A Rota-Baxter Leibniz algebra is a $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual of algebras. Next, we cohomology theory also study the infinitesimal formal deformation algebras show that our cohomology. Moreover, an abelian extension equivalence classes such extensions are related to groups.

In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The i local cohomol...

In this note, we use Connes’ theory of spectral triples to provide a connection between Manin’s model of the dual graph of the fiber at infinity of an Arakelov surface and the cohomology of the mapping cone of the local monodromy. Abridged English version In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ...

§0 Introduction. The Morava K-theories K(n)∗( ) (for p a prime and 0 < n < ∞) form a collection of multiplicative cohomology theories whose central rôle in homotopy theory is now well established. However, even though they have been intensively studied there are still many aspects of their structure which remain undeveloped. In this paper we give a construction for families of operations remini...

The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal ,4-modules is studied in the special case in which A is the ring of integers in the field obtained by adjoining pth roots of unity to Qp , the p-adic numbers. Information about these cohomology groups is used to give new proofs of results about the E2 term of the Adams spectral sequence based on 2-local...

This article is based on five lectures the author gave during the summer school, Interactions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Gr...

Local cohomology was discovered in the 1960s as a tool to study sheaves and their cohomology in algebraic geometry, but have since seen wide use in commutative algebra. An example of their use is to answer the question: how many elements are necessary to generate a given ideal, up to radical? For example, consider two planes in 4-space meeting at a point. The vanishing ideal I = (x, y) ∩ (u, v)...

We prove a duality theorem for graded algebras over a field that implies several known duality results: graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

We reprove Lazard’s result that every commutative n-bud is extendible to an n+ 1 bud, from an obstruction theoretic point of view. We locate the obstruction to extending an arbitrary n-bud in a certain cohomology group, and classify isomorphism classes of n-bud extensions for low degrees. 1 Lubin-Tate cohomology Definition 1. Let R be a commutative, unital ring and F a formal group law on R (as...