Motivic and $$\ell $$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

We study the motivic and $$\ell $$ ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the $$\ell $$ ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

The Étale locus in complete local rings

Let R R be a complete local ring and let Q Q be a prime ideal of R R . It is determined precisely which conditions on R R are equivalent to the existence of a complete unramified regular local ring A A and an element g ∈ A − Q g\in A-Q such that R R is a finite A A -module and A g ⟶ R g A_g\longrightarrow R_g is étale . A number of other properties of the possible embeddings A ⟶ R A\longrightarrow R are developed in the process including the determination of precisely which fields can be coefficient fields in the Cohen-Gabber Theorem.

Almost Gorenstein rings arising from fiber products

The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

Shtukas with one leg III

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following result, where Y = Spa Ainf REVERSE SOLIDUS {xk}. Theorem 14.2.1 posits that there is an equivalence of categories between finite free Ainf-modules and vector bundles on Y. One should think of this as being an analogue of a classical result: If (R, m) is a 2-dimensional regular local ring, then finite free R-modules are equivalent to vector bundles on (Spec R)REVERSE SOLIDUS {m}. The chapter then provides a proof of Theorem 14.2.1.

Free resolutions of Dynkin format and the licci property of grade $3$ perfect ideals

Recent work on generic free resolutions of length $3$ attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade $3$ perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.

Lyubeznik tables of linked ideals

In this paper, we examine the Lyubeznik tables of two linked ideals [Formula: see text] and [Formula: see text] of a complete regular local ring [Formula: see text] containing a field. More precisely, we prove that the Lyubeznik tables of two evenly linked ideals [Formula: see text] and [Formula: see text] are the same when [Formula: see text] and [Formula: see text] both satisfy one of the following properties: (1) canonically Cohen–Macaulay, (2) generalized Cohen–Macaulay and (3) Buchsbaum. Furthermore, we give some conditions for equality of Lyubeznik tables of two linked ideals of dimension 2.

Directed unions of local monoidal transforms and GCD domains

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.