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An extension of the Hartshorne theorem on thecharacterization of cofinite complexes
Ken‐ichiroh Kawasaki** Department of Mathematics, Nara University of Education.

Abstract

This report is the announcement of the result on the characterization of cofinitecomplexes (the first one is in the talk at the RIMS conferencet). Let A be a ho‐
momorphic image of a Gorenstein ring of finite Krull dimension, J an ideal of A,
and N^{\cdot} a bounded‐below complex of A ‐modules. Suppose that A is complete withrespect to a J‐adic topology. In this report, we introduce that N^{\cdot} is a J ‐cofinitecomplex if and only if the support of H^{i}(N^{\cdot}) is in V(J) for all i , and Ext^{j}(A/J, N^{\cdot})
is of finite type for all j.

1Introduction

In this report, we shall introduce the following theorem.

Theorem 1 Let A be a ring, which is a homomorphic image of a Gorenstein ring offinite Krull dimension, J an ideal of A. Suppose that A is complete with respect toa J ‐adic topology. Let N^{\cdot} be a complex of A ‐modules in \mathcal{D}^{+}(A) , where \mathcal{D}^{+}(A) is thederived category consisting of complexes bounded below. Then the following conditions areequivalent:(1) The complex N^{\cdot} is J ‐cofinite.

(2) The complex N^{\cdot} satisfies the following conditions:a) Supp H (N^{\cdot})\subseteq V(J) for each i,b) Ext^{j}(A/J, N^{\cdot}) is of finite type for each j.
We assume that alı rings are commutative and noetherian with identity throughout thisreport.
2000 Mathematlcs Subject Classification. Primary 14B15,13D03 , 18Gı5This work was supported by the Research Institute for Mathematical Sciences, a Joint Us‐*age/Research Center located in Kyoto University. The results included in the report will be submitted
as a paper to the other publication elsewhere.

keywords: Local Cohomology Cofinite Complex Cofinite Module Abelian category

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2Preliminaries

We shall recall the definitions on cofiniteness.

For the basic definitions, the reader isreferred to [15] and [16] (see [3], [4], [2] and [14] for the notation on derived categories
and derived functors).
Before defining theJ ‐cofinitenesson complexes, we introduce the definition of thedualizing functor (cf. [14, §4.3, p. 70]):Definition 1 Let A be a ring, equipped with a dualizing complex D for A and J an idealof A (see [19, (1.1) Definition, p. 215] for the definition of the (fundamental) dualizingcomplex). We denote by D_{J}(-) the functor \mathbb{R}Hom\cdot(- , \mathbb{R}\Gamma_{J}(D)) on the derived
category \mathcal{D}(A) . In this paper, we call this functor D_{J}(-) the J‐dualizing functor (cf.
[14, p. 70]). Note that the J ‐dualizing functor is defined over unbounded complexes by[ 2\theta, Theorem C, p. 125]. In the case J=(0) , then we simply denote D_{J}(-) by D(-) .
The J ‐cofiniteness on complexes is defined as follows (see [3, §2, p. 149] for the definition
over regular rings):Definition 2 Let A be a ring, equipped with a dualizing complex D for A and J an idealof A. Let N be an object of the derived category \mathcal{D}(A) . We say N^{\cdot} is J‐cofinite, if
there exists M^{\cdot}\in \mathcal{D}_{ft}(A) , such that N^{\cdot}\simeq D_{J}(M^{\cdot}) in \mathcal{D}(A) . Here D_{J}(-) is the J‐
dualizing functor on \mathcal{D}(A) defined as above and \mathcal{D}_{ft}(A) is the derived category consistingof complexes with cohomology modules of finite type over A.

Over regular rings R of finite Krull dimension, it is proved by Hartshorne (cf. [3,Theorem 5.1, p. 154]). Over the rings A which is a homomorphic image of a Gorensteinring of finite Krull dimension, the theorem as a variant of [3, Theorem 5.1, p. 154] isproved under the additional conditions (ii) and (ii) (cf. [11, Theorem 5, p. 318]). Our
aim in the report, we introduce an extension of the Hartshorne theorem over the ring A,

without the conditions (i) and (ii).

3Sketch proof of the main theorem

The purpose of this section is to give a sketch proof of the main theorem as an extensionof [3, Theorem 5.1, p. 154]) without regularity assumption. The argument on the way
out functors is used in [3, Theorem 5.1, p. 154]) over the regular rings. On the otherhands, we use the spectral sequences. First let us prove the lemma:

Lemma 2 Let A be a ring, which is a homomorphic image of a Gorenstein ring of finite
Krull dimension, J an ideal of A , and M^{\cdot} in \mathcal{D}_{ft}^{-}(A) . Suppose that A is complete withrespect to a J ‐adic topology. Then Ext^{j}(T^{\cdot}, D_{J}(M^{\cdot})) is of finite type for all j , for eachin \mathcal{D}_{ft}^{-}(A) with Supp H (T^{\cdot})\subset V(J) for all l . Here \mathcal{D}_{ft}^{-}(A) is the derived category
consisting of complexes bounded above with cohomology modules of finite type over A.

T^{\cdot}

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Proof.

We have an isomorphism in the derived category \mathcal{D}(A) using the Hom‐tensoradjunction (cf. [14, the Hom‐tensor adjunction, p. 53]):\mathbb{R}Hom'(T^{\cdot}, D_{J}(M^{\cdot}))) \simeq \mathbb{R}Hom'(M', D(T\cdot)).

Then we have the lemma, using the spectral sequences and by Yoneda’s lemma. See [12]for the detailed calculation.

Proof of Theorem 1: See [12] for the detailed proof.

4\square\squareAn application of the theoremWe recall the following claim (cf. [Kaw3, Claim 1]):Claim 3 Let R be a ring, and J an ideal of R. Let N^{\cdot}\in \mathcal{D}^{+}(R) be a bounded‐below
complex. Then the following conditions are equivalent:(i) Ext^{j}(R/J, N^{\cdot}) is of finite type over R for all j ;(ii) Ext^{j}(R/\sqrt{J}, N^{\cdot}) is of finite type over R for all j ;(iii) Ext^{j}(R/P, N\cdot) is of finite type over R for all j and for each P\in{\rm Min}(R/J) ;(iv) Ext^{j}(W, N^{\cdot}) is of finite type over R for all j and for each R ‐module W of finite
type over R such that Supp(W)\subseteq V(J)_{j}(v) Ext^{j}(W^{\cdot}, N^{\cdot}) is of finite type over R for all j and for each W^{\cdot}\in \mathcal{D}_{ft}^{b}(R) such that

Supp(H^{l}(W^{\cdot}))\subseteq V(J) for all l ;(vi) Ext^{j}(W\cdot, N^{\cdot}) is of finite type over R for all j and for each W^{\cdot}\in \mathcal{D}_{ft}^{-}(R) such that

Supp(H^{l}(W^{\cdot}))\subseteq V(J) for all l.

We can give the following corollary as an application form Theorem 1.

Corollary 4 Let A be a ring, which is a homomorphic image of a Gorenstein ring offinite Krull dimension, J an ideal of A. Suppose that A is complete with respect to a J‐adic topology. Let N be a complex of A ‐modules in \mathcal{D}^{+}(A) . Then the following condition\mathcal{S}are equivalent:(1) The complex NiSJ ‐cofinite.

(2) The complex N satisfies the following conditions:a) Supp H (N^{\cdot})\subseteq V(J) for each i,b) the equivalence conditions in Claim 3.

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\# The data of the talk in the conference:

Date: February 15, in 2018, 11:00∼12:00, Location: the room No. 111 in RIMS, Kyoto,in Japan, RIMS Conference Researches on theory of isometries and preserver problems
from a various point of view.