Regularity of ideals and their powers
Marc Chardin
Introduction
If R is a polynomial ring over a field k, M is a finitely generated graded R-module (for the standard
grading of R) and m := R>0 , the invariants
i
ai (M ) := inf{µ | Hm
(M )>µ = 0}
and
bi (M ) := inf{µ | TorR
i (M, R/m)>µ = 0}
both give a definition of the Castelnuovo-Mumford regularity of M (over R):
reg(M ) := max{ai (M ) + i} = max{bi (M ) − i}.
i
i
We will particularly focus in this article on the control of the invariants ai (M ) for i close to the dimension
of the support of M (recall the ai (M ) = −∞ for i > dim M ).
If X ⊆ Pnk = Proj(R) is a non empty projective scheme, reg(X) is the regularity of the unique
homogeneous quotient B of R with positive depth such that X = Proj(B), it is also easily seen that
reg(X) = maxi>0 {ai (B) + i} for any homogeneous quotient B of R such that X = Proj(B).
The regularity of a projective scheme depends on its embedding. It may also be defined in terms of the
ideal sheaf IX defining X in Pnk ; one has reg(X) = reg(IX ) − 1, with the standard definition of regularity
for sheaves (see e.g. [Ei, 20.20]).
The motivations for this work came from several sides. One was the search of bounds for the regularity of
powers of an ideal (or of an ideal sheaf) that generalizes the ones obtained by Bertram, Ein and Lazarsfeld for
ideal sheaves defining smooth equidimensionnal schemes in [BEL]. By the work of Kodiyalam and Cutkosky,
Herzog and Trung [Ko, CHT] we know that, if I is a homogeneous ideal in a polynomial ring over a field,
then reg(I m ) is eventually a linear function in m whose leading coefficient is at most the maximal degree of
a generator of I; our goal was then to find a bound of that type for any power of a homogeneous ideal under
geometric hypotheses on the corresponding scheme.
Another interesting challenge is the conjecture of Katzman on the regularity of Frobenius powers of a
homogenous ideal, or of a graded module, in positive characteristic. It turned out that this problem may
also be attacked by the argument we use for ordinary powers.
The works on regularity bounds in terms of degrees of defining equations and on regularity of the tensor
product of two modules also appeared to be candidates for a common treatment, by considering the tensor
product of any number of free resolutions and looking at the corresponding acyclicity defects : the (multiple)
Tor modules.
One central idea in our study is that from a complex D• which is not too far from being acyclic, one can
derive non trivial information on the cohomology of H0 (D• ) from a control on the cohomology of the Di ’s.
This idea was a key point in the proof of a regularity bound for reduced curves by Gruson, Lazarsfeld and
Peskine; our lemmas in the first section are variations from Lemma 1.6 in [GLP]. We state them in a ring
theoretic version. Let us point out that the simplest of these results, Lemma 1.1, is in many cases sufficient
to prove new interesting bounds on regularity.
A direct application is to bound the regularity of Frobenius powers of a module. Our result, which
solves in a non trivial series of cases conjectures of Katzman in [Ka], is the following:
Theorem 0.1. Let S be a standard graded ring over a field of characteristic p > 0 and M a
finitely generated graded S-module. Assume that dim(NonReg(S) ∩ Supp(M )) ≤ 1 and set bSi (M ) :=
max{j | TorSi (M, k)j 6= 0} if TorSi (M, k) 6= 0, and bSi (M ) := −∞ else. Then, denoting by F the Frobenius
functor, one has
reg(F e M ) ≤
max
0≤i≤j≤dim S
{pe bSi (M ) + aj (S) + j − i} ≤ reg(S) +
1
max
0≤i≤dim S
{pe bSi (M ) − i}.
Another application given in section 3 is to provide bounds on the regularity of tensor product of modules
in terms of the regularity of the modules.
A useful ingredient in the study of tensor product of modules is the Tor modules of a s-tuple of modules.
We give in an appendix several properties of these modules. The results are classical in the case of two
modules, we needed some extensions for an arbitrary number of modules. In particular, we prove the
following generalization of results of Auslander and Serre,
Theorem 0.2. Let R be a regular local ring containing a field and M1 , . . . , Ms be finitely generated
R-modules. Then,
R
(i) TorR
i (M1 , . . . , Ms ) = 0 implies Torj (M1 , . . . , Ms ) = 0 for all j ≥ i.
(ii) Let j := max{i | TorR
i (M1 , . . . , Ms ) 6= 0}. Then
pdimM1 + · · · + pdimMs = dim R + j − ε,
R
with 0 ≤ ε ≤ dim TorR
j (M1 , . . . , Ms ). Moreover, ε ≥ ε0 := mini {depth Torj−i (M1 , . . . , Ms ) + i} and the
equality holds if ε0 = depth TorR
j (M1 , . . . , Ms ).
As compared to earlier work of Sidman and others, we obtain bounds that mixes the invariants ai of
one module with the bi ’s of the others (see Proposition 3.1). This type of bounds is in several cases sharper
than the previously known ones. For example Proposition 3.1 (i) implies the following (see also Corollary
3.2),
Theorem 0.3. Let k be a field, Z1 , . . . , Zs be closed subschemes of a projective k-scheme Z of dimension
d. Assume that reg(Z1 ) ≥ · · · ≥ reg(Zs ). If the intersection of the Zi ’s is of dimension at most 1, then
min{d,s}
reg(Z1 ∩ · · · ∩ Zs ) ≤ reg(Z) +
X
reg(Zi ).
i=1
Even in the case where Z is a projective space and the Zi are hypersurfaces, such a bound was not
known without restrictions on the local number of defining equations of the intersection.
As another illustration we derive the following bound for powers of ideals that refines the results of
[Chan] and [GGP] (recall that b0 (I) is the maximal degree of the minimal generators of I and b1 (I) is the
maximal degree of minimal first syzygies),
Theorem 0.4. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ] such that dim(R/I) ≤ 1. Then,
for any m ≥ 1, a1 (R/I m+1 ) ≤ max{a0 (R/I), a1 (R/I) + mb0 (I)} and
a0 (R/I m+1 ) ≤ max{a0 (R/I) + b0 (I), a1 (R/I) + b1 (I)} + (m − 1)b0 (I),
in particular, reg(I m+1 ) ≤ reg(I) + mb0 (I) if I is linearly presented and reg(I m+1 ) ≤ 2reg(I) + (m − 1)b0 (I)
in any case.
We also use a natural map from the exterior algebra of the conormal module to the Tor algebra of the
quotient ring to give estimates for the regularity of powers of generically complete intersection ideals defining
a quotient of dimension 2, we get for example the following
Theorem 0.5. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ] such that dim(R/I) = 2 and
d1 ≥ · · · ≥ ds be the degrees of a minimal system of generators of I. Assume that Ip ⊂ Rp is a complete
intersection for every prime p such that dim R/p = 2. Then,
reg(I/I 2 ) ≤ max{reg(R/I) + max{b0 (I), b1 (I) − 1, b2 (I) − 2}, a2 (R/I) + d1 + d2 }.
In section 5 we study quite carefully the regularity of two dimensional quotients of a Gorenstein standard
graded algebra over a field. We first establish several lemmas that are useful for bounding the regularity
of ideals by inducting on the dimension and/or the number of generators. We next turn to some duality
2
results on the Koszul homology of ideals, that we use, in the case of two dimensional quotients, to give quite
sharp estimates of the regularity in terms of the degrees of generators and the minimal degree of a section
of the unmixed component of the corresponding scheme (a projective curve). In case this degree is zero, for
instance if the curve is reduced, this bound is the expected one. We also show by examples that there are
cases where the bad behavior described for curves with very negative sections do occur.
Proposition 5.10 gives a geometric description of the local cohomology in the case of an almost complete
intersection of dimension 2 in a Gorenstein standard graded ring and explains why sections of negative
degrees do influence the regularity :
Theorem 0.6. Let S be a standard graded Gorenstein algebra of dimension n over a field k, aS be
its a-invariant, I = (f1 , . . . , fn−1 ) be a graded S-ideal with dim(S/I) = 2, where fi is a form of degree
di . Assume that J := (f1 , . . . , fn−2 ) is a complete intersection strictly contained in I. Let C be the one
dimensional component of Proj(S/I) and C 0 be its residual in Proj(S/J). Then,
1
(S/IC )),
a0 (S/I) = aS + d1 + · · · + dn−1 − indeg(Hm
a1 (S/I) = aS + d1 + · · · + dn−1 − indeg(IC /I),
a2 (S/I) = aS + d1 + · · · + dn−2 − indeg(IC 0 /J),
(with indeg(0) = +∞).
Our results on the regularity of two dimensional quotients (Proposition 5.12, Theorem 5.13 and Proposition 5.15) have as corollaries estimates on the regularity of schemes of dimension two, for example we have
the following,
Theorem 0.7. Let S be a standard graded Gorenstein ring, Z := Proj(S), n := dim Z and I be a
graded S-ideal generated by forms of degrees d1 ≥ · · · ≥ ds . Set X := Proj(S/I) and ` := min{s, n}. Assume
that dim X = 2, the component of dimension two of X is a reduced surface S and µ(IX,x ) ≤ dim OZ,x for
x ∈ S except at most at finitely many points, then
reg(X) ≤ reg(S) + d1 + d2 + · · · + d` − `.
Without the hypothesis on the local number of generators, we are able to get a bound which is essentially
twice the bound above (see Proposition 5.12).
We then turn to another type of estimates for the regularity of powers of ideals that come from the
acyclicity properties of the Z-complex. As a preliminary, we have to study the cohomology of the Koszul
cycles, this is the subject of section 6. The Z-complex is, or approximates, a resolution by Koszul cycles
of the symmetric algebra of an ideal. We therefore get results on the local cohomology of the symmetric
powers. As a consequence, we also get results on the cohomology of powers. We prove, for instance,
Theorem 0.8. Let I ⊂ R = k[X0 , . . . , Xn ] be an ideal generated in degrees d1 ≥ · · · ≥ ds . Set
X := Proj(R/I) ⊆ Pnk , X j := Proj(R/I j ) for j ≥ 2 and ` := min{s, n}.
Assume that dim X ≤ 3, µ(Ip ) ≤ dim Rp for all p ∈ Supp(R/I) of codimension at most n − 2, and
further X is generically reduced if dim X = 3, then
reg(X j ) ≤ (j − 1)d1 + max{reg(X), d2 + · · · + d` − `}.
In the second part of section 7, we combine these results with the estimates of section 5, or other
estimates on the regularity, to give new results on the regularity of powers of ideal sheaves that corresponds
to projective schemes of dimension at most 3. Propostion 7.13 and Propostion 7.14 extends in small dimension
the estimate of [BEL] to schemes that are neither smooth nor unmixed, they hold in any characteristic.
In the last section we collect what we proved on the torsion in the symmetric algebra of an ideal in the
particular case of a polynomial ring, and derive an application to the implicitization problem, in the spirit of
the work of Busé and Jouanolou [BJ]. We treat cases where the base locus is of dimension one (see Section
8 for the definition of the graded pieces Z• [µ] in the result below) :
3
Theorem 0.9. Let φ : Pn−1 · · · →Pn be a rational map defined by n + 1 polynomials of degree d,
I be the ideal generated by these polynomials and X ⊂ Pn−1 be the scheme defined by I. Assume that
dim X = 1 and let C be the one dimensionnal component of X. If C has no section of degree < −d and for
any closed point x ∈ X
(
n − 1 if x ∈ C
µ(IX,x ) ≤
n
else,
then det(Z• [µ]) is a non-zero multiple of the equation of the closure of the image of φ in Pn for µ ≥
(n − 1)(d − 1).
Section 1. Some key lemmas
In this section, R is a polynomial ring over a field and D• a graded complex of finitely generated
R-modules with Di = 0 for i < 0.
The following lemmas give ways for controlling the local cohomology of H0 (D• ) in terms of local cohomologies of the Di ’s under restrictions on the dimension of the modules Hi (D• ) for i > 0.
The quantities δp := maxi≥0 {ap+i (Di )} and εq := maxi≥0 {ai (Dq+i )} are useful in these estimates.
Lemmas below all derive from the study of the two spectral sequences coming from the double complex
•
Cm
D• . For clarifying later references, we first state particular cases that are often used in the sequel before
giving more refined statements. The proofs of the six lemmas will be given after they are all stated.
We set Hi := Hi (D• ) and introduce the following conditions on the dimension of the Hi ’s :
D` (τ ) :
dim Hi ≤ max{τ, τ − ` + i}, ∀i > 0.
Lemma 1.1. ap (H0 ) ≤ δp for p ≤ τ − 1 if D1 (τ ) is satisfied.
τ −2
τ
Lemma 1.2. If D1 (τ ) is satisfied, there exists a natural map ξ : Hm
(H0 )−→Hm
(H1 ) such that ξµ is
onto for µ > δτ +1 and ξµ is into if D2 (τ ) is satisfied and µ > δτ .
Lemma 1.3. If D2 (τ ) is satisfied, there exists a diagram of natural maps,
0
/K
τ
/ Hm
(H2 )
²
/C
/ H τ −3 (H0 )
m
α
/ H τ −1 (H1 )
m
θ
τ −4
Hm
(H0 )
γ
/ H τ −2 (H1 )
m
β
/0
with K = ker(α) and C = coker(β), such that,
(a) αµ and θµ are onto for µ > δτ ,
(b) θµ is into and Im(γµ ) = ker(βµ ) if D3 (τ ) is satisfied and µ > δτ −1 .
Lemma 1.4. Assume that dim Hi ≤ 1 for i ≥ 1. Then ap (H0 ) ≤ δp for p ≥ 0. Also a0 (Hq ) ≤ εq and
a1 (Hq ) ≤ εq−1 for q > 0.
Lemma 1.5. If dim Hi ≤ 2 for i ≥ 2, then a1 (Hq ) ≤ εq−1 for q > 0 and there exists natural maps
p
p+2
α p : Hm
(H0 )−→Hm
(H1 ) and
0
2
φq : Hm
(Hq )−→Hm
(Hq+1 )
for p ≥ 0 and for q > 0 such that
(a) αµp is onto for µ > δp+1 and is into if µ > δp ,
(b) φqµ is onto for µ > εq−1 and is into if µ > εq .
Lemma 1.6. If dim Hi ≤ 3 for i ≥ 3, there exists complexes of natural maps
βp
αp
p
p+2
p+4
0→Kp →Hm
(H0 ) −→ Hm
(H1 ) −→ Hm
(H2 )→Cp →0
4
for p ≥ −1 that are exact except possibly in the middle, maps θp : Kp −→Cp−1 for p ≥ 0, and for q > 0 a
collection of maps
φq
0
2
Hm
(Hq ) −→ Hm
(Hq+1 ),
ψq
1
3
Hm
(Hq ) −→ Hm
(Hq+1 )
and
ηq
ker(φq ) −→ coker(ψ q+1 )
(with β −1 = ψ 1 ) such that:
(a) ker(βµp ) = im(αµp ) for µ > δp+1 ,
(b) θµp is onto for µ > δp+1 and is into if µ > δp ,
(c) φqµ is onto for µ > εq−1 ,
(d) ψµq is into for µ > εq−1 ,
(e) ηµq is onto for µ > εq−1 and is into if µ > εq .
We now turn to the proof of these statements. Notice that Lemma 1.6 implies Lemma 1.5 that in turn
implies Lemma 1.4.
Proof of the lemmas 1.1 to 1.6.
•
We choose the cohomological index of Cm
D• as line index and the homological index as column index.
The two spectral sequences coming from the horizontal and vertical filtrations of this double complex have
i
i
i
first terms: h1 Eji = Cm
(Hj ), h2 Eji = Hm
(Hj ) and v1 Eji = Hm
(Dj ).
v i
We therefore note that (1 Ej )µ = 0 for µ > δp if i − j = p ≥ 0 and for µ > εq if i − j = −q ≤ 0.
−q
•
•
D• ))µ =
This implies that H p (Tot(Cm
L0 for p ≥i 0 and µ > δp and H (Tot(Cm D• ))µ = 0 for q ≥ 0 and
• `
µ > εq (with the convention Tot(C• ) = i−j=` Cj ).
We will now combine these vanishing with the study of the further steps in the spectral sequences coming
from the horizontal filtration to derive the six lemmas.
For the first three lemmas, we assume that condition D1 (τ ) is satisfied and concentrate on the bottom
right part of the diagram at step 2 and relabel the second differentials that we will use by setting ξ := h2 dτ0 −2 ,
α := h2 dτ0 −3 , β := h2 dτ1 −2 , γ := h2 dτ0 −4 :
···
···
···
···
···
···
···
···
···
···
h τ −4
2 E0
©
©©
©
©©
©© h E τ −3
©
···
···
···
©γ 2 0
©
©©
©
©©
©
©
©
©
¥©©
©©
©
τ
−2
h
h τ −2
···
···
2 E1
©© α 2 E0
©
©
©
©
©©
©©
©©
©
©
©
©
©
¥©©
©© β τ −1
©©
©
©
h τ −1
h
···
···
©© 2 E1
©© ξ 2 E0
©
©
©
©
©
©©
©©
©©
©
©
©
©
©
¥©¡ ©
©© h τ¥©
©
h τ
h τ
©
···
2 E1
2 E0
2 E2
©©
©
©
©
©
©
©
©©
©©
©
©
¥
¡
¡
©© h τ +1
h τ +1
h τ +1
0
2 E3
2 E0
©© 2 E2
©
©
©
©©
¥©¡ ©
h τ +2
h τ +2
0
0
2 E3
2 E0
..
.
..
.
..
.
5
..
.
The dotted arrows show direction of differentials at step 3. We set θ := h3 dτ0 −3 .
τ +`+i−1
τ +`+i−1
τ +`+i−1
As for i ≥ 0 and ` ≥ 2, h2 E`−1
= Hm
(H`−1 ) = 0 it follows that also h` E`−1
= 0 in this
τ
+i−1
τ
+i−1
τ
+`+i−1
h
h
h
case, and therefore ` d0
: ` E0
−→` E`−1
is the zero map.
τ +i−1
This implies that h` E0τ +i−1 ' h2 E0τ +i−1 = Hm
(H0 ) for any ` ≥ 2 and i ≥ 0, and proves Lemma 1.1.
For Lemma 1.2 first notice that coker(ξ) ' h3 E1τ ' h∞ E1τ . Moreover if D2 (τ ) is satisfied, for ` ≥ 2,
τ +`−1
h τ +`−1
= Hm
(H` ) = 0 and therefore h3 E0τ −2 ' h∞ E0τ −2 , h3 E1τ −1 ' h∞ E1τ −1 and h4 E2τ ' h∞ E2τ . The
2 E`
first isomorphism finishes to prove Lemma 1.2, and the last two isomorphisms imply Lemma 1.3 (a).
Furthermore, if D3 (τ ) is satisfied then one also has h4 E0τ −3 ' h∞ E0τ −3 and h3 E1τ −2 ' h∞ E1τ −2 because
h τ +`−1
τ +`−1
= Hm
(H`+1 ) = 0 for ` ≥ 2, which implies Lemma 1.3 (b).
2 E`+1
We now turn to Lemma 1.6. At step 2 we have, after relabeling the maps :
···
···
···
···
···
···
···
0
0
0
0
Hm
(H1 )
Hm
(H0 )
Hm
(H3 )
Hm
(H2 )
¦
¦
¦
¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦
¦
¦
¦
¦
¦
¦
¦
1
1
1
1
¦¦ 3 Hm
···
(H3 ) ¦¦¦ 2 Hm
(H2 ) ¦¦¦ 1 Hm
(H1 ) ¦¦¦ 0 Hm
(H0 )
¦
¦ φ
¦ α ¦
¦ φ
¦ φ
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
£¦
£¦
£¦
£¦
¦¦ 2
ψ3 ¦
ψ2 ¦
ψ1 ¦
2
2
2
2
Hm
(H4 ) ¦¦¦ Hm
(H3 ) ¦¦¦ Hm
(H2 ) ¦¦¦ Hm
(H1 ) ¦¦¦ 1 Hm
(H0 )
α
¦
¦
¦¦
¦¦
¦¦
¦¦
¦
¦
¦
¦
¦
¦
¦
¦¦
¦¦
¦¦ 0 ¦¦
¦¦
¦¦
¦
£¦~ ¦
£¦~ ¦
£¦~ ¦ β ¦¦¦
£¦¦
¦
3
3
3
3
3
(H0 )
(H1 ) ¦¦¦ 2 Hm
(H2 ) ¦¦¦ Hm
(H3 )
Hm
(H4 )
Hm
Hm
α
¦
¦
§
§
¦
¦
§
§
¦
¦
§
§§
¦¦
¦¦
§§
§
£¦~ ¦ β 1 §§§
£¦¦
§
4
4
4
Hm
(H2 ) §§§ Hm
(H1 ) §§§ 3 Hm
(H0 )
0
0
α
§
§
¨
¨
§
§
¨
¨
§
§
¨
¨
§§
§§ 2 ¨¨
¨¨
§
§
¨
¨
β
£§
£§~
¨¨ 5
¨¨
5
..
Hm
(H2 ) ¨¨ Hm
(H1 ) ¨¨ 4
0
0
.
¨
¨ α
¨
¨
¨
¨
¨
¨
¨¨
¨¨
¤¨Ä ¨
¤¨¨
6
6
..
Hm
(H2 )
Hm
(H1 )
0
0
.
···
..
.
..
.
..
.
..
.
..
.
The dotted arrows shows direction of maps at step 3.
This spectral sequence abouts at step 4 because h` dij = 0 for any i, j if ` ≥ 4. The estimates we proved
on the degree where the graded components of the corresponding total complex vanishes therefore implies
Lemma 1.6.
¤
These lemmas will be used frequently in the case where D• is of the form F• ⊗R M , where F• is a graded
complex of finitely generated free R-modules, with Fi = 0 for i < 0 and M is a graded R-module. Each Fi
is of the form ⊕j R[−j]βij and we set bi (F• ) := max{j | βij 6= 0} if Fi 6= 0 and bi (F• ) := −∞ else. In this
context, the following two easy lemmas will be of use:
Lemma 1.7. Let F• be a graded complex of finitely generated free R-modules, with Fi = 0 for i < 0
and M a finitely generated graded R-module. Then
ap (Fi ⊗R M ) = ap (M ) + bi (F• ).
6
Lemma 1.8. Let F•1 , . . . , F•s be graded complexes of finitely generated free R-modules, with Fij = 0
for i < 0 and any j, and F• be the tensor product of these complexes. Then
b` (F• ) =
max
{bi1 (F•1 ) + · · · + bis (F•s )}.
i1 +···+is =`
Section 2. Regularity of Frobenius powers
In all this section R is a Noetherian ring of characteristic p > 0, F is the Frobenius functor and
NonReg(R) denotes the set of prime ideals of R such that Rp is not regular.
If R is a graded ring with a unique graded maximal ideal m (for instance R is local or standard graded
over a field), M is a finitely generated graded R-module and F• is a minimal graded free R-resolution of M ,
then any graded free R-resolution F•0 of M is isomorphic to F• ⊕ T• , where T• is a direct sum of trivial exact
id
sequences 0→R → R→0. Therefore
Hi (F e F• ) ' Hi (F e F•0 ).
because F preserves trivial exact sequences.
Notation 2.1. The isomorphism class of Hi (F e F• ) for F• a graded free R-resolution of the finitely
generated graded R-module M is denoted by DSi,e (M ).
Notice that DS0,e (M ) = F e (M ), and we have the following result,
Lemma 2.2. Let S be a standard graded ring over a field of characteristic p > 0 and M a finitely
generated graded S-module. Then DSi,e (M ) is supported in NonReg(S) ∩ Supp(M ) for any i > 0 and any e.
Proof. Let F• be a minimal graded free S-resolution of M . As F commutes with localization (see e.g.
[BH, 8.2.5]) and localization is flat, Hi (F e F• )p ' Hi (F e (F• ⊗S Sp )) ' DSi,ep (Mp ). If p is not in NonReg(S),
F is exact in Sp and therefore Hi (F e (F• ⊗S Sp )) = 0 for i > 0. On the other hand DSi,ep (Mp ) = 0 for any i
if p is not in the support of M .
¤
Theorem 2.3. Let S be a standard graded ring over a field of characteristic p > 0 and M a
finitely generated graded S-module. Assume that dim(NonReg(S) ∩ Supp(M )) ≤ 1 and set bSi (M ) :=
max{j | TorSi (M, k)j 6= 0} if TorSi (M, k) 6= 0, and bSi (M ) := −∞ else. Then,
reg(F e M ) ≤
max
0≤i≤j≤dim S
{pe bSi (M ) + aj (S) + j − i} ≤ reg(S) +
max
0≤i≤dim S
{pe bSi (M ) − i}.
Proof. Let F• be a minimal graded free S-resolution of M , therefore Fi = ⊕j S[−j]βij , where βij =
dimk TorSi (M, k)j . The complex F e F• is a complex of graded free S-modules and F e Fi = ⊕j S[−jpe ]βij .
The hypothesis implies that dim Hi (F e F• ) ≤ 1 for i > 0 by Lemma 2.2. Therefore it follows from Lemma
1.7 and Lemma 1.1 (or Lemma 1.4) that
a` (F e M ) ≤
which proves our claim.
max
{ai+` (S) + pe bSi (M )}, ∀` ≥ 0,
0≤i≤dim S−`
¤
This theorem gives a positive answer to question (2) in the introduction of [Ka] when dim(NonReg(S) ∩
Supp(M )) ≤ 1 by providing the bound
reg(F e M ) ≤ reg(S) + pe
max
0≤i≤dim S
{bSi (M )}.
Notice that Lemma 1.4 also shows that reg(DSi,e (M )) ¿ pe for any i under the hypotheses of Theorem
2.3, and Lemma 1.5 tells us that reg(F e M ) ¿ pe in the case where NonReg(S) ∩ Supp(M ) is of dimension
two if and only if a2 (DS1,e (M )) ¿ pe in this case. The question of Katzman may naturally be extended by
asking:
7
If F• is a graded complex of finite free S-modules, does one have reg(Hi (F e F• )) ¿ pe for any i ?
The theorem also gives an affirmative answer to Conjecture 4 in [Ka] in the special case below,
Corollary 2.4. Let I, J ⊆ R be two homogeneous ideals and Li be the ideal generated by Xn , . . . , Xn−i+1
for i ≥ 1. Assume that
(1) Proj(R/J) is regular outside finitely many points,
(2) for i ≥ 1, Proj(R/(J + Li )) is regular and Xn−i is not a zero divisor in R/(J + I + Li )sat .
e
Then there exists CI such that the Gröbner basis of I [p ] + J for the reverse lexicographic order is
generated in degrees at most pe CI for any e > 0.
e
e
Proof. Condition (2) implies that reg(I [p ] +J) = reg(inrev−lex (I [p ] +J)) so that we may apply Theorem
2.3 to S := R/J to get our claim.
¤
Notice that, when condition (1) is satisfied, Bertini theorem implies that, if k is infinite, there exists
Li ’s such that Proj(R/(J + Li )) is regular for i ≥ 1. These Li ’s being chosen, there exists a non empty
open subset of the linear group such that condition (2) is satisfied after applying a linear transformation
corresponding to a point in this subset.
Section 3. Regularity of tensor products of modules
To a collection of R-modules M1 , . . . , Ms is associated a collection of R-modules TorR
i (M1 , . . . , Ms ),
which are supported in TorR
0 (M1 , . . . , Ms ) = M1 ⊗R · · ·⊗R Ms , some properties of these modules are collected
in the appendix.
Proposition 3.1. Let R be a polynomial ring over a field and M, M1 , . . . , Ms be finitely generated
graded R-modules. Set Ti := TorR
i (M, M1 , . . . , Ms ), d := dim M , τ := dim T1 and
b` :=
max
i1 +···+is =`
{bi1 (M1 ) + · · · + bis (Ms )}.
Then,
(i) For p ≥ τ − 1,
ap (T0 ) ≤
max {ap+i (M ) + bi }.
0≤i≤d−p
τ −2
τ
(ii) Hm
(T0 )µ ' Hm
(T1 )µ for
µ>
max
{aτ +i−2 (M ) + bi }.
max
{aτ +i−3 (M ) + bi },
0≤i≤d−τ +2
(iii) For
µ>
0≤i≤d−τ +3
there is an exact sequence,
τ −4
τ −3
τ
τ −3
τ −1
Hm
(T0 )µ −→Hm
(T1 )µ −→Hm
(T2 )µ −→Hm
(T0 )µ −→Hm
(T1 )µ −→0.
(iv) If τ ≤ 1,
• ap (T0 ) ≤ max0≤i≤d−p {ap+i (M ) + bi } for p ≥ 0,
• a0 (Tq ) ≤ max0≤i≤d {ai (M ) + bq+i }, for q ≥ 0,
• a1 (Tq ) ≤ max0≤i≤d {ai (M ) + bq+i−1 }, for q ≥ 1.
Proof. Let F•i be a minimal finite free R-resolution of Mi and F• be the tensor product over R of the
complexes F•i . We now apply Lemma 1.7 and lemmas 1.1, 1.2, 1.3 and 1.4 to F• ⊗R M to get respectively
(i), (ii), (iii) and (iv), noticing that Hq (F• ⊗R M ) ' Tq by Lemma A.3 and condition D` (τ ) is satisfied for
any ` ≥ 1 by Theorem A.7 (i).
¤
8
Corollary 3.2. Let k be a field, Z1 , . . . , Zs be closed subschemes of Pnk and let Z := Z1 ∩ · · · ∩ Zs .
Assume that reg(Z1 ) ≥ · · · ≥ reg(Zs ) and there exists S ⊆ Z of dimension at most one such that, locally at
each point of Z − S, Z is a proper intersection of Cohen-Macaulay schemes. Then,
min{n,s}
reg(Z) ≤
X
reg(Zi ).
i=1
Proof. Let R be the polynomial ring in n +1 variables over k and I1 , . . . , Is be the defining ideals Z1 , . . . , Zs ,
respectively. Corollary A.10 shows that the hypothesis implies that dim Tor1 (R/I1 , . . . , R/Is ) ≤ 2. Therefore
by Proposition 3.1 (with M = R and Mi = R/Ii )
reg(Z) = max{ap (R/(I1 + · · · + Is )) + p}
p>0
≤ max
max
{bi1 (R/I1 ) + · · · + bis (R/Is ) + p − n − 1}
≤ max
max
{²i1 reg(R/I1 ) + · · · + ²is reg(R/Is )}
p>0 i1 +···+is =n+1−p
p>0 i1 +···+is =n+1−p
where ²j = 0 if j ≤ 0 and ²j = 1 else (note that b0 (R/Ii ) = 0 for any i). The bound follows.
¤
Corollary 3.3. Let R be a polynomial ring over a field, M a finitely generated graded R-module and
f1 , . . . , fs be forms of degrees d1 ≥ · · · ≥ ds . Set M 0 := M/(f1 , . . . , fs )M , δ 0 := dim M 0 and δ := dim M ,
then
aδ0 (M 0 ) ≤ max{aδ0 (M ), aδ0 +1 (M ) + d1 , . . . , aδ (M ) + d1 + · · · + dδ−δ0 },
and
aδ0 −1 (M 0 ) ≤ max{aδ0 −1 (M ), aδ0 (M ) + d1 , . . . , aδ (M ) + d1 + · · · + dδ−δ0 +1 }.
For instance, if dim M 0 = 1,
reg(M 0 ) ≤ max{a0 (M ), a1 (M ) + d1 , . . . , aδ (M ) + d1 + · · · + dδ }.
and if dim M 0 = 2,
0
reg(M 0 /Hm
(M 0 )) ≤ max{a1 (M ), a2 (M ) + d1 , . . . , aδ (M ) + d1 + · · · + dδ−1 } + 1.
Proof. Note that TorR
i (M, R/(f1 ), . . . , R/(fs )) ' Hi (K• (f1 , . . . , fs ; M )) by Corollary A.3 and [BH,
1.6.6], and apply Lemma 1.1.
¤
Example 3.4. Let I and J be two homogeneous ideals of R := k[X1 , . . . , Xn ] with codim(I + J) ≥
min{n − 1, codimI + codimJ}. Assume that Rp /Ip and Rp /Jp are Cohen-Macaulay for every homogeneous
prime p supported in V (I + J) such that dim R/p ≥ 2. Then the estimate given by Corollary 3.2 gives for
any p
ap (R/(I + J)) ≤ max {bi (R/I) + bn−p−i (R/J)} − n
0≤i≤n−p
and Corollary 3.3 gives
ap (R/(I + J)) ≤
max {bi (R/I) + ap+i (R/J)}.
0≤i≤n−p
The first formula is more symmetric, but on the other hand in the second the roles of I and J may be
permuted to have another estimate, and there is fewer terms in the second maximum as ap+i (R/J) = 0
for i > dim R/J − p. More significatively for proving bounds, if a0 (M ) > a1 (M ) > · · · > adim M (M ) then
bn−p−i (M ) − n ≥ ap+i (M ) for any i, so that in this case the second estimate is always sharper then the first.
Note also that the first estimate for p = 0 is at least max{a0 (R/I), a0 (R/J)} and in the second we can take
the minimum of these two terms as the only term concerning the m-primary components of I and J.
Section 4. Regularity of powers of ideals
9
m
We will here apply the results of section 3 to control the regularity of I m /I m+1 = TorR
1 (R/I, R/I ),
for m ≥ 1 in terms of invariants attached to I. We start with an example which refines previous results in
[Chan] and [GGP].
Example 4.1. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ] such that dim(R/I) ≤ 1. Note
that b0 (R/I) = 0, b1 (R/I) = b0 (I) and b2 (R/I) = b1 (I). By Proposition 3.1 (iv) applied to M := R/I m and
M1 := R/I, one has
a0 (I m /I m+1 ) ≤ max{a0 (R/I m ) + b0 (I), a1 (R/I m ) + b1 (I)}.
Now one also has a1 (I m /I m+1 ) ≤ max{a0 (R/I), a1 (R/I) + mb0 (I)}, by Proposition 3.1 (iv) applied to
M := R/I and M1 := R/I m . Therefore by an immediate recursion on m,
a0 (R/I m+1 ) ≤ max{a0 (R/I) + b0 (I), a1 (R/I) + b1 (I)} + (m − 1)b0 (I),
in particular, reg(I m+1 ) ≤ reg(I) + mb0 (I) if I is linearly presented and reg(I m+1 ) ≤ 2reg(I) + (m − 1)b0 (I)
in any case.
The next lemma will be useful to study the cases of dimension two and three.
Lemma 4.2. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ] generated in degrees d1 ≥ · · · ≥ ds
and d := dim(R/I). Assume that Ip ⊂ Rp is a complete intersection for every prime p ⊆ Supp(R/I) of
maximal dimension. Then,
ad (TorR
m (R/I, R/I)) ≤ ad (
m
^
(I/I 2 )) ≤ ad (R/I) + d1 + · · · + dm .
ψ
2
Proof. Set B := R/I. The alternating algebra structure on TorR
• (B, B) and the identification I/I −→
gives rise to a B-algebra homomorphism:
TorR
1 (B, B)
^
ψ:
^
(I/I 2 )−→TorR
• (B, B)
which is an isomorphism
at every prime p such that dim R/p = d by [BH, 2.3.9]. Therefore
Vm locally
ad (TorR
(I/I 2 )). Also if I = (f1 , . . . , fs ) with deg fi = di , there is a natural onto map
m (B, B)) ≤ ad (
Tm :=
m
m
O
^
(fi , . . . , fs )/I(fi , . . . , fs )−→ (I/I 2 )
i=1
Vm
(I/I 2 )) ≤ ad (Tm ). By
of graded modules supported in dimension at most d, which shows that ad (
Proposition 3.1 (i) we have ad (Tm ) ≤ ad (R/I) + d1 + · · · + dm , because (fi , . . . , fs )/I(fi , . . . , fs ) is generated
in degree at most di . This concludes the proof.
¤
Theorem 4.3. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ] such that dim(R/I) = 2. Assume
that Ip ⊂ Rp is a complete intersection for every prime p such that dim R/p = 2. Set ai := ai (R/I), bi := bi (I)
and let d1 ≥ · · · ≥ ds be the degrees of a minimal system of generators of I. Then a0 (TorR
n (R/I, R/I)) ≤
a0 + bn−1 and for any m > 0,
R
a0 (TorR
m (R/I, R/I)) ≤ max{a0 + bm−1 , a1 + bm , a2 + bm+1 , a2 (Torm+1 (R/I, R/I))}
≤ max{a0 + bm−1 , a1 + bm , a2 + bm+1 , a2 + d1 + · · · + dm+1 },
a1 (TorR
m (R/I, R/I)) ≤ max{a1 + bm−1 , a2 + bm }
and
a2 (TorR
m (R/I, R/I)) ≤ a2 + d1 + · · · + dm .
10
Note that TorR
m (R/I, R/I) = 0 and bm = 0 if m > n.
Proof. By Lemma 4.2, a2 (TorR
m (B, B)) ≤ a2 + d1 + · · · + dm and the result follows from Lemma 1.5
(b).
¤
Corollary 4.4. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ] such that dim(R/I) = 2 and
d1 ≥ · · · ≥ ds be the degrees of a minimal system of generators of I. Assume that Ip ⊂ Rp is a complete
intersection for every prime p such that dim R/p = 2. Then,
reg(I/I 2 ) ≤ max{a0 (R/I) + b0 (I), a1 (R/I) + b01 (I), a2 (R/I) + b02 (I), a2 (R/I) + d1 + d2 }
≤ max{reg(R/I) + max{b0 (I), b1 (I) − 1, b2 (I) − 2}, a2 (R/I) + d1 + d2 }
where b0i (I) := maxj≤i bj (I).
Proof. Recall that I/I 2 ' Tor1 (R/I, R/I) and apply Theorem 4.3.
¤
Theorem 4.5. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ] such that dim(R/I) = 3. Assume
that Ip ⊂ Rp is a complete intersection for every prime p such that dim R/p = 3. Set ai := ai (R/I),
bi := bi (I) and let d1 ≥ · · · ≥ ds be the degrees of a minimal system of generators of I. Then,
0
2
(i) there exists a natural map ψ : Hm
(I/I 2 )−→Hm
(Tor2 (R/I, R/I)) such that ψµ is onto for µ >
max{a1 + b0 , a2 + b1 , a3 + b2 } and into for µ > max{a0 + b0 , a1 + b1 , a2 + b2 , a3 + b3 , a3 + d1 + d2 + d3 },
1
(ii) Hm
(I/I 2 )µ = 0 for µ > max{a1 + b0 , a2 + b1 , a3 + b2 , a3 + d1 + d2 },
2
(I/I 2 )µ = 0 for µ > max{a2 + b0 , a3 + b1 },
(iii) Hm
3
(iv) Hm
(I/I 2 )µ = 0 for µ > a3 + b0 .
Proof. Let F• be a minimal free resolution of R/I. As I ⊂ R is a generically a complete intersection,
a3 (TorR
¤
m (R/I, R/I) ≤ a3 + d1 + · · · + dm by Lemma 4.2. The result then follows from Lemma 1.6.
Corollary 4.6. Let I be an homogeneous ideal of R := k[X1 , . . . , Xn ], B := R/I and S := Proj(B)
be the corresponding projective scheme. Assume that S is of dimension two and generically a complete
intersection. Set ai := ai (B), bi := bi (I) and let d1 ≥ · · · ≥ ds be the degrees of a minimal system of
generators of I. Then if ΩB is the module of Kähler differentials of B (over k), one has
(i) a3 (ΩB ) ≤ a3 + 1, a2 (ΩB ) ≤ max{a2 + 1, a3 + b0 },
(ii) a1 (ΩB ) ≤ max{a1 + 1, a2 + b0 , a3 + b1 } if S is generically reduced,
(iii) a0 (ΩB ) ≤ max{a0 + 1, a1 + b0 , a2 + b1 , a3 + b2 , a3 + d1 + d2 } if S is a reduced complete intersection
outside finitely many points.
Proof. We consider the exact sequence presenting ΩB and defining K :
ψ
0−→K−→I/I 2 −→ B[−1]n −→ΩB −→0,
where ψ is given by the jacobian matrix on a minimal system of generators of I. Taking cohomology on the
sequence we get (i) from Theorem 4.5 (iv). If S is generically reduced, then K is supported in dimension at
most two, and therefore (ii) follows from Theorem 4.5 (iii). In the situation of (iii) dim K ≤ 1, and Theorem
4.5 (ii) proves the claim.
¤
Section 5. Regularity results in small dimensions
In this section S is a standard graded ring over a field.
The following lemmas are useful for bounding regularity of ideals by inducting on the dimension of the
quotient ring and/or number of generators:
Lemma 5.1. Let I be a graded S-ideal and f ∈ S be a form such that depth (S/(I : (f ))) > 0. Then,
0
0
Hm
(S/I) ⊆ Hm
(S/I + (f )).
11
Proof. I ∩ (f ) = (f ).(I : (f )) so that depth (S/I ∩ (f )) > 0. Therefore the exact sequence
0−→S/I ∩ (f )−→S/I ⊕ S/(f )−→S/I + (f )−→0
0
0
0
shows that Hm
(S/I) ⊕ Hm
(S/(f )) ⊆ Hm
(S/I + (f )) and proves the assertion.
¤
Lemma 5.2. Let I be a graded S-ideal and f a form of degree δ in S and i an integer. Then,
ai (S/I + (f )) ≤ max{ai (S/I), ai+1 (S/I) + δ, ai+2 (S/I) + δ, ai+1 (S/I : (f )) + δ}.
Proof. We first consider the exact sequence,
/ ((I : (f ))/I)[−δ]
0
/ S/I[−δ]
×f
/ S/I
/ S/(I + (f ))
/ 0.
Taking cohomology, it implies that
ai (S/I + (f )) ≤ max{ai (S/I), ai+1 (S/I) + δ, ai+2 ((I : (f ))/I) + δ}
and taking cohomology on the canonical exact sequence 0→(I : (f ))/I→S/I→S/(I : (f ))→0 implies that
ai+2 ((I : (f ))/I) ≤ max{ai+1 (S/I : (f )), ai+2 (S/I)},
and concludes the proof.
¤
Lemma 5.3. Let f1 , . . . , fs be a sequence of forms in S, fi := (f1 , . . . , fi ) and let Ii be the S-ideal
generated by f1 , . . . , fi . Assume that Ii−1 and Ii coincide locally in dimension `. Then if p > `,
ap (Hj (fi ; S)) ≤ max{ap (Hj (fi−1 ; S)), ap (Hj−1 (fi−1 ; S)) + di },
∀j
where di := deg fi .
Proof. The exact sequence of [BH, 1.6.13] gives rise to exact sequences
0
/ Mj
/ Hj (fi−1 ; S)
/ Hj (fi ; S)
/ Hj−1 (fi−1 ; S)[−di ]
/ Nj
/0
where Mj and Nj have a support of dimension < ` by [BH, 1.6.21]. Therefore we get an exact sequence
/ H `+1 (Hj (fi ; S))
/ H `+1 (Hj−1 (fi−1 ; S))[−di ]
m
m
ee
e
e
e
e
e
e
e
e
eeeeee
eeeeee
reeeeee
`+2
/ H `+2 (Hj−1 (fi−1 ; S))[−di ]
/ H `+2 (Hj (fi ; S))
Hm
(Hj (fi−1 ; S))
m
dm
d
d
d
dddd
d
d
d
d
d
d
ddddddd
ddddddd
d
d
d
d
d
d
rd
···
dddd
ddddddd
d
d
d
d
d
d
ddddddd
ddddddd
d
d
d
d
d
d
rd
d
/ H d (Hj−1 (fi−1 ; S))[−di ]
/ H d (Hj (fi ; S))
Hm
(Hj (fi−1 ; S))
m
m
`+1
Hm
(Hj (fi−1 ; S))
from which the result follows.
/ 0.
¤
If I is a S-ideal and t an integer, we will denote by I ≤t the intersection of primary components of I of
codimension at most t. We will need the following result on subideals generated by general elements of a
fixed ideal:
12
Lemma 5.4. Assume that k is infinite and let I be a graded S-ideal of codimension r generated by
forms of degrees d1 ≥ · · · ≥ ds . Consider the conditions:
(1) µ(Ip ) ≤ dim Sp for every p ⊇ I of codimension r,
(2) µ(Ip ) ≤ dim Sp for every p ⊇ I ≤r of codimension ≤ r + 1,
(3) µ(Ip ) ≤ dim Sp for every p ⊇ I ≤r+1 of codimension ≤ r + 2,
(4) µ(Ip ) ≤ dim Sp + 1 for every p of codimension ≤ r + 1.
(5) µ(Ip ) ≤ dim Sp + 1 for every p of codimension ≤ r + 2.
For µ ∈ Z, let (Hj,µ )j∈Sµ be a k basis of the degree µ part of I, and set fi :=
Ii := (f1 , . . . , fi ) for 1 ≤ i ≤ s.
P
j∈Sdi
λij Hj,di and
There exists Zariski open subset Ω` of Spec(k[λij ]) such that Ω` 6= ∅ if condition (`) holds and:
(1) λ ∈ Ω1 implies I = Is and codim(I + (Ir : I)) ≥ r + 1,
(2) λ ∈ Ω2 implies I = Is , codim(Ir+1 : I) ≥ r + 1 and if λ ∈ Ω2 ∩ Ω4 , codim(I + (Ir+1 : I)) ≥ r + 2,
(3) λ ∈ Ω3 implies I = Is , codim(Ir+2 : I) ≥ r + 2 and if λ ∈ Ω3 ∩ Ω5 , codim(I + (Ir+2 : I)) ≥ r + 3,
(4) λ ∈ Ω4 implies I = Is and codim(Ir+2 : I) ≥ r + 2,
(5) λ ∈ Ω5 implies I = Is and codim(Ir+3 : I) ≥ r + 3.
Proof. See [U, 1.4] or [CEU, 2.5 (b)].
¤
Corollary 5.5. Let f1 , . . . , fs be a sequence of forms in S of degrees d1 ≥ · · · ≥ ds and I the S-ideal they
generate. Set d := dim S/I and r := codimI. If µ(Ip ) ≤ dim Sp + 1 for every p of codimension ≤ r + 2, and
µ(Ip ) ≤ dim Sp for every p ⊇ I ≤r+1 of codimension ≤ r+2, then ad (H1 (f ; S))) ≤ max{ad−2 (S/I), σr+1 +aS },
ad (Hs−r (f ; S))) = ad (ωS/I ) + aS + σs and
ad (Hj (f ; S)) ≤ max{ad−2 (S/I) + σr+j+1 − σr+2 , ad (ωS/I ) + aS + σr+j },
∀j ≥ 1.
Proof. We induct on s. Note that ad (H1 ) ≤ max{ad−2 (S/I), σr+1 + aS } by Lemma 1.2 and Hs '
ωS/I [σs + aS ], so that we may assume s ≥ r + 3.
We choose generators as in Lemma 5.4 (3) and we write I = Is−1 ∩J where Is−1 is the ideal generated by
the first s − 1 ≥ r + 2 new generators of I. By Lemma 5.4 (3), as Ir+2 ⊆ Is−1 , J is such that codimJ ≥ r + 2
and codim(Is−1 + J) ≥ r + 3. We have an exact sequence,
0−→S/I−→S/Is−1 ⊕ S/J−→S/(Is−1 + J)−→0,
and as dim S/(Is−1 + J) ≤ d − 3 we get an onto map
d−2
d−2
d−2
Hm
(S/I)−→Hm
(S/Is−1 ) ⊕ Hm
(S/J)
which shows that ad−2 (S/Is−1 ) ≤ ad−2 (S/I) and the result follows from Lemma 5.3 by recursion on s.
¤
Theorem 5.6. Let S be a standard graded algebra of dimension n and I be a graded S-ideal generated
by s forms of degrees d1 ≥ · · · ≥ ds ≥ 1. Assume that dim S/I = 2 and set ` := min{s, n − 1}, then
reg(S/I sat ) ≤ reg(S) + d1 + · · · + d` − `.
Proof. Apply Corollary 3.3 to M := S and notice that reg(S) ≥ maxi>0 {ai (S) + i}.
¤
Lemma 5.7. Let S be a standard graded Cohen-Macaulay algebra of dimension n over a field k,
I = (f1 , . . . , fs ) be a graded S-ideal of codimension r. Set —∨ := ExtrS (—, ωS ) and σ := d1 + · · · + ds . Then,
Hi (f1 , . . . , fs ; ωS )∨∨ [σ] ' Hs−r−i (f1 , . . . , fs ; S)∨ .
13
Proof. Set —∗ := HomgrS (—, k) and f := (f1 , . . . , fs ). Comparing the two spectral sequences coming
•
i
n
from Cm
K• (f ; S) gives rise to filtrations F0i ⊆ F1i ⊆ · · · ⊆ Fn−r
= Hs−i (f ; Hm
(S)) ' Hi (f ; ωS )[σ]∗ such that
n−r−j
i
(Fji /Fj−1
)∗ is supported in codimension at least r + j (it is a subquotient of Hm
(Hs−r−i−j (f ; S)))∗ '
r+j
ExtS (Hs−r−i−j (f ; S), ωS )) and exact sequences
n−r
0−→Ni −→Hm
(Hs−r−i (f ; S)))−→F0i −→0
such that Ni∗ is supported in codimension at least r + 2. Taking graded k-duals, we get exact sequences
0−→(F0i )∗ −→Hs−r−i (f ; S)∨ −→Ni∗ −→0
and
0−→Mi −→Hi (f ; ωS )[σ]−→(F0i )∗ −→0
where Mi is supported in codimension at least r + 1. We therefore get
Hs−r−i (f ; S)∨ ' ((F0i )∗ )∨∨ ' Hi (f ; ωS )[σ]∨∨
which proves our claim.
¤
If further S is Gorenstein, then this can be rewritten
Hi (f1 , . . . , fs ; S)∨∨ [σ + aS ] ' Hs−r−i (f1 , . . . , fs ; S)∨ .
In the case r = n − 1, we have
Lemma 5.8. Let S be a standard graded Cohen-Macaulay algebra of dimension n over a field k,
I = (f1 , . . . , fs ) be a graded S-ideal of codimension n − 1. Set —∗ := HomgrS (—, k), —∨ := Extn−1
(—, ωS )
S
and σ := d1 + · · · + ds . Then,
Hi (f1 , . . . , fs ; ωS )∨∨ [σ] ' Hs−n−i+1 (f1 , . . . , fs ; S)∨
and
0
0
Hm
(Hi (f1 , . . . , fs ; ωS ))[σ] ' Hm
(Hs−n−i (f1 , . . . , fs ; S))∗ .
Proof. Following the steps of the proof of Lemma 5.7, we notice that in this case Ni = 0 and Mi '
0
Hm
(Hs−n−i (f ; S))∗ , from which the result follows.
¤
In the case of quotients of dimension 2, the situation is fairly more complicated, we will restric ourselves
to the Gorenstein case for simplicity.
If M is a finitely generated S-module, we will set M unm := M/Hb0 (M ), where b is the intersection of
prime ideals associated to M that are not of minimal codimension. Therefore M unm has all its associated
primes of codimension equal to the codimension of the support of M . If M = S/I then M unm = S/I top ,
where I top is the intersection of primary components of I of minimal codimension.
Lemma 5.9. Let S be a standard graded Gorenstein algebra of dimension n over a field k, aS be its
a-invariant, I = (f1 , . . . , fs ) be a graded S-ideal with dim(S/I) = 2, where fi is a form of degree di . Set
—∗ := HomgrS (—, k) and σ := d1 + · · · + ds , then
2
Hs−n+2 (f1 , . . . , fs ; S)[aS + σ] ' ωS/I ' Hm
(S/I)∗ ,
0
1
Hm
(Hs−n+1 (f1 , . . . , fs ; S))[aS + σ] ' Hm
(S/I top )∗ ,
and there is exact sequences
1
0−→Hm
(S/I)∗ −→Hs−n+1 (f1 , . . . , fs ; S)[aS + σ]−→Hs−n+1 (f1 , . . . , fs ; S)unm [aS + σ]−→0
14
and
1
0
0−→Hm
(Hs−n+1 (f1 , . . . , fs ; S)unm )[aS + σ]−→Hm
(S/I)∗ −→Hs−n (f1 , . . . , fs ; S)[aS + σ].
Proof. The first isomorphimes are classical, and follows from Lemma 5.7. We set —∨ := Extn−2
(—, ωS ).
S
First notice that if s = n − 2 the other assertions are trivial. Therefore we may assume that s ≥ n − 1
and set f := (f1 , . . . , fs ).
•
Comparing the two spectral sequences coming from Cm
K• (f ; S) gives rise to a filtration F0 ⊆ F1 ⊆
n
∗
0
2
F2 = Hs (f ; Hm (S)) ' (S/I)[σ + aS ] with F0 ' coker(Hm (Hs−n+1 (f ; S))→Hm
(Hs−n+2 (f ; S))), F1 /F0 '
1
0
2
Hm (Hs−n+1 (f ; S)) and F2 /F1 ' ker(Hm (Hs−n (f ; S))→Hm (Hs−n+1 (f ; S))).
2
This shows first that Hm
(Hs−n+2 (f ; S))∗ ' (S/I)∨∨ [σ + aS ], and provides, by duality, three exact
sequences:
/ (S/I)[σ + aS ]
/ F1∗
/0
/ (F2 /F1 )∗
0
/ H 1 (Hs−n+1 (f ; S))∗
m
0
/ F0∗
0
/ (S/I)∨∨ [σ + aS ]
/ F1∗
/ F0∗
/0
/ H 0 (Hs−n+1 (f ; S))∗
m
/ 0.
0
These in turn identifies Hm
(Hs−n+1 (f ; S))∗ as the cokernel of the map (S/I)−→(S/I)∨∨ ' (S/I top )∨∨
shifted in degree by σ + aS and proves the second isomorphism.
n
∗
We also have a filtration G0 ⊆ G1 = G2 = Hn+1 (f ; Hm
(S)) ' Hs−n−1
(f ; S)[σ + aS ] with G0 =
0
2
2
coker(Hm
(H0 (f ; S))→Hm
(H1 (f ; S))). This implies that Hm
(H1 (f ; S))∗ ' (Hs−n−1 (f ; S))∨∨ [σ + aS ], and
provides the exact sequences
0
and
0
/ G∗0
/ H 1 (S/I)∗
m
/ Hs−n−1 (f ; S)∨∨ [σ + aS ]
/ H 0 (S/I)∗
m
/ Hs−n−1 (f ; S)[σ + aS ]
/0
/ G∗0
unm
[σ + aS ] and thereby giving the first exact sequence.
therefore proving that G∗0 ' Hs−n−1
n
∗
(S)) ' Hs−n
(f ; S)[σ +
For the last exact sequence we consider the filtration G00 ⊆ G01 ⊆ G02 = Hn (f ; Hm
0
2
0
0
aS ] with G2 /G1 = ker(Hm (H0 (f ; S))→Hm (H1 (f ; S))). By duality, and using the identification of G0 above,
it gives an exact sequence
0
/ H 1 (Hs−n+1 (f1 , . . . , fs ; S)unm )[aS + σ]
m
/ H 0 (S/I)∗
m
/ (G02 /G01 )∗
and the filtration gives a natural injection (G02 /G01 )∗ ⊆ Hs−n (f1 , . . . , fs ; S)[aS + σ].
/0
¤
This gives a fairly geometric descritption of the local cohomology in the case of an almost complete
intersection of dimension 2,
Proposition 5.10. Let S be a standard graded Gorenstein algebra of dimension n over a field k, aS
be its a-invariant, I = (f1 , . . . , fn−1 ) be a graded S-ideal with dim(S/I) = 2, where fi is a form of degree
di . Assume that J := (f1 , . . . , fn−2 ) is a complete intersection strictly contained in I. Let C be the one
dimensionnal component of Proj(S/I) and C 0 be its residual in Proj(S/J). Then,
1
a0 (S/I) = aS + d1 + · · · + dn−1 − indeg(Hm
(S/IC )),
a1 (S/I) = aS + d1 + · · · + dn−1 − indeg(IC /I),
a2 (S/I) = aS + d1 + · · · + dn−2 − indeg(IC 0 /J),
(with indeg(0) = +∞). Therefore,
0
(i) Hm
(S/I) = 0 if and only if C is arithmetically Cohen-Macaulay.
1
(ii) Hm (S/I) = 0 if and only if I is unmixed.
(iii) a0 (S/I) ≤ aS + d1 + · · · + dn−1 if C is reduced. Further if C is geometrically reduced, the inequality
is strict if and only if C is geometrically connected.
15
(iv) If η := max{µ | H 0 (C, OC (−µ)) 6= 0} is positive or C is geometrically disconnected, then
reg(S/I) = a0 (S/I) = aS + d1 + · · · + dn−1 + η.
Proof. The values for a0 and a1 directly follows from the isomorphismes
0
1
Hm
(S/I)[aS + d1 + · · · + ds ] ' Hm
(S/I top )∗
and
1
Hm
(S/I)∗ ' (IC /I)[aS + d1 + · · · + ds ]
given by Proposition 5.9, and the value of a2 is given by liaison (see e.g. [CU, 4.1 (a)]). Properties (i) and
(ii) are direct corollaries of the isomorphismes above. For (iii) and (iv) recall the exact sequence
1
0→(S/IC )µ →H 0 (C, OC (µ))→Hm
(S/IC )→0,
note that (S/IC )µ = 0 for µ < 0 and (S/IC )0 = k. Also ⊕µ H 0 (C, OC (µ)) is a finitely generated graded
S-module and therefore has no element of negative degree if it is reduced, and if C is geometrically reduced,
dimk H 0 (C, OC ) is the number of components of C over an algebraic closure of k.
For (iv) notice that a1 (S/I) + 1 ≤ d1 + · · · + dn−1 + aS , a2 (S/I) + 2 ≤ d1 + · · · + dn−2 + aS + 1 ≤
1
d1 + · · · + dn−1 + aS , and that Hm
(S/IC )0 6= 0 if C is geometrically disconnected.
¤
Cm
Example 5.11. Let bm := (y m z − xm t, z m+1 − xtm ) ⊂ Im := (z m , tm , y m z − xm t) ⊂ S := k[x, y, z, t],
:= Proj(S/Im ) and Jm ⊃ bm be the defining ideal of the monomial curve (1, m2 , m(m + 1)).
top
sat
(1) Im
= Im
= (z, t)m + (y m z − xm t) = bm : (Jm ∩ (x, z)) have regularity 2m − 1,
(2) reg(Jm ) = m2 and reg(Jm ∩ (x, z)) = m2 + 1,
(3) Γ(Cm , OCm (`)) = 0 if and only if ` < −(m − 1)2 ,
j
(4) reg(S/Im
) = (m + j − 1)(m + 1) − 2 (if m > 1).
Proposition 5.12. Let S be a standard graded Gorenstein algebra of dimension n over a field k, aS be
its a-invariant, I = (f1 , . . . , fs ) be a graded S-ideal with dim(S/I) = 2, where s ≥ n − 1 and fi is a form of
degree di , with d1 ≥ · · · ≥ ds . If the one dimensionnal component C of Proj(S/I) is generically a complete
intersection, then,
a0 (S/I) ≤ 2(d1 + · · · + dn−2 + aS + 1) + dn−1 + η,
with η := max{µ | H 0 (C, OC (−µ)) 6= 0}, unless n = 2 and d2 > aS + η + 2, in which case a0 (S/I) ≤
d1 + d2 + aS .
Proof. We may change the generators of I to assume that the fi0 s satisfies property (1) of Lemma 5.4.
Set r := n − 2. We may assume that Ir 6= I, because otherwise a0 (S/I) = −∞, and set J := Ir : I. As
2
S is Gorenstein, (Ir : I)/Ir ' ωS/I [−σ − aS ] is perfect of dimension 2, σ = d1 + · · · + dr and Hm
(ωS/I ) is
0
the graded k-dual of End(ωS/I ) ' ⊕µ∈Z H (C, OC (µ)).
It follows that reg(J/Ir ) = σ + aS + reg(ωS/I ) = σ + aS + a2 (ωS/I ) + 2 = σ + aS + η + 2.
This implies that J is generated modulo Ir in degrees ≤ σ + aS + η + 2. As J and IC do not share any
associated prime, we may choose f ∈ J of degree at most σ + aS + η + 2 which is not a zero divisor modulo
IC so that Ir : (f ) = I : (f ) have positive depth.
By Lemma 5.1, a0 (S/I) ≤ a0 (S/I + (f )). Now σ + aS + η + 2 ≥ dn unless d1 = 1 and S is a polynomial
0
ring (in which case Hm
(S/I) = 0) or n = 2 and dn > aS + η + 2. Therefore, in any of the two remaining
cases, Corollary 3.3 (with M := S as a quotient of a polynomial ring R) shows the asserted inequality.
¤
With a slightly stronger condition on the local number of generators of Proj(S/I), we have the following
result,
Theorem 5.13. Let S be a standard graded Gorenstein algebra of dimension n over a field k, aS be its
a-invariant, I = (f1 , . . . , fs ) be a graded S-ideal with dim(S/I) = 2, where s ≥ n and fi is a form of degree di ,
16
with d1 ≥ · · · ≥ ds . Let C be the one dimensionnal component Proj(S/I) and η := max{µ | H 0 (C, OC (−µ)) 6=
0}. If µ(Ip ) ≤ dim Sp for every homogeneous prime ideal p such that m 6= p ⊃ IC , then
a0 (S/I) ≤ aS + d1 + · · · + dn−1 + max{dn , η}.
Proof. We may assume that the fi0 s satisfies conditions (1) and (2) of Lemma 5.4.
We set fi := (f1 , . . . , fi ), σi := d1 + · · · + di , Ii for the ideal generated by f1 , . . . , fi , and induct on i to
bound a0 (S/Ii ).
For i ≤ n − 2, a0 (S/Ii ) = −∞ as Ii is perfect, and a0 (S/In−1 ) ≤ aS + d1 + · · · + dn−1 + η by Proposition
5.10.
If i ≥ n,
a0 (S/Ii ) ≤ max{a0 (S/Ii−1 ), a1 (S/Ii−1 ) + di , a2 (S/I) + di , a1 (S/Ii−1 : (fi )) + di }
≤ max{σn−1 + aS + max{η, dn }, σn−1 + aS + di , σn−2 + aS + di , a1 (S/Ii−1 : (fi )) + di }
≤ max{σn−1 + aS + max{η, dn }, a1 (S/Ii−1 : (fi )) + dn }
by Lemma 5.2, the induction hypothesis, Theorem 5.6 and the inequality di ≤ dn .
It remains to show that a1 (S/Ii−1 : (fi )) ≤ σn−1 + aS .
As i ≥ n, Lemma 5.4 (2) implies that for every p ⊃ IC of codimension n − 1, (In−1 )p = Ip , so that
(Ii−1 )p = (Ii )p . If codim(Ii−1 : (fi )) = n, a1 (Ii−1 : (fi )) = −∞, else Ii−1 : (fi ) = J ∩ K with J pure of
1
1
dimension one and codim(I + J) = n. Now Hm
(S/J ∩ K) ' Hm
(S/J) and a1 (S/J) ≤ σn−1 + aS by [CP, §3,
Corollaire 2 (i)].
¤
For applications it may also be of interest to bound the local cohomology of all the Koszul homology
modules. This is sometimes possible with the help of Corollary 5.5 and Lemma 1.5,
Proposition 5.14. Let S be a standard graded Gorenstein algebra of dimension n over a field k, aS
be its a-invariant, I = (f1 , . . . , fs ) be a graded S-ideal with dim(S/I) = 2, where s ≥ n and fi is a form of
degree di , with d1 ≥ · · · ≥ ds . Then for 0 < i < s − n + 2,
a1 (Hi (f ; S)) ≤ aS + d1 + · · · + dn+i−1 .
Let C be the one dimensionnal component Proj(S/I) and η := max{µ | H 0 (C, OC (−µ)) 6= 0}. If
µ(Ip ) ≤ dim Sp for every homogeneous prime ideal p such that m 6= p ⊃ IC and µ(Ip ) ≤ dim Sp + 1 for every
homogeneous prime ideal p such that m 6= p ⊃ I, then a2 (H1 (f ; S)) ≤ aS + d1 + · · · + dn−1 + max{dn , η} and
for 1 < i < s − n + 2
a2 (Hi (f ; S)) ≤ aS + d1 + · · · + dn−1 + max{dn , η} + dn+1 + · · · + dn+i−1 ,
and for 0 < i < s − n + 1
a0 (Hi (f ; S)) ≤ aS + d1 + · · · + dn−1 + max{dn , η} + dn+1 + · · · + dn+i .
Proof. The bound for a1 (Hi (f ; S)) follows directly from Lemma 1.5. By Corollary 5.5,
a2 ((H1 (f ; S)) ≤ max{a0 (S/I), aS + d1 + · · · + dn−1 + η},
and for 1 < i < s − n + 2,
a2 ((Hi (f ; S)) ≤ max{a0 (S/I) + dn+1 + · · · + dn+i−1 , aS + d1 + · · · + dn+i−2 + η},
But by Theorem 5.13,
a0 (S/I) ≤ aS + d1 + · · · + dn−1 + max{dn , η}.
17
which gives the estimate for a2 ((Hi (f ; S)). Lemma 1.5 (b) then provides the bound on a0 (Hi (f ; S)).
¤
By liaison we can give estimates on the regularity in the case where the one dimensionnal component is
generically a complete intersection,
Proposition 5.15. Let S be a standard graded Gorenstein algebra of dimension n, multiplicity eS
and a-invariant aS and I be a graded S-ideal generated by s forms of degrees d1 ≥ · · · ≥ ds . Let C be the
component of dimension one of X := Proj(S/I) and let C 0 be the residual of C in a complete intersection Y
of degrees d1 , . . . , dn−2 and assume that C 0 6= ∅. Then
η = max{0, reg(C 0 ) − reg(Y ) + 1}.
Moreover, if C 0 is reduced, or if S is geometrically reduced and C is generically a complete intersection, then
η ≤ max{0, deg(C 0 ) − reg(Y )}
= max{0, eS d1 · · · dn−2 − deg(C) − (d1 + · · · + dn−2 + aS + 2)}.
Proof. The first equality is [CU, 4.2].
To prove the second inequality, we may assume that k is algebraically closed and use Bertini theorem
to find a complete intersection b ⊂ I of degrees d1 , . . . , dn−2 such that J := b : I is radical (see for instance
[CU, 4.4 (f)] with c = r and c0 = c − 1 together with [CU, 4.5]).
Now, as C 0 or Proj(R/J) is reduced over a perfect field, and both have the same regularity and degree
eS d1 · · · dn−2 − deg(C), reg(C 0 ) ≤ deg(C 0 ) − 1 by [GLP, 1.1 and Remark (1) p. 497], applied either to C 0 or
to Proj(R/J). The asserted inequality follows.
¤
Proposition 5.16. Let S be a standard graded Gorenstein ring, Z := Proj(S), n := dim Z and I
be a graded S-ideal generated by forms of degrees d1 ≥ · · · ≥ ds . Set X := Proj(S/I). Assume that
dim X = 2 < n and the component of dimension two of X is a reduced surface S. Then,
reg(X) ≤ 2(d1 + · · · + dn−2 + aS + 2) + dn−1 ,
if s 6= n − 2.
If moreover µ(IX,x ) ≤ dim OZ,x for x ∈ S except at most at finitely many points, then
reg(X) ≤ reg(S) + d1 + d2 + · · · + d` − `,
` := min{s, n}.
Proof. We may assume that k is infinite and s ≥ n − 1. Recall that reg(S) = aS + n + 1 because S is
Cohen-Macaulay of dimension n + 1.
Let h ∈ S1 that is not in any associated prime of I except possibly m and such that S ∩ {h = 0} is
reduced (see [FOV, 3.4.14]). The ideal (I, h) satisfies the hypotheses of Proposition 5.12, and one has an
exact sequence
0
0
0
1
1
0−→(I : (h)/I)[−1]−→Hm
(S/I)[−1]−→Hm
(S/I)−→Hm
(S/I + (h))−→Hm
(S/I)[−1]−→Hm
(S/I)
so that a1 (S/I) + 1 ≤ a0 (S/I + (h)). Notice that aS/(h) = aS + 1. As the component of dimension 1 of
X ∩ {h = 0} is S ∩ {h = 0}, which is reduced and therefore have no section of negative degree, Proposition
5.12 implies that
a0 (S/I + (h)) ≤ 2(d1 + · · · + dn−2 + aS/(h) + 1) + dn−1 ,
and if further the extra hypothesis on the local number of generators is satisfied, then by Theorem 5.10 or
Theorem 5.13,
a0 (S/I + (h)) ≤ d1 + · · · + d` + aS/(h) .
These estimates together with the inequality a1 (S/I) + 1 ≤ a0 (S/I + (h)) above proves our claim.
18
¤
Proposition 5.17. Let S be a standard graded Gorenstein algebra of a-invariant aS and multiplicity
eS and I be a graded S-ideal generated in d1 ≥ d2 ≥ · · · ≥ ds . Let X := Proj(S/I) and assume that
dim X = 2 < n, X is generically a complete intersection and Proj(S) is geometrically reduced of dimension
n. If s 6= n − 2,
reg(X) ≤ d1 + · · · + dn−1 + aS + 2 + max{d1 + · · · + dn−2 + aS + 2, eS d1 · · · dn−2 − deg X}.
Proof. The proof follows along the same lines as for the proof of Proposition 5.16, using Proposition
5.15 in place of Proposition 5.12.
¤
Section 6. Local cohomology of Koszul cycles
A very useful tool in our study of regularity of powers of an ideal will be the Z-complex introduced by
Herzog, Simis and Vasconcelos that approximates a free resolution of the symmetric algebra of an ideal, in
a sense that will be clarified in the next section. We will now explain results on the local cohomology of the
Koszul cycles in the spirit of [BC, 4.1].
Proposition 6.1. Let S be standard graded algebra of dimension n over a field and M be a finitely
generated graded S-module. Let K• := K• (f ; M ) be the Koszul complex on M given by a sequence of forms
f := (f1 , . . . , fs ) of degrees d1 ≥ · · · ≥ ds and denote by Zi and Hi the i-th cycles and homology modules of
Pi
p
K• . Set d := dim H0 and σi := j=1 dj . Then Hm
(Zq )µ = 0 if either,
p+i
(Hq+i )µ = 0 for i ≥ 0; or
(i) µ > max0≤i≤min{s−q−1,d−p} {ap+i (M ) + σq+i+1 }, and Hm
p−i−1
(Hq−i )µ = 0 for 1 ≤ i ≤ p − 1.
(ii) µ > maxmax{0,p−d}≤i≤q {ap−i (M ) + σq−i } and Hm
•
K>q
Proof. We consider, as in [BC, 4.1], the double complex Cm
• where
K>q
: 0→Ks → · · · →Kq+1 →Zq →0.
•
It gives rise to two spectral sequences, one of which have as first terms,
 p

 Hm (Ki ) if s ≥ i > q,
0 p
p
Hm
(Zq ) if i = q,
1 Ei =


0
else.
p+i−1
As ap (Ki ) = ap (M ) + σi , (1 0 E q+i )µ = 0 if i ≥ 1 and µ > max0≤i≤min{s−q−1,d−p} {ap+i (M ) + σq+i+1 }
p
p
so that for such a µ, Hm
(Kq )µ ' (∞ 0 E q )µ . The other spectral sequence has as second terms:
(
p
2 Ei
00
and (i) follows as
2
00
=
p+i
p
Hm
(Hi ) if s ≥ i ≥ q,
0
else,
p+i
E q+i = 0 if i < 0 and (2 00 E q+i )µ = 0 for i ≥ 0 by hypothesis.
For (ii) we consider the complex
K≤q
: 0−→Kq −→Kq−1 −→ · · · −→K0 −→0.
•
•
The first terms of the spectral sequences derived from Cm
K≤q
• are now
(
0 p
1 Ei
=
p
Hm
(Ki )
0
if 0 ≤ i ≤ q,
else,
19
p−i
for the first one, and therefore (1 0 E q−i )µ = 0 for any i and µ > maxmax{0,p−d}≤i≤q {ap−i (M ) + σq−i }
 p

 Hm (Hi ) if i < q,
00 p
p
Hm
(Zq ) if i = q,
2 Ei '


0
else,
p−`
p−i−1
for the second one. If Hm
(Hq−i )µ = 0 for 1 ≤ i ≤ p − 1, then (` 00 E q−`+1 )µ = 0 for ` ≥ 2 so that
00 p
00 p
(2 E q )µ ' (∞ E q )µ vanishes if further µ > maxmax{0,p−d}≤i≤q {ap−i (M ) + σq−i }.
¤
p
Corollary 6.2. Hm
(Zq )µ = 0 if one of the following conditions holds:
(i) q = 0 and µ > ap (M ),
(ii) q = 1 and µ > max{ap−2 (H0 ), ap−1 (M ), ap (M ) + d1 },
(iii) p ≤ 1 and µ > max0≤i≤q {ap−i (M ) + σq−i },
p
(iv) p = d, q ≥ s or µ > max0≤i<s−q {ap+i (M ) + σq+i+1 }, and Hm
(Hq )µ = 0,
(v) p > d and µ > max0≤i<s−q {ap+i (M ) + σq+i+1 }.
Notice that the estimates for µ in (i), (iii) and (v) only depends on M and the degrees d1 , . . . , ds .
≤q
•
•
The study of the double complexes Cm
K>q
• and Cm K• as in the proof of 6.1, when (R, m) is Noetherian
local gives the following,
Proposition 6.3. Let (R, m) be a Noetherian local ring and f1 , . . . , fs elements of m.
p
(Zq (f ; M )) = 0 if either
Then Hm
p+i
i
(Hq+i (f ; M )) = 0 for i ≥ 0; or
(i) Hm
(M ) = 0 for p ≤ i ≤ p + s − q − 2 and Hm
p−i−1
(Hq−i (f ; M )) = 0 for 1 ≤ i ≤ p − 1.
(ii) depth M > p and Hm
p+i
(Hq+i (f ; M )) = 0 for i ≥ 0 and (ii) is
Note that (i) is satisfied if depth M > p + s − q − 2 and Hm
satisfied if q = 1, depth M > p and depth (M/(f )M ) > p − 2.
Section 7. Approximation complexes and the regularity of powers
In this section, R is a standard graded over a homomorphic image of a Gorenstein ring, n its dimension
and m its graded maximal ideal.
L
Let (f1 , . . . , fs ) be a s-tuple of forms in R and I the ideal they generate. The algebras RI := j≥0 I j
and Sym(I) have a natural bigrading given by the following diagram:
R[T1 , . . . , Ts ]
KK
p
KK
p
pp
KK
p
p
p
KK
p
p
KK
p
wp
%
/ RI
SymR (I)
where we set deg Ti := (di , 1) and deg x := (deg x, 0) for x ∈ R.
We let S := R[T1 , . . . , Ts ] and consider the approximation complex Z• , which is bigraded with Zi =
Zi ⊗R S, where Zi is the i-th module of cycles of the complex H• (f1 , . . . , fs ; R), or equivalently Zi is the
i-th module of cycles of the complex H• (f1 , . . . , fs ; S).
The piece of bidegree (µ, j) of Z• is:
···
/
L
|e|=j−q (Zq )µ−e·d T
/
e
L
|f |=j−q+1 (Zq−1 )µ−f ·d T
f
/ ···
where the map is given by:
Te
P
i1 <···<iq
fi1 ,...iq ei1 ∧ · · · ∧ eiq 7−→ T e
Pp
l+1
l=1 (−1)
20
P
i1 <···<iq
dil ∧ · · · ∧ eiq
fi1 ,...iq Til ei1 ∧ · · · ∧e
with the abbreviated notations: e := (e1 , . . . , es ), |e| := e1 + · · · + es , T e := T1e1 · · · Tses and e · d :=
e1 d1 + · · · + es ds .
Also we will consider the graded pieces Z•j of Z• of degrees (∗, j):
···
/
L
µ,|e2 |=j−2 (Zq )µ−e2 ·d T
e2
/
L
µ,|e1 |=j−1 (Z1 )µ−e1 ·d T
/
e1
L
µ,|e0 |=j (Z0 )µ−e0 ·d T
e0
/ 0.
Notice that this complex is of length at most min{j, s} and that there is a degree zero isomorphism
H0 (Z•j ) ' Symj (I). Also naturally (Z•j )µ = (Z• )µ,j .
•
We now look at the bigraded double complex Cm
Z• which gives rise to two spectral sequences, one of
which have as first terms:
M
0 p
p
Hm
(Zq )[−e · d]T e
1 E q (Z• ) =
|e|=j−q
and the second have as first terms
00 p
1 E q (Z• )
(
p
2 E q (Z• )
00
=
p
= Cm
(Hq (Z• )) so that
p
Hm
(Sym(I)) if q = 0,
p
Hm (Hq (Z• )) if q > 0.
Lemma 7.1. Assume (R, m) is a Noetherian local ring, f1 , . . . , fs ∈ R and ² is an integer. If Z• (f ; M )
²−i−1
i
is acyclic outside V (m), depth M ≥ max{s − q − 1, ²}, Hm
(Hq−i ) = 0 for 1 ≤ i ≤ ² − 1 and Hm
(Hq+i ) = 0
0
for i ≥ ², then Hm (Hq (Z• )) = 0.
p
p
Proof. As Z• is acyclic outside V (m), 2 00 E q (Z• ) ' ∞ 00 E q (Z• ).
i
Hm
(Zq+i ) = 0 for i ≥ 0, so that the result follows from Proposition 6.3.
0
Therefore, Hm
(Hq (Z• )) = 0 if
¤
Corollary 7.2 Assume (R, m) is a Noetherian local ring, I = (f1 , . . . , fs ) a R-ideal and M is a finitely
generated R-module. If I contains a non zero divisor on M and for every prime ideal p, there exists εp ≥ 0
such that µ(Ip ) ≤ depth Mp + 1 − εp and depth (Hs−j (f ; Mp )) ≥ depth Mp − j − εp , then Z• (f ; M ) is acyclic.
Proof. If Z• (f ; M ) is not acyclic, let p be a minimal prime in the support of Z• (f ; M ). We may localize
at p to assume that m = p and Z• (f ; M ) is acyclic outside V (m). The homology of Z• (f ; M ) is independant
of the system of generators, as well as the depth condition on Koszul homology and the existence of a non
zero divisor in I, so that we may assume that s = µ(I).
We have by hypothesis p := depth M ≥ s + εm − 1 and depth (Hi (f ; M )) ≥ p − s + i − εm ≥ i − 1.
For q ≥ 2, p ≥ s + εm − 1 ≥ s − q − 1 and depth (Hq+i (f ; M )) ≥ p + q + i − s − εm ≥ q + i − 1 ≥ i + 1 and
0
i
(Hq (Z• (f ; M ))) = Hq (Z• ) = 0 by Lemma
(Hq+i (f ; M )) = 0 for i ≥ 0 which implies that Hm
therefore Hm
7.1.
i
Now notice that Hm
(Zi+1 (f ; M )) = 0 unless i = s − 1 and εm = 0 by Proposition 6.3 because either
i
p > i and Hm (Hi+2 (f ; M )) = 0 for 0 ≤ i ≤ p − 2 or p ≤ i which gives i + 1 ≥ s + εm .
i
s−1
s−1
If i = s − 1 and εm = 0, Hm
(Zi+1 (f ; M )) = Hm
(Zs (f ; M )) = Hm
(Hs (f ; M )) but depth M ≥ s − 1
by hypothesis and therefore depth M > 0 which shows that Hs = 0 unless s = 1 because I contains a non
zero divisor on M . Finaly, if s = 1, Z• is always acyclic because its only differential is the restriction to
Z1 ⊗R R[T ] of the multiplication by T from M ⊗R R[T ] to itself.
¤
Lemma 7.3. Let R be a Cohen-Macaulay standard graded algebra of dimension at least 2 over a field
and I = (f1 , . . . , fs ) be a homogeneous R-ideal generated in degrees d1 ≥ · · · ≥ ds . Set di := 0 for i > s. If
Z• is acyclic outside V (m), and
µ > (j − 2)d1 + max{d1 + · · · + dq+n+1 + aR , max ai (Hq+i )}
i≥2
then
0
Hm
(Hq (Z•j ))µ = 0.
0
Notice that if q > 0, Hm
(Hq (Z• )) = Hq (Z• ) under the hypothesis of the lemma.
21
`
Proof. We first remark that Hm
(Zq+` ) = 0 for ` < 2 by Proposition 6.3 (ii). By Proposition 6.1 (i),
`
if ` ≥ 2 and |e| = j − ` ≤ j − 2 then Hm
(Zq+` )µ−e·d = 0. Therefore the conclusion follows from Lemma
1.4.
¤
The following variant of Lemma 5.9 will give an improvement of the acyclicity results for approximation
complexes, over Gorenstein rings, for ideals of deviation at most three:
Lemma 7.4. Let (R, m) be a Gorenstein local ring of dimension n. Let I be an ideal of codimension
n − 2 and I top its unmixed part. The following are equivalent,
(i) I satisfies sliding depth,
(ii) R/I top is Cohen-Macaulay.
Proof. Set —∗ := HomR (—, E), where E is an injective hull of R/m and let I = (f1 , . . . , fs ).
1
0
(R/I top ). The proof
(Hs−n+1 (f1 , . . . , fn+1 ; R))∗ ' Hm
Then the result follows from the isomorphism Hm
of this isomorphism follows along the same lines as in the proof of Lemma 5.9, replacing graded k-duals by
Matlis duals.
¤
Remark 7.5. If in the preceeding lemma, we only assume that R is Cohen-Macaulay, the same proof
shows that I satisfies sliding depth if and only if the canonical map from ωR /IωR to its S2 -ification is onto.
Proposition 7.6. Let R be a standard graded Cohen-Macaulay algebra of dimension n, I be a graded
R-ideal generated by s forms of degrees ≤ d1 , d := dim R/I and ` ≥ 0. Then,
(i) If for any prime p of dimension ≥ `, Z• ⊗R Rp is acyclic, then
ai (Symj (I)) ≤
max
0≤p≤min{j,n−i}
{ai+p (Zp ) + (j − p)d1 }
for i ≥ ` − 2.
(ii) If for any prime p of dimension ≥ `, M• ⊗R Rp is acyclic, then
ai (Symj (I/I 2 )) ≤
max
0≤p≤min{j,d−i}
{ai+p (Hp ) + (j − p)d1 }
for i ≥ ` − 2.
(iii) If for any prime p of dimension ≥ `, Ip ⊆ Rp is of linear type,
a`−1 (I j ) ≤ a`−1 (Symj (I)) and
ai (I j ) = ai (Symj (I)) ∀i ≥ `,
a`−1 (I j /I j+1 ) ≤ a`−1 (Symj (I/I 2 )) and
ai (I j /I j+1 ) = ai (Symj (I/I 2 ))
∀i ≥ `.
p
p
(Hq ) = 0 for p > d.
and Hm
Proof. We first note
L that Hm (Zq ) = 0 for p j> n L
j
Recall that Zp = |e|=j−p Zp [−e·d] and Mp = |e|=j−p Hp [−e·d]. By assumption, condition D1 (`−1)
is satisfied for the complex Z•j in (i) and the complex Mj• in (ii). Therefore Lemma 1.1 applied to Z•j and
Mj• proves respectively (i) and (ii).
We now turn to (iii), and notice that the hypothesis implies that the natural surjections
Symj (I)−→I j
and Symj (I/I 2 )−→I j /I j+1
have kernels supported in dimension at most ` − 1. Therefore,
i
i
Hm
(Symj (I)) ' Hm
(I j )
i
i
and Hm
(Symj (I/I 2 )) ' Hm
(I j /I j+1 )
`
`
`
`
for i ≥ ` and the natural maps Hm
(Symj (I))→Hm
(I j ) and Hm
(Symj (I/I 2 ))→Hm
(I j /I j+1 ) are onto. This
provides the equalities and upper bounds of (iii).
¤
Let us recall the following fact,
Proposition 7.7. If Z• is acyclic, then I is of linear type if and only if M• is acyclic.
We now collect in a proposition the acyclicity criterions that we will use for the Z-complex,
22
Proposition 7.8. Let I = (f1 , . . . , fs ) be an ideal in R. Then,
(1) Assume that ` ≥ dim R/I − 1, then
(1.1) if µ(Ip ) ≤ dim Rp + 1 for every p ⊇ I such that dim R/p ≥ ` then Z• ⊗R Rp = 0 is acyclic for every
p such that dim R/p ≥ `,
(1.2) if µ(Ip ) ≤ dim Rp for every p ⊇ I such that dim R/p ≥ ` then M• ⊗R Rp = 0 is acyclic for every
p such that dim R/p ≥ `,
(2) Assume that ` = dim R/I − 2, then
(2.1) if µ(Ip ) ≤ dim Rp for every p ⊇ I such that dim R/p ≥ ` then Z• ⊗R Rp = 0 is acyclic for every p
such that dim R/p ≥ `,
(2.2) if µ(Ip ) ≤ dim Rp + 1, Rp is Gorenstein, and (R/I top )p is Cohen-Macaulay for every p ⊇ I such
that dim R/p ≥ ` then Z• ⊗R Rp = 0 is acyclic for every p such that dim R/p ≥ `,
(2.3) if µ(Ip ) ≤ dim Rp , Rp is Gorenstein and (R/I top )p is Cohen-Macaulay for every p ⊇ I such that
dim R/p ≥ ` then M• ⊗R Rp = 0 is acyclic for every p such that dim R/p ≥ `.
(3) Assume that ` = dim R/I − 3, then if µ(Ip ) ≤ dim Rp , Rp is Gorenstein and ωRp /Ip is CohenMacaulay for every p ⊇ I such that dim R/p = ` then Z• ⊗R Rp = 0 is acyclic for every p such that
dim R/p ≥ `.
Proof. (1.1) and (1.2) follows from Corollary 7.2 with εp = 0. (2.1) follows from Corollary 7.2 with
εp = 1. (2.2) and (2.3) follows from Lemma 7.4 and Corollary 7.2 with εp = 0.
For proving (3), by (2.1) it suffices to consider the case of a deviation 3 ideal I = (g1 , . . . , gn ) in a local
ring (R, m) with dim R/I = 3.
0
(H2 (g1 , . . . , gn ; R)) =
In this case, Corollary 7.2 with εp = 1 implies that it suffices to show that Hm
0
2
2
0. But we have an isomorphism Hm (H2 (g1 , . . . , gn ; R)) ' Hm (H3 (g1 , . . . , gn ; R)) ' Hm
(ωR/I ), and by
assumption ωR/I is a Cohen-Macaulay module of dimension 3.
¤
The next result gives a control on the four highest non trivial local cohomology modules of the symmetric
powers of an ideal,
Theorem 7.9. Let R be a standard graded algebra
P` of dimension n and I be a graded R-ideal generated
by s forms of degrees d1 ≥ · · · ≥ ds . Set σ` :=
i=1 di and ηp := max2≤i≤s−1 {ap+i (R) + σi+1 } (and
ηp := −∞ if s ≤ 2) then
(1) For p ≥ d − 1,
ap (Symj (I)) ≤ max{ap (R) + jd1 , max{ap−1 (R/I), ap−1 (R)} + (j − 1)d1 , ηp + (j − 2)d1 }.
(2) If ad−1 (Sym2 (I)) > max{ad−1 (R) + 2d1 , ad−2 (R) + d1 , ηd−1 } then
ad−1 (Symj (I)) = ad (H1 ) + (j − 2)d1 = ad−2 (R/I) + (j − 2)d1 ,
∀j ≥ 2.
(3) If µ(Ip ) ≤ dim Rp + 1 for every minimal prime p of I of maximal dimension, then
ad−2 (Symj (I)) ≤ max{ad−2 (R) + jd1 , max{ad−3 (R/I), ad−3 (R)} + (j − 1)d1 , max{ηd−2 , ad (H2 )} + (j − 2)d1 }.
Proof. We will apply the lemmas of section 1 to the complex D•j := Z•j (f ; R), where f is a s-tuple of
generators of I of degrees d1 ≥ · · · ≥ ds .
We first estimate δpj (i.e. the value of the constant δp as defined in section 1 for the complex D•j ) for
L
p
max{0, d−2} ≤ p ≤ d+1 using Lemma 6.2. To this end, remember that Zqj (f ; R) = |e|=j−q Hm
(Zq )[−e·d]
j
and notice that e · d ≤ (j − q)d1 for q ≤ j (and Zq (f ; R) = 0 otherwise). Therefore Corollary 6.2 (i), (ii)
and (v) gives for max{0, d − 1} ≤ p ≤ d + 1,
δpj ≤ max{jd1 + ap (R), (j − 1)d1 + max{ap−1 (R/I), ap−1 (R)}, (j − 2)d1 + ηp }.
23
Also Corollary 6.2 (i), (ii), (iv) and (v) shows that
j
δd−2
≤ max{jd1 + ad−2 (R), (j − 1)d1 + max{ad−3 (R/I), ad−3 (R)}, (j − 2)d1 + max{ηd−2 , ad (H2 )}}.
As the homology of Z•j (f ; R) is supported in V (I), condition D` (d) is satisfied for any `. Therefore
Lemma 1.1 implies (1). Also if µ(Ip ) ≤ dim Rp + 1 for every minimal prime p of I of maximal dimension,
then Proposition 7.8 (1.1) shows that D` (d−1) is satisfied for any `, so that Lemma 1.1 implies the inequality
in (3).
¤
Proposition 7.10. Let I ⊂ R = k[X1 , . . . , Xn ] be an ideal generated in degrees d1 ≥ · · · ≥ ds , such
that that dim R/I = 3 and µ(Ip ) ≤ dim Rp for all p such that dim R/p ≥ 2, then for j ≥ 2,
(1) If s = n − 3, a0 (R/I j ) = −∞.
(2) If s = n − 2, a0 (R/I j ) ≤ (j − 1)d1 + a0 (R/I).
(3) If s = n − 1, a0 (R/I j ) ≤ (j − 1)d1 + max{a0 (R/I), a3 (ωR/I ) + d2 + · · · + dn−1 − n}.
(4) If s ≥ n,
a0 (R/I j ) ≤ (j − 2)d1 + max{a0 (R/I) + d1 , a1 (R/I) + dn , d1 + · · · + dn−1 + max{dn , a3 (ωR/I )} − n}.
In particular if further the unmixed part of I is a radical ideal,
a0 (R/I j ) ≤ (j − 1)d1 + max{a0 (R/I), a1 (R/I), d2 + · · · + d` − n},
where ` := min{s, n}.
Proof. Case (1) is clear, so that we will assume that s ≥ n − 2. By Proposition 7.6 (iii) and Proposition
7.8 (1.2), we have a0 (R/I j ) = a1 (I j ) ≤ a1 (Symj (I)). Therefore, Theorem 7.9 (3) shows that
a0 (R/I j ) ≤ (j − 2)d1 + max{a0 (R/I) + d1 , a3 (H2 ), η1 }.
If s < n then η1 = −∞ and further H2 = 0 if s < n − 1. If s = n − 1, a3 (H2 ) = a3 (ωR/I ) + d1 + · · · + dn−1 − n
and by Corollary 5.5 a3 (H2 ) ≤ max{a1 (R/I) + dn , a3 (ωR/I ) + d1 + · · · + dn−1 − n} if s ≥ n. The estimates
follows.
¤
Proposition 7.11. Let I ⊂ R = k[X1 , . . . , Xn ] be an ideal generated in degrees d1 ≥ · · · ≥ ds . Assume
that dim R/I = 4 and µ(Ip ) ≤ dim Rp for all p such that dim R/p ≥ 3, then for j ≥ 2,
(1) If s = n − 4, a1 (R/I j ) = −∞.
(2) If s = n − 3, a1 (R/I j ) ≤ (j − 1)d1 + a1 (R/I).
(3) If s = n − 2, a1 (R/I j ) ≤ (j − 1)d1 + max{a1 (R/I), a4 (ωR/I ) + d2 + · · · + dn−2 − n}.
(4) If s ≥ n − 1,
a1 (R/I j ) ≤ (j − 2)d1 + max{a1 (R/I) + d1 , a2 (R/I) + dn−1 , d1 + · · · + dn−2 + max{dn−1 , a4 (ωR/I )} − n},
In particular if further the unmixed part of I is a radical ideal,
a1 (R/I j ) ≤ (j − 1)d1 + max{a1 (R/I), a2 (R/I), d2 + · · · + d` − n},
where ` := min{s, n − 1}.
Proof. By Proposition 7.6 (iii) and 7.8 (1.2), we have a1 (R/I j ) = a2 (I j ) ≤ a2 (Symj (I)). Therefore,
Theorem 7.9 (3) shows that
a1 (R/I j ) ≤ (j − 2)d1 + max{a1 (R/I) + d1 , a4 (H2 ), η2 }
If s < n−1 then η2 = −∞ and further H2 = 0 if s < n−2. If s = n−2, a4 (H2 ) = a4 (ωR/I )+d1 +· · ·+dn−2 −n
and by Corollary 5.5, a4 (H2 ) ≤ max{a2 (R/I) + dn−1 , a4 (ωR/I ) + d1 + · · · + dn−2 − n} if s ≥ n − 1. The
estimates follows.
¤
24
Theorem 7.12. Let I ⊂ R = k[X0 , . . . , Xn ] be an ideal generated in degrees d1 ≥ · · · ≥ ds . Set
` := min{s, n}, X := Proj(R/I) ⊆ Pnk and X j := Proj(R/I j ) for j ≥ 2.
Assume that dim X ≤ 3 and µ(Ip ) ≤ dim Rp for all p ∈ Supp(R/I) of codimension at most n − 2. In
the case dim X = 3, assume further that the component of dimension 3 of X is reduced, then
reg(X j ) ≤ (j − 1)d1 + max{reg(X), d2 + · · · + d` − n}.
Proof. If dim X ≤ 1, this follows from Theorem 7.9 (1). The case dim X = 2 and X is generically a complete
intersection follows from Proposition 7.6 (iii) and Theorem 7.9 (1). These results also gives the bound for the
three highest non trivial local cohomology modules in the case dim X = 3. Finally the bound for a1 (R/I j )
in the case dim X = 3 is given in Proposition 7.11.
¤
Combining the results with the bounds obtained in section 5, we get for instance the following result,
Proposition 7.13. Let R be a standard graded Gorenstein ring, aR its a-invariant, Z := Proj(R) and
I be a graded R-ideal generated by forms of degrees d1 ≥ · · · ≥ ds and let X j := Proj(R/I j ).
Assume that Z is geometrically reduced of dimension n, dim X = 2, s 6= n − 2 and the component of
dimension two of X is a reduced surface S. Then, dn ≤ aR + 2 or n > 2,
reg(X j ) ≤ (j − 1)d1 + 2(d1 + · · · + dn−2 + aR ) + dn−1 + 3,
and else reg(X j ) ≤ jd1 + d2 + aR + 1.
If moreover µ(IX,x ) ≤ dim OZ,x for x ∈ S except at most at finitely many points, then
reg(X j ) ≤ reg(R) + jd1 + d2 + · · · + d` − `.
Also these results may be combined with the bounds in [CU] or [Char], giving for example the following
result in the spirit of [BEL] for schemes of dimension at most three:
Proposition 7.14. Let R be a standard graded Gorenstein algebra and I be a graded R-ideal of
codimension r > 0 generated in degrees d1 ≥ d2 ≥ · · · ≥ ds ≥ 1. Assume that Proj(R) is smooth of
dimension n, dim X ≤ 3, X is unmixed and either
(i) char(k) = 0, X is locally a complete intersection oustide finitely many points and dim Sing(X) ≤ 1;
or
(ii) char(k) = p ≥ dim X, dim Sing(X) ≤ 0, X is locally a complete intersection and lifts to W2 (k).
Then, setting X j := Proj(R/I j ) and ` := min{s, n}, one has
reg(X j ) ≤ reg(R) + (j − 1)d1 + max{d1 + · · · + dr − r, d2 + · · · + d` − n}.
Proof. This is immediate from Theorem 7.12 and [CU, 4.7 (a)].
¤
Section 8. Control on torsion in the symmetric algebra of an ideal and applications
We will now give results on the torsion in the symmetric powers of a homogeneous ideal in a polynomial
ring that extends the ones of [BJ] and [BC] to the case of a two dimensionnal quotient. As a consequence,
we also generalize the implicitization algorithm to some cases where the base locus of the rational map is of
dimension one.
Theorem 8.1. Let I ⊂ R := k[X1 , . . . , Xn ] be an homogeneous ideal generated in degrees d1 ≥ d2 ≥
· · · ≥ ds ≥ 1. Set µ0 := d1 + · · · + dmin{s,n+1} − n.
(1) If dim R/I ≤ 1,
0
Hm
(Symj (I))µ = 0, ∀µ > (j − 2)d1 + µ0 .
25
(2) If dim R/I = 2, assume that µ(Ip ) ≤ dim Rp + 1 for every p ∈ Supp(I) of maximal dimension. Then,
0
Hm
(Symj (I))µ = 0,
∀µ > (j − 2)d1 + max{µ0 , a2 (H2 )}.
0
2
0
Moreover Hm
(Sym2 (I))>µ0 ' Hm
(H2 )>µ0 ' Hm
(H1 )>µ0 .
Proof. (1) follows from Corollary 3.3 (with M := R) and Theorem 7.9 (1), and (2) follows from Theorem
7.9 (3).
¤
We will denote by Z• [µ] the following subcomplex of Z• :
···
/
L
|e2 |=2 (Z2 )µ+e2 ·d
⊗R S
/
L
|e1 |=1 (Z1 )µ+e1 ·d
⊗R S
/ (Z0 )µ ⊗R S
/0
Notice that the maps are homogeneous if we set deg Ti := di , but that Z• [µ] is not the complex obtained
by taking the components of bidegree (µ, ∗) of Z• . Nevertheless, this complex inheritates of the bigrading of
Z• and the part of bidegree (ν, j) of Z• [µ] is obtained by setting deg Ti := di and taking the degree ν part
of Z• [µ] for this new graduation.
L
For instance, the part of bidegree (ν, j) of (Z0 )µ is |e|=j,ν=µ+e·d (Z0 )µ T e .
(s+q−1
)
q
Also in the case where all the di ’s are equal to a same number d one has Zq [µ] = (Zq )µ+qd
⊗R S.
As a consequence of the acyclicity criterions of the last section and the above result we get the following,
that has an application to the implicitization problem,
Proposition 8.2. Let I = (f1 , . . . , fn+1 ) ⊂ k[X1 , . . . , Xn ] be an homogeneous ideal of codimension
n−2 generated in degrees d1 ≥ · · · ≥ dn+1 . Then Z• is acyclic outside V (m) if and only if µ(Ip ) ≤ dim Rp +1
for every homogeneous prime ideal p 6= m, and in this case
(1) Z• is acyclic if and only if R/I top is Cohen-Macaulay,
(2) Z• [µ] is acyclic for µ > d1 + · · · + dn − 2dn+1 + ` − n if C := Proj(R/I top ) has no section of degree
smaller than −`.
Proof. Corollary 7.2 shows that Z• is acyclic outside V (m) if and only if µ(Ip ) ≤ dim Rp + 1 for every
homogeneous prime ideal p 6= m and (1) follows from Lemma 7.4, which also shows that
0
1
Hm
(H2 )µ ' Hm
(R/I top )−µ+d1 +···+dn+1 −n ,
so that a0 (H2 ) ≤ d1 + · · · + dn+1 + ` − n if C has no section of degree less than −`.
2
Applying Proposition 6.1 (ii) we get that Hm
(Z3 )µ = 0 for µ >La0 (H2 ), and by Corollary 6.2 (v)
i+2
Hm (Z3+i )µ = 0 for i > 0 and µ > d1 + · · · + dn+1 − n. Now Zp [µ] = |ep |=p (Zp )µ+ep ·d ⊗R S. Therefore
i+2
2
Hm
(Z3+i [µ]) = 0 for µ > d1 + · · · + dn − (i + 2)dn+1 − n if i > 0 and Hm
(Z3 [µ]) = 0 for µ > d1 + · · · + dn −
2dn+1 − n.
i
It follows that Hm
(Zi+1 [µ]) = 0 for µ > d1 + · · · + dn − 2dn+1 + ` − n and any i, which in turn implies
that H1 (Z• [µ]) = 0.
On the other hand, Hi (Z• [µ]) = 0 for i ≥ 2 by Lemma 7.1 and therefore Z• [µ] is acyclic for µ >
d1 + · · · + dn − 2dn+1 + ` − n.
¤
Corollary 8.3. Let φ : Pn−1 · · · →Pn be a rational map defined by n + 1 polynomials of degree d,
I be the ideal generated by these polynomials and X ⊂ Pn−1 be the scheme defined by I. Assume that
dim X = 1 and let C be the one dimensionnal component of X. If C has no section of degree < −d and for
any closed point x ∈ X
(
n − 1 if x ∈ C
µ(IX,x ) ≤
n
else,
then det(Z• [µ]) is a non-zero multiple of the equation of the closure of the image of φ in Pn for µ ≥
(n − 1)(d − 1).
L
Proof. First notice that H0 (Z• [µ]) = j Symj (I)µ+jd .
26
0
0
0
As Hm
(R) = Hm
(Sym1 (I)) = 0, we have Hm
(H0 (Z• [µ])) = 0 for µ > (n − 1)d − n if a2 (H2 ) ≤
(n + 1)d − n by Theorem 8.1. Thus it follows from Proposition 5.14 and Proposition 8.2 that Z• [µ] is acyclic
0
and Hm
(H0 (Z• [µ])) = 0 for µ ≥ (n − 1)(d − 1). The conclusion follows along the same lines as in the proof
of [BC, 4.1].
¤
Appendix. Some properties of Tor modules.
Definition A.1. Let A be a ring, M1 , . . . , Ms be A-modules and F• be the tensor products over A of
the canonical free resolutions of M1 , . . . , Ms . Then
TorA
i (M1 , . . . , Ms ) := Hi (F• ).
0
If F•0 is the product of any choice of flat resolutions of M1 , . . . , Ms , then TorA
i (M1 , . . . , Ms ) ' Hi (F• ).
Proposition A.2. Let A be a ring, M be an A-module and F• be a complex of flat A-modules. If L•
is a resolution of M , there is a natural isomorphism,
Hi (F• ⊗A L• ) ' Hi (F• ⊗A M ).
1
2
1
Proof. Consider the spectral sequence with Eij
= Hj (Fi ⊗A L• ) = Fi ⊗A Hj (L• ) and Eij
= Hi (E•j
)
that abouts to Hi (F• ⊗A L• ) and note that by hypothesis Hj (L• ) = 0 for j 6= 0 and H0 (L• ) = M .
¤
Corollary A.3. Let A be a ring, M, M1 , . . . , Ms be A-modules and F• be the tensor products over A
of free resolutions of M1 , . . . , Ms . Then,
TorA
i (M1 , . . . , Ms , M ) ' Hi (F• ⊗A M ).
If (R, m) is local regular of dimension n, this result implies that TorR
i (M1 , . . . , Ms ) = 0 for i > n(s − 1).
(M
,
.
.
.
, Ms ) ' R/m 6= 0.
Also notice in this case that if M1 = · · · = Ms = R/m, TorR
1
n(s−1)
Proposition A.4. Let A be a ring and M1 , . . . , Ms , N1 , . . . , Nt be A-modules. There exists a spectral
sequence,
A
A
2
= TorA
Eij
i (M1 , . . . , Ms , Torj (N1 , . . . , Nt )) ⇒ Tori+j (M1 , . . . , Ms , N1 , . . . , Nt ).
Proof. Let F• and L• be the tensor products over A of free resolutions of M1 , . . . , Ms and N1 , . . . , Nt ,
respectively. The double complex Cij := Fi ⊗A Lj gives rise to two spectral sequences both abouting to
2
H• (Tot(C•• )) ' TorA
• (M1 , . . . , Ms , N1 , . . . , Nt ). One of them have as second terms Eij = Hi (F• ⊗A Hj (L• ))
and the result follows from Corollary A.3.
¤
Proposition A.5. If A−→B is a flat map and M1 , . . . , Ms are B-modules, then
B
TorA
i (M1 ⊗B A, . . . , Ms ⊗B A) ' Tori (M1 , . . . , Ms ) ⊗B A
for every i.
Proof. This is a special case of [EGA III, 6.9.2].
¤
Remark A.6. If I1 , . . . , Is are ideals in a ring A, then TorA
1 (A/I1 , . . . , A/Is ) admits the following
(A/I,
A/J)
' (I ∩ J)/IJ.
description that generalizes the classical isomorphism TorA
1
Let M be the submodule of I1 ⊕ · · · ⊕ Is of tuples x = (x1 , . . . , xs ) such that x1 + · · · + xs = 0. Then
M contains the submodule P generated by the tuples x such that x` = 0 except for two indices i and j and
xi = −xj ∈ Ii Ij . Then,
TorA
1 (A/I1 , . . . , A/Is ) ' M/P.
Theorem A.7. Let R be a regular local ring containing a field and M1 , . . . , Ms be finitely generated
R-modules. Then,
27
R
(i) TorR
i (M1 , . . . , Ms ) = 0 implies Torj (M1 , . . . , Ms ) = 0 for all j ≥ i.
(ii) Let j := max{i | TorR
i (M1 , . . . , Ms ) 6= 0}. Then
pdimM1 + · · · + pdimMs = dim R + j − ε,
R
with 0 ≤ ε ≤ dim TorR
j (M1 , . . . , Ms ). Moreover, ε ≥ ε0 := mini {depth Torj−i (M1 , . . . , Ms ) + i} and the
equality holds if ε0 = depth TorR
j (M1 , . . . , Ms ).
Proof. First we complete R for the m-adic filtration (where m is the maximal ideal of R), and use Cohen
structure theorem to reduce to the case of a power series ring over k := R/m. Notice that R−→R̂ is flat and
as the Mi ’s are finite M̂i = Mi ⊗R R̂, dimR Mi = dimR̂ M̂i and depth R Mi = depth R̂ M̂i for every i.
We let n := dim R, and may assume that R = k[[X1 , . . . , Xn ]].
Consider S the completed tensor product over k of s copies of R. Notice that S ' k[[Xij | 1 ≤ i ≤
n, 1 ≤ j ≤ s]], if we set Xij := 1 ⊗ · · · ⊗ 1 ⊗ Xi ⊗ 1 · · · ⊗ 1 where Xi is at the j-th position. The ring S is
local regular with maximal ideal mS = (Xij ) = dS + m/dS , where dS defined by the exact sequence
mult
0−→dS −→S −→ R−→0.
Notice that dS is generated by a regular sequence, for instance f := (Xij − Xi(j+1) | 1 ≤ i ≤ n, 1 ≤ j < s)
is a minimal generating n(s − 1)-tuple. Considering R as an S-module via the above exact sequence, for any
tuple L1 , . . . , Lt of free R-modules, there is a natural isomorphism
∼
ˆk ···⊗
ˆ k Lt ) ⊗S R.
L1 ⊗R · · · ⊗R Lt −→ (L1 ⊗
(i)
(1)
(s)
ˆk ···⊗
ˆ k F•
ˆk ···⊗
ˆ k Ms choose a minimal free R-resolution F• of Mi , set F•k := F• ⊗
We put P := M1 ⊗
(s)
(1)
and F•R := F• ⊗R · · · ⊗R F• .
Notice that F•k is a minimal free S-resolution of P and K• (f ; S) is a minimal free S-resolution of
R = S/dS . We therefore have isomorphismes,
S
R
k
TorR
i (M1 , . . . , Ms ) ' Hi (F• ) ' Hi (F• ⊗S R) ' Tori (P, R) ' Hi (K• (f ; P )),
and (i) follows by [BH, 1.6.31] as (S, mS ) is local Noetherian and dS ⊂ mS .
•
let T • be the total complex associated to the double complex Cm
K• (f ; P ), with T i :=
S
L For (ii)
p
p−q=i CmS Kq (f ; P ) and choose the upper index as line index. We will estimate in two ways the number θ := min{i | H i (T • ) 6= 0}.
We have noticed above that the projective dimension of P (over S) is equal to the sum of the projective
dimensions of the Mi ’s (over R). Equivalently the depth δ of P is equal to the sum of the depths of the
Mi ’s.
The homology of T • is the aboutment of two spectral sequences associated to the horizontal and vertical
filtrations of the double complex.
p
p
(n(s−1)
) . Therefore
q
For the vertical filtration, we have as first terms v1 Eqp = Hm
S Kq (f ; P ) ' HmS (P )
v δ
v
δ
v p
1 Eq = 0 for q > n(s−1) and for p < depth P = δ which shows that θ ≥ δ−n(s−1). Also 2 En(s−1) ' ∞ En(s−1)
is Matlis dual to Extns−δ
(P, ωS )/dS which is not zero by Nakayama’s lemma. We deduce that θ = δ−n(s−1).
S
p
R
The other spectral sequence has second terms h2 Eqp ' Hm
S (Torq (M1 , . . . , Ms )), and an easy computation
h p
h
p
gives ε0 − j = min{p − q | 2 Eq 6= 0} so that θ = min{p − q | ∞ Eq 6= 0} =: ε − j ≥ ε0 − j and it is clear form
ε0
h ε0
h
its definition that ε0 ≥ 0. Also if ε0 = depth TorR
j (M1 , . . . , Ms ) we have 2 Ej ' ∞ Ej which implies that
θ ≤ ε0 − j and shows that ε = ε0 in this case. It remains to prove that ε ≤ d := dim TorR
j (M1 , . . . , Ms ).
R
(M
,
.
.
.
,
M
),
ω
)
To show this notice that h2 Ejd is Matlis dual to E := Extns−d
(Tor
1
s
S which is a module
j
S
h
d
of dimension d, and ∞ Ej is Matlis dual to a submodule of E that coincides with E in dimension d − 1,
d−`
for ` ≥ 2 is Matlis dual to a module supported in dimension at most d − `.
¤
because h` Ej−`+1
Corollary A.8. Let R be a regular local ring containing a field, M1 , . . . , Ms be finitely generated
R-modules. The following are equivalent,
28
(i) TorR
1 (M1 , . . . , Ms ) = 0 and M1 ⊗R · · · ⊗R Ms is Cohen-Macaulay,
(ii) the codimension of M1 ⊗R · · · ⊗R Ms is the sum of the projective dimensions of the Mi ’s,
(iii) the intersection of the Mi ’s is proper and every Mi is Cohen-Macaulay.
Proof. Let M := M1 ⊗R · · · ⊗R Ms . Krull’s intersection theorem gives,
codimM ≤ codimM1 + · · · + codimMs .
If (i) holds, Krull’s intersection theorem, Theorem A.7 (i) and (ii) implies that
pdimM1 + · · · + pdimMs = n − depth M = codimM ≤ codimM1 + · · · + codimMs
which implies (ii) and (iii).
If (ii) holds, Krull’s intersection theorem implies that (iii) holds.
Now assume that (iii) holds. We then have codimM = pdimM1 + · · · + pdimMs = n + j − ε, with the
notations of Theorem A.7. Therefore ε = dim M + j which implies that j = 0 and ε = dim M as ε ≤ dim M
by Theorem A.7. This proves (i), and conclude the proof.
¤
References.
[BEL] A. Bertram, L. Ein, R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations
defining projective varieties. J. Amer. Math. Soc. 4 (1991), 587–602.
[BH] W. Bruns, J. Herzog, Cohen-Macaulay rings. Cambridge Stud. in Adv. Math., 39. Cambridge
Univ. Press, Cambridge, 1993.
[BC] L. Busé, M. Chardin, Implicitizing rational hypersurfaces using approximation complexes, preprint
math.AG/0301238.
[BJ] L. Busé, J.-P. Jouanolou, On the closed image of a rational map and the implicitization problem.
J. Algebra 265 (2003), 312–357.
[Chan] K. Chandler, Regularity of the powers of an ideal. Comm. Algebra 25 (1997), 3773–3776.
[Char] M. Chardin, Cohomology of projective schemes: From annihilators to vanishing , J. Algebra 274
(2004), 68-79.
[CEU] M. Chardin, D. Eisenbud, B. Ulrich, Hilbert functions, residual intersections, and residually S2
ideals. Compositio Math. 125 (2001), no. 2, 193–219.
[CU] M. Chardin, B. Ulrich, Liaison and Castelnuovo-Mumford regularity, Amer. J. Math. 124 (2002),
1103–1124.
[CHT] S. D. Cutkosky, J. Herzog, N. V. Trung, Asymptotic behaviour of the Castelnuovo-Mumford
regularity. Compositio Math. 118 (1999), no. 3, 243–261.
[Ei] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in
Mathematics, 150. Springer-Verlag, New York, 1995.
[FOV] H. Flenner, L. O’Carroll, W. Vogel, Joins and intersections. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1999.
[GLP] L. Gruson, R. Lazarsfeld, C. Peskine, On a theorem of Castelnuovo, and the equations defining
space curves. Invent. Math. 72 (1983), 491–506.
[GGP] A. Geramita, A. Gimigliano, Y. Pitteloud, Graded Betti numbers of some embedded rational
n-folds. Math. Ann. 301 (1995), 363–380.
[HSV] J. Herzog, A. Simis, and W.V. Vasconcelos, Koszul homology and blowing-up rings, in Commutative Algebra, eds. S. Greco and G. Valla, Lecture Notes in Pure and Appl. Math. 84, Marcel Dekker,
New York, 1983, 79-169.
29
[Ka] M. Katzman, The complexity of Frobenius powers of ideals. J. Algebra 203 (1998), 211–225.
[Ko] V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Amer. Math.
Soc. 128 (2000), no. 2, 407–411.
[S] J. Sidman, On the Castelnuovo-Mumford regularity of products of ideal sheaves. Adv. Geom. 2
(2002), 219–229.
[U] B. Ulrich, Artin-Nagata properties and reductions of ideals. Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), 373–400, Contemp. Math. 159, Amer. Math.
Soc., Providence, RI, 1994.
Marc Chardin,
Institut de Mathématiques de Jussieu, CNRS & Université Paris 6,
4, place Jussieu, F–75252 Paris cedex 05, France
[email protected]
30

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