M , N ) = P ev − P odd = length Tor R 2 j ( M , N ) − length Tor R 2 j +1 ( M , N ) is simply Hochster’s original invariant θ ( M , N ). S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Dao’s invariant � length Tor R if length Tor R j ( M , N ) < ∞ and j ( M , N ) β j ( M , N ) = 0 otherwise, � n j =0 ( − 1) j β j ( M , N ) η c ( M , N ) = lim . n c n →∞ Lemma Under the same assumptions as above with c > 0 , we have η c ( M , N ) = θ c ( M , N ) 2 c · c ! . S. Spiroff, University of Mississippi An Invariant for Complete Intersections
History and Background Isolated Singularities Higher Codimension Example (Revisited) If c = 1, then P ev and P odd are constant polynomials, whose values are length Tor R 2 j ( M , N ) and length Tor R 2 j +1 ( M , N ), respectively, for j ≫ 0. Thus, θ 1 ( M , N ) = P ev − P odd = 2 η 1 ( M , N ) is simply Hochster’s original invariant θ ( M , N ). S. Spiroff, University of Mississippi An Invariant for Complete Intersections
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