Aside

Church Service

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Yesterday afternoon, I went to sing hymns at a nursing home with a group from my church, and sang Amazing Grace solo.  This was my second time singing alone for church (the first time was three days ago, on Valentine’s Day), and I was terrified to begin, but once I got started, it wasn’t about me anymore; it was about reaching people and glorifying God.  We had an excellent short sermon on forgiveness.  The service went pretty well, and was made more interesting by the resident who appointed himself our new pianist near the end and accompanied us by bashing his fists forcefully on the piano keys.  The other, less lively residents sat in their chairs and listened, and have to say I was impressed by the number of people who showed up.  I am very much looking forward to doing it again.

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Sheaves

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Sheaves

Let C be a category with objects ob(C), let X be a topological space, and define Top(X) to be the category whose objects are the open subsets of (X) and whose arrows are the inclusion maps between open subsets of (X). A sheaf on X of objects of C consists of

i) for every open subset U of X, an object F(U) of C, and

ii) for every inclusion of an open subset V of X in an open subset U of X, an arrow pUV:F(U)->F(V), such that

condition 1) pUU is the identity arrow of F(U),

condition 2) pUW=pVWpUV where pVW and pUV are defined,

condition 3) for any open covering {Vi} of any open subset U of X, and for any element s of F(U) such that pUVi(s)=0 for all i, s=0 and

condition 4) for any open covering {Vi} of any open subset U of X, and for any set {si} of elements of F(U) such that pUViintVj(si)=pUViintVj(sj) and si is in F(Vi) for all i, there exists an element s of F(U) such that pUVi(s)=si for each i.

Presheaves

The definition of a presheaf is weaker than that of a sheaf.  A preasheaf on X consists of

i) for every open subset U of X, an object F(U) of C, and

ii) for every inclusion of an open subset V of X in an open subset U of X, an arrow pUV:F(U)->F(V), such that

condition 1) pUU is the identity arrow of F(U) and

condition 2) pUW=pVWpUV where pVW and pUV are defined.

Algebraic Varieties

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Preliminary Definitions

Let k be an algebraically closed field.  Affine n-space over k is the set of all sets (a1,…,an) of n elements of k, and is denoted by Ank or just An. Each element P=(a1…,an of An is a called a point, and the ai are called its coordinates.

Suppose f is a member of the polynomial ring A in n variables over k and P=(a1,…,an) is a point in An. Then f(P) is defined to be equal to P(a1,…,an).

Let T be some subset of A. The set Z(T) of zeros of T is defined to be the greatest subset of elements P of An such that f(P)=0 for all f in T. The sets Z(T) are called algebraic sets, and the Zariski topology on An is defined by taking the closed sets to be the algebraic sets.

A subset of a topological space is irreducible if it is nonempty and contains no two proper closed subsets of which it can be written as the union.

Projective n-space Pn over an algebraically closed field is defined as the set of equivalence classes of elements of An+1-{0,…,0} defined by the following equivalence relation: P1 is in the same equivalence class as P2 iff P1=LP2 for some L in k-{0}. For any member of the polynomial ring in n+1 variables over k and any member e of Pn, define f(e) thusly: select some point P in An+1 whose equivalence class is e. Then compute f(e). This value will be determined up to multiplication by a nonzero element of k, which means that nonzero values of f(e) are undefined. Convention remedies this unfortunate situation by defining the nonzero values to be equal to 1.

Varieties
Varieties fall into four types, as detailed below.

Let An have the Zariski topology. The closed irreducible subsets of An are known as affine algebraic varieties or just algebraic varieties, and quasi-affine algebraic variety or quasi-affine variety is an open subset of an affine variety.

Let T be some subset of the polynomial ring in n+1 variables over k. Z(T) is then defined to be be the set of all equivalence classes in Pn such that f(e)=0 for all f in T, and the Zariski topology can be defined in the same way as for affine space. Let us endow Pn with the Zariski topology, and define a projective algebraic variety or projective variety (respectively, quasi-projective algebraic variety or quasi-projective variety) to be an irreducible closed (respectively, an open subset of an irreducible closed) subset of Pn.

[Note: there also exists the (obviously) less concrete notion of an abstract variety.]