By David Eisenbud

This is a accomplished assessment of commutative algebra, from localization and first decomposition via measurement idea, homological equipment, unfastened resolutions and duality, emphasizing the origins of the information and their connections with different components of arithmetic. The booklet provides a concise therapy of Grobner foundation idea and the optimistic tools in commutative algebra and algebraic geometry that stream from it. Many workouts included.

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

**Example text**

Fng = 0, for some elements h,···, fn E R, then the module is R/(h, ... , fn). Here is an exact sequence view: An element m of a module M corresponds to a homomorphism from R to M, sending 1 to m. Thus, giving a set of elements {m a }aEA E M corresponds to giving a homomorphism cp from a direct sum G := RA of copies of R, indexed by A, to M, sending the a;th basis element to mao If the ma generate M, then cp is a surjection. The relations on the ma are the same as elements of the kernel of the map G ----+ M.

We set 7f(x) = y. 10, 7f is actually a polynomial map. In addition, we claim that 7f factors through the set X/G. Indeed, since R is invariant under g, we have R n mx = g(R n mx) = R n g(mx). Since g(mx) = mg-lx, this says that 7f(g-lX) = 7f(x); that is, 7f : X -+ Y factors through a map X/G -+ Y. Under good circumstances, the map X/G -+ Y is surjective. If we are in situation where R is a summand of A(X), as in the cases Hilbert treated, then for any maximal ideal n of R we have nA(X) =I- A(X).

To this end note that if X c pr is an algebraic set defined by homogeneous polynomial equations Fi(xo, ... i(XI, ... , x r ) = Fi(l, Xl, ... , Xr ) = 0; thus X n U is naturally an algebraic set in Ar. Every polynomial ! (Xl, ... , Xr ) may be written in the form F(l, xl, . , x r ) for some homogeneous polynomial F(xo, ... , xr). , and let F be the result of multiplying each homogeneous component of ! by a power of Xo to bring up its degree to d. More formally, we may write It follows that F(l,XI, ...