Adjoint associativity: an invitation to algebra in infinity-categories

There appeared not long ago a Reduction Formula for derived Hochschild cohomology, that has been useful e.g., in the study of Gorenstein maps and of rigidity w.r.t. semidualizing complexes. The formula involves the relative dualizing complex of a ring homomorphism, so brings out a connection between Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses homotopical considerations via a number of noncanonical projective and injective resolutions of differential graded objects. Recent efforts aim at more intrinsic approaches, hopefully upgradable to "higher" contexts--like bimodules over algebras in infinity-categories. This would lead to wider applicability, for example to ring spectra; and the methods might be globalizable, revealing some homotopical generalizations of aspects of Grothendieck duality. (The original formula has a geometric version, proved by completely different methods coming from duality theory.) A first step is to extend Hom-Tensor adjunction--adjoint associativity--to the infinity-category setting.Comment: 23 page

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, I\omega_Y), where f:Y -> Spec(R) is a proper birational map with Y nonsingular and IO_Y invertible, and \omega_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,\omega_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.Comment: 12 pages, Amstex 2.

A vanishing theorem for finitely supported ideals in regular local rings

A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed characteristic.) It follows that the adjoint of a finitely supported ideal I of order a has order sup(a+1-d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, the adjoint of I is also finitely supported. Also: if this I is a normal ideal, then its reduction number is the least integer > d(1-1/a).Comment: 12 pages. Main results and proofs similar to previous version, but with numerous enhancement

Equisingularity and simultaneous resolution of singularities

The purpose, mainly expository and speculative, of this paper---an outgrowth of a survey lecture at the September 1997 Obergurgl working week---is to indicate some (not all) of the efforts that have been made to interpret equisingularity, and connections among them; and to suggest directions for further exploration. Zariski defined equisingularity on an n-dimensional hypersurface V via stratification by ``dimensionality type," an integer associated to a point by means of a generic local projection to affine n-space. A possibly more intuitive concept of equisingularity can be based on stratification by simultaneous resolvability of singularities. The two approaches are known to be equivalent for families of plane curve singularities. In higher dimension we ask whether constancy of dimensionality type along a smooth subvariety W of V implies the existence of a simultaneous resolution of the singularities of V along W. (The converse is false.) The underlying idea is to follow the classical inductive strategy of Jung--begin by desingularizing the discriminant of a generic projection--to reduce to asking if there is a canonical resolution process which when applied to quasi-ordinary singularities depends only on their characteristic monomials. This appears to be so in dimension 2. In higher dimensions the question is quite open.Comment: 4 figure

Integrally closed ideals in two-dimensional regular local rings are multiplier ideals

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field, there is in fact no difference between "multiplier" and "integrally closed" (or "complete.") However, among multiplier ideals arising from an integer multiplying constant (also known as adjoint ideals) the only simple complete ones primary for the maximal ideal are those of order one.Comment: 12 pages. In a two-dimensional regular local ring with algebraically closed residue field, every complete fractionary ideal is a multiplier ideal. In the original posting this was proved only for ideals primary for the maximal ideal. A reexamination of the proof, prompted by work of Favre and Jonsson, revealed that with minor modifications it applies--as presented here--to the general cas

Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor

For a map f: X -> Y of quasi-compact quasi-separated schemes, we discuss quasi-perfection, that is, the right adjoint f^\times of the derived functor Rf_* respects small direct sums. This is equivalent to the existence of a functorial isomorphism f^\times O_{Y} \otimes^L Lf^*(-) \to f^\times (-); to quasi-properness (preservation by Rf_* of pseudo-coherence, or just properness in the noetherian case) plus boundedness of Lf^* (finite tor-dimensionality), or of the functor f^\times; and to some other conditions. We use a globalization, previously known only for divisorial schemes, of the local definition of pseudo-coherence of complexes, as well as a refinement of the known fact that the derived category of complexes with quasi-coherent homology is generated by a single perfect complex.Comment: Thanks to a theorem of Bondal and van den Bergh, previously unknown to us, we can now prove our main results for quasi-compact quasi-separated (rather than just separated) schemes. To appear in Illinois J. Math. 27 page

On the fundamental class of an essentially smooth scheme-map

Let f: X -> Z be a separated essentially-finite-type flat map of noetherian schemes, and \delta: X --> X \times_Z X the diagonal map. The fundamental class C_f (globalizing residues) is a map from the relative Hochschild functor L\delta^*\delta_* f^* to the relative dualizing functor f^! A compatibility between this C_f and derived tensor product is shown. The main result is that, in a suitable sense, C_f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f^! to relative d-forms.Comment: Minor corrections. 31 pages. To appear in Algebraic Geometr

Duality and flat base change on formal schemes

We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about "Brown Representability," is indicated as well. A section on applications and examples illustrates how these theorems synthesize a number of different duality-related results (local duality, formal duality, residue theorems, dualizing complexes...). A flat-base-change theorem for pseudo-proper maps leads in particular to sheafified versions of duality for bounded-below complexes with quasi-coherent homology. Thanks to Greenlees-May duality, the results take a specially nice form for proper maps and bounded-below complexes with coherent homology.Comment: 89 pages. Change from published version: in section 2.5, about dualizing complexes on formal schemes, a weakening of one flawed Lemma is proved, and shown adequate for the several applications made of the original. For another correction, see math.AG/010623

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(\R\Gamma_Z\O_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.Comment: DVI file pub/lipman/homology.dvi (214776K, 38 pages) available via anonymous ftp (binary) at ftp.math.purdue.edu, AMSLaTeX v 1.

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.Comment: 42 pages, minor revisions and reorganized introduction, PLAIN Te