A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Maturation of humoral immune responses : Studies on the effects of antigen type, apoptosis and age

The humoral immune response is dependent on the formation of antibodies. Antibodies are produced by terminally differentiated B cells, plasma cells. Plasma cells are generated either directly from antigen challenged B cells, memory cells or from cells that have undergone the germinal center (GC) reaction. The GC is the main site for class switch, somatic hypermutation and generation of memory cells. Different factors, both internal and external, shape the outcome of the immune response. In this thesis, we have studied a few factors that influence the maturation of the humoral response. We have studied how age affects the response, and we show that responses against thymus dependent antigens (TD) are more affected than responses to thymus independent (TI) antigens, in concordance with the view that the T cell compartment is more affected by age than the B cell compartment. Furthermore, we demonstrate that priming early in life have a big influence on the immune response in the aged individual. Priming with a TI form of the carbohydrate dextran B512 (Dx) induces a reduction of IgG levels in later TD responses against Dx. We have evaluated possible mechanisms for this reduction. The reduction does not seem to be caused by clonal exhaustion or antibody mediated mechanisms. We also showed that the reduced TD response after TI priming can be induced against another molecule than Dx. With the hypothesis that TI antigens induce a plasma cell biased maturation of the responding B cells, we examined the presence of Blimp-1, a master regulator of plasma cell differentiation, in GCs induced by TD and TI antigen. Blimp-1 was found earlier in GCs induced by TI antigen and the staining intensity in these GCs was stronger than in TD antigen induced GCs, indicating that plasma cells might be continuously recruited from these GCs. B cells undergoing the GC reaction are thought to be under a strict selection pressure that removes cells with low affinity for the antigen and also cells that have acquired self-reactivity. We investigated the effect of apoptotic deficiencies on the accumulation of somatic mutations in GC B cells. In mice lacking the death receptor Fas, lpr mice, the frequency of mutations was increased but the pattern of the mutations did not differ from wild type mice. In contrast, mice over-expressing the anti-apoptotic protein Bcl-2, had a lowered frequency of mutations and the mutations introduced had other characteristics.