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
R
R
T
· · · → TorT
n (M, N) → Torn (M, N) → Torn−2 (M, N) → Torn−1 (M, N) → · · ·
one obtains
R
∼
TorR
j (M, N) = Torj+2 (M, N)
for j 0.
Example
R = C[[X , Y , U, V ]]/(XU − YV)
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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 Np has
finite projective dimension over Rp for every p 6= m.
Definition (Hochster, 1981)
Define θ : M × N → Z by
R
θ(M, N) = length(TorR
2j (M, N)) − length(Tor2j+1 (M, N))
where j 0.
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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
h
0o
Mo
Ro
x
y
i
R2 o
u
4
−v
3
−y 5
x
2
R2 o
x
4
v
3
y5
u
...
Tensor with N = R/(u, v):
2
h
0o
Co
C[[X , Y ]] o
X
Y
i
4
C[[X , Y ]]2 o
3
0 Y5
0 X
R
∼
TorR
even (M, N) = C and Torodd (M, N) = 0, hence θ(M, N) = 1.
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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
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 )) 6= 0.
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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.
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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[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,
R
length(TorR
2j (M, N)) − length(Tor2j+1 (M, N)) = 0,
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An Invariant for Complete Intersections
j 0.
History and Background
Isolated Singularities
Higher Codimension
Definition
A pair of modules (M, N) is rigid if for any integer i ≥ 0,
R
TorR
i (M, N) = 0 implies Torj (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.
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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 6= 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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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 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 , . . . , m` } of R
such that the R-module TorR
j (M, N) is supported on {m1 , . . . , m` }
for all j 0. Then the graded components of the Koszul complex
for the sequence χ1 , . . . , χc acting on TorR
∗ (M, N), i.e.,
M
0 → TorR
TorR
j+2c (M, N) → · · · →
j+4 (M, N)
1≤i1 <i2 ≤c
→
M
R
TorR
j+2 (M, N) → Torj (M, N) → 0,
1≤i≤c
are exact for j 0.
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An Invariant for Complete Intersections
History and Background
Isolated Singularities
Higher Codimension
Proposition (Prop 2)
In the situation of Proposition 1, there are polynomials
R (M, N) and P
R
Pev = Pev
odd = Podd (M, N)
of degree at most c − 1 so that, for all j 0,
R
length TorR
2j (M, N) = Pev (j) and length Tor2j+1 (M, N) = Podd (j).
Definition
For R, M, and N as above, let mc,ev (M, N) denote (c − 1)! times
R (M, N), and likewise define
the coefficient of j c−1 in Pev = Pev
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:
c
c c
an − can−1 +
an−2 + · · · + (−1)
an−c = 0, n ≥ c;
2
c
P
p(x)
• assoc. gen. fun. H(x) := n≥0 an x n satisfies 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
X
bi
H(x) =
(1 − x)c−i
i=0
are given by a polynomial Q(j) of degree at most c − 1;
p(1)
(c−1)! ;
•
the coefficient of j c−1 in Q(j) is
•
set Pev (j) = Q(j − E /2); length TorR
2j (M, N) = Pev (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 Pev and Podd are constant polynomials, whose values
R
are length TorR
2j (M, N) and length Tor2j+1 (M, N), respectively, for
j 0. Thus,
R
θ1 (M, N) = Pev −Podd = length TorR
2j (M, N)−length Tor2j+1 (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
Dao’s invariant
(
R
length TorR
j (M, N) if length Torj (M, N) < ∞ and
βj (M, N) =
0
otherwise,
Pn
j
j=0 (−1) βj (M, N)
.
ηc (M, N) = lim
n→∞
nc
Lemma
Under the same assumptions as above with c > 0, we have
ηc (M, N) =
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θc (M, N)
.
2c · c!
An Invariant for Complete Intersections
History and Background
Isolated Singularities
Higher Codimension
Example (Revisited)
If c = 1, then Pev and Podd are constant polynomials, whose values
R
are length TorR
2j (M, N) and length Tor2j+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
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c > 1.
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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An Invariant for Complete Intersections
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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An Invariant for Complete Intersections
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
0
1
0 v
B
C
B−u x C
@
A
y
0
R 2 −−−−−−−→ R 3 .
The pair (M, N) is not 2-Tor-rigid, and hence, θ2 (M, N) 6= 0.
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An Invariant for Complete Intersections
History and Background
Isolated Singularities
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In general, one only has (c + 1)-Tor-rigidity for pairs of modules
over 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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An Invariant for Complete Intersections
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
R
TorR
i (M, N), . . . , Tori+c−1 (M, N)
all vanish for some i ≥ 0, then TorR
j (M, N) = 0 for j ≥ i.
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An Invariant for Complete Intersections
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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An Invariant for Complete Intersections