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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

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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

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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 · · ·→TorT

n (M, N)→TorR n (M, N)→TorR n−2(M, N)→TorT n−1(M, N)→· · ·

• ne obtains

TorR

j (M, N) ∼

= TorR

j+2(M, N)

for j ≫ 0.

Example

R = C[ [X, Y , U, V ] ]/( XU − YV)

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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 Np has finite projective dimension over Rp for every p = m.

Definition (Hochster, 1981)

Define θ : M × N → Z by θ(M, N) = length(TorR

2j(M, N)) − length(TorR 2j+1(M, N))

where j ≫ 0.

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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): M

• R
• R2

h

x y

i

• R2

2 4 u

−y −v x

3 5

• . . .

2 4x

y v u

3 5

• Tensor with N = R/

( u, v): C

• C[

[X, Y ] ]

• C[

[X, Y ] ]2

h

X Y

i

• . . .

2 40

Y X

3 5

• TorR

even(M, N) ∼

= C and TorR

• dd(M, N) = 0, hence θ(M, N) = 1.
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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

R = C[ [X, Y , U, V ] ]/( XU − YV), 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(

R/ (x, y), R/ (u, v) ) = 0.

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Isolated Singularities

If the ring R is an isolated singularity, then θ is defined on any pair

• f 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.

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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

• f finitely generated R-modules M and N.

Theorem (Moore, Piepmeyer, S., Walker, 2009)

Let k be a field and let R = k[x0, . . . , xn] /( f (x0, . . . , xn) ), where deg xi = 1 for all i and f is a homogeneous polynomial of degree d and m = (x0, . . . , xn) 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(TorR

2j(M, N)) − length(TorR 2j+1(M, N)) = 0,

j ≫ 0.

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History and Background Isolated Singularities Higher Codimension

Definition

A pair of modules (M, N) is rigid if for any integer i ≥ 0, TorR

i (M, N) = 0 implies TorR 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

• dd. If M is a finitely generated R-module with θ(M, M) = 0,

then M is rigid.

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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 f1, . . . , fc ∈ T, and Rp is regular for all p = m. For any pair (M, N) of finitely generated R-modules, the Tor modules TorR

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.

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History and Background Isolated Singularities Higher Codimension

Proposition (Prop 1)

Let R be the quotient of a Noetherian ring T by a regular sequence f1, . . . , fc and let M and N be finitely generated R-modules. Suppose the T-module TorT

j (M, N) vanishes for all

j ≫ 0 and there is a finite set of maximal ideals {m1, . . . , mℓ} of R such that the R-module TorR

j (M, N) is supported on {m1, . . . , mℓ}

for all j ≫ 0. Then the graded components of the Koszul complex for the sequence χ1, . . . , χc acting on TorR

∗ (M, N), i.e.,

0 → TorR

j+2c(M, N) → · · · →

• 1≤i1<i2≤c

TorR

j+4(M, N)

→

• 1≤i≤c

TorR

j+2(M, N) → TorR j (M, N) → 0,

are exact for j ≫ 0.

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History and Background Isolated Singularities Higher Codimension

Proposition (Prop 2)

In the situation of Proposition 1, there are polynomials Pev = PR

ev(M, N) and Podd = PR

• dd(M, N)
• f degree at most c − 1 so that, for all j ≫ 0,

length TorR

2j(M, N) = Pev(j) and length TorR 2j+1(M, N) = Podd(j).

Definition

For R, M, and N as above, let mc,ev(M, N) denote (c − 1)! times the coefficient of jc−1 in Pev = PR

ev(M, N), and likewise define

mc,odd(M, N). Define θc(M, N) = mc,ev(M, N) − mc,odd(M, N). Equivalently, θc(M, N) = (Pev − Podd)(c−1).

First difference: q(1)(j) = q(j) − q(j − 1), and recursively one defines q(i) = (q(i−1))(1).

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An Invariant for Complete Intersections

History and Background Isolated Singularities Higher Codimension

Proof of Proposition 2-Sketch

• define a sequence an = length TorR

E+2n(M, N), n ≥ 0;

• linear recurrence relation via Proposition 1:

an − can−1 + c 2

• an−2 + · · · + (−1)c

c c

• an−c = 0, n ≥ c;
• assoc. gen. fun. H(x) :=

n≥0 anxn satisfies H(x) = p(x) (1−x)c ,

where p(x) is a polynomial of degree at most c − 1;

• coefficients of the power series expansion of

H(x) =

c−1

• i=0

bi (1 − x)c−i are given by a polynomial Q(j) of degree at most c − 1;

• the coefficient of jc−1 in Q(j) is

p(1) (c−1)!;

• set Pev(j) = Q(j − E/2); length TorR

2j(M, N) = Pev(j), j ≥ E/2.

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History and Background Isolated Singularities Higher Codimension

Example

If c = 1, then Pev and Podd are constant polynomials, whose values are length TorR

2j(M, N) and length TorR 2j+1(M, N), respectively, for

j ≫ 0. Thus, θ1(M, N) = Pev−Podd = length TorR

2j(M, N)−length TorR 2j+1(M, N)

is simply Hochster’s original invariant θ(M, N).

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History and Background Isolated Singularities Higher Codimension

Dao’s invariant βj(M, N) =

• length TorR

j (M, N)

if length TorR

j (M, N) < ∞ and

• therwise,

ηc(M, N) = lim

n→∞

n

j=0(−1)jβj(M, N)

nc .

Lemma

Under the same assumptions as above with c > 0, we have ηc(M, N) = θc(M, N) 2c · c! .

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History and Background Isolated Singularities Higher Codimension

Example (Revisited)

If c = 1, then Pev and Podd are constant polynomials, whose values are length TorR

2j(M, N) and length TorR 2j+1(M, N), respectively, for

j ≫ 0. Thus, θ1(M, N) = Pev − Podd = 2η1(M, N) is simply Hochster’s original invariant θ(M, N).

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An Invariant for Complete Intersections

History and Background Isolated Singularities Higher Codimension

New Assumptions

• k is a field.
• R = k[x0, . . . , xn+c−1]/(f1, . . . , fc), where deg xi = 1

for all i and the fj’s are homogeneous polynomials

• f1, . . . , fc forms a regular sequence.
• X = Proj(R) is a smooth k-variety.
• m = (x0, . . . , xn+c−1) is the only non-regular prime of R.
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An Invariant for Complete Intersections

History and Background Isolated Singularities Higher Codimension

Recall: θc(M, N) = mc,ev(M, N) − mc,odd(M, N) θc(M, N) = (Pev − Podd)(c−1)

Theorem (Moore, Piepmeyer, S., Walker, 2011)

Under the assumptions above with k separably closed, let M and N be finitely generated graded R-modules. Then θc(M, N) vanishes when c > 1.

Corollary

Under the assumptions above, for every pair of finitely generated, but not necessarily graded, R-modules M and N, θc(M, N) vanishes when c > 1.

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An Invariant for Complete Intersections

History and Background Isolated Singularities Higher Codimension

Example (Revisited)

R = C[X,Y ,U,V ]/( XU−YV); c = 1 M = R/ ( x, y), N = R/ ( u, v), L = R/ ( x, v) Then θ1(M, N) = 1; θ1(M, M) = 1; and θ1(M, L) = −1.

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History and Background Isolated Singularities Higher Codimension

Definition

We say that the pair (M, N) is r-Tor-rigid if whenever r consecutive Tor modules vanish, then all subsequent Tor modules vanish too. Dao proved that the vanishing of ηc(M, N) implies that the pair (M, N) is c-Tor-rigid.

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History and Background Isolated Singularities Higher Codimension

Example (D. Jorgensen, O. Celikbas)

Let R = C[ [X, Y , U, V ] ]/( XU, YV). R is a local complete intersection of codimension two with positive dimensional singular

• locus. Let M = R/

( x, y), and let N be the cokernel of the map R2

B B @

v −u x y

1 C C A

− − − − − − − → R3. The pair (M, N) is not 2-Tor-rigid, and hence, θ2(M, N) = 0.

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History and Background Isolated Singularities Higher Codimension

In general, one only has (c + 1)-Tor-rigidity for pairs of modules

• ver a codimension c complete intersection.

Upshot of our main result: all pairs of modules over rings of the above form having an isolated singularity are c-Tor-rigid, provided c > 1.

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History and Background Isolated Singularities Higher Codimension

Corollary

With the assumptions above, let M and N be finitely generated, but not necessarily graded, R-modules. Then for c > 1, the pair (M, N) is c-Tor-rigid. That is, if c consecutive torsion modules TorR

i (M, N), . . . , TorR i+c−1(M, N)

all vanish for some i ≥ 0, then TorR

j (M, N) = 0 for j ≥ i.

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History and Background Isolated Singularities Higher Codimension

Motivated by the main theorem of this paper, we conjecture that θc(M, N) vanishes for all pairs of modules (M, N) over an isolated complete intersection singularity of codimension c > 1, and hence that all pairs of modules over such a ring are c-Tor-rigid.

Conjecture

Suppose R = T/(f1, . . . , fc) with T a regular Noetherian ring and f1, . . . , fc a regular sequence, with c > 1. If the singular locus of R consists of a finite number of maximal ideals, then θc(M, N) = 0 for all finitely generated R-modules M and N.

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An Invariant for Complete Intersections

History and Background Isolated Singularities Higher Codimension

Key References

• 1. H. Dao, Asymptotic behavior of Tor over complete

intersections and applications, preprint.

• 2. H. Dao, Some observations on local and projective

hypersurfaces, Mathematical Research Letters, 2008.

• 3. O. Celikbas, Vanishing of Tor over complete intersections, J.
• Comm. Alg. Volume 3, Number 2, (2011).
• 4. Hochster, Melvin, The dimension of an intersection in an

ambient hypersurface, Lecture Notes in Math., 1981

• 5. D. Jorgensen, Complexity and Tor on a complete intersection,
• J. Algebra 211 (1999), 578-598.
• 6. W. F. Moore, G. Piepmeyer, S. Spiroff, M. E. Walker,

Hochster’s theta invariant and the Hodge-Riemann bilinear relations, Advances in Math., 226 (2010), no. 2, 1692-1714.

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