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. In this article we are going to concentrate on the canonical element conjecture due to M. Hochster as well as several of its ramifications. In [17] Hochster introduced a number of equivalent forms of this conjecture and proved it in the equicharacteristic case. One of the earliest forms, the direct summand conjecture, was proved by Hochster [15] a decade earlier under the same hypothesis (see also [16]). In 1980 Evans and Griffith [10] gave an affirmative answer to the syzygy problem for equicharacteristic local rings. In the course of their proof, they implicitly established a result for finite free complexes [10] that Hochster explicitly isolated in his article [17]. He referred to the new result as the “improved new intersection theorem” (henceforth INIT) since it is clear that INIT implies the new intersection theorem ([17]). Of course INIT remains a “conjecture” in the case of mixed characteristic. In the same article [17] Hochster pointed out that INIT was a consequence of the canonical element conjecture and later the first author [3] showed that the two conjectures are equivalent. Over the years several special cases of the canonical element conjecture have been proved and new equivalent forms have been introduced ([2], [3], [5], [6], [7], [8], [14]). The four main equivalent versions of this conjecture, i.e., the direct summand conjecture, the monomial conjecture, the canonical element conjecture and the

Intersection multiplicity, canonical element conjecture and the syzygy problem