arXiv subjects

Joseph Lipman

Publications and source records attributed to Joseph Lipman.

21 records · Page 2Linked to original sources

arXiv1996

The multiple-point schemes of a finite curvilinear map of codimension one

Let X and Y be smooth varieties of dimensions n-1 and n over an arbitrary algebraically closed field, f:X-> Y a finite map that is birational onto its image. Suppose that f is curvilinear; that is, at every point of X, the Jacobian has rank at least n-2. For r at least 1, consider the subscheme N_r of Y defined by the (r-1)st Fitting ideal of the O_Y-module f_*O_X, and set M_r:=f^{-1}N_r. In this setting --- in fact, in a more general setting --- we prove the following statements, which show that M_r and N_r behave like reasonable schemes of source and target r-fold points of f. Each component of M_r and N_r is empty or has dimension at least n-r. If each component of M_r, or equivalently of N_r, has dimension n-r, then M_r and N_r are Cohen--Macaulay, and their fundamental cycles satisfy the relation, f_*[M_r]=r[N_r]. Now, suppose that each component of M_s, or of N_s, has dimension n-s for s=1,...,r+1. Then the blowup Bl(N_r,N_{r+1}) is equal to the Hilbert scheme Hilb^r_f, and the blowup Bl(M_r,M_{r+1}) is equal to the universal subscheme Univ^r_f of Hilb^r_f x_Y X; moreover, Hilb^r_f and Univ^r_f are Gorenstein. In addition, the structure map h:Hilb^r_f->Y is finite and birational onto its image; and its conductor is equal to the ideal J_r of N_{r+1} in N_r, and is locally self-linked. Reciprocally, h_*O_{Hilb^r_f} is equal to Hom(J_r,O_{N_{r}}). Moreover, h_*[h^{-1}N_{r+1}]=(r+1)[N_{r+1}]. Furthermore, similar assertions hold for the structure map h_1:Univ^r_f->X if r>1.

alg-geom↗

arXiv1995

Local Homology and Cohomology on Schemes

We prove a sheaf-theoretic derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendieck's local duality theorem): for a quasi-compact separated scheme X and a "proregular" subscheme Z---for example, any separated noetherian scheme and any closed subscheme---there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, the i-th left-derived completion functor is the "local homology" sheaf $Ext^i(\RΓ_ZØ_X, -)$. Sheafified generalizations of a number of duality theorems scattered about the literature result, e.g., the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes.

alg-geom↗

arXiv1994

Adjoints of ideals in regular local rings

The adjoint of an ideal I in a regular local ring R is the R-ideal adj(I):=H^0(Y, Iω_Y), where f:Y -> Spec(R) is a proper birational map with Y nonsingular and IO_Y invertible, and ω_f is a canonical relative dualizing sheaf. (Such an f is supposed to exist.) The basic conjecture is that I.adj(I^n)=adj(I^{n+1}) whenever n is >= the analytic spread of I. This is a strong version of the Briancon-Skoda theorem, implying all other known versions. It follows from the (conjectural) vanishing of H^i(Y,ω_Y) for all i>0. When R is essentially of finite type over a char. 0 field, that vanishing has been deduced by Cutkosky from Kodaira vanishing. It also holds whenever dim.R = 2; and in this case we can say considerably more about adjoints, tying in e.g., with classical material on adjoint curves and conductors.

alg-geom↗