Closed Sets in

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

If k is finite, the Zariski topology is discrete and contains no information. Consequently we assume k infinite in the rest of this chapter and in all subsequent references to closed sets in fc". We have then one simple fact to observe ... 

Theorem. A closed set in k" is irreducible iff its ring of functions is an integral domain. 

Corollary. A closed set in Id1 has only finitely many connected components, each a union of irreducible components. 

We want to extend the concept of connectedness to group schemes more general than matrix groups. To do this we will associate with each of them some topological space. This space will usually have more points than just those of a closed set in k", and before we go on it is worth observing that even in our current material there are indications that we do not have all the points we should have. 

Let A be finitely generated over a field. Show that X -+ X n T is a bijection from closed sets in Spec A to closed sets in the subspace of maximal ideals. 

If the base ring k is not plain from context, we write explicitly 2 /t. Clearly we can also construct 2 for 4 not finitely generated just by extending the preliminary computation to polynomial rings in infinitely many variables. When S is a closed set in k", the elements of 2 s] are the (algebraic) differentials defined on S—combinations of the dxt multiplied by functions. In general therefore we call 2 the module of differentials of 4. 

The smoothness of algebraic matrix groups is a property not shared by all closed sets in /c". To see what it means, take fc = fc and let 5 fc" be an arbitrary irreducible closed set. Let s be a point in S corresponding to the maximal ideal J in k[S]. If S is smooth, n si k = O si /J us) has fc-dimension equal to the dimension of S. (This would in general be called smoothness at s.) If S is defined by equations fj = 0, the generators and relations for OUS] show that S is smooth at s iff the matrix of partial derivatives (dfj/dXi)(s) has rank n — dim V. Over the real or complex field this is the standard Jacobian criterion for the solutions of the system (f = 0) to form a C or analytic submanifold near s. For S to be smooth means then that it has no cusps or self-crossings or other singularities . 

Suppose for illustration that A and B are rings of functions on closed sets in k", with k = k. The maximal ideals P in A then correspond to points x in the set. If PB 4 B, some maximal ideal of B contains P, and the corresponding point maps to x. Thus when A - B is flat, the extra condition involved in faithful flatness is precisely surjectivity on the closed sets. Condition (2) is the generalization of that to arbitrary rings. 

Cartan subgroup 77 Cartier duality 17 Center of a group scheme 27 Central simple algebra 145 Character 14 Charater group 55 Clopen set 42 Closed embedding 13 Closed set, closure 156 Closed set in k 28 Closed set in Spec A 42... 

Theorem. Every closed set in k" is in a unique way a finite irredundant union of irreducible closed sets. 

Open sets, closed sets, relative complement, and continuity of functions. In a metric space Y, a subset A of Y is regarded as an open set if around every point of A there exists some (perhaps very tiny) ball that is still within set A. Informally, an open set does not contain its boundary points (e.g., an open potato is the potato without its skin where the skin is thought to be infinitely thin). The relative complement of A in space Y is the set Ac of all points of Y which are not in A. The relative complement can be written as Ac=Y A. A. subset C of set Y is a closed set in Y if the relative complement of C is an open subset in Y. The closure clos(A) of a set A is the smallest closed set that contains A. 

The deterministic proliferation of HEX cylinder orientations during repeated heating-cooling cycles is a unique phenomenon and is a consequence of the fact that the HEX produces a twinned BCC. Indeed, if the HEX-to-BCC transition yielded only a simple BCC structure, then although the first reverse transition would give four cyhnder orientations, repeated heat-ing/coohng would not yield any new cyhnder orientations since these four directions form a closed set. In contrast, in the LAM -> 111 LX transition, the epitaxial relationship only constrains the cylinders to be imbedded in the layers of the minority component of the LAM, but the orientation of the HEX is determined by random fluctuations. 

We want to define the product X x Y of any two prevarieties X,Y. Now we will certainly want to have An x Am = An+m. But the product of the Zariski topologies in An and Am does not give the Zariski topology in An+m in A1 x A1, for instance, the only closed sets in the product topology are finite unions of horizontal and vertical lines. The only reliable way to find the correct definition is to use the general category-theoretic definition of product. 

To get a proper closed set, we take a finite number of irreducible curves, generic points and all, plus a finite number of closed points. Clearly, adding the non-closed points has not affected the topology much here given a closed subset of the set of closed points in the old topology, there is a unique set of non-closed points to add to get a closed set in our new plane. 

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