Eisenbud D., Huneke C. (Eds.) Free Resolutions in Commutative Algebra and Algebraic Geometry: Sundance 90

CRC Press, 1992. — xii, 148 p. — (Research Notes in Mathematics; 2). — ISBN 0-86720-285-8.Free resolutions arise from systems of linear equations over rings other than fields. A system of linear equations in finitely many unknowns over a field has a basic set of solutions in terms of which all others can be expressed as linear combinations, and these basic solutions can be chosen to be linearly independent. Over a polynomial ring, or more generally any Noetherian ring R, a system of linear equations in finitely many unknowns still has a finite system of solutions in terms of which all others may be expressed, but now these solutions cannot in general be taken to be linearly independent.
To find the dependence relations on a given system of solutions requires solving a new system of linear equations. Iterating this process, one gets a whole series of systems of equations, which makes up a free resolution of the original problem. If one considers the cokernel M of the matrix expressing the original system of equations as a module over ii, one speaks of a free resolution of M. The free resolution expresses certain properties which are implicit in, but not at all obvious from, the original system of equations.
Free resolutions and questions related to them occur in many areas of commutative algebra and algebraic geometry. In May of 1990 there was a small and informal conference in Sundance, Utah, organized by David Eisenbud, Craig Huneke, and Robert Speiser, on the topic of free resolutions and their uses in commutative algebra and algebraic geometry.
A good deal of the conference was devoted to discussions of the current state of work on some of the central problems in the area. These discussions seemed worth transmitting to a broader audience, and we were able to convince a number of the participants to write up accounts of areas in which they are expert. Some of these writeups develop groups of current problems which seem likely to influence future development of the field. Others are basic expositions of areas of current interest; and some contain new research, not otherwise published.Introduction
Structure and Size of Free Resolutions
Problems on infinite free resolutions
L. Avramov
Problems on Betti numbers of finite length modules
H. Charalambous and E.G. Evans, Jr.
Multiplicative structures on finite free resolutions
M. Miller
Wonderful rings and awesome modules
G. Kempf
Green's Conjecture
Green's conjecture: an orientation for algebraists
D. Eisenbud
Some matrices related to Green's conjecture
D. Bayer and M. Stillman
Other Topics
Problems on local cohomology
C. Huneke
Recent work on Cremona transformations
S. Katz
The homological conjectures
P. Roberts
Remarks on residual intersections
B. Ulrich
Some open problems in invariant theory
J. WeymanФайл: отскан. стр. (b/w 600 dpi) + OCR + закладки