212 research outputs found
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
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