Abstract :
LetRbe a commutative noetherian ring and :F → Gbe a homomorphism of freeR-modules where rank F = fand rank G = g. Fix an elementbg + 1 g + 1Fand a generator ωG*for gG*. The module action of • F* on • Fproduces the elementb1 = [( g *)(ωG*)](bg + 1) inF. LetJdenote the image ofb1:F* → R. Assume that gradeJ = f − g, which is the largest grade possible and is attained in the generic case. The idealJmay be interpreted as the defining ideal of the degeneracy locus of a regular section of a rankf − greflexive sheaf. It may also be interpreted as the order ideal of an element in a second syzygy module of rankf − g. Also,Jmay be interpreted as the defining ideal for the symmetric algebra of a module of projective dimension two. Migliore and Peterson have studied the idealJunm, which is the unmixed part ofJ. Under geometric hypotheses, they have shown thatR/Junmis a Cohen–Macaulay ring and they have resolved this ring. Furthermore, iff − gis odd, thenJunmis a Gorenstein ideal and is not equal toJ. On the other hand, iff − gis even, thenJunm = J. In the present paper, we produce the resolution ofR/Jby freeR-modules in the case thatf − gis even and (f − g − 2)! is a unit inR. Our resolution is minimal whenever the data are local or homogeneous. Our resolution is built from the differential graded algebra ( • F* X1,…,Xg , d), where the restriction ofdto • F* is the Koszul complex associated tob1:F* → Rand the degree two divided power variablesX1,…,Xghave been adjointed in order to kill the cycles *(G*) 1F*. The acyclicity lemma is used to prove exactness. Ifg = 1, then the idealJis equal to the Huneke–Ulrich almost complete intersection idealI1(yX), whereyis a 1 × fmatrix andXis anf × falternating matrix. The resolution of this ideal is already known.
