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In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.

In this paper, we investigate some open questions regarding perinormal domains posed by Neal Epstein and Jay Shapiro in [6] . More specifically, we focus on the ascent/descent property of perinormality between “canonical” integral domain extensions, in particular, A ⊂ A [ X ] and A ⊂ A ˆ . We give special conditions under which perinormality ascends from A to the polynomial ring A [ X ] in the case of an universally catenary domain A. Whereas we have a characterizing result for when perinormality descends from A [ X ] to A, the sufficient condition for the descent is cumbersome to check. For this reason, we turn to special cases for which perinormality descends from A [ X ] to A. In the case of an analytically irreducible local domain ( A , m ) and its m -adic completion ( A ˆ , m ˆ ) , we refer to a technique for generating examples in which perinormality fails to ascend. When A ˆ is perinormal, we explore hypotheses under which A must be normal, perinormal, or weakly normal. Finally, we make connexions between the concepts of semi-normality, weak-normality, relative and global perinormality, and normality.

Let ( C , ⊗ , 1 ) be an abelian closed symmetric monoidal category satisfying certain conditions. The purpose of this paper is to prove three descent results for schemes over ( C , ⊗ , 1 ) that are not necessarily Noetherian.

Using Nisnevich coverings and a Hilbert stack of stacky points, we prove etale devissage results for non-representable etale and quasi-finite flat coverings. We give applications to absolute noetherian approximation of algebraic stacks and compact generation of derived categories.

We show that ideal-length defines a length function on almost-Noetherian integral domains. This length function is a sum of multiplicities and in favourable cases, a linear combination on N of discrete valuations. As a consequence, Σ 1 -Noetherian domains have length functions. Infra-Krull domains are weakly Krull almost-Noetherian domains. We characterize these integral domains and establish their properties, placing emphasis on integral closures and length functions. Descent properties are shown. Applications to the computation of elasticities and factorization properties are given.

In this paper, the Shintani descent of irreducible characters of finite special linear groups is discussed. We parametrize irreducible characters of SL n ( F q ) by making use of (modified) generalized Gelfand–Graev characters. Shintani descent of irreducible characters of SL n ( F q m ) is determined by investigating the Shintani descent of generalized Gelfand–Graev characters. Our results hold for arbitrary p and q .

The main purpose of this article in some sense is to illustrate the manner in which the classical methods of Galois descent work to impose constraints on the algebraic behavior of local rings (for contrast, see A. Grothendieck, “Seminaire de Geometrie Algebrique” 1962 I.H.E.S., fasc. I and II, 1963) . By way of example, we show in Section 2 that a factorial local domain which arises as a normal extension (see definition) of a regular local ring is necessarily a complete intersection. This is in sharp contrast to the situation for general normal domains (e.g., see M.-J, Bertin, C.R. Acad. Sci. Paris 264 (1967) , 653–656; P. Griffith, J. Pure Appl. Algebra 7 (1976) , 303–315; and R. Fossum and P. Griffith, Ann. Sci. Ecole Norm. Sup. (4) 8 (1974) , 189–200).

The analysis of T,(R) is by a “descent” from the ring S, and so we are led to considering T,(S). Since S contains a pth root of unity, namely E, the computation of T,(S) follows easily from standard arguments of Kummer theory. In this section we recall the part of the theory which we need to compute T,(R) [S]. The map Sx + T,(S) sending s to [S[y]/(yp - s), a], where o(y) = EJJ, is a group homomorphism; (S x )” is the kernel, and we obtain an injection j: sx/(s”)p’ T,(S). Given [A, a] E Z”,,(S), consider

For a given triple (monad) U : C → C in the category C , we develop a theory of descent for U. We start by introducing the basic constructions associated to a triple: descent data, symmetry operators, and flat connections. The main result of this section asserts that the sets of these objects are bijectively equivalent. Next we construct a monoidal category C (U) such that U is an algebra in C (U) . If C is abelian, we define Amitsur cohomology of U with coefficients in a functor F : C (U)→ D . As an application of this construction, in the case where U is faithfully exact, we describe those morphisms that descend with respect to U. In the last part of the paper we classify all U-forms of a given object C 0 ∈ C . We show that there is a one-to-one correspondence between the set of equivalence classes of U-forms and a certain noncommutative Amitsur cohomology. Let A/B be an extension of associative unitary rings and let C be the category of right B-modules. Then (−)⊗ B A : C → C is a triple which is faithfully exact if and only if the extension A/B is faithfully flat. Specializing our results to this particular setting, we recover faithfully flat descent theory for extensions of (not necessarily commutative) rings.

Solomon's descent algebra is generated by sums of descent classes corresponding to certain hook shapes. This particularly implies that the ring of class functions of any finite symmetric group S n is generated by the irreducible characters corresponding to certain hook partitions of n . As another consequence, a second generating set of Solomon's descent algebra (and of the ring of class functions of S n ) is obtained related to the major index of permutations.

Elements of the hyperoctahedral group Bn can be represented as lists of integers π = π1π2 … πn, where the list of absolute value of the πk's is a permutation of the integers from 1 to n. The descent set of such a π is Des(π) = {kϵ {0, …, n − 1} ¦ πk > πk + 1}, where k = 0, we set π0 = 0. A descent class AS of Bn, is the formal sum of all elements of Bn with a given descent set S ⊆{0, 1, …, n − 1}. In this paper, we shall give a combinatorial proof of a result of Solomon [6] stating that the product, in the group algebra Q[Bn], of two descent classes is a linear combination of descent classes. We refer to the linear span of these descent classes (it is a subalgebra ∑Bn of Q[Bn], as Solomon's Hyperoctahedral descent algebra. We derive from our combinatorial multiplication rule (for ∑ Bn) that there is a family of ideals Jv(n) of ∑ Bn such that J0(n) ⊆- J1(n) … ⊆- Jn(n) , with ∑ Bn − v + 1 ∼- ∑ BnJv(n) . In [4], Garsia and Reutenauer relate a similar decomposition of the multiplicative structure of the descent algebra of the symmetric group Sn, to the action of the symmetric group on the free Lie algebra. It develops that one can introduce a Bn-analog of the free Lie algebra, and give similar results for ∑ Bn.

DESCENT THEOREM. Let C be a simple algebra with center E, and let G be a $nite group of automorphism of E with fixed field F. If F can be faithfully lifted to a group G of E-semilinear automorphisms of C (i.e. for CT E G, L? is an automorphism of C such that O(xv) = a(x) Z(v) for x E E and v E C), then C g C’ OF E where C’ is the algebra of fixed elements of G. Moreover, C’ is simple with center F, and if [C : E] < 00, then [C : E] = [C’ : F].