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On the minimal free resolution of \(n+1\) general forms. (English) Zbl 1053.13005
Let \(R=K[X_1,\dots,X_n]\) be a polynomial ring over a field \(K\), and \(I\) an ideal generated by \(n+1\) generically chosen forms of degrees \(d_1\leq\cdots\leq d_{n+1}\). (In particular, \(R/I\) is artinian.) The authors determine the graded Betti numbers of \(R/I\) in quite a number of cases, for
(1) \(n=3\),
(2) \(n=4\), \(\sum d_i\) even,
(3) \(n\) is even, \(d_i\) constant and even, (4) \(d_2+\cdots+d_n<d_1+d_{n+1}+n\), \(d_1+\cdots+d_n-d_{n+1}-n\gg0\).
(This condition can be omitted if \(n=4\) or \(n\) is odd.)
In many other cases rather sharp bounds are obtained.
The main technique of the paper is to link \(I\) to a Gorenstein ideal \(G\) via the complete intersection defined by the first \(n\) forms and to exploit results on the resolution of \(R/G\). Via linkage one obtains a free resolution of \(R/I\) and has to control its potential non-minimality. A notorious difficulty in the study of graded Betti numbers is presented by “ghost terms” in the minimal free resolution that cancel each other in the Hilbert function, and this phenomenon is studied in some detail. The authors present an example that disproves Iarrobino’s “thin resolution conjecture”.
The paper makes interesting contributions to the theory of Hilbert functions and graded Betti numbers, touching (and using) the work of many other mathematicians.

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C40 Linkage, complete intersections and determinantal ideals
Software:
Macaulay2
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References:
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