Connectedness

and local cohomology

Uli Walther joint with

Anurag Singh Tokyo, August 2005

The schedule

1. The main theorem.

2. Local cohomology and Frobenius.

3. The history of the theorem.

4. Some comments on the proof.

5. How to compute with the theorem.

6. Remarks on an alternative approach.

Statement of main theorem

Notation: (R, 𝔪, 𝕂) local or standard graded 𝕂-algebra of positive characteristic. 𝔞 ⊆ R an ideal. Throughout 𝕂 separably closed.

X = Spec (R/𝔞), X^{∘}= X \ {𝔪}.

Theorem: X^{∘} connected ⟺ Frob ^{∞}(H_{𝔪}^{1}(R)) = 0. If 𝕂 is perfect (, hence 𝕂 = 𝕂̄) then

#(π_{0}(X^{∘})) = dim_{𝕂}(Frob ^{∞}(H_{𝔪}^{1}(R))) + 1.

Local cohomology and Frobenius

Notation: R =any commutative ring,

(y_{1}, …, y_{t}) = 𝔟 ⊆ R an ideal.

Consider Č_{i}^{•} = (R → R[y_{i}^{−1}]) and

Č_{y}^{•} = t ; ⨂ ; 1 Č_{i}^{•}

Definition: H^{i}(Č_{y}^{•}) = H_{𝔟}^{i}(R). Assumption: R of characteristic p > 0 Frob : R → R, y → y^{p}, R^{p}-linear, induces

Frob : Č_{y}^{•} → Č_{y}^{•}

and hence H_{𝔟}^{i}(R) → H_{𝔟}^{i}(R), R^{p}-linear.

Since M = H_{𝔪}^{i}(R) is Artinian,

M ⊇ R Frob (M) ⊃ R Frob ^{2}(M) ⊇ ⋯ stabilizes. If 𝕂 is perfect, this is a 𝕂-space. Definition:

H_{st}^{1}(R) := _{e}^{⋂} Frob ^{e}(H_{𝔪}^{1}(R)).

Fact: dim (M_{st}) < ∞ if M Artinian. In the graded case,

Frob (M_{i}) ⊆ M_{pi}, so M_{st} ⊆ M_{0}.

Some history

Frob comes from Peskine–Szpiro. One of their results: Theorem: S regular local 𝕂-algebra, R = S/𝔞, dim (R) = n. Then

X^{∘} connected ⟺ H_{𝔞}^{n}(S) = H_{𝔞}^{n−1}(S) = 0.

Lyubeznik developed F -modules, one result: Theorem: S regular local 𝕂-algebra, R = S/𝔞, dim (S) = d. Then: H_{𝔞}^{i}(S) = 0 ⟺ Frob ^{e}(H_{𝔪}^{d−i}(R)) = 0 for e ≫ 0. Combined, one obtains the first part of our main theorem. We give a direct proof.

When R is a complete domain, dim> 1

Theorem: H_{𝔪}^{1}(R) is Frob -torsion. Let η ∈ H_{𝔪}^{1}(R), which is finite length(!). Finite length gives an e, k: Frob ^{e+k} (η)+r_{1} Frob ^{e+k−1} (η)+⋯+r_{e} Frob ^{k}(η) = 0. Pick e such that #(r_{i} ≠ 0) → min .

Apply Frob ^{≫0} to see: r_{e} unit. Replace η ⇝ Frob ^{k}(η), get

(1) \underbrace{Frob ^{e}(η) + r_{1} Frob ^{e−1}(η) + ⋯ + r_{e}η}_{=: ξ ∈ H_{𝔪}^{1}(R)} = 0. x an s.o.p.

H_{𝔪}^{1}(R) = H^{1} (R → _{i}^{⨁} R[x_{i}^{−1}] → _{i,j}^{⨁} R[x_{i}^{−1}x_{j}^{−1}]).

r ↦ (r, …, r).

0 = ξ ∈ H_{𝔪}^{1}(R) means: ξ ∈ R.

Since η = [(y_{1}/x_{1}, …, y_{d}/x_{d})], and (1), each y_{i}/x_{i} is a root to

f(T) = T^{p^{e}} + r_{1}T^{p^{e−1}} + ⋯ + r_{e}T + r_{e+1} ∈ R[T]

As f′(T) = r_{e} unit, f(T) is separable, splits in R[T].

R domain, so all roots in R; y_{i}/x_{i} = y_{j}/x_{j} ∈ R,

⟹ η = 0.

The general case:

1. Reduce to complete local reduced with separably closed coefficient field. (Technical things about stable parts.)

2. X^{∘} connected iff H_{𝔪}^{1}(R) is Frob -torsion. (Use separable closure and locality.)

3. X^{∘} disconnected, use Rung’s display: R = A/𝔄 ∩ 𝔅,

0 → R → A/𝔄 ⊕ A/𝔅 → A/𝔄 + 𝔅 → 0

and (H_{𝔐}^{0}(A/𝔄 + 𝔅)) _{st} = 𝕂.

Computing the stable part

The point is: cannot compute with 𝕂 = 𝕂̄^{sep} .

A = ℤ/pℤ[\underbrace{x_{1}, …, x_{n}}_{𝔪}], R = A/𝔞.

Want: #π_{0}(Spec (R_{𝔪}) \ 𝔪).

1. Saturate 𝔞. Let E = Ext _{A}^{n−1}(R, A).

2. 𝔞_{+} = 𝔞 : ann (E), 𝔞_{1} = ann (E) : 𝔞_{+}^{∞}.

3. 𝔞_{1} ↔ the 1-d components.

Count: 𝔭 ∈ ass (𝔞_{1}) with 𝔭 ⊂ 𝔪.

5. 𝔞_{+} is the part of X of dimension ≥ 2.

6. Let R_{+} = A/𝔞_{+} , then

ℓ = dim_{ℤ/pℤ}(H_{𝔪}^{1}(R_{+} ))
= dim_{ℤ/pℤ}(Ext _{A}^{n−1}(R_{+} , A))_{𝔪} < ∞.

7. Now: H = H^{1}(R_{+} , x^{pe}), Koszul homology.

8. e ≫ 0 gives dim_{ℤ/pℤ}(H) = ℓ.

9. Find the Frob -matrix and its iterates on H.

10. The stable rank is dim (H_{𝔪}^{1}(R_{+} )_{st}).

Alternative method

𝔞 homogeneous, Y the projective variety. If 𝔞 is radical and 𝕂 algebraically closed,

dim_{𝕂}(H^{0}(Y, O_{Y})) = #π_{0}(Y) = #π_{0}(X^{∘}).

There are methods by G. Smith for H^{i}. Problem: 𝕂 never closed—

how to make sure that 𝔞 stays radical?