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, for all$xεX$, the Jacobian ϱ$f(x)$has rank at least$n$−2. For$r$≥1, consider the subscheme$N$_{$r$}of$Y$defined by the ($r$−1)th Fitting ideal of the $$\mathcal{O}_Y $$ -module $$f_ * \mathcal{O}_X $$ , 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$f$.

We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagrangians and more generally coisotropic submanifolds with elements of a primitive cohomology, which dualizes to a homology on coisotropic chains.

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Homology theory relative to classes of objects other than those of projective or injective objects in abelian categories has been widely studied in the last years, giving a special relevance to Gorenstein homological algebra. We prove the existence of Gorenstein flat precovers in any locally finitely presented Grothendieck category in which the class of flat objects is closed under extensions, the existence of Gorenstein injective preenvelopes in any locally noetherian Grothendieck category in which the class of all Gorenstein injective objects is closed under direct products, and the existence of special Gorenstein injective preenvelopes in locally noetherian Grothendieck categories with a generator lying in the left orthogonal class to that of Gorenstein injective objects.

In this paper, a structure-preserving algorithm is developed for the computation of a semi-stabilizing solution of a Generalized Algebraic Riccati Equation (GARE). The semi-stabilizing solution of GAREs has been used to characterize the solvability of the (J, J ′ )-spectral factorization problem in control theory for general rational matrices which may have poles and zeros on the extended imaginary axis. The main difficulty in solving such a GARE lies in the fact that its associated Hamiltonian/skew-Hamiltonian pencil has eigenvalues on the extended imaginary axis. Consequently, it is not clear which eigenspace of the associated Hamiltonian/skew-Hamiltonian pencil can characterize the desired semi-stabilizing solution. That is, it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute. Hence, the well-known generalized eigenspace approach for the classical algebraic Riccati equations cannot be employed directly. The proposed algorithm consists of a structure-preserving doubling algorithm (SDA) and a postprocessing procedure to determine the desired eigenvectors and principal vectors corresponding to the purely imaginary and infinite eigenvalues. Under mild assumptions, linear convergence of rate 1/2 for the SDA is proved. Numerical experiments illustrate that the proposed algorithm performs efficiently and reliably.