Intersection theory in algebraic cobordism

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf.[4]).(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf.[4]).(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf.[6], [2], [4]).

Pointed irreducible bialgebras

The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers

The main purpose of this paper is to provide a survey of results concerning the $\mathbb{Z}$-gradings on the infinite-dimensional Grassmann algebra $E$ over a field of characteristic zero. First, we provide graded identities and central polynomials for $E$ equipped with fine gradings on $E$ by the semigroup $(\mathbb{Z}^\ast,\times)$. We also describe briefly techniques in order to illustrate some important methods to exhibit graded identities and central polynomials of $E$ for other abelian groups. In particular, over a field of characteristic zero, so-called $2$-induced gradings of full support were considered. In order to obtain these descriptions, we strongly use elementary number theory as a tool, providing an interesting connection between this area and PI-Theory.

Let M be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units G. The multiplicative semigroup .4 (F) , where F is a finite field, is a particular example. Using HarishChandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of M is completely reducible. Using this we characterize the representations of G extending to irreducible representations of M as being those induced from the irreducible representations of certain parabolic subgroups of G. We go on to show that if F is any field and S any multiplicative subsemigroup of .4 (F) , then the semigroup algebra of S over any field of characteristic zero has nilpotent Jacobson radical. If S = .4 (F) , then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of .4 (F) over a field of characteristic zero is regular if and only if ch(F) = p > 0 and F is algebraic over its prime field.

It is shown that every compact group G is a Q-simplicial space where Q is any field of characteristic zero. As a consequence it follows that G satisfies a variation of the Lefschetz fixed point theorem. It has been known for some time that the Lefschetz fixed point theorem applies to a few spaces other than just ANR spaces, especially if some care is taken to use coefficients in certain fields [2]. The case of all compact groups provides a broad class of spaces which may not have local connectivity of any order. It is shown that every compact group G satisfies the Lefschetz fixed point theorem when coefficients for the homology groups are taken in a field of characteristic zero.

Let A be an abelian group and let K be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in A. In this thesis we prove foundational properties of the A-fibered Burnside ring functor B^A_K as an A-fibered biset functor over K. This includes the determination of the lattice of subfunctors of B^A_K and the determination of the composition factors of B^A_K. The results of the paper extend results of Coşkun and the author for the A-fibered Burnside ring functor restricted to p-groups and results of Bouc in the case that A is trivial, i.e., the case of the Burnside ring functor over fields of characteristic zero.

We study log varieties, over a field of characteristic zero, which are generically logarithmically smooth and fs in the Kummer normally log etale topology. As an application, we prove an analog of Abramovich-Temkin-Wlodarczyk’s log resolution of singularities of fs log schemes in the Kummer fs setting.

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

On the finite generation of additive group invariants in positive characteristic

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with \(n \geqq 2\), let H(n) be the group of all n x n matrices having determinant \(\pm 1\). Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants LB(n) of B(n), with \(B(n) \in \{ H(n), {\rm SL}(n)\} \), is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let \(\langle {\rm Sem}(n) \rangle \) be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that \(\langle {\rm Sem}(n) \rangle \) is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].

Let A be a commutative Noetherian ring of characteristic zero and R = A [ X 1 , … , X d ] be a polynomial ring over A with the standard N d -grading. Let I ⊆ R be an ideal generated by elements of the form aU, where a ∈ A (possibly nonunit) and U is a monomial in X i ‘s. We call such an ideal as a ‘ C -monomial ideal’. It was observed earlier that for u ¯ ∈ Z d , the u ¯ th components of local cohomology modules supported on monomial ideals depend only on which coordinates of u ¯ are negative. This article shows that similar statements hold in our general setting, even for certain invariants associated to the components. We mainly focus on Bass numbers, injective dimensions, dimensions of supports, associated primes, Bernstein-type dimensions and multiplicities of the components. Under the extra assumption that A is regular, we describe the finiteness of Bass numbers of each component and bound its injective dimension by the dimension of its support. Finally, we present a structure theorem for the components when A is a power series ring in one variable over a field of characteristic zero.

Let \({\mathbb{K}}\) be a field of characteristic zero or, more generally, a \({\mathbb{Q}}\)-algebra. A formal power series \({F(x,y)=x+y+ \sum_{i,j \geq 1} a_{i,j}x^iy^j \in \mathbb{K}{[\![} x, y{]\!]}}\) is called a one-dimensional formal group law if F(F(x, y), z) = F(x, F(y, z)). Using some elementary methods, we prove that for every one-dimensional formal group law F(x, y) there exists a formal power series \({f(x)=x+\sum_{n\geq 2}f_nx ^n \in \mathbb{K}{[\![} x, y{]\!]}}\) so that F(x, y) = f−1(f(x) + f(y)).

Let k be a field of characteristic zero. Let V be a k-scheme of finite type, i.e., a k-variety, which is integral. We prove that if the associated arc scheme \({\mathcal{L}_{\infty}(V)}\) is reduced, then the \({\mathcal{O}_{V}}\)-Module \({\Omega_{V/k}^{1}}\) is torsion-free. Then if the k-variety V is assumed to be locally a complete intersection (lci), we deduce that the k-variety V is normal. We also obtain the following consequence: for every class \({\mathfrak{C}}\) of integral k-curves which satisfies the Berger conjecture, and for every \({\mathscr{C} \in \mathfrak{C}}\), the k-curve \({\mathscr{C}}\) is smooth if and only if \({\mathcal{L}(\mathscr{C})}\) is reduced.

Let A be a normal local ring which is essentially finite type over a field of characteristic zero. Let I⊂A be an ideal such that the Rees algebra RA(I) is Cohen–Macaulay and normal. In this paper we address the question: “When does RA(I) have rational singularities?” In particular, we study the connection between rational singularities of RA(I) and the adjoint ideals of the powers In (n∋ℕ).