By David Eisenbud

ISBN-10: 1461253500

ISBN-13: 9781461253501

ISBN-10: 3540781226

ISBN-13: 9783540781226

This is a finished evaluate of commutative algebra, from localization and first decomposition via measurement thought, homological tools, loose resolutions and duality, emphasizing the origins of the guidelines and their connections with different elements of arithmetic. The ebook supplies a concise remedy of Grobner foundation thought and the confident tools in commutative algebra and algebraic geometry that stream from it. Many routines included.

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**Extra resources for Commutative Algebra: with a View Toward Algebraic Geometry**

**Sample text**

Reason: The case of one variable is the statement that a polynomial in one variable can have only finitely many roots, and follows from Euclid's algorithm for division. In the general case we think ° 32 1. Roots of Commutative Algebra of a nonzero polynomial f (Xl, ... , xn) in n variables as a polynomial in n - 1 variables with coefficients that are polynomials in one variable. ) If follows that if k is infinite, then distinct polynomials define distinct functions. Thus we may regard the polynomial ring k[XI, ...

But there is another, coarser, topology on X that is defined over any field. Polynomial functions on X will play the role of continuous functions, even when the fields we are working over have no topology, and by analogy with the continuous case it is natural to think of an algebraic subset Y as a closed subset of X. Since we obviously have niZ(Ji) = Z(U;)i), the intersection of any collection of algebraic subsets is algebraic. Furthermore, if we define II~=1 Ji to be the set consisting of all products of one function from each J i , then Ui=l Z (Ji ) = Z(IIi=1 Ji ), so any finite union of algebraic subsets is algebraic.

1. Let S = k[Xl, ... ,xrl be the polynomial ring, and let ~ be the symmetric group of all permutations of {I, ... , r}. The group ~ acts on S as follows: If (T E ~ and I E S, we define The group ~ then acts as a group of k-algebra automorphisms of S. The set of invariants which in this case is called the ring of symmetric functions, is therefore a subring of S. I(Xl, ... ,Xr) :=Xl+···+ Xr, h(Xl, ... ,Xr):= L XiXj, l~i

### Commutative Algebra: with a View Toward Algebraic Geometry by David Eisenbud

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