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1.

We find necessary and sufficient conditions for a complete local ring to be the completion of a reduced local ring. Explicitly, these conditions on a complete local ring with maximal ideal are (i) or , and (ii) for all , if is an integer of , then .

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2.
We give restrictions on the cardinality of compact Hausdorff homogeneous spaces that do not use other cardinal invariants, but rather covering and separation properties. In particular, we show that it is consistent that every hereditarily normal homogeneous compactum is of cardinality . We introduce property wD(), intermediate between the properties of being weakly -collectionwise Hausdorff and strongly -collectionwise Hausdorff, and show that if is a compact Hausdorff homogeneous space in which every subspace has property wD( ), then is countably tight and hence of cardinality . As a corollary, it is consistent that such a space is first countable and hence of cardinality . A number of related results are shown and open problems presented.

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3.
Let be a commutative Noetherian ring with non-zero identity, and ideals of with , and a finitely generated -module. In this paper, for fixed integers and , we study the finiteness of and in several cases.

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4.
It is a well-known result of M. Brodmann that if is an ideal of a commutative Noetherian ring , then the set of associated primes of the -th power of is constant for all large . This paper is concerned with the following question: given a prime ideal of which is known to be in for all large integers , can one identify a term of the sequence beyond which will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

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5.
Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

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6.
    
Answering a question of Arhangel'skii, we show - under GCH - that for most cardinals there exists an -compact space such that but does not embed in a closed fashion into the product of copies of .

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7.

It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra is at least , where is the center of . We improve this result by showing that a better lower bound is , where and is a complement of in . Furthermore, we provide evidence that this is the best possible bound of the form .

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8.
Let be a Noetherian homogeneous ring with one-dimensional local base ring . Let be an -primary ideal, let be a finitely generated graded -module and let . Let denote the -th local cohomology module of with respect to the irrelevant ideal 0} R_n$"> of . We show that the first Hilbert-Samuel coefficient of the -th graded component of with respect to is antipolynomial of degree in . In addition, we prove that the postulation numbers of the components with respect to have a common upper bound.

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9.
About a year ago Angus Macintyre raised the following question. Let and be complete local noetherian rings with maximal ideals and such that is isomorphic to for every . Does it follow that and are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.

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10.
It is known that the sets of extreme and exposed points of a convex Borel subset of are Borel. We show that for there exist convex subsets of such that the sets of their extreme and exposed points coincide and are of arbitrarily high Borel class. On the other hand, we show that the sets of extreme and of exposed points of a convex set of additive Borel class are of ambiguous Borel class . For proving the latter-mentioned results we show that the union of the open and the union of the closed segments of are of the additive Borel class if is a convex set of additive Borel class .

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