Generalization of a connectedness result to cohomologically complete intersections

TL;DR

This paper extends a classical connectedness result from set-theoretic complete intersections to cohomologically complete intersections, showing that disconnected punctured spectra imply low depth in a broader algebraic context.

Contribution

It generalizes a key connectedness theorem from set-theoretic to cohomologically complete intersections using local cohomology properties.

Findings

Disconnection of punctured spectra implies depth at most 1 for cohomologically complete intersections.

Endomorphism ring of local cohomology modules is the ring itself, indicating indecomposability.

The generalization is smooth and relies on properties of local cohomology in complete local rings.

Abstract

It is a well-known result that, in projective space over a field, every set-theoretical complete intersection of positive dimension in connected in codimension one (Hartshorne [H1,3.4.6] or [H2, Theorem 1.3]). Another important connectedness result is that a local ring with disconnected punctured sprectrum has depth at most $1$ ([H1, Proposition 2.1]). The two results are related, Hartshorne calls the latter "the keystone to the proof" of the former (loc. cit). In this short note we show that the latter result generalizes smoothly from set-theoretical to cohomologically complete intersections, i. e. to ideals for which there is in terms of local cohomology no obstruction to be a complete intersection ([HeSc1], [HeSc2]). The proof is based on the fact that, for cohomologically complete intersections over a complete local ring, the endomorphism ring of the (only) local cohomology cohomology module is the ring itself ([HeSt, Theorem 2.2 (iii)]) and hence indecomposable as a module.

Equations

$$\mathrm{depth}(R)=\mathrm{depth}(m,R)\le 1$$

$$\mathrm{depth}(I+J,R)\le 1$$

$$\mathrm{depth}(I+J,R)\le h+1$$

$$\frac{{\mathrm{End}}_R\left(H_I^h(R)\right)\text{buildrel}\text{canonically}}{=R,}$$

$$\mathrm{depth}(I+J,R)\le h+1.$$

$$0\to H_I^h(R)\oplus H_I^h(R)\to H_{I\cap J}^h(R)\to 0$$

$$H_I^h(R)\oplus H_J^h(R)=H_{I\cap J}^h(R)$$

$$H_{I\hat{R}}^h(\hat{R})\oplus H_{J\hat{R}}^h(\hat{R})=H_{(I\cap J)\hat{R}}^h(\hat{R}).$$

Full text

In this short note we show how the latter result generalizes smoothly from set-theoretical to cohomologically complete intersections, i. e. to ideals for which there is in terms of local cohomology no obstruction to be a complete intersection ([HeSc1], [HeSc2]).

The proof is based on the fact that, for cohomologically complete intersections over a complete local ring, the endomorphism ring of the (only) local cohomology cohomology module is the ring itself ([HeSt, Theorem 2.2 (iii)]) and hence indecomposable as a module.

michael.hellus@mathematik.uni-regensburg.de, Fakultät für Mathematik, Universität Regensburg, D-93040, Germany

††MSC2010: 13D45 (primary); 14M10, 13C40 (secondary)††Keywords: Local cohomology, complete intersections, connectedness

1 Introduction

In [H1, 2.1], Hartshorne proved (then by elementary means)

Theorem 1 Let $(R,m)$ be a noetherian local ring whose punctured spectrum $\operatorname{Spec}(R)\setminus\{m\}$ is disconnected.

[TABLE]

$\square$

Hartshorne’s result can be generalized:

Theorem 2 Let $R$ be a noetherian ring, indecomposable as a module over itself (e. g. local, as a ring), $I,J\subsetneq R$ ideals of $R$ such that $\sqrt{I\cap J}=\sqrt{0}$, but $\sqrt{I},\sqrt{J}\neq\sqrt{0}$.

[TABLE]

$\square$

To the best of my knowledge this was first observed by Irving Kaplansky (no reference). It is nowadays well known that theorem 2 can be proved quickly using local cohomology by a straightforward adaption of the method used in [I, proof of Proposition 15.7].

It is natural to formulate Kaplansky’s result for set-theoretical complete intersections (trivially, the case $h=0$ gives back Theorem 2, in the local case):

Corollary Let $(R,m)$ be a local noetherian ring, $I,J\subsetneq R$ ideals of $R$, $\underline{x}=x_{1},\ldots,x_{h}$ a regular sequence in $R$ such that $\sqrt{I\cap J}=\sqrt{\underline{x}R}$, but $\sqrt{I},\sqrt{J}\neq\sqrt{\underline{x}R}$.

[TABLE]

$\square$

Though Kaplansky’s result can be proved quickly using local cohomology, it is in some sense strong: For example, it implies immediately (by working in the graded instead of in the local situation) that two ‘skew’ (i. e., not contained in a plane) lines in $\mathbb{P}^{3}$ are not a set-theoretical complete intersection, cf. [I, Example 15.10] and note that this specific example is also covered both by Faltings’ connectedness theorem (e. g. as given in a simple form in [I, Theorem 15.11]; the full statement [F, Cor. 4] is more sophisticated) and by Hartshorne’s connectedness theorem stating that, in projective space over a field, every set-theoretical complete intersection of positive dimension in connected in codimension one ([H1, 3.4.6] or [H2, Theorem 1.3]; see [R, Satz 2.6’ on p. 20] for a stronger version).

And though it is strong in the aforementioned sense, Kaplansky’s result generalizes smoothly from set-theoretical complete intersections to cohomologically complete intersections: The sole content of this note is to prove this somewhat surprising fact (theorem in section 2 below).

The proof makes significant use of the fact that, for a cohomologically complete intersection ideal $I$ in a local complete ring $R$, one has

[TABLE]

where $h$ denotes the codimension of $I$ ([HeSt, Theorem 2.2 (iii)]).

2 Result

An ideal $I$ of a local noetherian ring $R$ is called cohomologically complete intersection ([HeSc1], [HeSc2]) whenever $H^{h}_{I}(R)=0$ for all $i\neq h$ for some $h$ (which is then necessarily the height of $I$). In [HeSc1, Theorem 3.2] it was shown that for $R$ Gorenstein this condition is completely encoded in homological properties of $\operatorname{H}^{\operatorname{height}(I)}_{I}(R)$. ([HeSc2, Theorem 4.4] is a related result for modules.)

This condition is clearly weaker than being a set-theoretical complete intersection; the actual difference between these two notions can be described in terms of regular sequences on the Matlis dual of the (only) local cohomology module ([He, 1.1.4 Corollary]).

Theorem Let $(R,m)$ be a local noetherian ring, $I,J\subsetneq R$ ideals of $R$ such that $\sqrt{I}\nsubseteq\sqrt{J}$, $\sqrt{J}\nsubseteq\sqrt{I}$ and $I\cap J$ is a cohomologically complete intersection of depth $h$.

[TABLE]

Proof: Assume that $\operatorname{depth}(I+J,R)\geq h+2$. The exact Mayer-Vietoris sequence

[TABLE]

then leads to a canonical isomorphism

[TABLE]

and hence, by application of the functor $\_\otimes_{R}\hat{R}$, to a canonical isomorphism

[TABLE]

Note that $(I\cap J)\hat{R}$ is a cohomologically complete intersection of depth $h$.

But the endomorphism ring of $\operatorname{H}^{h}_{(I\cap J)\hat{R}}(\hat{R})$ is canonically isomorphic to $\hat{R}$, by [HeSt, Theorem 2.2 (iii)]. Therefore, the direct sum decomposition of $\operatorname{H}^{h}_{(I\cap J)\hat{R}}(\hat{R})$ must be trivial. In particular, either of the two endomorphism rings of $\operatorname{H}^{h}_{I\hat{R}}(\hat{R})$ and $\operatorname{H}^{h}_{I\hat{R}}(\hat{R})$ must be the zero ring, w. l. o. g. we assume this happens for the former one.

This means of course that $\operatorname{H}^{h}_{I\hat{R}}(\hat{R})$ is zero, i. e. $\operatorname{H}_{I}^{h}(R)=0$. But this is only possible if all prime ideals minimal over $I$ have height greater than $h$.

Therefore, no prime ideal minimal over $I$ is minimal over $I\cap J$, i. e. to each prime ideal $p$ minimal over $I$ there exists a prime ideal $p_{0}$ containing $J$ and such that $p\supseteq p_{0}$, i. e. $\sqrt{I}\supseteq\sqrt{J}$, contradiction. $\square$