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Let be a Tychonoff space and be an intermediate subalgebra of , i.e., . We show that such subrings are precisely absolutely convex subalgebras of . An ideal in is said to be a -ideal if , and imply that . We observe that the coincidence of -ideals and -ideals of is equivalent to the equality . This shows that every -ideal in need not be a -ideal and this is a point which is not considered by D. Rudd in Theorem 4.1 of Michigan Math. J. 17 (1970), 139–141, or by G. Mason in Theorem 3.3 and Proposition 3.5 of Canad. Math. Bull. 23:4 (1980), 437-443. We rectify the induced misconceptions by showing that the sum of -ideals in is indeed a -ideal in . Next, by studying the sum of -ideals in subrings of the form of , where is an ideal in , we investigate a wide class of examples of subrings of in which the sum of -ideals need not be a -ideal. It is observed that, for every ideal in , the sum of any two -ideals in is a -ideal in or all of if and only if is an -space. This result answers a question raised by Azarpanah, Namdari and Olfati in J.Commut. Algebra 11:4 (2019), 479–509.
Let be a commutative local Noetherian ring, a -regular sequence in , and . Given a complex of finitely generated free -modules, we give a construction of a complex of finitely generated free -modules having the same homology. A key application is when the original complex is an -free resolution of a finitely generated -module. In this case our construction is a sort of converse to a construction of Eisenbud and Shamash which yields a free resolution of an -module over given one over .
For a finite-type star operation on a domain , we say that is -super potent if each maximal -ideal of contains a finitely generated ideal such that (1) is contained in no other maximal -ideal of and (2) is -invertible for every finitely generated ideal . Examples of -super potent domains include domains each of whose maximal -ideals is -invertible (e.g., Krull domains). We show that if the domain is -super potent for some finite-type star operation , then is -super potent, we study -super potency in polynomial rings and pullbacks, and we prove that a domain is a generalized Krull domain if and only if it is -super potent and has -dimension one.
Let be a commutative Noetherian ring and be a self-dual acyclic complex of finitely generated free -modules. Assume that has length four and has rank one. We prove that can be given the structure of a differential graded algebra with divided powers; furthermore, the multiplication on exhibits Poincaré duality. This result is already known if is a local Gorenstein ring and is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that is local, is Gorenstein, and is minimal.
The -vector of a graph is defined in terms of its clique vector by the equation where is the largest cardinality of a clique in . We study the relation of the -vector of a chordal graph with some structural properties of . In particular, we show that the -vector encodes different aspects of the connectivity and clique dominance of . Furthermore, we relate the -vector with the Betti numbers of the Stanley–Reisner ring associated to clique simplicial complex of .
The aim of this paper is to give a connection between the -pure threshold of a polynomial and the height of the corresponding Artin–Mazur formal group. For this, we consider a quasihomogeneous polynomial of degree equal to the degree of and show that the -pure threshold of the reduction is equal to the log-canonical threshold of if and only if the height of the Artin–Mazur formal group associated to , where is the hypersurface given by , is equal to 1. We also prove that a similar result holds for Fermat hypersurfaces of degree greater than . Furthermore, we give examples of weighted Delsarte surfaces which show that other values of the -pure threshold of a quasihomogeneous polynomial of degree cannot be characterized by the height.
Let and be schemes of finite type over and let be a finite map. We show the following holds for all sufficiently large primes : If and are any splittings on and , such that the restriction of is compatible with and , and is any compatibly split subvariety of , then the reduction is a compatibly split subvariety of . This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.
We study the sets of semistar and star operations on a semilocal Prüfer domain, with an emphasis on which properties of the domain are enough to determine them. In particular, we show that these sets depend chiefly on the properties of the spectrum and of some localizations of the domain; we also show that, if the domain is -local, the number of semistar operations grows as a polynomial in the number of semistar operations of its localizations.
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