This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d’Estudis Catalans, July 15–26, 1996

Infinite Free Resolutions

In [6], S. Bloch conjectures a formula for the Artin conductor of the l-adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The formula, which we call the conductor formula of Bloch, enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms. In this paper, we prove the formula in arbitrary dimension under the assumption that the reduced closed fiber has normal crossings.

/pdf/on-the-conductor-formula-of-bloch-4pq0rump49.pdf

On the conductor formula of Bloch

Classical deflnitions of locally complete intersection (l.c.i.) homomor- phisms of commutative rings are limited to maps that are essentially of flnite type, or ∞at. The concept introduced in this paper is meaningful for homo- morphisms ': Ri! S of commutative noetherian rings. It is deflned in terms of the structure of ' in a formal neighborhood of each point of SpecS .W e characterize the l.c.i. property by difierent conditions on the vanishing of the Andre-Quillen homology of the R-algebra S. One of these descriptions estab- lishes a very general form of a conjecture of Quillen that was open even for homomorphisms of flnite type: If S has a flnite resolution by ∞at R-modules and the cotangent complex L(SjR) is quasi-isomorphic to a bounded complex of ∞at S-modules, then ' is l.c.i. The proof uses a mixture of methods from commutative algebra, difierential graded homological algebra, and homotopy theory. The l.c.i. property is shown to be stable under a variety of operations, including composition, decomposition, ∞at base change, localization, and com- pletion. The present framework allows for the results to be stated in proper generality; many of them are new even with classical assumptions. For in- stance, the stability of l.c.i. homomorphisms under decomposition settles an open case in Fulton's treatment of orientations of morphisms of schemes.

/pdf/locally-complete-intersection-homomorphisms-and-a-conjecture-1iloo4kozx.pdf

Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology

At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.

p −1-Linear Maps in Algebra and Geometry

The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S , for some local homomorphism φ: R → S . Various techniques are developed to study the new invariants and to establish their basic properties. In some cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R -modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R . A third is to obtain criteria for a ring equipped with a "contracting" endomorphism—such as the Frobenius endomorphism—to be regular or complete intersection; these results represent broad generalizations of Kunz's characterization of regularity in prime characteristic.

/pdf/homology-over-local-homomorphisms-qhwzi0zhwk.pdf

Homology over local homomorphisms

Written to complement standard texts on commutative algebra, this short book gives complete and relatively easy proofs of important results, including the standard results involving localisation of formal smoothness (M. Andre) and localisation of complete intersections (L. Avramov), some important results of D. Popescu and Andre on regular homomorphisms, and some results from A. Grothendieck's EGA on smooth homomorphisms. The authors make extensive use of the Andre–Quillen homology of commutative algebras, but only up to dimension 2, which is easy to construct, and they deliberately avoid using simplicial methods. The book also serves as an accessible introduction to some advanced topics and techniques. The only prerequisites are a basic course in commutative algebra and the first definitions in homological algebra.

/pdf/smoothness-regularity-and-complete-intersection-15it7rv264.pdf

Smoothness, Regularity and Complete Intersection

On the acyclicity of the Tate complex

Projective exterior Koszul homology and decomposition of the Tor functor

Sur les Morphismes d'Intersection Complète en Caractéristiquep☆

Commutative algebras of finite hochschild homological dimension

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Javier Majadas

Author Tools

Chat about Author