Morphism Of Schemes Research Articles

Koszul complexes over Cohen-Macaulay rings

We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring A is Cohen-Macaulay, and a1,…,an is any sequence of elements in A, then the Koszul complex K(A;a1,…,an) is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring A, by finding a Cohen-Macaulay DG-ring B such that H0(B)=A, and using the Cohen-Macaulay structure of B to deduce results about A. As application, we prove that if f:X→Y is a morphism of schemes, where X is Cohen-Macaulay and Y is nonsingular, then the homotopy fiber of f at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.

Morphisms of Rational Motivic Homotopy Types

We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776 ) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070 ). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite etale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.

On Suslin homology with integral coefficients in characteristic zero (with an appendix by Bruno Kahn)

We show that the Suslin homology group with integral coefficients of a scheme $X$ separated of finite type over an algebraically closed field of characteristic 0 is a direct sum of a uniquely divisible group, finite copies of $\mathbb{Q}/\mathbb{Z}$, and a finitely generated group. We also study the possible type of homomorphisms between such groups induced by the morphisms of schemes. An appendix written by Bruno Kahn is included, which simplifies the proofs and generalizes the results.

Finite spaces and schemes

A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed space, endowed with a finite open covering, produces a ringed finite space. We introduce the notions of schematic finite space and schematic morphism, showing that they behave, with respect to quasi-coherence, like schemes and morphisms of schemes do. Finally, we construct a fully faithful and essentially surjective functor from a localization of a full subcategory of the category of schematic finite spaces and schematic morphisms to the category of quasi-compact and quasi-separated schemes.

On the semicontinuity problem of fibers and global F-regularity

ABSTRACTIn this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and F-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global F-regularity.

Infinitesimal criterion for flatness of projective morphism of schemes

Infinitesimal criterion for flatness of projective morphism of schemes

Recent Progress on Submersions: A Survey and New Properties

This paper is a survey about recent progress on submersive morphisms of schemes combined with new results that we prove. They concern the class of quasicompact universally subtrusive morphisms that we introduced about 30 years ago. They are revisited in a recent paper by Rydh, with substantial complements and key results. We use them to show Artin-Tate-like results about the 14th problem of Hilbert, for a base scheme either Noetherian or the spectrum of a valuation domain. We look at faithfully flat morphisms and get “almost” Artin-Tate-like results by considering the Goldman (finite type) points of a scheme. Bjorn Poonen recently proved that universally closed morphisms are quasicompact. By introducing incomparable morphisms of schemes, we are able to characterize universally closed surjective morphisms that are either integral or finite. Next we consider pure morphisms of schemes introduced by Mesablishvili. In the quasicompact case, they are universally schematically dominant morphisms. This leads us to a characterization of universally subtrusive morphisms by purity. Some results on the schematically dominant property are given. The paper ends with properties of monomorphisms and topological immersions, a dual notion of submersions.

A generalization of the Contou-Carrère symbol

Using techniques of Algebraic Geometry, the aim of this paper is to give a generalized definition of the Contou-Carrere symbol as a morphism of schemes. In fact, from formal schemes and Heisenberg groups, we provide a new definition of the Contou-Carrere symbol and a generalization of it associated with a separable extension . Moreover, a reciprocity law is proved and classical explicit reciprocity laws are recovered from it.

Descent Theory for Schemes

We give a complete characterization of the class of quasi-compact morphisms of schemes that are stable effective descent morphisms for the SCHEMES-indexed category given by quasi-coherent sheaves of modules.

Universal log structures on semi-stable varieties

Given a morphism of schemes which is flat, proper, and "fiber-by-fiber semi-stable'', we study the problem of extending the morphism to a morphism of fine log schemes, which is log smooth, integral, and vertical. The problem is rephrased in terms of a functor on the category of fine log schemes over the base, and the main result of the paper is that this functor is representable by a fine log scheme whose underlying scheme maps naturally to the base by a monomorphism of finite type. In the course of the proof, we also generalize results of Kato on the existence of log structures of embedding and semi-stable type.

The local pro-p anabelian geometry of curves

Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner automorphisms) by the property that the category of finite, discrete sets equipped with a continuous π1(X)-action is equivalent to the category of finite etale coverings of X. Moreover, the assignment X → π1(X) is a functor from the category of connected schemes (and morphisms of schemes) to the category of profinite topological groups and continuous outer homomorphisms (i.e., continuous homomorphisms of topological groups, where we identify any two homomorphisms that can be obtained from one another by composition with an inner automorphism).

Platitude d'une adherence schematique et lemme de Hironaka generalise

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied: (i) OY,y is reduced. (ii) f is universally open at the generic points of the components of Xy which contain x. (iii) For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to Xy, we have dimx, (Xy')=dimx(Xy)=d. (iv) Xy is reduced at the generic points of the components of Xy which contain x and (Xy)red is geometrically normal over K(y) in x. Then there exist an open neighbourhood U of x in X and a subscheme U0 of U which have the same underlying space as U such that f0:U0\arY is normal (i.e. f0 is a flat morphism whose geometric fibers are normal).