A SIMPLE PROOF FOR A THEOREM OF NAGEL AND SCHENZEL

The aim of this short note is to give a simple proof for Theorem 1.1 based on standard argument on local cohomology [2].. Proofs2[r]

A SIMPLE PROOF FOR A THEOREM OF NAGEL AND SCHENZEL

DUONG THI HUONG

Abstract In this short note, we present a simple proof for a theorem of Nagel and

Schen-zel.

1 Introduction

Throughout this paper, let R be a commutative Noetherian ring, M a finitely genrated

R-module and I an ideal of R Local cohomology H i

I (M ), introduced by Grothendieck, is an

important tool in both algebraic geometry and commutative algebra (cf [2]) Moreover, the

notion of I-filter regular sequences on M is an useful technique in study local cohomology.

In [4] Nagel and Schenzel proved the following theorem (see also [1])

Theorem 1.1 Let I be an ideal of a Noetherian ring R and M a finitely generated R-module.

Let x1, , x t an I-filter regular sequence of M Then we have

H I i (M ) ∼=

{

H i

(x1, ,x t)(M ) if i < t

H I i −t (H t

(x1, ,x t)(M )) if i ≥ t.

The most important case of Theorem 1.1 is i = t, and H I t (M ) ∼ = H I0(H (x t

1, ,x t)(M )) is a submodule of H t

(x1, ,x t)(M ) Recently, many applications of this fact have been found [3, 5].

It should be noted that Nagel-Schenzel’s theorem was proved by using spectral sequence theory The aim of this short note is to give a simple proof for Theorem 1.1 based on standard argument on local cohomology [2]

2 Proofs

Firstly, we recall the notion of I-filter regular sequence on M

Definition 2.1 Let M be a finitely generated module over a local ring (R, m, k) and let

x1, , x t ∈ I be a sequence of elements of R Then we say that x1 , , x t is a I-filter regular

sequence on M if the following conditions hold:

Supp(

((x1, , x i −1 )M : x i )/(x1, , x i −1 )M)

⊆ V (I)

for all i = 1, , t, where V (I) denotes the set of prime ideals containing I This condition

is equivalent to x i ∈ p for all p ∈ Ass / R M/(x1, , x i −1 )M \ V (I) and for all i = 1, , t.

Remark 2.2 It should be noted that for any t ≥ 1 we always can choose a I-filter regular

sequence x1, , x t on M Indeed, by the prime avoidance lemma we can choose x1 ∈ I and x1 ∈ p for all p ∈ Ass / R R \ V (I) For i > 1 assume that we have x1, , x i −1, then we choose

x i ∈ I and x i ∈ p for all p ∈ Ass / R R/(x1, , x i −1)\ V (I) by the prime avoidance lemma

again For more details, see [1, Section 2]

1

2 DUONG THI HUONG

The I-filter regular sequence can be seen as a generalization of the well-known notion of

regular sequence (cf [4, Proposition 2.2])

Lemma 2.3 A sequence x1, , x t ∈ I is an I-filter regular sequence on M if and only if for all p ∈ Spec(R) \ V (I), and for all i ≤ t such that x1 , , x i ∈ p we have x1

1, · · · , x t

1 is

an Mp-sequence.

Corollary 2.4 Let x1, , x t ∈ I be an I-filter regular sequence on M Then H i

(x1, ,x t)(M )

is I-torsion for all i < t.

Proof For each p ∈ Spec(R) \ V (I) we have either (x1 , x t )Rp = Rp or x1, , x t is an

Mp-regular sequence by Lemma 2.3 For the first case we have

(H (x i

1, ,x t)(M ))p∼ = H i

(x1, ,x t )Rp(Mp) = 0

for all i ≥ 0 For the second case we have

(H (x i

1, ,x t)(M ))p∼ = H i

(x1, ,x t )Rp(Mp) = 0

for all i < t by the Grothendieck vanishing theorem [2, Theorem 6.2.7] Therefore we have (H i

(x1, ,x t)(M ))p∼ = 0 for all i < t and for all p ∈ Spec(R)\V (I) So H i

(x1, ,x t)(M ) is I-torsion

It is well-known that local cohomology H i

(x1, ,x t)(M ) agrees with the i-th cohomology of

the ˇCech complex with respect to the sequence x1, , xt

0→ M d0

i

M x i → d1 ⊕

i<j

M x i x j → · · · d2 d t −1

→ M x1 x t → 0 (⋆)

The following simple fact is the crucial key for our proof

Lemma 2.5 Let x ∈ I be any element of R Then H i

I (M x ) = 0 for all i ≥ 0.

Proof Obviously the multiplication map M x → M x x is an isomorphism It induces

isomor-phism maps H i

I (M x) → H x i

I (M x ) for all i ≥ 0 But H i

I (M x ) is I-torsion, so it is (x)-torsion since x ∈ I Therefore H i

We are ready to prove the theorem of Nagel and Schenzel

Proof of Theorem 1.1 We set (x) the ideal generated by x1, , x t Let C j the j-th chain

of ˇCech complex (⋆), and set L j := Im(d j −1 ) and K

j := Ker(d j ) for all j ≥ 1 We split the

ˇ

Cech complex (⋆) into short exact sequences

0→ H0

0→ L1 → K1 → H1

(x) (M ) → 0 (B1)

· · ·

0→ L j → K j → H j

(x) (M ) → 0 (B j)

0→ K j → C j → L j+1 → 0 (A j)

· · ·

THE NAGEL-SCHENZEL ISOMORPHISM 3

0→ L t −1 → K t −1 → H t −1

(x) (M ) → 0 (B t −1)

0→ K t −1 → C t −1 → L t → 0 (A t −1)

0→ L t → M x1 x t → H t

(x) (M ) → 0 (B t)

By Lemma 2.3 we have H i

I (C j ) = 0 for all i ≥ 0 and all j ≥ 1 Since L j and K j are

submodules of C j for all j ≥ 1, we have H0

I (L j ) ∼ = H I0(K j ) = 0 for all i ≥ 1 We also

note that H (x) j (M ) is I-torsion for all j < t by Lemma 2.4, so H I0(H (x) j (M )) = H (x) j (M ) and

H i

I (H (x) j (M )) ∼ = 0 for all j < t and for all i ≥ 1.

Now applying the functor H i

I(−) to the short exact sequence (A0) and using the above observations we have

H I0(M ) ∼ = H (x)0 (M )

and

H I i (M ) ∼ = H I i (L1) (1)

for all i ≥ 1.

For each j = 1, , t − 1, applying the local cohomology functor H i

I(−) to the short exact

sequence (A j ) we have H1

I (K j) = 0 and the isomorphism

H I i (L j+1 ) ∼ = H I i+1 (K j) (C j)

for all i ≥ 1 Furthermore, if we apply H i

I(−) for the short exact sequence (B j), then we get the short exact sequence

0→ H j

(x) (M ) → H1

I (L j)→ H1

I (K j)→ 0,

and the isomorphism

H I i (L j ) ∼ = H I i (K j) (D j)

for all i ≥ 2 Note that H1

I (K j) = 0 as above, so

H (x) j (M ) ∼ = H I1(L j ). (2)

We next show that H I i (M ) ∼ = H (x) i (M ) for all i = 1, , t − 1 Indeed, using isomorphisms

(1), (2), (C j ) and (D j) consecutively, we have

H I i (M )

(1)

∼

= H I i (L1)

(D1 )

∼

= H I i (K1)

(C1 )

∼

= H I i −1 (L2)D ∼=2 · · · (C i −1)

∼

= H I1(L i)

(2)

∼

= H (x) i (M ).

Therefore, we have showed the first case of Nagel-Schenzel’s isomorphism H i

I (M ) ∼ = H (x) i (M ) for all i = 0, , t − 1 Finally, for i ≥ t by similar arguments we have

H I i (M )

(1)

∼

= H I i (L1)

(D1 )

∼

= H I i (K1)

(C1 )

∼

= H I i −1 (L2)D ∼=2 · · · (C t −1)

∼

= H I i −t+1 (L t ).

On the other hand, by applying the functor H i

I(−) to the short exact sequence (B t) we have

H I i −t (H (x) t (M )) ∼ = H I i −t+1 (L t)

for all i ≥ t Thus H i

I (M ) ∼ = H I i −t (H t

(x) (M ))for all i ≥ t, and we finish the proof. 

4 DUONG THI HUONG

References

[1] J Asadollahi and P Schenzel, Some results on associated primes of local cohomology modules, Japanese

J Mathematics 29 (2003), 285–296.

[2] M Brodmann and R.Y Sharp, Local cohomology: an algebraic introduction with geometric applications,

Cambridge University Press, 1998.

[3] H Dao and P.H Quy, On the associated primes of local cohomology, Nagoya Math J., to appear [4] U Nagel and P Schenzel, Cohomological annihilators and Castelnuovo-Mumford regularity, in

Com-mutative algebra: Syzygies, multiplicities, and birational algebra, Contemp Math 159 (1994), Amer.

Math Soc Providence, R.I., 307–328.

[5] P.H Quy and K Shimomoto, F -injectivity and Frobenius closure of ideals in Noetherian rings of

char-acteristic p > 0, Adv Math 313 (2017), 127–166.

Department of Mathematics, Thang Long University, Hanoi, Vietnam

E-mail address: duonghuongtlu@gmail.com