Faltings’ annihilator theorem and t-structures of derived categories

概要

ˇ
Let R be a commutative noetherian ring. Following Cesnaviˇ
cius [5], we say that R is CM-excellent if it
satisfies the three conditions below, which have been studied deeply by Kawasaki [14, 15, 16].
• The ring R is universally catenary.
• The formal fibers of the localization of R at each prime ideal are Cohen–Macaulay.
• The Cohen–Macaulay locus of each finitely generated R-algebra is Zariski-open.
Typical examples of a CM-excellent ring include an excellent ring, more generally an acceptable ring in the
sense of Sharp [22], and a homomorphic image of a Cohen–Macaulay ring [16]. In particular, the ring R is
CM-excellent if it possesses a dualizing complex, since the existence of a dualizing complex is equivalent to
the condition that the ring is a homomorphic image of a Gorenstein ring of finite Krull dimension [14].
Let Db (R) stand for the bounded derived category of finitely generated R-modules. The first main result
of this paper is the following theorem, which is Faltings’ annihilator theorem for complexes.
Theorem 1.1 (Theorem 3.5). Let R be a CM-excellent ring. Let Y and Z be specialization-closed subsets
of Spec R, and let n be an integer. Then the following two conditions are equivalent for each X ∈ Db (R).
(1) For all prime ideals p and q of R with Z 3 p ⊇ q ∈
/ Y , one has the inequality ht p/q + depth Xq ⩾ n.
(2) There exists an ideal b of R such that V(b) ⊆ Y and b H Z (X) = 0.
If we restrict Theorem 1.1 to the case where the complex X is a module, then it is the same as the main result
of [15], which extends a lot of previous results with additional assumptions, including Faltings’ original one
[8]; see [15] for more details. The main result of [7] shows the assertion of Theorem 1.1 under the stronger
assumptions that Y contains Z and that R possesses a dualizing complex. The latter assumption is to use
the local duality theorem; it does play an essential role in the proof of the result of [7].
As an application of Theorem 1.1, we obtain the second main result of this paper: the following theorem
provides a complete classification of the t-structures (in the sense of Be˘ılinson, Bernstein and Deligne [3]) of
the triangulated category Db (R) in terms of certain filtrations by specialization-closed subsets of Spec R.
Theorem 1.2 (Theorem 5.5). Let R be a CM-excellent ring with finite Krull dimension. Then the aisles in
Db (R) bijectively correspond to the sp-filtrations of Spec R satisfying the weak Cousin condition.
The notion of sp-filtrations satisfying the weak Cousin condition, which appears in the above theorem, has
been used by Deligne, Bezrukavnikov and Kashiwara [2, 4, 13], and explictly introduced by Alonso Tarr´ıo,
Jerem´ıas L´opez and Saor´ın [1]. This is a generalized version of the notion of codimension functions in the sense
of Grothendieck [11, Chapter V, §7]. The mutually inverse bijections giving the one-to-one correspondence
in the above theorem can be described explicitly; see Theorem 5.5. The main result of [1] shows the assertion
of Theorem 1.2 under the stronger assumption that the ring R admits a dualizing complex. This assumption
is, again, to apply the local duality theorem, and in fact, local duality plays a key role in the proof of the
result of [1].
2020 Mathematics Subject Classification. 13D09, 13D45, 13F40.
Key words and phrases. CM-excellent, derived category, Faltings’ annihilator theorem, local cohomology, specialization-closed
subset, sp-filtration, t-structure, weak Cousin condition.
The author was partly supported by JSPS Grant-in-Aid for Scientific Research 19K03443. ...