Local Cohomology and Matlis Duality Joint work with J. Stückrad Notation R (commutative, noetherian) ring, I ⊆ R an ideal (R − mod) :=category of all R-modules Notation R (commutative, noetherian) ring, I ⊆ R an ideal (R − mod) :=category of all R-modules Γ_{I} : (R − mod) → (R − mod) Γ_{I}(M) := {m ∈ M|I^{n} · m = 0∀n ≫ 0} ”local section functor” Γ_{I} is additive, left exact Notation R (commutative, noetherian) ring, I ⊆ R an ideal (R − mod) :=category of all R-modules Γ_{I} : (R − mod) → (R − mod) Γ_{I}(M) := {m ∈ M|I^{n} · m = 0∀n ≫ 0} ”local section functor” Γ_{I} is additive, left exact ⇒ has right-derived functors H_{I}^{l} = ”local cohomology functors with support in I”

(R, m) local ring E := E_{R}(R/m) a fixed R-injective hull of R/m D := Hom_{R}(·, E) = ”Matlis – dual – functor”

(R, m) local ring E := E_{R}(R/m) a fixed R-injective hull of R/m D := Hom_{R}(·, E) = ”Matlis – dual – functor” I (set-theoretic) complete intersection iff it can be generated by height(I) many elements (up to radical)

Example k a field, d ∈ ℕ, d ≥ 3 Let C_{d} ⊆ ℙ^{3} be the curve given by

(u^{d} : u^{d−1}v : uv^{d−1} : v^{d})

Example k a field, d ∈ ℕ, d ≥ 3 Let C_{d} ⊆ ℙ^{3} be the curve given by

(u^{d} : u^{d−1}v : uv^{d−1} : v^{d})

if char(k) > 0: Every C_{d} is a complete intersection (Hartshorne)

Example k a field, d ∈ ℕ, d ≥ 3 Let C_{d} ⊆ ℙ^{3} be the curve given by

(u^{d} : u^{d−1}v : uv^{d−1} : v^{d})

if char(k) > 0: Every C_{d} is a complete intersection (Hartshorne) if char(k) = 0: Open question, even for d = 4 Easy: I complete intersection ⇒ H_{I}^{l}(R) = 0 (∀l > height(I)) (use Cech-cohomology)

Easy: I complete intersection ⇒ H_{I}^{l}(R) = 0 (∀l > height(I)) (use Cech-cohomology) But: Reversed statement is not true, set R = k[[x, y, z, w]] f = xw − yz g_{1} = y^{3} − x^{2}z, g_{2} = z^{3} − u^{2}w I := \sqrt{(f, g_{1}, g_{2})R} is a height two prime ideal of R (corresponds to the curve C_{4}) ⇒ I/fR ⊆ R/fR has height one

• H_{I/fR}^{l}(R/fR) = 0 (∀l > 1)

• minimal number of generators of I/fR up to radical is > 1, I/fR is not a complete intersection Question I complete intersection ⟺ ⟺ H_{I}^{l}(R) = 0 (∀l > height(I)) and ???

Theorem h := height(I) f_{1}, …, f_{h} ∈ I an R-regular sequence Theorem h := height(I) f_{1}, …, f_{h} ∈ I an R-regular sequence TFAE:

(i) \sqrt{\underline{f}R} = \sqrt{I}; in particular I is a complete intersection Theorem h := height(I) f_{1}, …, f_{h} ∈ I an R-regular sequence TFAE:

(i) \sqrt{\underline{f}R} = \sqrt{I}; in particular I is a complete intersection (ii) H_{I}^{l}(R) = 0 (∀l > h) and \underline{f} is a D(H_{I}^{h}(R))-regular sequence (iii) H_{I}^{l}(R) = 0 (∀l > h) and \underline{f} is a D(H_{I}^{h}(R))-quasiregular sequence One might try to define depth(I, D(H_{I}^{h}(R))) (by using regular sequences), but:

One might try to define depth(I, D(H_{I}^{h}(R))) (by using regular sequences), but: Let I ⊆ k[[x, y, z, w]] be the ideal corresponding to the curve C_{4}, assume positive characteristic We know: I is a complete intersection One might try to define depth(I, D(H_{I}^{h}(R))) (by using regular sequences), but: Let I ⊆ k[[x, y, z, w]] be the ideal corresponding to the curve C_{4}, assume positive characteristic We know: I is a complete intersection ⇒ depth(I, D(H_{I}^{2}(R))) = 2 One might try to define depth(I, D(H_{I}^{h}(R))) (by using regular sequences), but: Let I ⊆ k[[x, y, z, w]] be the ideal corresponding to the curve C_{4}, assume positive characteristic We know: I is a complete intersection ⇒ depth(I, D(H_{I}^{2}(R))) = 2 take f = xw − yz ∈ I f is a D(H_{I}^{2}(R))-regular sequence (of length 1)

One might try to define depth(I, D(H_{I}^{h}(R))) (by using regular sequences), but: Let I ⊆ k[[x, y, z, w]] be the ideal corresponding to the curve C_{4}, assume positive characteristic We know: I is a complete intersection ⇒ depth(I, D(H_{I}^{2}(R))) = 2 take f = xw − yz ∈ I f is a D(H_{I}^{2}(R))-regular sequence (of length 1) there is no h ∈ I such that \sqrt{(f, h)R} = \sqrt{I} (because I/fR is not a complete intersection)

One might try to define depth(I, D(H_{I}^{h}(R))) (by using regular sequences), but: Let I ⊆ k[[x, y, z, w]] be the ideal corresponding to the curve C_{4}, assume positive characteristic We know: I is a complete intersection ⇒ depth(I, D(H_{I}^{2}(R))) = 2 take f = xw − yz ∈ I f is a D(H_{I}^{2}(R))-regular sequence (of length 1) there is no h ∈ I such that \sqrt{(f, h)R} = \sqrt{I} (because I/fR is not a complete intersection) ⇒ f is a maximal D(H_{I}^{2}(R))-regular sequence of length 1 (in I)

Study D(H_{I}^{h}(R))-regular sequences in I

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Study Ass_{R}(D(H_{I}^{h}(R))) (associated primes)

Observation (in a special case)

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Observation (in a special case) Set R = k[[X, Y]], I = XR Observation (in a special case) Set R = k[[X, Y]], I = XR H_{I}^{1}(R) = k[[Y]][X^{−1}] (compute using Cech-cohomology)

Observation (in a special case) Set R = k[[X, Y]], I = XR H_{I}^{1}(R) = k[[Y]][X^{−1}] (compute using Cech-cohomology) D(H_{I}^{1}(R)) = k[Y^{−1}][[X]] ( ≠ k[[X]][Y^{−1}]) (calculate!)

Observation (in a special case) Set R = k[[X, Y]], I = XR H_{I}^{1}(R) = k[[Y]][X^{−1}] (compute using Cech-cohomology) D(H_{I}^{1}(R)) = k[Y^{−1}][[X]] ( ≠ k[[X]][Y^{−1}]) (calculate!) ⇒ {0} ∈ Ass_{R}(D(H_{I}^{1}(R))) (because the element Y^{−1}X + Y^{−4}X + Y^{−9}X^{3} + ⋯ ∈ D(H_{I}^{1}(R)) has trivial annihilator)

More generally p ⊆ R prime ideal, x_{1}, …, x_{i} part of a system of parameters of R/p ⇒ p ∈ Ass_{R}(D(H_{(x_{1},...,x_{i})R}^{i}(R)))

More generally p ⊆ R prime ideal, x_{1}, …, x_{i} part of a system of parameters of R/p ⇒ p ∈ Ass_{R}(D(H_{(x_{1},...,x_{i})R}^{i}(R))) In general, the reversed statement is not true.

Conjecture (∗) Ass_{R}(D(H_{(x_{1},...,x_{i})R}^{i}(R))) = {p ∈ Spec(R)|H_{(x_{1},...,x_{i})R}^{i}(R/p) ≠ 0} Conjecture (∗) Ass_{R}(D(H_{(x_{1},...,x_{i})R}^{i}(R))) = {p ∈ Spec(R)|H_{(x_{1},...,x_{i})R}^{i}(R/p) ≠ 0} ⊆ holds.

Conjecture (∗) Ass_{R}(D(H_{(x_{1},...,x_{i})R}^{i}(R))) = {p ∈ Spec(R)|H_{(x_{1},...,x_{i})R}^{i}(R/p) ≠ 0} ⊆ holds. (∗) is equivalent to: Y := Ass_{R}(D(H_{(x_{1},...,x_{i})}^{i}(R))) is stable under generalization for every local ring R Conjecture (∗) Ass_{R}(D(H_{(x_{1},...,x_{i})R}^{i}(R))) = {p ∈ Spec(R)|H_{(x_{1},...,x_{i})R}^{i}(R/p) ≠ 0} ⊆ holds. (∗) is equivalent to: Y := Ass_{R}(D(H_{(x_{1},...,x_{i})}^{i}(R))) is stable under generalization for every local ring R There are other equivalent formulations of (∗)

Special case R = k[[X_{1}, …, X_{n}]] i ∈ {1, …, n} I = (X_{1}, …, X_{i})R Special case R = k[[X_{1}, …, X_{n}]] i ∈ {1, …, n} I = (X_{1}, …, X_{i})R i = n: (∗) holds (trivial)

Special case R = k[[X_{1}, …, X_{n}]] i ∈ {1, …, n} I = (X_{1}, …, X_{i})R i = n: (∗) holds (trivial) i = n − 1: (∗) holds Special case R = k[[X_{1}, …, X_{n}]] i ∈ {1, …, n} I = (X_{1}, …, X_{i})R i = n: (∗) holds (trivial) i = n − 1: (∗) holds i = n − 2: (∗) holds for all primes with height(p) ≠ 1 and we also know that it holds for many primes with height(p) = 1 Theorem Let J ⊆ R be an ideal such that dim(R/J) = 1 and H_{J}^{dim(R)}(R) = 0. Then Assh(D(H_{J}^{dim(R)−1}(R))) = Assh(R) (Assh = associated primes of highest dimension)

Theorem Let J ⊆ R be an ideal such that dim(R/J) = 1 and H_{J}^{dim(R)}(R) = 0. Then Assh(D(H_{J}^{dim(R)−1}(R))) = Assh(R) (Assh = associated primes of highest dimension) More precise result if R is complete:

Theorem (R complete) Let J ⊆ R be an ideal such that dim(R/J) = 1 and H_{J}^{dim(R)}(R) = 0.

Ass_{R}(D(H_{J}^{dim(R)−1}(R)) = {P | dim(R/P) = dim(R) − 1, dim(R/(P + J)) = 0} ∪ Assh(R)